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-^^
^Ic^
^
THE
MESSENGER OF
MATHEMATICS.
EDITED BT
J.
W.
L.
GLAISHER,
So.D.,
P.E.8.,
niLIiOW
or TRINITT
OOLLiail,
OAKBBIDSH.
VOL. XVII.
[Mat,
1887
Apkil,
1888.]
MAOMILLAN
AND
00.
Honlron
anlr
Cambtfiige.
1888.
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irr
7;
j/^^z .^
0^
lii'h
^fjc,
G^
CAMBRIDGE)
:
PRINTED
BT
W.
MBTOALPK
AND
SON,
TRINITY
STREET
AND ROSE
CRESCENT.
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CONTENTS.
ARITHMETIC,
ALGEBRA,
AND
TRIGONOMETRY.
On
the
order
of
proof
of
the
principal
equations
of
spherical
trigonometry.
By
M.
Jenkins
.------
Note
on
a
theorem
in
Higher
Algebra.
By
H.
G.
Dawsok
On
a
theorem
of
Prof.
Klein's
relating
to
symmetric
matrices.
By
A.
BnCHHEIM
.......
A
new
method
for
the
graphical representation
of
complex quantities.
By
J.
Brill
---.---
Note
on
the
anharmonic
ratio
equation.
By
Prof.
Catlet
-
Note
on
the
multiplication
of
nonions.
By
G. G.
Morricb
.
An
extension
of
a
certain
theorem
in
inequalities.
By
L.
J.
Roobrs
PAGE
30
69
79
80
95
104
145
GEOMETRY
OP
TWO
AND
THREE
DIMENSIONS.
System
of
equations
for
three circles
which
cut
each other
at
given angles.
By
Prof. Catlet
.......
18
On
plane
cubics
which
inflect
on
crossing
their
asymptotes.
By
F,
Morlbt
61
Note
on
certain
theorems
relating
to
the
polar
circle
of
a
triangle
and
Feuerbach's
theorem
on
the
nine-point
circle.
By
S. Roberts
-
57
On
the
system
of
three circles which
cut
each
other
at
given angles
and
hare their centres in
a
line.
By
Prof.
Catlet
- - -
60
On
systems
of
rays.
By
Prof. Catlet
-
-
- - -
78
Note
on
the
two
relations
connecting
the
distances
of
four
points
on
a
circle.
By
Prof,
Catlet
-
-
-
- - -
94
The
cosine
orthocentres
of
a
triangle
and
a
cubic
through
them.
By
R.
Tttokbr
--.-.--
Geometry
on a quadric
surface.
By
Prof.
Mathews
-
97
-
151
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IV
CONTENTS,
DIFFERENTIAL
AND INTEGRAL CALCULUS
AND
DIFFERENTIAL
EQUATIONS.
.
PAGE
Note
on
the
Legendrian
coefficients
of
the
second
kind.
By
Prof. Cayley
-
21
The
transformation
of
multiple
surface
integrals
nto
multiple
line
integrals.
ByJ.
Larmor --------23
Depression
of
differential
equations.By
Lt.-Col.
Allan
Cunningham
-
118
nomographic
inyariants
and
quotient
deriyatiyes.
By
A.
R.
Forsyth
-
154
THEORY OF ELLIPTIC FUNCTIONS.
On the
tn^nsformation
and
derelopments
of the
twelve
elliptic
unctions
and
the
four
Zeta functions.
By
J.
W.
L.
Glaishhr
-
-
-
1
On the
second
solution
of
the
differential
equation
of
the
hypergeometric
series,
nd the
series
for
K'f
E',
c.,
in
Elliptic
unctions.
By
Brof.
W. Woolsby
Johnson
-------
35
Expressions
for
8
(a;)
s
a
definite
integral.
By
J. W. L. Glaishbr
-
152
APPLIED MATHEMATICS.
Yortices
in
a
compressible
luid.
By
C. Chrbb
....
105
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n
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MESSENGER
OF MATHEMATICS.
ON
THE
TRANSFORMATION
AND
DEVELOP-ENTS
OF
THE
TWELVE
ELLIPTIC
FUNCTIONS
AND
THE
FOUR ZETA
FUNCTIONS.
By
J,
W.
L.
Olaisher.
The
principalbject
f
this
paper
is
to
give
the transfor-ations
of
the
elliptic
nd
Zeta
functions,
which
are
due
to
the
change
of
q
into
-
q^
^
and
q^
;
and
also the firstfew
terms
of
the
expansions
of these
functions
in
ascending
powers
of the
arguments.
I omit the
demonstrations,iving
only
the
resultsin
a
form
convenient
for
reference.
This
paper
(in
so
far
as
it
relates
to
the
elliptic
unctions)
may
be
regarded
as
a
continuation
of
two
papers
^^
On
[Elliptic
unctions,
hich
were
published
n
vol.
XI.
of
the
Messenger,
pp.81-95,
120-138).
In
17 19
I have
given
tables
of the
values of
the
elliptic
unctions
when the
argument
is
increased
by Jf,
iK*
or
K-\-iK\
and
also the values of
the
functions
for certain
special
alues of the
argument.
I
have
found these tables of
such
constant
use
in
working
ith
the
twelve
elliptic
unctions
that
I
have
been
tempted
to
include
them in this
paper.
The
firsttable
in
17
was
given
in vol.
XI.,
p.
88,
but
the
functions
were
there
arranged
in
an
inconvenient order.
For the sake of
completeness
he
g'-series
or
the
elliptic
and
Zeta
functions
are
also
given
(
2,
3).
Notation^
1.
The letters
p
and
u
are
used
to
denote
2Jr
,
^Kx
and
IT IT
respectively
so
that
u
=
px.
The
argument
x
is
supposed
o
be
independent
f
k.
The
lettersh and Ji
are
used
to
denote
k^
and
A'*
respectively,
VOL, XVII.
B
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2
MB.
QLAISHER,
TBANSFQBMATION
AND
DEVELOPMENTS
The
q^series
for
Icp
nu,
c.,
and
pZ{u)^
2.
2.
The
j-series
or the
twelve
elliptic
unctions
are
:*
hp
snw
=
2*
^
^n-i
sin
(2n
1)
a?,
kp
cnM=
2
y^7-5n=i
os
(2 i
1)
a?,
pdnttsa
27
.
,n
co82yia?;
p
nsM=
-:;
+
2*
T-^-i =i
in
(2n
1)
x.
'^
sina;
*
1
-
^*^
^ ' '
p
dsw
=
-:
2
,
^
n-i
sin
(2n 1)
a?,
^
sina?
*
1+2'
'
k'p
ncu^
2r
(-p/^'-i
os
(2n
1)
a?,
^
cosa?
'
^
'
1
+
2
Ap
cdtt=
Sr
(-) -
j^^,
os
(2n
1)
x,
Mp
8d
u
=
Sr (-r
^^
n
(2n
1)
x,
A;
nd
tt
=
1
-
2^
(-) -*
j-^
os
2nar
;
Meittnger,
oI,
xvi.,
pp.
187,
188.
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OF
THE TWELVB
ELLIPTIC
AND
FOUR
ZETA
FUNCTIONS.
3
and
those
for the
four
Zeta
functions
are
;*
pz
(u)
sr
Yif^
^
'^'^^
i
to
th
2Van /brmatu M
of
kp
snu,
c.,
3,
4.
3.
The
changes
n the functions
hp
sn
u,
c irhich
are
dae
The
changes
in
the
functions
which
are
due
to
the
change
of
2
into
^
and
^
are
shown
in
the
table
on
the
next-
page:
t
Messenger
f
yoI.
xv., p.
146.
b2
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4
MR.
GLAISHER,
TRANSFORMATION
AND
DEVELOPMENTS
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OF THE TWELVE
ELLIPTIC
AND
POUR
ZETA
PUNCTIONS.
5
The
quantities
n'^u
h'
which
occur
in
the
firstcolumn
of
results
may
be
expressed
n the
forms
(l *')(cn'iu *'sn'iM),
and
the
quantities\ 1c)
(\^h
snV)
which
occur
in
the
second column
may
be
expressed
n the forms
dn'tf
k
cn'tf.
4.
In
the
following
ases
the
transformed
quantities
are
expressible
ery
simply
n
terms
of
sums or
differences
of
elliptic
r
Zeta functions.
Transformationsf
pZJ(u)^
5.
5.
The
changes
in
pZ{u) P^,{^)t
c
dae
to
the
change
of
2
into
j*
are :
?.
-?.
pZ( ),
pZ^u),
pZM,
pz,{u\
pZ,{u),
pZ{u),
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6
MR.
GLAISHER,
TRANSFORMATION
AND
DEVELOPMENTS
The
changes
hich
are
dne to
the
change
of
q
into
q*
or
q^
are
shown in
the
following
able
:
The
first
two
quantities
n
the
second
column
may
be
expressed
n the forms
\{pZ{^u)^.pZl\u)],
\\pZSMu)-^pZi\u)}
respectively
and
the second and third
quantities
n
the
third
column
may
be
expressed
n the
forms
pZ[u)-^pZ,{u),
Z^{u)+pZJ,u)
respectively.
Transformations
f
sum,
c.,
6.
6.
It
seems
desirable
to
give
also
the
transformations
of the functions
snte,
cnu,
c.
The
.changes
ue
to
the
change
of
q
into
q^x^x
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OF
THE
TWELVE
ELLIPTIC
AND
POUR ZETA FUNCTIONS,
7
The
changes
ue
to
the
change
of
q
into
^
and
^
are :
snu
cnu
dnu
nsu
dsu
csu
dcu
ncu
scu
cdu
sdu
niu
^
'^
dn^u
dn^u
dn^u
1
dn^M
l
+
k'
sn^ucn^t^
1
cn'^M-f-
:'8n'^^
l+k'
sn^i^cQ^u
1
cn'^M
k'
sn^u
cn'^M
+
i'sn'^M
cn^^w
Aj'sn'^M
dn^M
cn'^u
k'
sn'^M
( +*')::::?
n^ucn^t^
cn*iu
k'
sn^'^u
cn'^M
k'
sn'^M
cn'^M
+
i'
sn*^M
a+^')rrf
sn
^i
n
iu
cn*iw+
A;'Bn*^w
dnj^u
cn^^M
+
k'
sn*^u
1
-
A
sn^u
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8
MR.
QLAISMER,
TRANSFORMATION AND
DEVELOPMENTS
Transformations
fZ{u)y
c.,
7.
7.
It
18
UDDecessaiy
to
give
the
tranaforinations
of
-2^(m),
as
they
are
deducible
at
sight
from
those
of
pZJ^u)
given
in
5.
To deduce the
transformations of
Z,{u)
from
those
of
pZ,{u)
it
suffices
to
replace,
n the transformed
results,
p
by
p
in
the
g
column,
1
2
Transformationsf
p,
A:p,
'p,AA'p,
8.
8.
In
connexion
with the
preceding
ystems
of
formulas
it
is
convenient
to
give
the
transformations of
p,
kpy
c.|
which
are
as
follows
:
Transformations
f
sna;,
c.,^{^),
c.,
9.
9.
The transformations of the sixteen functions
snar,
c.
Z[x)^
c.,
are
easily
deducible
from those
of
sum,
c.,
Z(w),
c.,
in
6
and
7
by simply
multiplying
he
trans-ormed
arguments
in
the columns
headed
g,
g^
and
g*
by
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OP
THE
TWELVE
ELLIPTIC
AND FOUR
ZETA
FUNCTIONS.
9
For
example,
by
the
change
of
q
into
-;,
q*
and
^S
sn^
becomes
respectively.
The
transformations
xpressed
s
equatwnSy
10.
10.
By
the
change
of
q
into
^q
the
modalas
k
is
converted into
p
and
JTinto
k'K.
By
the
changes
f
q
into
^
and
of
2
into
^
the modulus
X;
is
converted
into
X
and
7^
and
K is
converted
into A
and
F,
where
Denoting
and
by
px
^'^^
Py
^^^ ^^^
^7
^
and
Wj
the
transformations
given
in
the
last
two
sections
may
be
expressed
s
equations
n
the
form
:
ikp
sn
f
A'w,
77
)
=
ilck'p
d
u^
c.,
,
_.
--,
sn^Mcniw
Xpx8n(t;,
)=^A;p
^^^^
,
c.,
7P
en
[w.
7)
=
2A:*p ; /
^ ,
c.,
sn(x/|)=*'sd(|),
c.,
X
X
sn
;
j7
en
sn
(a:,)
=
(1
+
A:')
,
c.,
Bn(ar,7)=
^
C.
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10
MR.
aLAISHER,
TRANSFORMATION
AND
DEVELOPMENTS
Expansions
f
the
elliptic
unctions
n
powers
of
x^%\\.
11.
The
expansions
of the
twelve
elliptic
unctions
in
ascending
owers
of
x
are
as
follows
:
8na?
=
;c-(l
+
A)|^
(l
+
14A
+
A')^
a;'
-
(1
+
135A
+
1
35A'
+
A')
y7
+
c.,
cna:
=
1
-
1^
(1
+
4A)
^
-
(1
+
44*
+
16A')
1^
c.,
dna
=
l-A^j
A(A
+
4)|^j-A(A'
44A4l6)^
c.;
n8a
=
^
+
i(l
+
A)a
+
^(7-22A
+
7A')|^
+
tk
(31
ISA
-
15A'
+
31A')
jj
+
c,,
dsa;
=
i
-
J
(A
A')
+
sV
(7A'
22AA'
+
TA )
J^.
-
X
Jy
(31A
15A'A'
-
15AA'
-
3lA )
^j
c.,
C8
=
^
i
(1
+
A')
;
+
j'ty
7
22A'
+
7A )
^j
-
Ti,
(31
15A' -
15A
+
31A'')
+
*c-
J
dca:
=
1
+
A'
^
+
A'
(A'
4)
|^,
A'
(A
+
44A'
+
16)
|^,
c.,
nca;
=
1
+
^
+
(1
+
4A')
2l
+
(1
+
44A'
+
ICA )
^
+ c.,
Bca
=
+
(1
+
A')^
{1
+
14A'
+
A'')^j
+
(1
+
135A'
+
1
35A'
+
A )
^j
fec.
;
.-
cd
=
1
-
A'
Jy
+
A'
(A'-
4A)
Jy
A'
(A -44AA'+
16A')
J-j
c.,
sdar
=
+
(A A')
^j
(A'
14AA'
+
A )
|^
4
(A*
135A'A'
+
135AA
-
A )
,
+
c.,
nda
=
1
+
A
^j
A
(A
4A')
1^
A
(A*
44AA'
+
16A )
1^
c.
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12
MR.
GLAISHEfi,
TRANSFORMATION
AND
DEVELOPMENTS
gz,aj
=
- ;i'
{^
2
^i
2X1
+
A')^j
2X2+13A'+2r)f^
c.l
,
gz,:p
=
AA'|2
j
+2'(A-A')|^j
2*(2A'-13AA'+2r)
c.|
14. The other
Zeta
functions
iz^e,
c.,
differ from
gz cc
only
by
multiples
f
a?,
so
that
with
the
sole
exception
f
the
term
involving
the
expansions
of these functions
are
the
same as
those of
gz/c.
The
accompanying
able
gives
the
value of the
term
involving
for
each
of
the
twelve
functions
iz,a;,
z^,
ez;r.
The
corresponding
erms
in
the
expansions
f
Z^
{x)
and
Z
{x)
are
shown in the
next
table.
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OF THE TWELVE
ELLIPTIC AND POUR
ZETA FUNCTIONS.
13
The fanctions
^{x)
are
the
functions
so
desig^nated
n
vol.
XV.
pp.
92-102.
The
quantities
and
H' denote
J
(7+
G
+
JS)
and
J
(/'
+
ff
'
+
')
respectively.
15.
It
will
be
noticed
that
the
series
for
ffza;,
gz,ar,gz,x
contain
A,
A',
hh'
respectively
s
factors.
The
only
other
series
which contain
a
factor
common
to
all
the
terms
are
those for iz^
and
ez^a;,
neither of
which
contains
a
term
in
x.
The three series
which contain
no
term
in
x
are
:
izur=-
aJ2J-2 (1
A)^
2*(2
+13A
+2A )
^j- c.l
ez^=-
*'J2^
2-(l+A')|^+2^(2
13A'
+2A )|^j+i
gz,a?=
AA'
|2
^4
'{A-A')|'j
2*(2A -
13AA'+2A' )
|^+ c.l
16.
The
values of
Z^(x),
when
x
is
small,
are
often
required
in
verifying
formulae
and for
other
purposes.
I therefore
add
for
the sake of
reference
the
expansions
of
Z,
{x)
as
far
as
the
terms
involving
^
:
Z{x)=
-^a?-iAa:',
Increase
of
the
argument
by
K^
iK\
K-\-
{K\
17.
17.
In
working
with
the
twelve
elliptic
unctions
it
is
convenient
to
have for
reference
in
a
tabular
form the
complete
system
of
changes
which
ar^
produced
in
the
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14
MR.
GLAISHER,
TRANSFORMATION
AND
DEVELOPMENTS
functions
by
the increase of
the
argument
by
K^
iK and
These
changes
re
shown
in the
following
able
:
The
next
table,
hich
is
deducible
at
once
from
the
last,
gives
the
correspondinghanges
of
A;sna;,
A;
en
a;,
c.
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OF THE
TWELVE
ELLIPTIC
AND
FOUR
ZETA
FUNCTIONS.
15
These twelve
fanctions form
a
group
complete
n
itself,
iz.
each
function
is
transformed
into
another
member
of the
group
multiplied
y
1
or
i.
Valttes
of
the
elliptic
unetions
hen
the
argument
is
OjKjiK'jK+iK'y
18.
18.
The
following
table
gives
the
values of the
twelve
elliptic
unctions for the
arguments
0,
Ky iK',
K+
iK*.
The letter
a
denotes
zero
and A
infinity,
ut the
following
more
precise
ignifications
ay
be
attached
to
these
letters^
viz.
the
arguments
0,
K^ iK'^
Jr+
iK'
may
be
regardedas
denoting
a,
K-^a^ iK'+a,K-\-iK'-\-a^
here
a
is
infinitesimal,
and
A
denotes
-
a
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16
MR.
GLAISHER,
TRANSFORMATION
AND
DEVELOPMENTS
Thus the column
headed
0
shows
that,
beinginfinitesimaly
sna
=
a,
n8a
=
-,
dsa
=
-,
c.
The column
headed
K
shows
that
en
(Jr+
a)
=
-
A?'a,
dc(^+a)
=
--,
c.,
and
80
on.
Digiti
zed
by
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Goo
gle|'
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18
PROF.
CATLBT,
SYSTEMS
OF
EQUATIONS
In
the
last column the values
of
the functions
for
x^a,
a
being
infinitesimal,
re
added.
By
the aid
of
the three
previous
olumns the values of
the
functions
2K- ay
2iir'+a,
2^4
2iK'
+
a
for
a
infinitesimal
may
be written
down
at
sight
by affixing
he
proper
sign
;
for
example
sn(2^+a)
=
-a,
sn(2iX'4a)=
+
a,
sn(2Z'+2iX'+a)=-a,
cn(2Jr+a)=:-l,
cn(2iJS:'
a)=-l,
cn(2JS:+2tJr'+a)=+lj
c.,
c.
SYSTEM
OF
EQUATIONS
FOR THREE
CIRCLES
WHICH
CUT
EACH
OTHER
AT
GIVEN
ANGLES.
By
Prof.
Cayley.
Consider
a
triangleBC^
angles.4,'B,
7
(-4
B-\-
C^ir)
:
to
fix
the
absolute
magnitude,
ssume
that
the
radius
of
the
circumscribed
circle is
=^1,
the
lengths
of
the
sides
are
therefore
=
2
sin^,
2
sin ,
2
sin (7
respectively.
n
the
three
sides
as
bases,
outside
of
each,
describe
isosceles
triangles
aBGy
bGAj cABy
the base
angles
whereof
are
=
a,
/8,
7
respectively.
f
we
draw
a
circle
touching
aJ5,
a
(7
at
the
points
-B,
G
respectively;
circle
touching
iC,
IA
at
the
points
(7,
respectively
and
a
circle
touching-4,
B
at
the
points
y
B
respectively
then
these
circles
form
a
curvilinear
triangle
BGy
the
angles
whereof
are -4+)8
+
7,
jB
+
7
+
a,
(7+a
+
)8
respectively.
aking
as
origin
the
centre
of the
circumscribed
circle,
nd
through
this
point,
or axis of
^,
an
arbitrary
ine,
ts
position
etermined
by
the
angle
dy
I
write
for convenience
F=0-]-2By
r^e-^Ay
^'
=
^
+
/S
+ 7,
?
=
^
+
2jB+2(7,
G^e^By
B^B+y
+
OLy
H=0y
E'=^0^2B+Cy
G'^C
+
a+fi;
then the
coordinates of the
angularpoints
Ay By
G
are
(cos-F,
injp'),cos
G,
sin
G^)
(cosS,
ini7)
respectively;
nd
the
equations
f
the
three
circles
are
(sin
-4
a)
A' /
sin
(-4
a)
.
-,A
sin'-4
x
+
\
^cos^')+{y
+
^^
ism^'j
=-t-^,
sma
/
V
sin a
/
sin a
'
/
8in(5-/S)
^Y
,
/
^sin(j5-/3)
^V
sin'5
[a
+
V--5
'cos ?
+
(y
+
\
^
^
sin
g
)
=-^^
\
sinjS
/
V sm/tf
/
sin'/S
/
sin(C-7)
TT'V
.
f
8in((7-7)
.
^V
sin'O
{x
+
-.
i^cosJ?
+
y
+
-'
^
sinJ?
=-:r-j-,
V
sin7
J
V
am
7
/ Bin'7
respectively.
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FOR
THREE
CIRCLES.
19
In
verification
observe
that
we
have
0'H^2ir-2A,
?'-J?=-w+^,
(7-^'=27r-j4,
-F^A^
H^F^
-25,
W^F^
ir-^-B,
^-O'^
-5,
F^ ^B,
F^G^
-2(7,
^'-(?'=-,r+(7,
^H^
-C7,
Q^H^Oj
hence
(cos
7-
co8fi^)'
(sin
7-
sm ^)'-
2
-
2
cos((7-fi^,
=
2
(1
cos
2^),
=
4
sInM
;
and
we
thus
see
that
the
sides
are
2
sin^,
2
sin
5,
2
sin
0
respectively.
The
firstcircle
should
pass
through
he
points
cos
O^
sin
CF)j
(cosjB',
in
J?)
we
ought
tnerefore to
have
tor
the
first
of
these
points
^^^BinM- )
_
.inV- )^BJn^l
Sin
a
^
'
sin
a
sin
a
'
that
isy
.
.sin
(-4
-a)
.
.
sin (-4-a)
sinM
1+2
^cos-4
+
Vt
i
as
.
,
sin
a
sma
sin
a
Bnd for
the
second
of
the
points
he
same
equation.
Write
for
a
moment
^
vAuA
. sinf^-a)
^
-
X
=
:
,
then
:
'-
9
JC
cosa
*
cos^,
sin
a
'
sina
'
and the
equation
is
1
+2'(Xcosa-
cos-4)
cos^
+
(Xcosa-
cos-4)'=s-r,
that
is,
1
cosM =
-3l*
sin*a,
which
is
right.
The
second
and
third
circles
should
intersect
at
the
angle
A\
that
is
we
ought
to
have
\
smp
sin7
/
\
sin/i^
sin7
/
siu^JS
sin'C
^
sin
-B
sin
(7
.,
=
-^-fo
+
^ T
+
2
.
^
.
COS^
,
Sin
p
sin
7
sinp8in7
.
'
or
reducing
and
for
cos
(
O'
H')
substitutfng
ts
value,
=
cos^,
the
equation
s
sin'(5-/3)
.
8in'((7-7)
.
,sin(5-/3)sin((7~7)
.
jTB
*
5
T
^
i
7i
eos
^
sin
p
siu
7
sin
p
sin
7
sin*j5 sin'*
^
sin
sin
(7
=
-T-^TD
+
+2
.
^
.
cos-4.
sin
p
sin
7
sin
p
sin
7
C2
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20
PROF.
CAYLET,
SYSTEMS OF
EQUATIONS
1
Writing
here
sm-B
^
sinJ?
'
^
Binp
'
siny
'
the
equatiop
s
(
Ycos^
cos
j5)*
(Zcos7
cos
Cy
+
2
(ycos/8-
co8j5)
(Zcos7-
cos
C)
cos-4
=
r'
+
Z'
+
2yZcos4',
Tiz.
this
is,
Y*
cos'^
+
Z*
cos*7
2
rZcos^ cos
7
cos^
-
2
Fcos^
(cos
5+
cos
Ccos.4)
2Zcos7
(cos
+
cos-4
cos
5)
+
cos^jB^-cos'C+
2
cos
-4
cos
5
cos
(7
4=r'
+
Z
+
2rZcosJ'.
Keducing
by
the
relation
-4
4
5+
(7=
ir,
this
becomes
2
FcosyS
sin-4
sin
(7
2^cos7
sin-4
sin-B+
1
cos-M
=
Y'
sin*i8
Z*
sln'7
2
rZ(cosud'
co3/3
cos
7
cos-4)*
Here
-4'
=
-4
+
^
+
7,
and thence
cos^' =
GOB
A
(cos^
cos
7
-
sin
^
sin
7)
Bin-4
(sin7
os^S
+
sin^S
C0S7),
and hence the
right
and
is
r'8in /3
+
Z'sin 7
2
yZ(cos-4
sin^
sin
7
+
sin
A
sin
7
cos 13
+
sin
A
sin^ cos
7)
or
reducing
by
FBin)8
=
sinJ5,sin7=sin(7,
this
is
=
sin'-iB
sin*
(7 2
sin B
sin
C
cos
A
2
y
cos)8
sln-^
sin
C
2Zcos7
sin-4
sinjB,
and the
terras
in
Yj
Z
are
equal
to
the
like
terms
on
the
left-hand
;
the whole
equation
hus
becomes
-
1
+
cosM
+
8in''jB+
in'
(7-
2
cos^
sin
j5
sin
(7
=
0,
where
the
last
term
Is
=
2
cos
-4
{cos
54-
(7)
cos
B
cos
(7},
=
2
cos*-4
-
2
cos
-4
cos
J5
cos
(7,
=
-
2
cosM
+
(cosM
+
cos'
+
cos'(7-
1),
;
=
cosM
+
cos*
J?
+
cos'
(7
1
;
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PROF.
CAYLEY,
LEQENDRIAN
COEFFICIENTS.
21
the
equation
s
thus
-
1
+
cos*^
+
sin'J?+
sin'(7- co^'A
+
cos'5+
cos'C-
1
=0,
or
finally
t is
-1
+
1+1
1=0,
which
is
an
identity.
he
formulae
for the intersection
of
the
third
and
first
circles,
nd
for that
of the first
and
second
circles,
re
of
course
precisely
similar
to
the
above formula
for the intersection of
the
second
and
third
circles;
nd
the
verifications
are
thus
completed.
Cambridge,April
7,
1887.
NOTE
ON THE LEGENDRIAN
COEFFICIENTS
OF
THE
SECOND KIND.
By
Prof.
Cayley.
As
regards
he
integration
f the
equation
(l-*')g-2x|+ (
l)y-0
(n
a
positiventeger),
t
seems
to
me
that
sufficient
prominence
is
not
given
to
the
solution
where
P^
is the
Legendrian
integral
of
the
first
hind,
a
rational
and
integral
unction
of
x
of
the
degree
tz,
and
Z^
is
a
rational and
integral
unction of the
degree
n
1
;
viz.
we
have here
a
solution
containing
o
transcendental
funqtion
other
than
the
logarithm,
nd
which should
thus
be
adopted
as a
second
particular
ntegral
n
preference
o
the form
y
=
C
in
which
we
have
the
infinite
series
Q^
which is
an
unknown
transcendental function.
Moreover,
the
expression
sually
iven
for
Z^j
viz.,
^_2n-lp
,
2n-5
,
2n-9
*
l.n
- '* 3(n-l)
- ^5(n-2)
^'^
(to
term
in
P,
or
PJ,
is
a
very
simple
and
elegant
one;
but
the
more
natural
definition
(and
that
by
which
Z^
is
most
readily
alculated)
s
*c
+
1
Z^
isthe
integral
art
of
iP log
-,
when
the
logarithm
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22
PBOP.
CAYLEY,
LEGENDBIAN
COEFFICIENTS.
is
expanded
in
descending
owers
of
x^
viz.
it
is
the
integral
part
of
W,h-\^--)
(whence
also
Q^
is
the
portion
ontaining
egative
owers
only
of
this
same
series).
The
expressions
for
P^,
Pj,...Pj
re
given
in
Ferrers*
Elementary
Treatise
on
Spherical
armonica^
cfec.
London
1877,
pp.
23-25.
Reproducing
these,
and
joining
o
them
the
values
of
^,
Z^...Z^^
e
have
as
follows
:
read
P,
=
fee'
i,
and
so
in
other
cases.
p.
=( ...
I)
\v
-
W
+
W
-
A
,
P,=( '...x)
vy.
_
^
+
a^ys
_
ij
,
P,
=(* ... )
Vi^
-
AJiit
+
ifji
-
Vi/'
+
P.
=(as'...a )i.Hgi-
Jfi
+
io^
-
Xj5i
+
IJI
,
II
,
V
+
A
,
z.
=( '...x)
vv
-
-4^
+
W
,
Z,
=(a/'...l)
^
-
iji
+
w
-
H
.
z.
=(x\..x)
v^
-
AV
+
V^ -^4Vy^
,
Z,
=(x'...l)ifi i-
-tflli
+
m^
-H\l^+
iff
.
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MR.
LABMOB}
TBAKSFOBMATION
OF SUBFACB
INTEOBALS.
23
I
notice that
the
numerical values
of
P
Pj
...,
P,,
for
a;
=
0*00,
O'Ol,
...,
I'OO
are
given
[Report
of
the
British
Association
for
1879,
Report
on
Mathematical
Tables
)
;
as
the
functions
contain
only
powers
of
2
in
their
denominatorS|
the decimal values
terminate,
nd
the
complete
values
are
given.
The
functions
Z have
not
been
tabulated,
^
the
denominators
contain
other
prime
factors,
nd
the
decimal
values
would
not
terminate.
Cambridge,
Mareh
29,
1887.
THE
TRANSFORMATION
OF
MULTIPLE
SURFACE INTEGRALS INTO
MULTIPLE
LINE
INTEGRALS.
,
By
J.
Larmor.
An
integral
xtended
throughout
volume
can
in
various
ways
be
expressed
s a
surface
integral
ver
its
boundary.
Many
elegant
theorems
of
this
kind
have
been
givenby
Gauss.*
L
But
in
order
that
the
integral
ver
a surface,
f
a
vector
function,
eaning
thereby
the
integral
f
its
normal
component
over
the
surface,
ay
be
expressible
y
a
line
integral
ver
its
contour,
the
function
must
satisfy
certain
condition.
Li
fact the
integrals
ver
any
two
surfaces
abutting
n
the
same
contour would
then be
equal,
and
the
two
together
would
form
a
closed
surface,
uch
that the
integral
aken
in
the
same
sense over
the
whole
of it
would
be
equal
to
zero.
Now if
B
denote
the
vector,
X, Y^Zita
componentd
parallel
o
the
axes,
and
B
cose
its
normal
component,
//iieo... .///(f.f.f).
,)
Therefore
if
this
condition of
zero
integral
s
to
hold for
all
closed
surfaces,
e
must
have
identically,
hroughout
he
space
considered,
dX
.
dY
.
dZ
^
,-
iJ+^+5?= '
(')
as
the
condition
required.
*
Theoria
AUraetionit...,
omm,
Soe,
GUtvng.^
I,1813,
or
W^rke,
Band
V.
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24
MB.
ULBMOBy
THE TRANSFORMATION OF
MULTIPLE
The truth of
the
formola
(1)
requires
hat
the
vector
should
not
become
discontinuous
or
its
differential
coefficients
infinite
anywhere
in the
space
in
question
for
if
that
were
not
provided
or,
he
integration
f its
ri^ht-hand
side
might
introduce other
terms:
cf.
Maxwell's
ElectricitVy
b.
I.
The
proposition
ust
therefore
be
applied
n its
simple
form^
only
when
the
region
in
question
oes
not
contain
places
where
the
vector
is
discontinuous
or
its
differential
coefficients
infinite.
If
Xy Yj
Z
are
the
components
of
a
flux
R^
the
condition
(2)
is
the well-known
Equation
f
Continuity,
hich
secures
that
the
flux is
that
of
an
incompressible
ubstance.
Thus
in
continuous motion
of
incompressible
luids
the
flux
through
any
ideal
aperture
is
expressible
s a
line
integral
ound its
contour
;
the
reason
for which is
obvious.
To
determine the
form
of the
integral
elation in
question,
we
may
firsttake
the
case
of
a
small
plane
surface.
Then
j{ada,
/3 7y)
^jjdxdy
g
|)
(3)
by
immediate
integration,
he rule of
signs
being
that
the
line
integralroceeds
ound
the
contour
in
the direction
from
^
to
y
in
the first
quadrant.
In
the
same
way,
for
areas
in
the
planes
of
yz
and
zx^
we
have
/0Si3,
, ).//* (|-f)
4)
/(r +.i.)-//.i (|.-|)
5)
By
what
precedes,
xpressions
o
be
integrated
n
the
right-hand
re
to
be
taken
as
the
components
normal
to
the
coordinate
planes
of the
vector
function
B]
and
we
remark
that
they
satisfy
2).
We
are
entitled
therefore
to
assert
for
any
small
plane
contour,
that
/(ada5^dy
+
7cfe)
^JJdS.BcoBB
(6)
where
the
components
of
B
are
^ dy^dz'
dz
dx'^^d^'Ty^' ^^
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26
lOL
LAXXOB,
THB
TSAaSTOUULTIOV GP ITULTIFLS
We
nutjr
obtain
odmr
fonns
fiirAe theotem
hy
miiXmg
to
the
rigfat-hfliid
de
of
(9)
the line
int^nl
of
anj
exact
differential|
wbidi
will
add
nothmg
when taken loond
the
drcnit.
This
fomrala
(9)
expnaMs
as
a
line
int^^
the
flax
dae
to
a
single
sonroe
o
floid
at
the
orig:in
f
eoordinates,
r
the
induction doe
to
a
angleattractingarticle
tnated
there
;
and
from
it
any
more
general
case
might
be
deduced
by
summation.
Bat
dcTclopment
n this direction
amply
leads
to
the
well-Juiown
theory
of
the
Tector
potential
n
Electro-ynamics*
IL There
is
another
class
of
int^irsls
elated
to
Mathe-atical
Physics
n
which the
intends
are
extended
oyer
two
contoms.
For
instance,
uniformly
uminous
open
surface
emits
a
quantity
f
radiation
through
a
giren
aperture
which
depends
nly
on
the
contours
of
the
sur ce
and
apeiturei
are
being
taken
that
all
parts
of
one
contour
are
yisible
from
all
parts
of
the
other.
Again,
the
mutual
energy
of
two
closed
electric
currents
may
be
expressed
either
as
an
integ^
extended
over
their
circuits,
r as a
surface
integral
erived
from
the
equivalentagnetic
shells.
We
propose
now
to
investigate
he
general
forms of
such
relations.
If
a
line
integral
ound
a
contour
is
to
be
expressible
s a
surface
integral
ver
a
sheet
bounded
by
the
contour,
by
means
of
(6),
it
must
involve
the elements of the
contour
linearly.
Therefore
the
most
general
type
of
double
line
integral
n
question
ust
involve
both
contours
linearly.
he
function
to
be
integrated
an
only
involve the distance
between
two
elements of
the
contours
and
the mutual inclinations of
the
distance
and these elements.
If
r
denote
the distance
of
the elements
dsjds'y
nd
i^,
'
the
angles
t makes
with
these
elements,
and
e
the
angle
between
the directions
of
the
elements,
he
most
genersd
orms
therefore involve
only
//c c 7(r)cose,
(10)
and
JSdsda'
(r)
cos^ cos^'
(11)
Of
these
the latter
clearly
anishes when either circuit
is
complete.
The
former
where
V^
m',
vl
are
the
directioncosines
of da'
;
^Jds'ffdB{X\'\-
fi
+
Zv),by
(6),
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SURFACE INTEGRALS
INTO
MULTIPLE
LINE
INTEGRALS.
27
where
\
fi^
v
are
the
direction
cosines
of
the
normal
to
dSj
and
r-/'(r)?^,
49,
y,
z
being
the
components
f
r,
the
origin
being
taken
temporarily
t
the
position
f
ds'.
Thus
changing
the order of
integration,
nd
transferrins
the
origin
o
the
position
f
dS^
so
that
we
write
^',
y
,
'
for
a;,
y, ,
we
have
jj,SJfir)
fj^a.'.^^^dy'
'^^^d.-)
^ffdsjjds'{X'\'
ry
+
v),
-;|:{;/'(r)|[-(y
O^V
+
x'X(y '
V)+...+
...]
-?/'(r)[XX'
/ /*'
' '']
^rj
|1/'
r)l
-
cos
*;
+
cos
^
cos
^
-
^/'
r)
os
*;
;|:{''/'(r)lcos,
r|^|i/'(r)|co. ?
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28
MR.
LABMOR,
THE
TRANSFORMATION
OF
MULTIPLE
where
17
is
the
angle
between
the normals
to
dS^
dS
each
drawn
towards
the
positive
ide
of
the
surface,
nd
d^
ff
are
the
angles
between
these
normals
and
r,
whose
direction
is
the
same
in
both
cases.
Therefore,inally,
\\d8\\dS
r
1^/'
}cos ?cos^-l
{r/'(r)}
cosi;]
^jdsjdsj^r)
ose,..
.(12)
where
the
positive
ide of
the surface
is
determined
by
the
rule that
a
right-handed
crew
in that
direction
corresponds
to
the direction of the line
integral
ound it.
We
have
proved
that
this result is
the
most
general
possible
f its
class.
Particular
cases
may
be
noted
as
follows
:
(i)
Make
the
two
circuits
coincide.
(ii)
Make the
two
open
surfaces
coincide,
nd
we
express
the
double surface
integral
by
a
double
line
integral
ound
the
contour.
To
avoid
infinities,
'[r)
must
not
contain
powers
of
r
lower
than the
inverse
firet.
(iii)
ake
the
surfaces
plane,
o
that
17
is
constant.
(iv)
Make
/'W
=
-^;
then
jjdSJjdS'
^'^^^r^^
-
i
Jdsjds'
ogr
cose.
...(13)
The
left-hand
side
is the
expression
or
the illumination
from
S
that
is
intercepted
y
8'
when
the
brightness
f 8
is
unity;
and
it follows from
elementary
opticalrinciples
that
this
quantity
ust
be
expressible
s a
line
integral
round the
contours
of 8 and
8\
When S
and 8'
coincide,
e
have
27r5= -
^Jdsjdslogr
cose,
(14)
true
only
when
8 is
plane
;
for
when
8
is
not
plane
he real
optical
nterpretationails,
he
parts
of
the surface
not
being
in
full
view of
each
other.
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8UBFACE IMTEOBALS
INTO
MULTIPLE
LINE
INTEOBALS.
29
(v)
Make
/(r)
=
-^,
o
that
/'(r)
-
J;
then
which
is
Neumann's
well-known
expression
or
the mutual
energy
of
two
simplemagneticshellsy
r
of
two
linear
electric
currents.
(vi)
Make
/'(r) Or;
then
/^5/(?/S'co8i7
-l/t /( Vco8e,
(16)
thus
giving
double
line
integral
orm for
JWdS^
where
n'
denotes the
area
of the
projection
f 8*
on
the
tangent
plane
at
dS, It
was
clear
h
priori
that
such
a
form
must
exist,
for
this
integral
epends
only
on
S
and
the
contours
of
8'^
while the
other
form
jUd8'
shows
that it
depends
only
on
the
contour
of
fi^;
hus
the form
of
the
function of
r
that
multi-lies
COSE
is
all that
remained
h
priori
o
be
determined,
nd
that
might
have been
found
from the
simplest
particular
ase.
When
one
surface
8
is
plane,
e
have
fi^n'
-i/i5/A'r'cose, (17)
where n' denotes
the
projection
f
/8'
on
the
plane
of 8.
Where
5,
8 coincide
in
one
plane,
e
have
8''
=
-ifdsJd8r^
OBB
(18)
And
comparing
this with
(iv)
e
deduce
(47r/8)
{Jdsjds
ogr
cose}
-
lir'
Jds
Jds
r*
cose...
(19)
for
any
plane
circuit
;
a
striking
esult.
The theorems
justgiven
may
be verified
by
direct
integra-ion
when the
surfaces
are
plane
circles,
and(18)
without
much
difficultly
or the
general
surface]
by applying
them
to
surfaces
bounded
by
other
curves,
we
obtain
evaluations
of
a
crop
of
definite
integrals
f somewhat
unusual form.
III.
If
elements of
three surfaces
enter
into
a
triple
integral,
he
components
of
the elements of their
three
contours
must
enter,
each
linearly,
nto the
corresponding
ine
integral.
The
most
general
form
of
such line
integral,
independent
f
special
oordinate
systems,
which
gives
finite
Talue
when
taken
over
completecircuits,
s
m{r,r\r )
dx
^
dy
J
dz
dx
,
dy'
,
e '
d7 \
dy'\
dz
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30
MR.
JENKINS,
ON
THE
OBDER OF
PROOF
OF
THE
where
r,
r',
are
the
mutaal
diatances of the
three
elements
of
contour
;
and the
determinant is
equal
o
30
dsdi
dd\
where
*=sini(a
+
J+c)8ini(i+c-a)sini(c+a
J)
flinj(a+i-c),
a,
i,
c
ieing
the
sides
of
the
sphericalriangle
etermined
by
the
directions
of
rf ,
fo\
fo .
The
integral
ay
therefore
hj
application
f
the
method
of
II
be
expressed
s
a
symmetricalriple
urface
integral
the
general
formulae
are
long,
ut
the
degenerate
ases
would
probably
e
interesting.
Finally,
here does
not
seem
to
be
any
reason
why
the
considerations
on
which
these theorems
are
founded should
be
restricted
to
the three dimensions
a;,
y,
z
of
ordinary
pace
;
but the
more
general
results
would
probably
be of
only
analytical
nterest.
ON
THE OBDER OF
PROOF
OF
THE
PRINCIPAL
EQUATIONS
OF SPHERICAL
TRIGONOMETRY.
By
ilf.
Jenkins^
M.A.
The
principal
ormulas of
spherical
rigonometry
re
made
to
depend,
holly
or
partially,
n
three
independent
quations,
which
are
of
a
less
simple
haracter than
most
of those which
are
derived from
them
;
that
is
to
say
on
cosa
~
COS
cose
+
sin
h sine
cos^
with the
two
other
equations
f
the
same
form.
Independent
proofs
re
given
of the
more
simpleequations
sin-4
^
sin-B
_
sin G
^
sina
sinft
sine
^
but
as
these
constitute
only
two
independent
quations,
third
is
needed
for
the
complete
investigation
f
the
properties
f
a
spherical
riangle.
propose
to
take the
equation
sin
(A
+
B)
_
cosg+cosft
sin 0
1
+
cose
(which
in
the usual
order would be
obtained
by
the
multi-lication
of
the
expressions
f
two
of
Gauss's
theorems),
ive
an
mdependent
proof
of
it,
se
the
properties
f
colunar
and
of
polar
triangles
o
obtain
the
equations
f similar
form,
and
then
show
how
these
may
be
applied
o
prove
other
formulas.
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PAINCIPAL
EQUATIONS
OF
SPHERICAL
TRIGONOMETRY.
31
Ab
a
leroma
take
the
equation
-C08a+C0R
cosm
2cos^c,
m
being
the median drawn from
G
to
the
mid-point
f AB,
Let
ABG
be
a
spherical
riangle,
the
raid-point
f the
arc
AB^
0
the
centre,
Ca,
00,
CS the
perpendiculars
n
OAj
OBj
OD
respectively,
/i,
v
perpendiculars
n
OA,
0B\
then
because
CaO,
G0O
and
CiO
are
rightangles,
he
sphere
on
OG
SLB
diameter
passes
through a,
^8,
S,
Therefore
O,
a,
ffy
S
are
on
a
circle;
nd
because
aOS
=
/905,
Sa^Sff,
also
Bfi^Sv]
hence
fia^v/B]
and
/a,
v
if
distinct
from
a
and
0
respectively
coinciding
hen
GA
GB)
must
be
on
opposite
ides
of
those
points
with
regard
to
0,
because
the
angles
OaS,
O0B
are
supplementary.
herefore
cosa
+
cos
J
OoL+ 00
20fi
^
,
=
j^
=
-7^=2cosic.
coswi
Oo
OS
*
We
may
note
that
the
same
proof
applies
f 2 be
not
the
mid-point
f
AB]
except
that,
instead
of
Sa
being
equal
to
S0J
we
have
a0
^
aS
^
0S
sin
c
sin
AD
sin
BD
'
also
OS.a0^
O0.a8+
Oa./3S,
whence
we
have
cos
OD
,
sin
c
=
cos
a
.
sin
^2)
+
cos
b
.
sin
BD.
Proceeding
o
the
equation
sin
(-44-
B)
cos
a +
cos
sin
C
1
+
cosc
'
let
D
be
the
mid-point
f
the
arc
AB^
join
OD]
producei
making
DK^
GDy
and
join
KA^
KB
so
as
to
obtain
the
spherical
homboid
A
GBK.
In
this,
s
in
plane
geometry,
opposite
ides
and
angles
re
equal,
and
alternate
angles
re
equal;
therefore the
angle
GAK=iA
+
B.
If
-4
4-5 7r,
CAK
is
a
proper
spherical
riangle
nd
denoting
D
by
m,
Bin
OAK
^
sin2m
__sin2m
Bin-4(7jr'
sin-^JT
sina
'
sin
A
CK
sin
^c
,
sin
A
sin
a
-
A~
-
-^-^
}
^^^
-^ n
=
^'
;
sm^
sm/7i
'
BiuU
smc
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32
MR.
JENKINS,
ON
THB
ORDER OF
PROOF
OF
THE
whence,
by
multiplication,
Bin
{A-{-B)
2
cosm
2
cosm
cos^
Bin
(7
2
cos
Jc
~
2
cob'^c
__
cosa+
cos
J
1
+
cosc
If
A
+
B Wj
CAK is
not
a
proper
sphericalriangle;
but
since
CK is also
tt,
if
we
cut
off
from
it
CO'
=
tt,
and
treat
the
triangle
AG'
as
before,
and
note
that
sin
{A
+
B)
is of the
same
sign
as
sin
OJT,
we
see
that the
previous
quation
till
holds.
Next,
by
means
of
a
colunar
triangle
y
a
being
unaltered|
B
changed
into
ir
Bj c.,
we
obtain
sin
[A^B)
_
cos
J
-
cosa
^
sin
0
1
-
cose
'
from
the
polar
triangle,
hangingA^ir-^ay c.,
e
have
sin
(q
+
h)
_
oo^A
+
cos-B
sine
1
-
cos
0
'
and from another
colunar
triangle,
sin
(a
I)
cos-B
-
cos^
,
sin^
+
sin B
sinG
\
'-
=
7=
,
also
-T
r-j-
=
-:
.
sine
1
+
cosU
'
BmaH-Bin6
sine
Hence
X
i/ i.r x
sin^+sinB sina+sini
.
^
sine
tani(^+5)
^j -^
=
inrr
sm
G.
cos^-FcosjB
sine
'8iu(a+6)(l
08(7)
C08i(a-i)
.,^
co8i(a
+
6)
* '
. ^it/ i
.
px_(cos^-co8^)(sin^
+
sin^
which,
n
substitution
and
reduction,
ives
cos'^(q-ft)_,,
^
j-^
cos
*
O.
cos^c
*
Similarly
e
may
obtain
the
rest
of
Napier's
analogies
and Gauss's
theorems,
he
sign
in
taking
the
square
root
being
determined
by
the
consideration
that the
greater
side
is
opposite
he
greater
angle
in
a
colunar
triangle
s
well
as
in
the
original
riangle,
hat is
^
+
-7r of
the
same
sign
as
a
+
6
-
TT,
as
well
as
J
-
JB
of
the
same
sign
as
a
-
6,
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CONTENTS.
PA
OB
On
the
transformation
and
developments
of
the
twelve
elliptic
nd the
four
Zeta
functions
(continued).
By
J.
W. L. Glaisher
-
-
-
17
System
of
equations
for three
circles which
cut
each
other
at
given
angles.
By
Prof.
Catlby
..-----
18
Not
on
the
Legendrian
coefficients
of
the second
kind.
By
Prof.
Catley -
21
The
transformation of
multiple
surface
integrals
nto
multiple
line
integrals.
By
J. Larmor
- -
-
-
--23
On the order
of
proof
of
the
principalequations
of
spherical
rigonometry.
By
M.
Jenkins
--------30
The
following
apers
liave
been
received
:
Major
Allan
Cunningham,
On
the
depression
of
differential
equations.
Prof. W.
Woolsey
Johnson,
On
the
second
solution of
the
differential
equation
of the
hypergeometric
series,
and
the
series
for
K\
',
Ac,
in
Elliptic
Functions.
Mr. F,
Morley,
On
plane
cubics
which
inflect
on
crossing
the
asymptotes.**
Articles
for
insertion will
be received
by
the
Editor,
or
by
Messrs.
W,
Metcalfe
and
Son,
Printing
Office,
Trinity
Street,Cambridge
NoTioR.
A
plate
will be
given
whenever
sufficient
diagrams
have been
received.
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No.
CXCV.]
NEW
SERIES.
s^
iL
JUL
18
1887
[July,
1887.
THE
-MESSENGER
OP
MATHEMATICS.
EDITED
BT
J, W.
L.
GLAISHER,
So.D.,
F.R.S.,
FELLOW
OP
TRINITY
COLLEGE,
CAMBRIDGE.
VOL.
XVH.
NO.
3.
|
MAOMILLAN
AND
00.
1887.
r.
MBTOALPB
\
ASTD
BON,
/
Price-One
Shilling.
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9
PBINCIPAL
EQUA^teKSL^OffieitKIAL
BIOONOMETBT.
33
To
obtain
cos
Cy
^
28in^co8(7 smB
sm(A+C)
+
sin(A-
C)
cos
(7=
T
;
:i
s=
r-: r
^:
5
-
2sm^
2
Bin
A
sin//
_
sinh fcosa
+
cose
cose
cosa|
cose
cosa
cosJ
2sina(
1
+
cosft
1
cos6
j
sinasini
For
right-angled
riangles,
f
C=i7r,
sin^
may
be
obtained
direct
from the
sine-aquation
co8c=
cosa cos
6,
by
producing
^Cto
A\
so
that GA'^
CA;
then
CB
being
the
median of
the
triangle
GA\
COSC+
cosc=2
cos(
cosa,
cos^
.
H
=*
cosa,
from
the
sine-equation
y
a
modificationof
figure
or
otherwise,
cos^
_
sin
((7
^)
_
cose
-f
cosa
_
cosa
cosft
+
cosa
sin5
sinjS
l
+
cos6
1
+
co86
=
cosa,
.
_sin((7+^)
_
sinJg
sin(C-f
^)
sin
0
sin
C
sinjB
__
sinft
cose
+ cosa
__
sinb
cosa
sine
l
+
cos6
sine
^
sin
ft
cos
c
_
tan
ft
^
sin
c
cosft
~
tan
c
'
sin^
sin
a
1
+
cosft
sin
a
tana
tanil
=
sin((7+^)
sinft
cosc+cosa
sinftcosa
sinft
'
^A
^n
sin(0+^)
sin((7-f-
B)
cosc
+ cosa
cose
+
cosft
cot-acotJ?=
\
p
^
1
:;
^
=
-
7
r
siujB
sm-4
1
+
cosft
1
+ cosa
=
cosa
cosft
cose.
The
equations
or
a
quadrantal
riangle
ay
be
found
in
a
similar
manner.
By
eliminating
ose
between
Bin(-4+(7)
sinft
cosa+cose
^siniA C)
sinft
cose-cosa
'
A
=
r-,
T-and
\ -r
=
-,
r-
sm^
sma
1
+
cosft
sm^a
sma
1
cosft
we
should
obtain cosft
cos
(7s=
cot
a
sinft
cot^ sinG.
If
the
equation
containing
os-4
+
cos5
were
deduced
from that
containing
osa
+
cosft
analytically,
nstead
of
by
the
use
of
the
polar
triangle,
e
should have
to
determine
cos^,
cos
J?
separately
n
terms
of the
sides.
We
may,
however,
rove
the
former
equation
ndependently
hus
:
TOL.
XVII. 0
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34
MR.
JENKINS,
ON
SPHERICAL
TRIGONOMETRY.
In the
figure
to
the
lemma
proved
above
the
angalar
equation
orresponding
o
the
lemma is
cos
B
sin
D
CA
cos
A
sin
BCD
=
cos
CD A
sin
C,
the
signs
being
most
conveniently
emembered
by
producing
BA
to
C\
and
measuring
the
angles
all
one
way,
that is
cosBsinjDC^
+
cos
CAC
sin
B CD
=
cos
CD
A
smBCA.
To
prove
this,
draw
the
arc
CN
perpendicular
o
ABy
CE
perpendicular
o
ON;
jEa,JBJ8,
i
perpendicular
o
the
planes OBC^
OCA^
OGD
respectively,
nd
EK
perpen-icular
to
OG. Then the
five
points
jB,
S,
a,
K^
P are
on
a
circle
with
EK
for
diameter,
n
a
plane
perpendicular
o
0K\
a^
:
/8S
:
aS
=
sinaE^
:
smlSES
:
sinoM
=
sin
a
:
sin
A CD
:
sin
J?CZ ,
also
Ea=^ECcosCEa
=
ECcosB;
Efi^ECcosCE/3=^
EC
co
Ay
-BS
=
EC
cos
CES
=
EC
cos
CD
A,
and
J5z./8S=^/3.aS
+
^.a/8,
whence
cos
BeinDCA
cos^
sin
J?CZ)
=
cos
CD A
sin
C,
Let ABC be
a
sphericalriangle.
n
BC
produced
ake
CE=CA]
join
EA
and draw
the
arc
FCD
bisecting
he
angleECA^
bisecting
A
at
right
angles
n
Fj
and
cutting
AB
produced
n D.
Then
8in(a b)
_
sin BE
_
smBAE
_
sinBAE
_
nin
DAF
^
sine
~
s'lnBA
sin
BE
A
sin CAE
sin
CAF
*
but
cos^J9Fsin
CAF--
cosAFCsin
CAD
=
cos
A
CF
sin
DAF,
and
cos
AFC
=0y
therefore
sin
(a
+
J)
_
cos
ADF
_
cos
ADF
sinA CB
sine
cosAGF
cos
A
CF
sin
A
CB
__
cos5sin^
CZ 4co8i48in5
CZ
_
(co85+cos^)cos^
7
cos
(^Tr
i
(7j
in
C
sin
i
C
sin C
^
cosB+cos^
1
COS
0
In
a
similar
manner
it
could
be
proved
that
sin
(a J)
_
cosB-
cos-4
sine
~
lH-cos(7
'.
April,
11,
1887.
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(
35
)
ON
THE
SECOND
SOLUTION OF
THE
DIF-ERENTIAL
EQUATION
OF
THE HTPER-
GEOMETKIC
SERIES,
AND THE
SERIES FOR
K\
E\ c.,
IN
ELLIPTIC
FUNCTIONS.
By
Prof.
W.
Wbolsey
Johnton.
1. The
following
olution
of the
case
of
failure
of
one
of
the
ordinary
olutions
of
a
linear
differential
equation
of
the
second
order
in series
was
suggested
to
me
by
Mr.
Forsyth's
solution
of
the
corresponding
ase
in
Legendre'sequation,
Messenger
of Mathematics^
ol.
xvi.,
p.
162.
The
form
of
solution
is
interesting
s
giving
the
series
for
the
functions
K\
E
c.,
in
elliptic
unctions
at
once
in
the form
given
by
Mr.
Glaisher
{Gamh,
Phil.
Proc.j
vol.
v.,
p.
186),
and
accounting
or
the
factors
which
he
has
called the
adjuncts^
occurring
in
the coefficients
of
the
series
{Camb.
Phil.
Proc.^
vol.
v.,
p.
240),
2.
Denoting
x-^hy^^
e
suppose
the
differential
quation
*(^)y-a^ i(^)y=o
(1),
which
is
the
most
general
form
for which
a
relation exists
between
two
consecutive
coefficients of
the series.
The
equation
eing
of the
second
order
we
may
assume
and
i ^
(5)
=^
(5 c)
(^
-
rf),
where
^
is
a
positive
r
negative
onstant.
We
may
also
take
the
exponent
s
as
unity;
or
putting
=x'
and
z
-7-
=y,
we
have
sx ax
8
'
and
equation1)
becomes
which
is
of
the form
(^-a)(^-J)y-2^aj(5-c)(5-cZ;y
0
(2).
C2
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36
PROF.
JOHNSON,
ON
THE
SECOND
SOLUTION
OP THE
We
shalltake
equatioD
2)
as
the
standard
form,
recollecting
that
when
8
is
not
unity
the
roots
a,
b
and
c,
d
are
to
be
divided
by
,
and x' substituted
for
x
in the
final
result.
3.
Putting
in this
equation
we
have
2{(m
+
r-a)(m
+
r-5)^^aj*^
-p{m
+
r'-c)(m
+
r-d)
-4^*^*}
0,
and
since the
coefficient
of
a; *^
must
vanish,
{m+r a) (m+r 5)^, p( n+r
c
1)(m+r-rf l)^^j=0.
This
gives
the
relation
between consecutive
coefficients,
A
_
(m
c
+ r
l)(m-e? +
r-l)
.
and,
when
r
=
0,
(m-a)(m-J).4,
=
0,
whence
w
=
a or
w
=
J.
Let
a
^ 6,
then
taking
ti
=
a,
and
the
solution is
^
a
ft
.
( -c)(a-d)
(o-c)(a-c+l)(ffl-rf)(a- ;+l)
-]
^
1.2(a-6+l)(a-6
+
2)
^^*''
^-J
Again,
interchanging
and
b,
the
second
solution
is
R
r,
(b~c-)(b-d)
(b-c)(b-c
+
l)(b-d)(b-d+l)
1
^
1 .2
(6
-o+l)(i-o+2)
^^*''
^-
J
To
conform
to
the
notation
of the
hypergeometric
eries
let
us
put
a
c=a
c=a \
o-J+l=7
J
(3),
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38
PKOF.
JOHNSON,
ON
THE SECOND SOLUTION OF
THE
The
factor
a; **
a; .T*
a: (l
Alogaj+...),
nd
finally
he
remaining
actor
in
equation
^6)
is
a
function of
h which
we
shall
denote
by
^ (A),
nd
which is such that
by equation
4)
y,=ajXO).
The
complete
ntegral
6)
may
now
be
written
y
=
^^.+B,7;.,4~(l
+
Aloga:+...)a;''[t(0)At'(0)+...]
-^i^,
+
5,7;_.+5y,loga;
Sx '(0)+ (8)
in which
A
is
put
for
the
constant
-4^,
-r*
and
the omitted
terms
are
terms
which vanish with
h.
It
remains
to
express
^'
(0)
as
a
series
in
powers
of
x.
5.
Let
H^
denote
the
coefficient of
{pxy
in the series
-^
(h)
;
that
is,
let
i7 i .
^_(a+A)...(a
r-l+A)(/3-h^)...(/3+r-l-hA)
/i-i,anax2-
(i
+
A)...(r*)(7
+
A)...(7
r-
1 +
A)
Then
f
(A)=S.
S:
(pa;)',
nd
^'(A)=2^
^'
(pa;)'.
ow
_p.r
1
1 1 1 1
1
^^^L +A
^a+l+A'
+r-l+A
*
/3+A'^* '*'/3h-
-l+Aj
When
A
=
0,
this
becomes
^s^irj_
.
j^
1
L.1
La
+
s ^/3
s
1
+
5
7
+
5J
in
which
H^
now
denotes the
coefficient
of
(^pxY
in
the
series
FQi^
ffj
,
j?a;),
quation4).
Thus
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EQUA.TION
OP THE
HTPERaEOMETRIC
SERIES,
C. 39
and,
writing
the
completeintegral
n
equation
(8)
in the
forms
y
=
Ay^
+
JBy,,
e
have for
the
two
independent
nte-rals
when
7
is
an
integral,
and
,. ly
(7-2)1
(y-1)
y
,
,v
where
^-' Ll-7U
/i
1
7/^^
1.2.7(7+1)
/I
.
1
.
1
.
1
111
-
1
\
/
M
,
1
(10),
and
Ty_^
denotes
the
sum
of the first
7
1
terms
of
y,
in
equation(5).
When
7
=
1
we
have
of
course
31
=
0,
and
when
7
=
2,
r,=a;-^^^
a;-\
Thus the second solution consists of three
parts,
the first
of
which is
the
product
of
the
firstsolution
by
log
a?,
the
second is
a
finiteseries
beginning
ith
o?* ^**
nd
ending
with
the
power
a? *,
nd
the third
is
the
secondary
series
y',
which
isthe
same
as
y^
except
that each
coefficient
is
multiplied
y
a
factor which
we
shall callits
adjunct^
onsisting
f
the
sum
of the
reciprocals
f the factors in the
numerator
taken
positively
rid of those in the
denominator
taken
negatively.
The firstcoefficient
of
y^
which
is
1
must
be
considered
as
having
the
adjunct
ero.
6.
The law
of
the
adjuncts
ust
stated is
the
same
as
that
pointed
ut
by
Mr.
Glaisher
in
the
case
of the
series for
JT',
E^
c.,
in the
paper
cited in
1,
although
the
notation
differs
from that of
the
hypergeometric
eries.
The
reason
for
this
persistence
f
the
law
is
readily
explained
s
follows,
and
is
illustrated in
the
examplesgiven
in the
succeeding
sections.
In
the first
place,
f
a
and
^8are
fractions
having
the
same
denominator
tw,
say
a
=
,
/8
,
the
coefficients
n
y^,
m m
m m
m\m
)
m
\m
J
^TT '
*
1.2.7(7+1)
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40
PROF.
JOHNSON,
ON THE SECOND
SOLUTION
OF
THE
may
be
written
in
the
form
m.rny
'
9n.27n.7117
ray
+
tn)
If
the
adjancts
e
now
formed
by
the
same
law
as
before,
each
term
of
each
will
have
one
Tnth
of
its former
value,
and
the result
will
be
the
value
of
u\
so
that
the law
holds
good
for
the
integral
y^j
when the
coefficients
re
written
in
the
new
form.
Again,
if
we
have
occasion
to
introduce
new
factors
into
the
numerators
or
denominators of the coefficientsin
y^^
so
that,
for
instance,
e
write
j^y\
or
the first
solution,
hen each
adjunct
fformed
by
the
same
law
should
contain
the
additional
terms
7
.
If
then
we
take
t
y,
+
(
t)
i
for
the
second
integral,
he
law
of the
adjunct
ill still
hold
good.
7. We
proceed
o
apply
the formulae
given
above in the
case
of
the differential
equation
atisfied
by
K
and
K\
the
independent
ariable
being
the
modulus
k.
This
equation
s
*(l-*0g
(l-3*')|-%
O.
Multiplying
y
k
and
putting
-^=.9,
hence
this
becomes
yy-A; (y
+
2^+l)y
=
0;
and
putting
5=i*
to
reduce
it
to
the
standard
form,
equation
(2),
e
have
Here^^l,
and
a
=
0,
a
=
0,
6
=
0,
a
=
i,
whence
,
'
?=-i.
7
=
1-
:
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=yi=i
+
^.* +^ **+-,
EQUATION
OF THE
HTP RaEOMETRIC
SERIES,
0.
41
Equations(4)
and
(9)give
then for
our
two
integrals
l^.=yilogi +
^*(2+2-
-
1)
*
+
^^]l;P^(2+2
+
|-l-l-4-i)**+....
It is
cnstomaiy
to
write
the
first
series,
hieh
is
the
alue of
,
in
the
form
IT
and
accordingly
e
write the
value
of
^y,,
which is
in
which
the
law
of
the
adjuncts
is followed.
Now
jr'=y.
log4
iy,
so
that the
law
holds for
the
series
for
K'
which
Mr.
Glaisher
writes
in the
form
^'=
log|
j
(log~Hl)*'+^(logf-f+
8.
Consider
next
the
equation
atisfied
by
E and
/',
hieh is
*(i-A^) +(i- o|+*y=o,
or
3'y-k\y-l)y
= 0.
Fatting
=
A^,
this
becomes
where
again
^=
1,
and
a
=
0,
a
=
0
6
=
0,
a
=
-i,
whence
d
=
-i,
7=1.
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42
PROF.
JOHNSON,
ON THE
SECOND
SOLUTION
07 THE
Thus
y,=i+:itiA.+=iifc|iA*+
and
+
^^[^(-2+2+
+
1-1
-l-i-i)A*+....
Bat
y,
which
is
the
value
of
is
usually
ritten
2E_
_
1
r.3
1'.3'.5
,j,
in
which
we
have
dropped
from
the
numerators
of the
coeffi-ients
the
factor
1
which
corresponds
o
the
firstfactor
(
i)
in
the
coefficients
as
firstwritten.
Accordingly
f
we
write
-^(-i
i
+
i
+
J-i-i-i-i)*'
,
the
law
of
the
adjunct
s
not
followed
because
the
term
-
1
in
each
adjunct
is
now
superfluous;
ut
adding
y^
we
get
rid
of these
terms
and
have
the
integral
y.+iy,=i+iy.iogi'-^ (i-i-4)A'
-^(i
+
i+i-i-i-i-i)**-...,
in
which
the
law holds
good.
Now
/'
is
found
to
be
y^
log4
-(^^4-
iy,),
nd thus satisfies
the
law of the
adjuncts.
9. The
equation
atisfied
by
I
and
E^
is,
hen
aj
=
^,
in
which
a=l,
o
=
l,
6
=
0,
a
=
i,
,
whence
^
.
^=-i,
7
=
2.
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44
PROF.
JOHNSON,
ON
THE
SECOND
SOLUTION
OP THE
in which
a
=
2,
a
=
2,
i
=
0,
a=f,
whence
^
.
rf
=
J,
7=3.
Thus
y.=**[i+a*'+rlli
*+ ]
and
y.=y.log* --^^^
l+^fe*-^*
+i*[f|(*+*-^-* *'+iill4(
the
expression
or
Ty-i
in
equation9)
in
this
case
containing
two terms.
The
value of
-
^
^^'^
^
is
iy
thus
^
-iy.-2.4*+2'.4.6*+-'
and
if
we
write
accordingly
he
value
of
i^y,
we
have
i^ry.
iSy.
log
A'-2(l-ii')
+
ig(i
i-i-i)
* +...
In
which,
in
order
to
follow
the
law,
the
terms
1
+
I
-
i
-
i
are
needed
in
each
adjunct.
Adding
therefore
f
of
the
preceding
eries
we
have
iVy,+
/jyi=iVyt
log**-
2
+iA'
in
which
the law of
the
adjuncts
olds
for
the coefficient
of
k^
and
higher
owers,
but
not
for the coefficient
of A;'.
But
this
term,
as
we
have
seen
above,
is
part
of the
expression
or
T^
and
not
of
that for
y'.
The
value
of
JT
+
G^'
is
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46
PROF.
JOHNSON,
ON
THE
SECOND
SOLUTION
OF
THE
where
T^
denotes
the
sum
of
the
first
n
terms
of
the
preceding
series for
y
and
^-' [TFn-)('*^)* '
i.2(n
i)(/i
2)
(^
*
*
^^TTi
*
^TTa)
^ '^^
^ J
in
which
the
terms
in
the
adjuncts
corresponding
o
the
a-
and
/3-factor8
anish
because
these
factors
are
infinite. This
solution
agrees
with
that
of
Hankel,
referred
to
bj
Mr.
Forsyth
in
the
paper
cited
in
I.
12.
When
each
of
the
functions
4
nd
4 ^
is
of
the
second
degree
we
can
derive
a
series
either in
ascending
or
in
descending
owers
of
x.
For
example,
Legendre'squation,
(l-a,')g-2xg
n(n-H)y
=
0,
may
be written
in
which
p
=
1,
and
=
2
;
thus
a
=
i,
a
=
i,
*
=
^'
.
.hence
-=*(^- '
rf=-i(n+l),
7
=
1,
and
bj
equations
4)
and
(5)
the
integrals
re
,
r,
,
(1
- )(2
+
)
.
(l- )(3- )(2+ )(4+ )
-j
J^'-'L^ ^
2:3
^
2XI5
-e+'-j,
and
_
1
,
-n(\+n)
-n(2- )(l+ )(3
4- )
13.
But
if
we
write
the
equation
n the
form
(5-n)(5
+
n
+
l)y-i
(^-1)^
=
0,
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EQUATION
OF
THE
HTPERGEOHETBIC
SEBIES,
C.
47
we
may
take
=
-
2,
and therefore
J
=
-in,
a=^(n
+
l),
whence
^
,
,
c=0, /8
=
Kn
+
2),
and
the
integrals
re
y -^
L
2(2w
+
3)
^
(w
+
l)(r 2)(n
+
3)(n
+
4)
I
*
2.4(2n
+
3)(2n
+
5)
'^
'
y
and
_
fi
w
(n
-
1)
^
w(w-l)(n-2)(n-3)
^
1
^''-^
L^ 2(2n-1)'^
^
2.4(2n-l)(2n-3)
^-J
14. When
n
=
i,
7
=
1,
and
the
two
series
are
identical
;
and when
2n
is
a
positive
dd
integer
is
an
integer
reater
than
1,
so
that
the
series
y^
contains
infinite
coefficients.
In
these
cases,
therefore,
e
have
by equation
9)
or,
taking
\y^
as
the
integral,ecause,
s
explained
n
6,
\y'
will
follow the
adjunct
law
when the coefficients
are
written
as
in
the
expression
or
y^
above,
sy.-yiiogj,
{ ( -2)(n-4)...(l- )}'
^o*k^iy^
where
T^^^
denotes the
sum
of
the first
+
i
terms
of the
preceding
eries for
y,,
and
*^
L
2(2n
+
3)
U
+
1
+2
'
2n
+
3/
- )(
+
2)(n
+
3)
(n
2.4
(2 i
3)
(2n
+
5)
(
+
l)(
+
2)(n+3)(n
+
4)^^^^ ^
J
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48
PROP.
JOHNSON,
ON
THE
SECOND
SOLUTION
OP
THE
where
the
symbol(ad)
written
after
a
coeflSci^nt
enotes
its
adjunct.
This solution is the
same
as
that
given
by
Mr.
Forsyth
in
the
paper
before cited.
When
2/2
is
a
negative
odd
integer
he
solution
may
be
found
in
like
manner,
or
at
once
by putting
n
1
in
place
of
n
in the
solution
just
found,
he result
being
^
-^-
n
,
-n(l-n)
-n(l- )(2-n)(3-n)
1
Vi-'^
L^+2(l-2 )*
2.4(l-2n)(3-2n)
*
+-J
and
iv
-V
Wl
-
{(-2 -3)(-2 -5)...4.2}'(-2 -l)
in which
T_^^
denotes
^.
r
(n
+
l)(
+
2)
*
L
2(2
+
3)
*
+
+
(
+
l)(
2)...(- -3)
1
^2.4...(-2n-3)(2n
3)...(-4)(-2)
J'
and
*^
L2(l-2n)^'^^^
2.4
(1-2/1)
3-2/1)
^^^^
+-J
15.
The
functions
^
and
4 ^
in
equation
1)beingsupposed
real,
he
case we
are
considering,
amely
that
in
which
7
is
an
integer,
annot
arise
when
a
and
,
the
roots
of
^,
are
imaginary
but
c
and
J,
the
roots
of
ff ^^
ay
be
imaginary,
and
a
and
/3
will
then
take the
forms
a
=
/i
+
tV,
jS^fA--
tV.
In
this
case
equations4)
and
(5)
may
be
written
in
the
forms
yi
-A
1
+
/*'+ '_,,(m'
+
i^'XC/^O'+v'}
1.7
| J
+
1.2.7(7+1)
(px)'+...
W,
(^+1-7)'
+
^'
'
(2-7).
I
^
.
=
x--[l
+
(2-
7)
(3
-7)1.2
(i'^J+'-J-CS),
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and
for
the
case
in
wtucb
y
is
an
integerquation
9)
becomes
(
SEP
26
1887
)
EQUiLTXOM
OF
THl ^ K ^|[^ ^UHt5sEBIE8,
C.
49
for
the
nes
.f
ty
(7-l)l(7-2)t
y
,
,0^
^
^
i*'-'
(/*
-7)'+
''l-.K/i-O'+F')}
'^^
' ^
^'
where
*
1.2.7(7+1)
V^
v
(m+O'
+
v'
*
-^-7Ti)(i'-)'+-]
These
forms,
ith
a
change
f
sign
of
v',
are
also
con-
venient
when
a
and
/9
are
irrational
real
numbers.
16.
The
form
of the
solution
when
7
is
an
integer
undergoes
odification
when
either
a
or
/3
is
an
integer
less
than
7.
In
the
first
place,
uppose
one
of
them
to
be
a
positive
nteger
ess
than
7.
We
cannot
now
employ
equation
9),
ecause
the
coeflScient
of
Ty-i
is
infinite;
n
fact,
in
equation
7)
is
now
zero,
and
the
complete
ntegral
(8)
reduces
to
m
which
7
is
a
finite
series
containing
a
terms.
It
is
to
be
noticed
that,
omparing
his
with
the
general
ntegral,
T
is
not
equivalent
o
y,,
but
we
have
in
which
the
coeflScient
f
y,
takes
the
indeterminate
form,
but
has
a
determinate
value
when
7
and
a
are
functions
of
a
single
uantity
nd
become
integers
imultaneously.
This
case,
in
which
the
finite
seriesis
not
the
limiting
value
of the
infinite
series
from
which
it
is
derived,
ccurs
in
the
solution
of
Kiccati's
equation
see
the
Memoir
by
J.
W.
L.
Glaisher,
hil.
Trans.^
881,
p.
771).
YOL. XVII.
E
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50
PBOF.
JOHNSON,
ON
THE
SECOND
SOLUTION
C.
17.
In
the
next
place,
et
a
or
)8
be
zero or
a
negative
integer,
ay
a
=
n
;
then
y^
is
a
finite
series
of
which the
last
term
is
.
-^fi(-n+l)..,(^l)/3(/3-H) .(/3-hfi-l)
fil7(7
+
l)...(7+ -l)
^^^
In
this
case,
supposing
an
integer,
he
series
y
in
the
second
integral
s
not a
finite
series,
although
he
series
y^
from
which it
is
derived
is
finite.
Denoting
the
term
of
y^
just
written
by N{pxf^
the
coefficientis
.
.
/8(y8
l)...(g+ -l)
^^ ^
7(7+l)...(7
n-l)'
or,
when
fi
=
0,
-^^=1.
The
coefficient of the
next
term,
that
is
to
say
of
the
first
term
in
y^
which
vanishes,
s
the
correspondingdjunct
in
y
contains
the
term
,
which
is
infinite,
he
remaining
erms
being
finite.
Thus
the
entire coefficient
of
d^a-) *^
n
y'
is
(n+l)(7
+
7i)*
In
like
manner,
every
succeeding
djunct
ontains the
same
infinite
term,
and
the
entire
coefficient
in
y
is
the
same
that
it would be in y, with the omission of the
zero
factor.
Thus,
in
addition
to
the
part
of
y
formed
by
means
of
adjuncts
it m
the
actual
terms
in
y^,
we
have,
corresponding
o
the
vanishing
part
of
y^,
the infiniteseries
(
+
l)(7
+
)
L
(
+
2)(7
+ n
+
l)^
1.2.(/3 +l)(j8
+ w
+ 2)
(n+2)(n
+
3)(7
+
n+
l)(7+n+2)
ipxy+.
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52
MR.
F.
MOELEY,
ON
PLANE CUBICS
WHICH
INFLECT
With
the first
value
for
m
the cubic is
y
(/
+
^\^y
+
3a,^ )
36,
(y*
3a, ')
0
.
.
.
(3).
Equation
(1)
now
becomes
-
8^(y
+
36,:ry 3a/y)
-
3
(-
\ay
+
{^a^fi^a,')'y
+
2 Aa3^y^-.aAy)
0,
or
\
{\xy'
+
a,^'y)
hfl^x''
(2a, 3
a.)
x*y
+
2
J^or/
=
0,
or
3
JgOry'
Sajb^x^^
or
y'
=
V'
(4).
Since
for
3
real
inflexions
a,
must
be
positive,
rite
a^
=
a*.
Then
the
lines
to
the other
inflexions
are
^
=
axy
so
that
they
are
harmonic
to
the
axes.
Substituting
rom
(4)
in
(3),
or
4y
+
3J,ar+12J,
0
(5),
which
is
the
line
of
inflexions.
Let
PQR
be
the
inflexions,
being
on
the
asymptote,
and
let the
inflexional
tangents
form
a
triangle
qr.
Let
0
be
the node and
let
FQB
and
Fqr
meet
the
ar-axis
in
T
and
L
?
.,^
It
is
calculated
at
once
that the
tangent
at
P is
3 -ar
+
y
+
3J3
0
(6),
and
at
Q
and R
3J,x-8(y+3J3)
9^y.
.(7).
Hence
Itis
manifest that
p
is
on
the
x-axis
;
also
putting
y
=
0in(5),(6),(7),
Op= 2.0T=8.0t.
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ON
CROSSING
THEIR
ASYMPTOTES.
53
Also
from
(6),(7),
Or
and
Oj
are
y
Zax
......(8),
eo
that
if
OQ^
OR^ Oq^
Or
meet
the
asymptote
in
OQ,^OIt,^30q,
=
30r^.
The
equation
to
pQ^
is
1
=
?* ^
or
3J,
3J,x~8(y
+
3J^-^-,
(Box
-y)
=
2
(Sb^x
4y
+
12 J
J.
Hence
pQ^^
Or
and
similarly
?5
Og'
meet
on
P^J?.
These
are
of
course
not
all
indepenaent
esults.
2.
If
the
cubic
inflects
on
crossing
second
asymptote,
say
at
0,
then
Oq
is
parallel
o
the
asymptote,
o
that
y
+
Sax
is
a
factor
of
y'
+
3i,^y
+
Sa'x*.
Hence
the other
factor
is
j/
+
ax^
and
3 ,
4a.
Hence
y
+
ox
=
0
meets
the
curve
at
infinity^
r
H is
at
infinity.
The
metrical
relations
are
now
very
simple.
Let
OAy
OB
be
parallel
o
the
asymptotes
at
P,
Q.
Let
PFaqj
QFffp
be the
mflexional
tangents,
then,
mce
0p
=
20A=^S0a
and
Oq^20B=SO^f
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54
MR. F.
MOBLEY,
ON
PLANE
CUBXCS
WHICH
INFLECT
aP^pq^
AB
are
parallel,
nd
Pi
2-45=
34P=
35$,
therefore
BP=PQ=:
QA
;
hence
pB^
OQ
are
parallel,
nd also
qAy
OP.
Also
in
the
asymptotes
meet
in
E^
PDxDA^PEx
OA^PQxAB^l
:
3,
therefore
PD
=
\PQ
=
QD.
Also
PQ
is
cut
by
OD
in
a
ratio
=
op :
eJS?=a^
:
PQ^^F:
FQ,
therefore
-Flies
on
OE.
And
since
EDiDO=^EQ\BO^UZ
and
0F\
FE^Op:
QE^Z\4.
it
follows
that
DF\
FO^
OF:
FE
or
OF'^DF.EF.
.Clearly
ll
the
lines
in
the
figure,
nd
all
got
by
joining
the
points
n
the
figure,
re
cut
in
very
simple
atios.
(jriven
0,
P, Q,
the
construction
for
the
asymptotes
is
:
take
on
PQ
points
^
B such
that
BP=PQ^
QA^
then
OA^
OB
are
in
the
direction
of
the
asymptotes.
For
the
tangents
:
draw
Bp
parallel
o
OQ
to
meet
AO.
Then
pQ
is
the
tangent
at
Q.
The
cubic
cannot
inflect
on
crossing
ll
asymptotes
unless
each
inflexion
is
at
infinity,
nd
the
asymptotes
are
then
the
inflexional
tangents.
3.
A
non-singular
ubic
has
of
course
much
more
freedom.
But
if
its
inflects
on
crossing
ach
asymptote
the
line
of
inflexions
has
an
envelope
hich
it
is
proposed
o
investigate.
Let
J5r=ar+y+ =0
be
the
line
at
infinity,
=ax+by-\-cz=0
be
the
line
joining
he
points
here
the
asymptotes
meet
the
curve,
and
let
xyz
=
0
be the
asymptotes.
Then the
cubic
is
LK^A-^mxyz^O
(9).
The
first
differential
oeiBcients
are
aJ5r*+
22iir+
6my ,
...
.
The
second
are,
omitting
factor
2,
2aX'+i,
...,
(6
+
c)
/iL-t-i
+
3mx,
....
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ON
CROSSING
THEIR ASYMPTOTES.
55
The
Hessian formed
.from
these
lastsix
works
down
to
2^{(a-J)(a-c)x+...}
+
6mK{(h
+
c)yz''
ar'
-f
...}
+
37ni(2y -ar'+...)
+
18m'jry
0
(10).
Letar
=
0
and i
=
0,
therefore
2
(y
+
)'
{(ft-c)
ft-a)y
+
(c-a)
(c-
J)z]
+
6in(y
){(
c)y5?-
y'-c *}
0,
or
writing,
ince
iy
-f
c
=
0,
y
=
c,
=
6,
(J-c)'{-c(J-a)-J(c-a)}3m(6-c){-Jc(J+c)-Jc'-cJ'},
or
(b
c)'
{2 C
a
(6
+
c)}
GrnJc
(J
+
c)
0
...
(11),
i
cs=0
would
mean
an
inflexionat
infinity].
Suppose
two
such
conditionshold
:
(a-c)
{2ca-
J
(c+
a)}
+
6mca
(c
+
a)=:
0,
(a
-
by{2ba
-c(b +
a)}+
Qmab
(ai'b)=^
.
Subtracting
nd
dividingy
c
-
J,
ic(2a-J-c)
+
6ma(6
+
c)+6wa'
+
3a'~2a (2c
2J)
+
2(6VJc
+
o')a-aJc=:0.
Let
a,
i,
c
be
the
roots
of
x*-px^
+
qx-r=iO
(12).
Then
since
3aic+6ma(a+J+c)+3a'-4a (
3)f
0(^+0*)-
Jc( +c)=0,
therefore
3r
+
6map
+
3a*
4a'
(p
-
a)
+
2a(2 *-2j-a )-^(; -a
0,
and
if the
3
such
conditions
hold,
,
,
e
are
given
by
4rar
+
Gmp^'
+
5ar*
4;?x'
2ar*
(p
2
j)
pr
=
0.
Dividing
his
by (12)
the
remainder
is
3
(/
-
3j
+
2mp)
x'
+
(yr
-;?j)
r,
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66
ME. F.
MOBLET,
ON PLANE
CUBICS
WHic^
INFLECT.
and
therefore
the
only
conditions
which
the
constaats
must
satisfy
re
37
'
Jc
+ ca
+
ai-a'-
c'
2m
=
-*
^
s=
7
,
p
a-\-o-\-c
'
and
pq
=
dr
(13),
or
a(J-c)'
+
(c-a)*
+ c(a- )*=b.
Given
the
asymptotes,
he
envelope
f the Hne of inflexions
is
the
envelope
f
aar
-f
iy
-h
C2
=
0
subject
o
(13).
It
may
of
course
be
got
by
eliminating
,
but the
followingymmetrical
method
gives
the
result
in
a
better
form.
Equating
dif-erential
coefficients
with
regard
to
a,
J,
c,
and
neglecting
a
constant,
;r
s=
2
+
(J
+
c)p
-
9bc
or
pa*
(2?'
2
-
^)
a
+
9r
=
0,
and
a'-2?a'
ga-r
=
0.
From
these
we
must
eliminate
a.
Multiply
he
second
by
9
and
add
;
9a*-8pa-(p'-8g'-a:)
0,
therefore
{9
(p
+
2
-
a:)
8;?'}
(p*
j
-
x)
(^
-
Sg
-
x)
+
72pr}
{p
(
p*
82
-
ar) 81r}',
or-9a:^+18a:'(/-32)-^^(V+lS/?-9.152*
8p*2-722'=0,
or
x^'-Fx^+Qx^B^O,
where
i'=2(p*-32),
Q
=/
+
2/2
-
152' (/
-
32)
(; 52),
5
=
1
(/2-9A
therefore
y'-32
=
iP,
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MR. S.
ROBERTS,
NOTE
ON
CERTAIN
THEOREMS.
57
therefore
( -*^)^+ )
^4'
or
(4g-P )(12 ?
+
P0
=
288PH
(14).
The
envelope
s
therefore
a
quartic.
It
meetg
the
line
at
infinity,
=0,
on
^
=
0,
which
is
the
minimum
ellipse
bout
the
triangle.
It
meets
the
sides,
=
0,
on
4^
=
P*,
but
this touches
at
the
middle
points,
nd
also
on
12
Q
-i-
JF*
0.
Hence
it
touches
a:=0
at
the
middle
point
nd
cuts
its
where
12y
+
(y+i5)
0,
r
y
+
70
=
4^30.
Hence
the
shape
is
as
in
the
figure.
Bath
College.
NOTE
ON
CERTAIN
THEOREMS
RELATING
TO
THE
POLAR
CIRCLE
OF
A
TRIANGLE
AND
FEUERBACH'S
THEOREM
ON
THE
NINE-POINT
CIRCLE.
By
Samuel
BoberU.
1.
If with
reference
to
a
triangle
he
centres
of the
cir-umscribed
and
inscribed
circles
and
the
orthocentre
are
respectively
,
7,
P,
and
the
radii
of those circles
and
the
polar
ircle
are
respectively
B,
,
p,
then
the
following
ell-
known
relations
hold
:
0P;
=
i?'+2p ,
7P =.2r'
+
p'.
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60
PROF.
CAYLEY,
ON
THE
SYSTEM
OF
THREE
CIRCLES
It
is
easy
to
establish
also
without
direct
reference
to
conies
the
following
heorem
:
If
A^
By
C,
D,
E^
F
denote
six
points
or
lines,
nd
that
any
four
of the
points
onnect
equi-anharmonically
ith the
remaining
two,
or
any
four of the lines intersect
equi-
anharmonically
ith the
remaining
two,
and
if
any
five
of
the
points
ie
on,
or
any
five
of
the
lines touch
a
circle,
he
remaining
point
lies
on,
or
the
remaining
ine
touches
the
same
circle.
Now
let
ABG
be
a
triangle
elf-conjugate
ith
regard
to
a
circle
(/S).
Take
any
point
D
on
the
circle
circum-cribing
the
triangle
BG
and draw
the
polar
of
D
with
respect
to
(/S).Suppose
that
one
of
the
intersections
of
this
polar
ith
the
circle
circumscribing
he
triangle
s
E,
Then
take
F
the
pole
of
the line
joining
,
E
with
respect
to
(/8).
The
triangle
EF
will
be
self-conjugate
ith
regard
to
(5),
and
by
the
above
given
theorems
F
lies
on
the
circle
through
ABGDE^
and
must
be
the
second
intersection
of the
polar
of
D
with
that
circle.
This
is
theorem
(I),
nd
theorem
(II)
can
be
similarly
roved.*
ON
THE
SYSTEM
OF
THREE
CIRCLES
WHICH
CUT
EACH
OTHER
AT
GIVEN
ANGLES
AND
HAVE
THEIR
CENTRES
IN
A
LINE.
By
Prof.
Cayley,
In
the
system
considered
in
the
paper
System
of
equations
for three
given
circleswhich
cut
each other
at
given
angles,
Messenger
t.
XVII.
pp.
18-21.,
we
may
consider
the
particular
case
where
the
centres
of the
circles
are
in
a
line.
The
condition
in
order
that
this
may
be
so
is
obviously
sin
{A
a)
cosF\
sin
[A
a)
sini^',
in
a
=
0,
sin
{B
-
/3)
os
0\
sin
{B
-
/3)
sin
G\
sin
^
sin
((7-7)
cosjH',
in
((7
-7)
sini?',
in
7
that
is,
sin(5-iS)sin((7-7)sina8in((?'-5')+...=
;
*
See
Cremona's
Geometrie
t.
par
Dewulf,
p.
224.
The
bare
statement
that
if
one
triangle
an
be
circumscribed
abont
or
inscribed
in
a
circle and self
-conjugate
with
regard
to
a
second
circle,
n
infinity
f
such
triangles
an
be
drawn,
does
not
fullj
express
the
result.
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WHICH
CUT
EACH
OTHER
AT
GIVEN
ANGLES.
61
or
since
sin
(G'-S ),
^miW-^F')
^m[r
-
G')
are
^sin^,
sinjS,
in
(7
respectively,
his
is
sin(jB
8)
8in((7
)
sin-4
sina
+
...=
0,
viz.
this is
sin
A
sin
a
sin
B
sin
yS
sin
C^
sin
7
_
sin(4-a)
^
sin(jB-/3)
^
sin(a-
7)
'
or as
this
may
also be be
written
1
1
2
cot-4
cota
cot
5-
cot
)8
cot(7
cot7~
Bat
assuming
this
equation
o
be
satisfied,
t does
not
appear
that
there
is
any
simple
expression
or
the
equation
of
the
line
through
the
three
centres
;
nor
would
it be
easy
to
transform
the
equations
o
as
to
have
this
line
for
one
of
the
axes.
The
case
in
question
(which
is
a
very
important
ne
from
its
connexion
with
Poincar^'s
theory
f
the
Fuchsian
functions)
may
be considered
independently.
Taking
the line of
centres
for the axis of
ar,and
writing
a,
/8,
7
for
the
abscissae
of
the
centres,
and
P,
Q^
R for
the
radii,
hen
the
equations
f the
circles
are
(;r-a)' /=P',
(x-/S)
/= 2',
and then if
the
pairs
of circles
cut at
the
anglesA^ B^
C
respectively,
e
have
Q'
+
2QBcofiA
+
B':={l3^y)\
R
+
2iZPcos5
+
P'
=
(7
-
a)',
P'
+
2P(2co8O+0 =(a-.i8)',
which
are
the
equationsonnecting
,
/8,
7,
P,
Q,
B.
See
the
figure.
It
is
to
be
remarked
in
regard
hereto
that
if
-4,
P,
G
are
used
to
denote
the
interior
angles
of
the
curvilinear
triangle
ABCy
then
the
angles
yA/3^
olBj^
0Ca
are
=7r
-4,
P,
G
respectively
whence
if
P,
Q,
B
were
used
to
denote
the
three
radii
taken
positively,
he
first
equation
ould
be
g*
+
2
(^P
cos^
4-
P'
=
(/8
7)*,
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62
PROF.
CAYLET,
ON
THE
SYSTEM
OF
THBEE CIBCLES
as
aboTe
;
but the
other
two
equations
ould
be
i?-2i?Pcos5+P'
=
(7-a)',
P'-2P(2co8(7+(2
=
(a-^)';
hence,
in
order
that
the
equations
ay
be
as
above,
it
is
necessaiy
that
P
denote
the
radius
of the
circle,
entre
a,
taken
negatively
and it
in
fact
appears
that
in
a
limiting
ase
afterwards
considered
the
value
of
P
comes
out
negative.
Similarly
s
regards
the
curvilinear
triangleBC]
here
Ay
B{^B^)
and
C(= (T)
are
the
interior
angles
of
the
triangle;
and
the
radius
of
the
circle,
entre
o^,
must
be
regarded
s
negative.
Considering
^B^Oba
given,
e
have
an
equation
etween
the
radii
P,
$,
R. In
fact
this is
at
once
obtained
in
the
irrational
form
\/(-3r)
Vl^O
+
Vl-^j^O,
and
proceeding
o
rationalise
this,
e
obtain
-2V(rz)=r+z-x,
that
is
-V{(P'
+
2Pi?co85+i?)
(P'
+ 2PacosO+
^)}
=
P'
+
P(^
cos
(7+
^
cosP)
-
^5
cosX
Hence,squaring
nd
reducing,
e
find without
difficulty
0=
(2'i?sin ^
i2'P'sin'P+
PVsm'(7
+
2P'^5
(cos^
+
cosP
cos
O)
+
2P(^R
(cos^+
cos
(7
cos
-4)
+ 2PQR
(cos
C+
cos-4
cosP,
or
puttmg
herem
P, Q^
B^ r-
i
y-
,
*his
la
(cos^
+
cosBcosC cosP+cos(7cos4
^'^'^'
sini^ sin
0
'
sinCsin^
'
cosO+
cos-4cos
sin
^ sin
P
^(f,i7,5)'
Oj
and it
may
be
remarked
that
in
this
quadric
form
the
three
coefficients
are
each
less than
1,
or
each
greater
than
1,
according
s-4
+
P+
C7 7r,
r
-4+P+
C ir.
Suppose
r,
A-k-B-{-
C v\
the coefficients
are
here
B
cosX,
cos/i,
cosK,
the
form
is
(1,
1,1,
cosX,
cos
/A,
cosvXf
17,
?)*,
that
is
(f
+
17
cos
F
+
?
cos/i*)'
[if
smV
+
217?
cos\
+
?'
8in ,
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r{8m'
WHICH
CUT EACH
OTHER
AT
aiVEN
ANGLES.
63
namely
thisis
,- ^ ,,
(
.
^cosX-
cos/Ltcosv)*
(f
+
17COBf+?cos/ia)'+
|i7Bmv+?
^^
1
(cos
COB
/14
cos
V\
')
^y
Jp
where the
last
term
is
=
-^
fsinV
sinV
(cosX
cos
ll
cos
v)*]
sm
V
^
' /
J
I
where
the
coefficient
n
{
}
is
=
1
cos'X
cosV
cosV
+ 2
cosX
cos/x
cos
v,
namely,
substituting
or
cosX,
cos/i,
cosv
their
values,
his
is
=
.
,
^
.
iT
At
(l-cosM-
cos'-B-
cos'
(7-2COS-4
cos5cos
0)'.
sm
-4
sm
5sm C
'
It
thus
appears
that
the
form
is
the
sum
of
three
squares,
and
is
thus
constantlyositive
it
therefore
only
vanishes for
imaginary
values of
the radii
;
or
the
case
does
not
arise for
any
real
figure.
Hence,
2'',
f
the
figure
be
real,
4
+
5+
(7 7r,
that is
the
sum
of the
angles
of
the
curvilinear
triangle
s
less
than
two
right
angles:
the
radii
are
connected
as
above
by
the
equation
(
cos
^+
cos
J?
cos
(7
cos
g
+
cos
g
cos
.4
'^'^
sin^sinO
'
sinOsin^
cos
G
+
cos
A
cos
B\
/sin
A
sin5
sin
G\
sin.4
sinjB
g\
/sin^
sin5
sInC\*_
in
which
form
the
three
coefficients
re
each
greater
than
1.
Eestoring
herein
f
,
i;,
f
and
regarding
hese
as
rectangular
coordinates,
he
equationepresents
a
real
cone
which
might
be
constructed
without
difficulty,
nd
then
taking
f,
i;,
f as
the coordinates
of
any
particularoint
n
the
conicalsurface
1
T^ ^ ^^
9in-4
sinJ5
sin
CJ'
^m
.
,
we
have
P, Q^
-B
=
^,
,
u-
Obviously,
oints
on
the
same
generating
ine
of
the
cone
give
values
of
P, Q^
R
which
difier
in
their
absolute
magnitudesnly,
their
ratios
being
the
same
:
the
original
quations
n
fact
remain
unaltered
when
P,
Q,
jS,
a,
/3,
7
are
each
affected
with
any
common
factor.
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64
PROF.
CAYLEY,
ON
THE SYSTEM
OP
THREE
CIBCLES
Supposing
P, Qy
B
taken
so
as
to
satisfy
he
equation
n
question,
hen
taking
the radicals
'JiQ'
2
QR
cos
A
+
IT),
J{IP-{'2RPcosB'\-r)y
V(P*
+
2PCco8
0+^y
with
the
proper
signs
we
have
a
sum
=0,
and
these
give
the
values of
)8
7^
7
a,
a
^
respectively
and
the
construction
of
the
figure
ould be thus
completed.
I
look
at
the
question
rom
a
different
point
of
viewj
taking
Q^
E,
0
y
such
as
to
satisfy
he
first
equation
^'
4
2
5 cos
id
+
JB'=t
(/8
7)%
(that
i
starting
from the
two
circles
{x
-
^)'
+
y*= Q'f
{x-yY
+
y^^R^
which
cut
each other
at
a
given
angle-4),
then the
problem
is
to
find
a
circle
{x
a)'
y*
=
P*,
cutting
these
at
givenangles(7,
B
respectively
and
to
determine
the
coordinate
of
the
centre
a,
and
radius
P,
we
have th6
remarining
wo
equations
iZ''
2iilP
cos
P
+
P
=
(7
a) ,
P'+2P(2cosa+(2 =(a-/S) .
namely,
considering
,
P
as
the
coordinates
of
a
point
(io^
reference
to
the
foregoing
rigin
and
axes),
and
for
greater
clearness
writing
=
x,
P=y
we
have
y*
+
2yi?
cosP
+
i2
(x
-
7)'
0,
y'
+
2y(2cosa+(2*-(x-/S)'=a,
or a
these
may
be
written
{j
+
R
COB
By
-
(x
-
7)'
- R'
sin*P,
(y
4
0
cos
Cy
-
(x
-
P)*=
-
Q^
sin'G,
namely^
the
first
of
these
equations
denotes
a
rectangular
hyperbola,
oordinates
of
centre
(x
=
7,
y
=
P
cos
P),
transverse
semi-axes
=P8inP;
and
the
second
of
them
a
rectangular
yperbola
oordinates
of
centre
(x=/8,
y=-Qco8(7),
transverse
semi-axes
^QsinC:
as
similar and
similarly
situate
hyperbolas,
hese intersect in
two
points
nly
;
namely,
the
points
re
the
intersections
of either
of them
with
the
common
chord
2y
(R
cosP-
Q
cos
(7)
2
(7
^)
{x i
(7
+
/3)},+P'
Q'
=
0.
It
is
possible
o construct
a
circle
through
the
two
points
of
intersection,
nd
so
to
obtain
these
points
s
the
intersections
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%
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66
PROF.
CAYLEY,
ON
THE
SYSTEM
OF
THREE
CIRCLES
But
we
have
and
this
equation
hus
becomes
- =
16)8'
'W
{(cos^
cos
B
cos C)'
sin 5 sin'
G]
=16^8'
(^R\-
l+cosM+cos'^+cos'
(7+2cos^cos5cos
G).
We
have
therefore
2
{4/3*
[R
cos5-
e
cos
C)'}
+
{4/9*
5
cos5+
Q
cos
C)
-
{R
cos5-
^
cos
(7)
[R -
Q')}
=
4/8
Q5
V{-
(
1
-
cosM
-
cos*J? cos*
(7-
2
cos^ cos5
cos
C)
}
4(5cos5-
^cos(7)y
+
ii *-
G'
=
4i8x,
or
completing
he
reduction
by
the
substitution
of
the
value
of
4/3*,
his
is
y
{(^
sin (7-f
^
sin'5)
2Qfi(cos^
+
cos5cos
G)}
^
QR{Q
(cos5+
cos
G
cos^)
+
R
(cos
(7+
cos^
cos
5)}
=
4/SQ5V{-
(1-
cosM-
cos'J?-
cos'
(7-
2
cos^
cos5cos(7)},
viz.
we
have
thus
two
values
of
the
radius
y
(=P)
;
and
to
each
of
these
there
corresponds
single
value
of
the
abscissa
x,
given
by
4)8x
^'-
^
+ 2(5co3J5-.
QcosC)y.
The
two values
become
equal
if
-4
4
jB
0=7r;
in
this
case
the
three
circles
meet
in
a
pair
of
points
(x^,
y,),
(x,,
y^).
In
fact,
riting
4
+
5+
C=7r,
and
thence
C08-4
=
cos
(5+
(7),
cos
J?
cos
(7
+
sinjB
sin
(7,
c.,
we
find
{G
sin'0+
2^^
(cos^+
cosJ?cos
G)
+
iZ'
sin'P}
\^QR[Q
(cosjB
cos
C
cos-4)
R
(cos
C+
cos^
cos5)}
0,
that
is
(Q
sin
(7+5
sin
J?)'y
QR
(^
sin
0+
R
sin5)
sin-4
=
0,
or
throwing
ut
the factor
[QAnG-^-R
sin
J?)
this
is
[Q
sin
(7+
R
sin
5)
y
+
$5
sin^
=
0,
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WHICH
CUT
ACH
OTHER
AT
GIVEN
ANGLES.
67
and
we
then
have
-
2
(
COB^
-QcmO)
QB
fimA}.
The
term
in
{ }
is
here
.
B?{AnB)
\-B?Q{
8in(7-2
8in^co8J5)
+
5^
(-
8in5+
2
sin^
cos
C)
+
i^[-AnC),
which
Is
=
i?( sin5)
+
5'Q
(-
sin
(7+
2
sin
J?
C08-4)
+
- '(
8in5-2
8in(7cos-4)
=
(5*
+
0'
+
2i?(2
cos^)
[R
AnB^Q
sin
(7),
4^(^sin5-^sin(7),
or
finally
'^ ^sin^H-
sinC
_/3(Jg8in^-gsing)
^
^sinjB+QsinC?
*
In
these
equations
y,
x
should be
replacedby
P,
a
respectively;
nd
in
obtaining
hem
it
was
assumed
that
7
=
-
/3;
restoring
he
general
values
of
/8,
the
equations
become
p
^jRsin^
iZsinJS+Csina*
^
i^o..A_4(^-7)(^8mg- (g8ing)
-i(/S+7)
i23in +(2sinO
'
F2
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68
PROF.
CAYLEY,
ON THE SYSTEM
OF
THREE CIRCLES.
viz.
this last
equation
ecomes
*
EsinB-{-QainG
'
or
say
a
{B
sin5+
Q
sin
C)
-
I3B
smB-yQ
sin
(7=
0,
y^hich
by
means
of
the
first
equation
ecomes
a-^sin^
i8iisinjB+7(28ina=0.
It
thus
appears
that
the
two
equations
re
sin^ BinJ5 sin
(7
T
^
Q
^
B
=
0,
asin^
8
H\nB
y inC
^
-p-
* - -+
S-=^'
viz.
these
equations,
herein
-4
+
-B+C7=7r,
belong
to
the
case
where
the three circles
intersect
in the
same
pair
of
points;
ence if
the
coordinates
x, y
refer
to
the
points
f
intersection of the three
circles,
e
have
simultaneously
he
equations
f
the
three
circles,
nd
the three
equations
hich
determine
the
angles
at
which
they
intersect,
iz.
we
have
the
six
equations
(aj-a)'
y*=PV
Q'
+
B'-\-2QBcosA
=
{^-'
y)%
{x-j3Y+y'^Q\
fi'
+
P'
+
25Pcos5
=
(7-a)',
(aj-
yy'\-y'=-B\
P +
(? 2P(?
cos
C
=
(a
-/3)V
viz.
from these six
equations,
ith
the
condition
A
+
B-\-
G=irj
it
must
be
possible
o
deduce
the
last-mentioned
pair
of
equations.
In the
general
case,
where
A-\-B-\-
G ir,
and the
three
circles
do
not
meet
in
a
point,
then
taking
the
circles
(x
/8)*
y
=
Q^
(x
7)*
-f
y^^B*
to
be circles
cutting
ach
other
at
the
angleA,
or, what is the
same
thing,
he values
Qy
By
/8,
to
be
such
as
to
satisfy
he
relation
^ +
5'
+
2Q^cos^
=
(i8-7)%-
the
two
equations
or the
determination
of the
abscissa
of
the
centre
a,
and
the
radius
P
of
the
remaining
circle
give,
by
what
precedes
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MR.
DAWSON,
A
THEOREM IN
HIGHER
ALaEBRA.
69
2
{(yS
7)
{B
COS
B-
Q
cos
(7)*}P
+
{(/8-7)'(^co85+
CcosC)-
(Bcos5-
gco8C)(ii'-
C'j}
= 2(/8-7)C^V{-(l-co8*^-cos'5-cos (7-2cos^co85cos(7)}
4(i2cosB-(2oo8(7)P+(iZ'-
e2')
(i8-7)(2a-i8-7),
viz.
we
have thus the
two
circles
(x
a)*
+
y*
=
P*,
each
of
them
cutting
he
circles
(x-/8)*+y*=
Q*,
and
(x
-
7)*
+
y*
jB'
at
the
angles
C^
B
respectively.
NOTE
ON A THEOREM IN HIGHER ALGEBRA.
By
H. O,
Dawson,
B,A.
The
theorem
concerns
the
equivalence
f
the
operators
and
o^2a,^^
3a.A
c,
when
applied
o
a
function
of the
quantities
4
,
A^^
A^...A^j
where the
transformation
x=pX'^qYy
y=p'X
+
qY
trans-orms
the
binary
form
iaja.,..a)
x.
yY
into
the form
This
equivalence,
hicU
is
of
importance
n
the
theory
of
the
covariants of
the
form,
is
proved
somewhat
tediously
n
Faa-de-Bruno's
Theorie
des
formes
btnaires^
p. 189,
the
equivalence
aving
been
previously
tated in
Section
77.
The
proof
now
given
seems
interesting,
ot
only
as
being
expeditious,
ut
also
as
pointing
out
very
precisely
he
raison
(TStre of
the
equivalence
n
question,amely,
that it
is
a
necessary consequence
of the linear
character of
the
transformation.
The
proof
in
question
s
as
follows
:
Let
us
suppose
that the
transformation
x={p^Xp')X+{q-^\q')Y,
y=/J:+}'r
is
made,
then
we
get
a
new
form
whose
coefficients
re
dp
dq
,
X
being
supposed
o
be
indefinitely
mall.
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70
MR.
DAWSON,
NOTE
ON
A THEOREM
Now,
owing
to
the
linear
character
of
the
transformation,
we
might
have
proceeded
in
a
different
manner,
thus^
we
might
have
put
fl5
SB
u
+
Xv,
y
=
t?,
and
afterwards
transform
by
the
substitution
The
first
of
these
two
transformations
gives
s
a
form
whose
coefficients
re
a^,
a^
+
2\a^j
a
+
3\a^j
c.,
supposing
*
to
be
inappreciable,
hilst
the
second
^ives
-4/,
k/,
..,^/)
X,
F)*,
where
A^\
A^^...yA^
are
just
the
same
functions
of
a^,
a^
+
2Xa5,
a^
+
3Xag,
c.,
that
-4^,
^...A^
re
of
a^,
a^...a^.
But,
remembering
ur
former
view
of
the
transformation
we
have the theorem
in
question,
or
where
y|
+
j'|=8^.
The
proof
is
easily
pplied
o
other
forms;
take
for
an
example
the
ternary
n'
and
write
it
as
follows
:
Suppose
that
when
we
make
the
transformation
we
obtain
{A)
(Z,
F,
Z)\
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IN
HTQHEK ALaEBRA.
71
Let
as
now
make
the tranformatlon
=
(X,
+
^A,
+
f,X,)
+
(^,
+
^A*.
^At.)
+
(v, 0v,
+
ipv,),
we
thus
obtain
a
form
whose coefficients
re
where
g;
X.
A
+
^,^
^,^
S.= c.;
again,
the transformation
could
have
been
performed
in
another
order,amely,
by
first
using
the
substitution
and
afterwards
putting
Now
if
we
make
the
first
transformation,
e
easily
ee
that
the
coefficient
a^^a
changes
o
[This
is
easily
obtained,
for
0
and
^
are
of
course
course
supposed
o
be
very
small
thoughperfectlyrbitrary
we
are
therefore at
liberty
o
neglect
all
terms
higher
than
the
first
(in
0
and
0)
of such
a term
as
{U+0V+4 W)'^.
Eesuming,
if
we
perform
the second
transformation
we
obt iin
a
form
[A')
[XYZfy
whose
coefficients
re
the
same
functions
of the
quantities
that
A^, -4,0,..
c.
are
of
a^, r,
c.,
and therefore
dA
where the
2
applies
o
all such
values
of
Up^^
as
can
enter
into
Ap^tr*
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72
MR.
DAWSON,
A THEOBEM
IN
HIGHEB
ALGEBRA,
But
we
obtained
before
tbe
equation
HeQoe
we
have
an
equivalence
etwen
the
operators
and
between
for
this
reason,
that
6
and
^
are
perfectly
rbitrary.
By writing
he form in powers of y^
we can
obtain
two
operators
hich
will be
equivalent
o
^
d
d
d
^
^
d
d
d
similarly
e
can
escpress
the
operators
^
d
d
d
-,
d
d
d
in
terms
of
the
coefficients.
The
invariants
of
the
form
satisfying
he
six
equations
V
=
0,
8/
=
0,
8/'
0,
8/'
0,
8/''
0,
8/'^=
are
the-common
solutions
of
the
equations
hus
found.
If
we
take
the
case
of
the
quadric
orm
(a,J,
c,/,
^
h)
{ocy^Y
our
equations
re
d
.d
^
d
d
^^d
d
Interchange
?,
y
and
we
get
Interchange
,
ssj
'd/^^dh-'^^M^'^dh^^^di^'d^
mi
BQ
on
for
other
forms.
Qhivrt'B
ollege,
Cambridge,
Vtt/i/0,
1887.
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PROF.
CAYLET,
ON
SYSTEMS
OP
RAYS.
75
the
ra;^8
at
rightangles.Evidently
,
/8,
=
8
F,
8^F,
S^V
reBpectiveljy
nd the
function
V
satisfiesthe
partial
ifferential
equation
(8.F)'
+
(S,F)'
(8.F)'
1.
Hamilton
in effect considers
only
systems
of
rays
of
the
form in
question,
iz. those which
are
the normals of
a
surface
(or,
what is
the
same
thing,
the normals of
a
system
of
parallel
urfaces),
nd
it
is
such
a
system
which issaid
to
have
the
characteristic
function
V.
It
is
shown that
a
system
of
rays
originally
f
this kind
remains
a
system
of
this
kind
after
any
number of
reflexions
(or
ordinary
refractions)
in
particular
f the
rays
originally
manate
from
a
point,
hen,
after any
number
of
reflexions
at
mirrors of any form what-ver,
they
are
a
system
of
rays
cut
at
rightanglesby
a
surface.
And
moreover,
there
is
given
for
the surface
a
simple
construction,
iz.
starting
rom
any
surface
which
cuts
the
rays
at
rightangles,
nd
measuring
off
on
the
path
of
each
ray
(as
reflected
at
the
mirror
or
succession
of
mirrors)
ne
and
the
same
arbitrary
istance,
e
have
a
set
of
points
forming
surface
which
cuts
at
right
angles
the
system
of
rays
as
reflected
at
the mirror
or
last
of
the
mirrors.
The
ray-systems
considered
by
Hamilton
are
thus the
normals
of
a
surface F
c
=
0,
and
a
large
part
of
the
properties
f
the
system
are
thus
included
under the known
theory
of
the
normals
of
a
surface
;
it
may
be
remarked
that
the
analytical
ormulaB
are
somewhat
simplified
y
the
circum-tance
that
V
instead
of
being
(as
usual)
an
arbitrary
function
of
(a;,
y,
z)
is
a
function
satisfying
he
partial
differential
equation
(8 F)'+
(8^F)*+
Sr)*
=
l.
In
par-
ticular
we
have
the theorem that each
ray
is
intersected
by
two
consecutive
rays
in
foci
which
are
the
centres
of
curva-ure
of
the
normal
surface
;
the
intersecting
ays
are
rays
proceeding
rom the
curves
of
curvature
of
the
normal
surface,
c.
There
are
other
properties
asily
educible
from,
but
not
actually
ncluded
in,
the
theory
of the normals
;
for
instance,
the
intersecting
ays
aforesaid
are
rays
proceeding
from certain
curves
on
the
mirror,
analogous
o,
but
which
obviously
re
not,
the
curves
of
curvature
of the
mirror.
The natural mode of
treatment
of this
part
of the
theory
s
to
regard
the
rays
as
proceeding
ot
from the
normal
surface,
but
from
the
mirror,
and
(by
an
investigation
erfectly
analogous
o
that
for
the normals of
a
surface)
o
enquire
s
to
the
intersection
of
the
ray
by
rays
proceeding
rom
con-ecutive
points
f
the
mirror
;
it
would
thus
appear
that
there
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76
PBOP.
CAYLEY,
ON
SYSTEMS
OP
RAYS.
are
on
the
mirror
two
directions,
uch
that
proceeding
long
either of
them
to
a
consecutive
point,
he
ray
from
the
original
oint
is
intersected
by
the
ray
from
the
consecutive
point,
ut
that these
directions
are
not
in
general
t
right
angles,
c.
But
in
regard
to
such
an
investigation,
he
restriction
introduced
by
the
Hamiltonian
theory
is
altogether
nne-essary
;
there
is
no
occasion
to
consider
the
rays
which
proceed
rom
the
several
points
f
the
pairror
as
being
rays
which
are
the
normals of
a
surface,
nd
the
question
s
con-idered
from
the
more
general
oint
of
view
as
well
by
Malus
in his
Th^orie de
la
Double
Befraction,
c., Paris,
1810,
as
more
recently
by
Eummer
in
the
Memoir
AUgemeine
Theorie
der
gradhnigentrahlensysteme,
relUy
t.
57
(1860),
pp.
189
230y
viz.
we
have
in
Kummer
a
surface of
any
form
whatever
(defined
according
o
the
Gaussian
theory,
a;,
y,
z
S'ven
unctions of
the
arbitrary
arameters
m,
r),
and
from
e
several
points
hereof
rays
proceeding
ccording
o
any
law
whatever,
viz. the
cosine-inclinations
a,
/S,
7
(or
as
Eummer
writes them
f
,
17,
g) being
given
functions
(such
of
course
that
a*
+
^8
+
7*
=
1)
of the
same
parameters
m,
v.
It
may
be remarked: 1* that
Kummer,
while
considering
he
simplifications
f the
general
theory
presenting
hemselves
in
the
case
where the
rays
are
normals
of
the
surface,
oes
not
specifically
onsider
the
case
where,
not
being
such
normals,
they
are
(as
in
the
Hamiltonian
theory)
ormals of
a
surface.
2^
that
some
interesting
nvestigations
n
regard
to
the
shortest distances between
consecutive
rays,
while
naturally
connectinghemselves,
ith
the
normals of
the
surface,
r
with that of the rays which
are
normals of another
surface,
do
not
properly
belong
to
the
AUgemeine
Theorie
of
a
congruence,
which
is
independent
f
the
motion
of
rect-
angularity.
It
has
been
already
remarked
that
the
system
may
be
looked
at
in
the
two
ways
1'
and
2*,
and itis in the
former
of
these
that
the
question
s
considered
by
Kummer;
it is
interesting
o
work
out
part
of
the
theory
in the
latter
of the
two
ways.
Taking
X^
Y,
Z
as current
coordinates,
e
have,
for
a
line
through
the
point(a;,
,
is),
he
equations
Z,
F,
Z=aj
+
a/ ,
y
+
^p,
+
7p;
a,
/8,
7
are
functions of
(a?,
,
z\ satisfyingdentically
he
equation
*
+
i8^
7*
=
1
(and
therefore
the
derived
equations
in
regard
to
(c,
y,
z
respectively);
nd also
satisfying
he
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PBOP.
CATLET,
ON
SYSTEMS OP BATS.
77
eqaations
(a8.
i88^
7S.)a
0,
(aS.
)88^
7SJ
/S
=
0,
(aS.+
i8S^
7S.)7-0.
It should be remarked
that
if
these
equations
ere
not
satisfied,
hen
instead
of
a
congruence
there
would be
a
complex,
or
triply
nfinite
system
of
lines,
iz.
to
the
several
points
of
space
(x^
y,
z)
there would
correspond
ines
X,
F,
Z^
x
+
OLpy
y
+
i p,
S
4
7p
as
above,
which
lines
would
not
reduce themselves
to
a
doubly
infinite
system.
Suppose
that
the
line
through
the
point
y
y^
z
is
met
by
the
line
through
a
consecutive
point
{x
+
dx^
y
+
dt/jZ-^z)
;
then,
if
X, F,
Z
refer
to
the
point
of
intersection
of the
two
lines,
e
have
dx
+
adp
+
pdOf
dt/'\rl3dppd/S^
dZ'\-ydp
pdj=^Oi
and
consequently
dxy
duj
d
dy, dp,
fi
dzy
dy,
7
as
a
relation
connecting
he
increments
dx,
dy^
dz,
in order
that the
lines
may
intersect,
iz. this is
a
quadric
relation
(^ydxydy,
dzy
=
0
between the
increments.
In
the
case
of
a
complex
this
equationrepresents
cone
(passing
evidently
through
the
line dx
:
dy
:
dz^a:
fi
:
7),
but
in
the
case
of
a
congruence
the
cone
must
break
up
into
a
pair
of
planes
intersecting
n
the line in
quisstion
x
:
dy
:
dz
=sai
0
i
y.
To
verify
h
posteriori
hat
this
is
so,
observe that
the
differential
equations
atisfied
by
a,
/8,
give
as
above
8,7- A
8.a-S.7,
S^-S,a
proportional
o
a,
^,
7,
or
say
=
2Aa,
2A)8,
2A7
;
and
it
hence
follows
that
the
differentials
a,
S/3.By
can
be
expressed
n
the
forms
da
adx
+
hdy
+
gdz
+
JcdSdz ydy),
dl3^hdx+
bdy
+/dz
'\-
k
(y
dx
-
adz),
dy
gdx+
fdy
+
cdz
+
k(ady
-fidx),
where
0
=
aa
+
A^S
+
gy,
0
=
Aa+ i8+/7,
0
=
flra+//3
C7,
.
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(
80
)
A NEW
METHOD FOR
THE
GRAPHICAL
REPRESENTATION
OF COMPLEX
QUANTITIES.
By
jr.Brill.
1. As
it
is
now
more
than
eighty
years
since
Argand
introduced
the
well-known
method
for the
graphical
epresen*
tation
of
complex
quantities,
nd
the
principle
f
duality
as
long
been
recognized
y
geometers,
it
seems
strange
that
it
should
have
occun'cd
to
no
one
to
apply
the
same
idea
to
tangential
oordinates
;
in
other words
to
construct
a
diagram
in which
complex
quantities
hould
be
represented
y
lines
instead of
points.
It
might
be
anticipated
hat the theorems
obtained
by
this
method
would
be
none
other
than
the
polar
reciprocals
f theorems
similarly
btained
by
the older
method
;
however,
since
metrical
properties
flFer
considerable
diflSculty
o
reciprocation,
t
might
be
advisable
to
have
a
method which
would
yield
he
reciprocal
roperties
irectly*
I
have
endeavoured
to
supply
uch
a
method
in the
following
paper.
It
will
be
founa
that the
theory
is
not
an
exact
analogue
of that in
which
complexquantities
re
represented
by
points.
This
is
as
might
be
expected,
or
itis well
known
that
complete
duality
does
not
exist
in
the
planimetry
f
homaloidal
space.
2.
Before
proceeding
o
develope
he
theory
e
will
prove
a
theorem
which will be of
use
in
the
course
of the
investi-ation.
0
is
a
fixed
point,
nd
a
straight
ine is
drawn
through
meeting
n
fixed
lines
in
the
points
^,r,,
...y
r^.
A
pomt
R
is
taken
on
the
line
through
0,
so
that
OB
ft^ T, ^- ^ ^-
The
locus
of
S will
be
a
straight
ine.
To
prove
this take
two
rectangular
xes
through
0,
and
let
the
equations
f the
given
lines
referred
to
these
axes
be
Mjic
+
Vjy
-
1
=
0,
u^x
+
1?^
1
=
0.
..tf^sTj
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MR.
BRILL,
ON
COMPLEX
QUANTITIES.
81
Then,
if B
be
the
angle
made
with the
axis
of
x
by
the
line
through
0,
we
have
TT
=
u.
cos5
+
V.
sin
0.
Tr
=
^i
cos^
+
v,
sin
tf,
-pr
=
M
cos^
+
sin
0,
Therefore
the
equation
f the locus
of
B
is
'^ ^'^' * '^'^'*=ffl,(^,cosg+t^,8ing
r
Swim
.
Smv
^
.
We
shall call
this
the
mean
line
with
respect
to
0
of
the
n
given
lines
for
multiples
j, wi
...,
?n^.
3.
Let
the
equation
f
a
straight
ine
be
given
in
the
form
ux
+
vy l =0,
where
u
and
v
are
the
reciprocals
f
the
intercepts
n
the
axes.
We
shall denote
the
position
of this
line
by
the
expression
4-
iv,
or
if
we
desire
to
use
a
single
ymbol
by
w.
This
being
premised,
e
see
that
the
line at
infinity
ill
be
represented
y
zero,
and that
any
line
through
the
origin
ill be
represented
y
an
infinite
complex.
Beal
quantities
ill
represent
lines
parallel
o
the
axis of
y^
and
purely
imaginary
quantities
ill
represent
lines
parallel
to
the axis of
x.
Further,
ny line
parallel
o
the
given
ne
will be
represented
y
a
real
mjiltiple
f the
expression
which
denotes
the
given
line.
4.
Let
there
be two
given
straight
ines whose
equations
Bre
t aj
+
vy
1
=
0
and
ux
+
v'y
1
=
0.
Consider
the
line
m
{ux
+
vy
-
1)
+
n
[u'x v'y 1)
0,
or as
it may
be
written
mu^nu'
mv
+
nv*
x^
y-
1
=
0.
Let this
be
equivalent
o
Kb
+
T^
1
=
0,
then
we
have
{m
+
n)U=mu
+
nu*
and
(m
+
)
r
twv
+
v'.
VOL. XTII.
G
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82
MS.
BRILL,
A NEW METHOD
FOB
THE
GRAPHICAL
Hence,
if
W=V+iVj
w=u+tv^
and
w'=u'+w'^
we
have
(wt n)
W^mw-{-nw\
The line
W
is what
we
have
calledthe
mean
line of
the
lines
w
and
w'
for
multiples
and
n.
If
we
make
m
=
n,
then
we
have
2W=w
+
xo\
In
this
case
W
will
coincide
with the harmonic
polar
of
the
origin
with
respect
o
the
lines
w
and w'. Thus
to
obtain
the
line
which is the
sum
of
two
given
lines,
e
have
to
draw
the
harmonic
polar
f the
origin
ith
respect
to
them,
and then
to
draw
a
line
parallel
o
this
and
at
one-half the distance
from
the
origin.
Thus
the
idea of
the
mean
line
furnishes
us
with
an
interpretation
f
the addition
of
lines,
ust
as.
the
idea
of the
mean
centre
furnishes
us
with
an
interpretation
of the
addition of
points.
The
actual
position
f the
mean
Jine,
however,
depends
upon
the
position
f
the
origin,
o
ths^t
the
jinalogy
s
not
quite
complete.
If
we
write
??i
4
n
=
0,
then
TV
becomes
a
line
through
the
origin,
pd
all
that
we
can
say
is
that
w
vS
is
some
line
{parallel
o this.
It
will
therefore
be
necessary
to
investigate
he
case
oi
'{jo
yi
separately.
he
equation
f
this
line
ia
(u
w')
; +
(v v')
1
=?
0,
This is
ohvioqsly
arallel
o
(t*
w')
?
+
(v
v')
=
0,
the
line
joining
the
origin
to
the intersection
of
vo
and
v)\
In fact
this
latter line coincides
with
TT.
Further,
the
equation
f
to
to'
may
be
written
iB
+
vy
1
[yix v'y)
0,
which shews
that it
passes
through
the
intersection
of
w
with
H
Une
drawn
through
the
origin
parallel
o
w\
Thus,
to
construct
the line
v
w',
we
draw
through
the
origin
line
parallel
o
w\
and
through
the
intersection
of this
with
w we
draw
a
line
parallel
o
that
joining
he
origin
o
the
intersex*
tion
of
w
and
w\
.
If
TF be the
mean
of
the lines
^j,
w^,
...,
^
for
multiples
m^
w
.,.,
i^,
then,by
proceeding
s
in the
early
part
of
this
article,
e
should
obtain
(Wj
+
w,
+
. .
.+
?w
J
Tr=
m^^
+
m^^
+.
.
.+
Vijio^^
6,
Let
the
straight
ine
t aj
+
t?y
1
=0
cut
the axis
of
x
fit
the
point
-4,
and
that
of
y
at
the
point
5;
and let
OA^a
l^nd
QB ^b,
From
Q
draw
OL
perpendicular
o
-45.
Let
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BEPKESENTATION
OF
COMPLEX
QUANTITIES.
83
OL=p^
and
let
0
be
the
angle
which
OD
makes
with
the
axis of
X.
Then
we
have
1
CO8
0
,
1
sin^
tt=
-
=
,
and
v=T
=
.
a
p
^
op
Therefore
1
.
*
w
=
u
+
iv=i-
(cos5
+
isind)
P
V
This
formula
will
enable
us
to
discuss
the
question
f the
multiplication
f
two
lines.
For
let
wlc
=
WjW^
where
w,
fff^j^
denote
lines
and
c
denotes
some
length.
Substituting
for
Wj
w^
and
w^
their values
as
given
by
the above
formula
we
obtain
cosg
+
tsing
_
cos
0^
+
i
sin
0^
cos^^+
i
sinS,
^
cos
{0^ gj
+
i
sin
{0,
0,)
Hence,
we
have
co8g^cos(g^-f
,)
.
sing
_
sin
(g^ g,)
^
PiPt
*
cj
'
.
p,p,
'
from
which itfollows
that
^-P\P%
**^d
0^0^-\-0^.
In
a
similar
manner
it
can
be
proved
that
if
vf^vi^^
then
P*
'=^PxPt
and
2g
=
g,
+
g,.
6. As
a
first
example
of
the
utility
f the method
we
will
investigate
he
analogue
of the well-known
proof
by
Argand's
method
of uc. I.
47.
Taking
the
figure
f
the
last
article
produce
BO to
B, making
OJS'=
OB.
Then
the
line AB
will be
represented
y \\a-\-i\h^
nd the
line
AB^
by
\\a
ilh.
Further,
rom
the
last
article
we
have
1
t
_
*
a
0
p
Also,
the
perpendicular
rom
0
on
AB
is
equal
in
length
to
that
from
0
on
AB^
and
the
two
perpendiculars
ake
equal
angles
with
the
axis
of
x
on
opposite
ides
of
it;
therefore
ah
p
'
G2
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84
MB.
BRILL,
A
NEW
METHOD FOR THE
GRAPHICAL
Multiplyingogether
hese two
equations
e
obtain
1
+
1
=
1
which expresses
a
well-known
properly
of the
right-angled
triangle.
7.
We
will
now
proceed
to
develope
some
metrical
properties
onnected
with
the
theory
of
the
mean
line.
JLet
w
be
the
mean
of
the
lines
w^
and
w^
for
multiples
971
and
fij
then
we
have
{m
+
n)w^
mw^
+
nw^.
This
is
equivalent
o
the
two
metrical
relations
,
.cosS
cos^,
cos
9,
(m
+
n)
=
i
-+n
=
P
Pi
Pf
,
V
sin^
sin
0.
sin
6^
and
(m
+
n)
=
m
*
+
n
'
.
P Pi Pf
From these
we
easily
educe
the
two
following
p
Pi
p,
PiPf
and
^^^^^
=
-
cos(5
^,)
+
-
cos(^
0X
P Pi
'
Pf
*
Let
AXj
ABy
^40
be
the lines
respectively
enoted
by
to,
w^j w^.
From 0
draw
OL, OM^
01s
respectively
erpen-icular
to
AX,
ABj
AG.
Then,
if
0
lie within the
angle
BAG
or
the
vertically
pposite
ngle,
e
have
Im
+
)
Iff?
^
IP?
^
mn
t%
a
n
but}
if
0 lie
without
both these
angles,
e
have
(m
+
n)'_
m*
^
r?
^^
mn
^.^
or
^'OW'^ON^'^
OMTON^^
Through
L
draw
LH and
LK
respectivelyerpendicular
to
OM
and ON. The
second of
the
above
relationsbecomes
OH
OK
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86
MR.
BRILL,
A
NEW
METHOD FOR
THE
GRAPHICAL
If
we
split
his
into its
real
and
imaginary
parts,
multiply
the
first
by
cosr^
and
the
second
by
sinr^
and
add,
we
have
^ '+^=^;co8r(^.-5)+r^'^cos{(r-l)(
.-^}
V
K
' '
'
Pi
P
+...+
cosr
(5,-^).
p;
It
would
be
troublesome
to
enumerate
all the
cases
of
this
that
might
arise,
o we
will
content
ourselves
with
one
particular
ase.
Let
0
lie
without
both
the
angle
BA
O
and
the
vertically
pposite
ngle,
and let
m
and
n
be
both
of
the
same
sign
so
that AX liesbetween AB and A
C.
The above
property
becomes
(m + n)'
w'
(
-DA
V)
H-
.
.
.+
-^^r
cos
{r
(7-4Z}.
This is
the
generalizednalogue
of
a
property
of
the
triangle
published
by
me
in
the
Educational
Times for
October
1885
(No.
8290).
9.
Let
?,
and
w^
denote
the
lines
AB
and
-4(7,
the
origin
0
lying
withm
the
angle
BAG. Draw
AX the
harmonic
polar
of
0
with
the
respect
to
the
lines
AB snidAG,
On
the
opposite
ide of 0
from
AX^
and
at
a
distance
from
0
equal
to
one-half
of that of AX from
0,
draw
a
line BG.
Then,
if
this
line be
denoted
by
w^,
we
have
w^
+
w^
+
w^
=
0.
This is
equivalent
o
the
two
metrical
relations
lind
cos
5,
cos^, cosS,
^
i-+ 2.+
5=0
A
P.
Pz
8in^^sln^_^
lnj93^^
P^ P.
P.
'
Fi F,
Pz
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88
MR.
BRILL,
A
NEW
METHOD FOR
THE
GRAPHICAL
Then the
expressionjw^ s^/w^
s
equal
to
f
C08(^,-
,)
-^
C08,(^.-
j
+
.-If'
in(^,-,)
-^
8in(^,-d,
Stf
Sfa VSfi Sfa
J
and thereforeitsmodulus
is
Hence,
we
have
the
following
heorem
:
Let
ABC
and DEF
be
two
triangles
aving
a common
centroid
0,
and let
OL, OM,
ON,
OP,
OQ,
OR
be
the
respective
erpendiculars
rom
0
on
BC,
CA,
AB, EF,
FD
DE'y
then
we
have
fON\'
(0L\^ ^ON.OL
,o ,
fqL\\/OM^.*
^OL.OM
111
Hopj'^vooj'^op.oe'''''^^~^^--oP'oE^^'o^'
11. Let
ABC
be
a
triangle.
ake
a
point
0
within
the
triangle,
oin
OA, OB,
00,
and draw
OB,
OE,
Oi^
respec-
TIT
tivelyerpendicular
o
BG, GA,
AB.
Through
0
draw
OX
parallel
o
AB
to
meet
GA
in
X,
and
through
X
draw
JfiV
parallel
o
OA.
Through
0 draw
OY
parallel
o
J5(7
to
meet
AB
in
F,
and
through
Y draw
-A/i
parallel
o
OB.
^Through
0
draw OZ
parallel
o
GA
to
meet BG in
-2',
nd
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90
MB
BBILL|
A NEW
METHOD
FOR
THE
GRAPHICAL
Let
MNj
NL,
LM
meet
AB^
BCj
GA
respectively
n
X\ Y\
Z\
Then
this relation
becomes
cos(Zr^-ZZr')
cos(Jf^^X~J/ZZ^
Olf.OP
*
OE\OQ
cos(NX'Y^NYX')
_
1
OF'.OB
OROQ.OR^
Similarly
e
may
deduce from
the
second
formula
the
relation
co^{AY Y-^AZ Z)
co^(BZ Z^
BX X)
OE.OF.OP
*
OF.OD.OQ
co (GX X^CY Y)^
OD.OE.OB
OROQ.OB^
where
X .
Y'\
Z
are
the
respective
ntersections
of
MN^
NL,
LM
with
BG,
GA,
AB.
13.
We
could obtain
analogues
of
all
the
properties
f
rectilinear
figures
btained
in
my
paper
on
Argand's
method.*
We
will,owever,only
consider
one more
instance^
iz.
the
interpretation
f
the
formula
Let
ABG
be
a
triangle
nd
0
a
point
within
it.
Let
MN be
the
harmonic
polar
of
0
with
respect
to
GA
and
AB,
NL
that
of
0
with
respect
to
AB and
BG^
and LM that of O
with
respect
to
BG and
GA.
Draw
CD, OE, OF, OP^ OQ^
OR
respectively
erpendicular
o
BG^
GA^
ABj
MN^
NL^
LM.
Then
we
have
cos(LBG'-LGB}
cob(MGA^MAG)
cos(NAB-NBAy
OD\OP
*
OE\OQ
*
OF\OB
cosiMGA'-NBA)
cos(NAB-LGB)
coB(LBG-MAGy
*
OE.OF.OP
*
OF.OD.OQ
*
OD.OE.OB
*
14.
Let
0
be
the
origin
nd
A
a
fixed
point
distant
2a
from
0.
On
OA
as
diameter
describe
a
circle,
nd
through
A
draw
a
line
AB
cutting
the
circlein L.
Draw
a
tangent
LM
to
the
circle,
oin
OL^
and draw OM
perpendicular
Messenger
of
AfathenuUicSf
ol.
xvi.,
pp.
8-20,
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92
MR.
BRILL,
A
NEW
METHOD FOR THE
GRAPHICAL
on
the
tangent
at
L for
our
initial
direction.
We
have
ON'=:OM.OM',
and
the
angle
MOM'^^.MON.
Thus,
if
(?Jf=2c,
and
LL' be
represented
y
w^^
then
LM' is
represented
y
2cw\
Draw another hne
LL
throughi,
and
draw
ON'
per-endicular
to
LL .
Let
L M
be
the
tangent
at
L\
and
OM
the
perpendicular
n
it
from
the
origin.
Join
LL ^
and draw
UK
perpendicular
o
it.
Then
we
have
ON'^OM.OM'
and
ON'^^OM.OM ,
and
therefore
0N\ ON
=
0M\
OM'.
OM
=
0M\ OK'
;
that
is,
ON. ON'
=
OM. OK.
Further,
^
bisects
the
angle
M'OM ,
and therefore
2.MOK=:MOM'
+
MOM
=
2(MON-i-
MON')\
that
is,
MOK^
MON+
MON'.
Thus,
if
LL
be
representedy
w^,
and
LL
by
w
then
iX
is
represented
y
2cw^w^.
15.
As
a
first
example
of
the method
of
interpretation
developed
n
the
last
article
we
will
take the
formula
for the
product
f two
polynomials.
This
will
be
easily
een
to
yield
the
following
heorem
:
0 and
L
are
two
fixed
points
on
the
circumference of
a
circle.
A
series
of
m
fixed
points
s taken
on
the
circum-erence
of
the
same
circle,
nd
a
second
series
containing
n
points
s also
taken.
Let
LM
be
the
mean
line
for
equal
multiples
ith
respect
to
0 of the lines
joining
to
the
m
points,
nd let LN
be
the
mean
line for
equal
multiples
with
respect
to
0
of the
lines
joining
L
to
'the
n
points.
Let
LM and
LN
meet
the
circle
in
M
and N
respectively
then
MN
is
the
mean
line for
equalmultiples
ith
respect
to
0
of the
mn
linesthat
can
be obtained
by
joining
ny
one
of the
m
points
o
any
one
of
the
n
points.
16.
We
will
next
take the formula for the
square of
a
polynomial,
iz.
As
before,
et
0
be
the
origin
and
L
another
fixed
point
on
the circumference
of the
circle.
Take
n
other fixed
points
on
the
circle,
nd
let the
lines
joining
to
these
points
e
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(
94
)
NOTE
ON
THE
TWO RELATIONS
CONNECTING
THE
DISTANCES
OF
FOUR POINTS ON
A
CIRCLE.
By
Prof.
Cayley.
Consider
a
qaadrilateral
ACD
inscribed in
a
circle;
and let
the
sides
BA^ AG^ GD^
DB and
diagonals
G and
AD be
=
c,
J,
A,
/,
a,
/
respectively
/
is
for
convenience
taken
negative,
o
that
the
equation
connecting
the
sides
and
diagonals
ay
be
A,
=a/+J^
+
cA,
=0.
We have
between
the sides
and
diagonals
nother
relation
F,
=aJc
+
a^A
+
M/+c/i/, =0,
as
is
easilyroved
geometrically;
n
fact,
recollecting
hat
the
opposite
angles
are
supplementary
o
each
other,
the
doable
area
of the
qnadrilateiul
s
=
(ic
+ gK)
edn^,
and it
is
also
=
(bh
+
cg)
sin^;
hat
is,
e
have
(he
+gd)
sin
A
(bh
+
eg)
sin
jS=
0.
But
from
the
triangles
AD and
BAG^
in
which the
angles
D^
G
are
equal
to
each
other,
e
have
c
f
c a
sini
sinj?'
sin
7
sin^
'
that
is
/sin^
+
asin-B=0;
and thence
the
required
elation
a
(Jc
+
:7*)+/(**
+
y)
^
The
distances
of the four
points
n
the
circle
are
thus
connected
by
the
two
equations
=
0,
F=0.
Considering
a,
i,
c,
f^g^h
as
the
distances
from each other of
any
four
points
n
the
plane,
e
have
between them the relation
12,
=
ay(-a'-/'
+
6*
+
i7'
c'
+
A )
+
jy(
a'+Z'^J'-Z
+
c'
+
AO
-aVc'-ayA'-J'Ay*-cy^ ,
=0;
and it is clear that this
equation
should
be
a
consequence
of
the
equations
=
0,
F=
0.
To
verifyhis,orming
the
sum
fl+
F*,
we
have
a+F'=
(a*+/ )(-ay'
+
6y
+
c'A'
+
2J(7cA)-
+
(*'+i7*)(-
*y
+
o'A*
+
ay+ 2cAa/)
+
(c*
A')(-c'A'+ay +jy
+
2a/ft 7);
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96
PROP.
CAYLEY,
THE
ANHARMONIC
RATIO
EQUATION.
or,
multiplying
ut,
the
equation,
s
is
well-known,
takes
the
form
But
to
effectthe
multiplication
n the easiest
manner we
may
proceed
s
follows
:
writing
a,i,c=(a-S)(^-7),
(/3-S)(7-a),
iy-8)(a-fi),
80
that
a
+
i
+
c
=
0,
the
equation
s
The
product
f the
first
pair
of factors
is
thus the
equation
s
that
is,
and
recollecting
hat
a+
+
c
=
0,
and
writing^hc-k-ca-^-abj
r^ahc^
the
equation
ecomes
(a:+l)*-3(aj+l/a;+f3
^)(aj
l)'aj -l
0;
that
is
(^
+
aj
+
l)'+^(aj
l)'aj*
0.
But,writing
f
=
-
,
we
have
(fl*+tf+l) +^(^+l)'^=0;
or
finally
(^
+
a,
+
ly
-
^,|/;^^^?'
?(x^.
1)'=
0,
the
required
esult.
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CONTENTS.
PAGB
A
new
method for the
graphical
representation
of
complex
quantities
(continued).
y
J.
Brill
81
Note
on
the
two
relations
connecting
the
distances
of four
points
on a
dide.
By
Prof.
Catlky
-
94
Note
on
the
anharmonic ratio
equation.
By
Prof. Cayley
-
- -
95
The
following
apers
have been received
:
Major
Allan
Cunningham,
On the
depression
of
differential
equations.
Mr.
B.
Tucker,
The
cosine
orthocentres
and
a
cubic
through
them.*'
Mr. Q-. Q-.
Morrice,
Note
on
the
multiplication
of
nonions.
Mr.
C.
Chree,
**
Vortices
in
a
compressible
fluid.
Articles
for insertion
will be
received
by
the
Editor,
or
by
Messrs.
Metcalfe
and
Son,
Printing
Office,
Trinity
Street,
Cambridge.
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WA
No.
CXCIX.]
4
NEW
SERIES.
-rff^
[November,
887.
^^
MESSENGER
OF
MATHEMATICS.
EDITED
BY
J. W.
L.
QLA.ISUELI,
So.D.,
F.K.8.,
FELLOW OF TRINITY
COLLEGE,
CAMBRIDQE.
VOL.
XVII.
NO.
7.
MAOMILLAN
AND
00,
HonBon
aui
ffiambrfifge.
1887.
Price
One
Shilling.
MWrnAT.PR
AND
SON.
OAMBRIDOB.
_Eig']^trvv
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^
u^
-^6^
NOV
15
1887
^J E
COStNE^*
ORTHOCENTRES
OF
A
TRIANGLE
AND
A
CUBIC THROUGH THEM.
By
R,
Tttcksr,
,A.
ABG
is
a
triangle,
f which
AL^
BMj
CNsre the
altitudes
co-intersecting
n
the
orthocentre
H.
LFj
LE'
are
drawn
parallel
o
AG^
AB]
MD,
MF
AB,
BG',
tod
NE.NU
BG,
GA]
the
lines
AD^
JBlS^
F
cointersect
in
d-^,
and
the lines
AD\
BE\
GF'
cointersect
in
r,.
The
equationsIn
trilinear
coordinates)
o
AD,
BE^
GF
are
respectively
J)3cos-4
a7Cos(7,
7CosB=
Jacos^,
aacos(7=c)8cos5...(i)j
and
to
AD\ BE\
GF'
are
afi
cosB=
cy
cos
A
,
by
cos
(7=
aa
cosJBj
ol
cos
A
=
5^
cos
G.
,
(ii),
hence
^^
c^
are
givenby
the
equations
a
)8
7
4A
c
cos
jB
a
cos
O
b
cos
A
a*
+
i*
+
c'
a
0
y
^
r
7y
=
T
=
r,
tanto
0 cos
u
c
COS
A
a cosx
1
= tan
CD
(iii).
From
(lii)
t
is
at once
seen
that
the
Lemoine
point
It)
s
the
mid-point
f the
join
f
o-^ c^
*'(iv)
If
now
Z Zj,5
71
,
m^]
w
n
are
the
projections
f
o-j,
a
On
the
altitudes,
nen
Bl^,
Um^^
An^
cointersect
in
12,
ana
0Z
Am,,
Bn^
in
a'
(v).
Again
Gl^j
m^^
Bn^\Bl^^Gm,^
An,
cointersect
respec^
lively
n
Sj
(a
cot
5,
b
cot
C,
c
cot
-4)
and
6,
(a
cot
C,Jcot-4,
cot-B)
*
(vi).
The
mid-point
rf)
f the
join
f
e^,e,
is
(vil),
':
J'
and lies
on
the
cifcum-Brocard
axis
;
this
axis
therefore
bisects
cr, r
12i2'
and
SjE,.
VOL,
XVII.
H
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98
MR.
TUCKER,
THE
COSINE
ORTHOCENTRE
OF
A
We
readily
btain
the
following
esults
:
AE
=
V
cos
Ajc^
CE
=ab
cos
jB/c,
AE'
=
be
cos^/a,
CE'
=
b'
cos
0/a,
CD
=
a*
cos
(7/6,
^Z
=
ac
cosAjb,
CD'
=
ab
cosAjCj
BD'
=
a'
cos
J5/o,
AF
=
bc
cos
(7/
,
J5F
=
c'
cos
5/a,
AF'^e
cosA/bj
BF'^accosCjb.
..(viii).
If
7,Z,,
,/^
produced
eet
the side
AC
in
Aj,A,',
nd
^-B
in
A/,Ag,
then
if K
be
put
for
a*
+
**
c ,
e
obtain
cr,
Aj
2a*6
cos
(7/
ST,
cr^A,
2a*c
cos
/
Z,
whence
7,A,
o-,A,
2a7ir,
=(by
I.
x.)*
twice
the
intercept
made
on
jBC
by
the
T. R.
circle.f
Also
cTjAj'
2abc
cos^
/
JSr=
o-,Ag',
and therefore
a^h^'aji^'
s
a
parallelogram,/A^'
being
bisected
m
the
Lemoine
point
with like
results
for
the other
sides.
..(ix).
^
.
Ah/
rX'
Agam
^
=
BD'
AG
CD'
hence
Ah^.
AB
=
Ah^.
A
(7,
f.e.
A/A^'
s
an
anti-parallel
o
BC
(x).
Since
Ah;^WclK,
Ah^^2bc'IK,
therefore
A;A;
2abclK(^
2DF
=
2ED'
=
2FE
of
I)
...(xi).
Similarly
or
the
other
sides
0
jV,hlk^corresponding
o
A/A,'),
herefore
i.e,
these
lines
are
diameters
of
the
cosine-circle
.....
(xii);
the
equation
f
which
is
(/*
1
+
cos
-4
cos
j?
cos
(7),
=
2(aa-\-...+...)[abc inBsin
7
cos^
+.,,+.
..]...(xiii).
I
cite
The
Triplicate-ratio
ircle,
uar,
Jour.,
yol.
xix,,
No.
76,
as
I.
t
We
readily
obtain
-~^
=-^
*c.
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100
MR.
TUCKER,
THE
COSINE
ORTHOCENTRES
OF
A
I
collect
together
ere
a
few
results
of
interest.
The
equation
o
a^a^^
is,
f
/
=
a*
-f
6*
+
c*,
aa(a'K^v')
+
bff(
)
+
cy(
)
=
0*..
.(xix),
to
a^O
is
which
passes
through
5 /a,cVJ,
fl'/c,(fi,),
)....(xx),
and
l/(c a)a,
l/(a
+
J)6,
l/(J
c)c...
to
a-^O
is
which
passes
through
c'/a,
a75, 57c, (m,),
*..(xxi).
(a +J')/a,
(b'-hc')lb,
c'
a')lc,
and
l/(a
+
J)a,
l/(
+
c)i,
l/(c
a)c,
The
join
of
G^Z,
aa
(J*
c')
...+...
=
0,
evidently
asses
through
the
middle
point
of the
join
of
^t
/i,
(x^^Oj
and the
equation
o
fi^f^^s
aa(a*-JV)+...4...=0
(xxiii).
Since
AFLE',
c.,
are
parallelograms,
herefore
AL,
FE\
c.,
mutually
bisect
each other.
Since
Cj^
a
cot
C
tano),
Bj^
=
c
cosec5
tan
w,
we
get
the
perpendicular
rom
1
on
-4Z
=
tanft)(ccosecJBsina)),
.e.
X2
is
Brocard-point
f
one
cosine
triangle,
nd
SI'
of
the
other
triangle
(xxiv).
The
lines
tjY,aj^
intersectin
Jccos-4,
a
cos
(7,
a'cosjB,
(TTj),
which
is
evidently
n
AL
(xxv).
We
note
the
following
oints
n
the
figure,
hich will
be
of
use
in
the
sequel.
[AO^
jBcTj,
n
meet
in
7rj(ccosJB,
cos^,
icos^)
'
TT,
(c
cosjB,
cos
(7,
cos
jB)
TTg
(b
cos
(7,
cos
(7,
cos
-4)
7r/(6
os
(7,
cos^,
b
cos^)
7r,'(c
os
5,
c
cos
-4,
a
cosB)
7r^(b
os
(7,
cos
(7,
cos
5)
^
(xxvi),
It is
readily
provedby
rotating
he
figures
hat
OM
is
perpendicular
o
r, r,.
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TRIANGLE
AND
A
CUBIC
THROUGH THEM.
101
.
(xxviii).
IB r^j
T^
n
Tj(tan-4/a,
an
(7/
,
tan5/c)
C 7j,
4(7,
n
T^(tSLnCla^
an^B/J,
an-4/c)
-4 7
Aj^
in
T,(tanjB/a,
an
-4/
6,
tan
C/c)
ICcTj,
BcTj
n
T'j(cot-4/co8^,otC/cosjB,
ot
5/
cos
C)
4 rj,
C r,
in
T\(cot
Gj
cos
A
j
cot
B/
cos
5,
cot
-4/
os
C)
Ba-^y
a^
in
T',(cotB/cos-4,
ot
-4/
os
JB,
cot
Cj
cos
C)
( vii),
I-4 r
(?,
in
v^^cotAja,
ot
C7/6,
ot
A
jc)
Ba^j
COy
in
i;,(cotjB/a,
otjB/6,
otAjc)
Ca^^
AGj
in
v^(cotja^ cotC/6,
cot
Cjc)
iAa^j
Gj
in
i;/(cot-4/a,
ot
-4/
6,
cotJB/c)
jB r
G^,
in
i;,'(cot
/a,
cot
5/
J,
cot
Bjc)
Ca^,
BG,
in
i;;(cot/a,
cot^/J,
cot
C/c)
If
-4,5j(7
s the
firstBrocard
triangle,
hen
-4^^,
jBB,,
CC,
bisect
i^i^',
^/ ',
J5'
respectively
and
intersect
as
is
well-known
on
Kiepert's
yperbola)
(xxix).
The
pole
of o-^o-,ith
regard
to
the circum-circle is
a'[ c-a*cos(jB
C)],
...,
...,
'
and
therefore lies
on
the well-known
line
....
(xxx),
5ca
+
ca^
+
a
Jy
=
0
and
of
e^6
lies
on
a
(6
c*)/a
...+...
=0,
i.e.
on
the
radical
axis
of
the
circum-
and
T.R
circles
(xxxi).
Assume
then
whence
and
therefore
and
CO
=
tan
^
tan
tan
sin
(B
^i)
_
i
cos-4
sin ^,
acosJS*
cot^j
tanjB+
cotjB-
cot
G,
cot
^1
=
tan
(7
+
cot
C7-
cot
B
;
cot
(f ^
cot
^j'
tanB
+
tan
(7,
cot
9^
+
cot
9j
= taniJ
+
tan
O,
^
t
^1
+
cot^,
cot^3=
tan-4+tanjB+
tan
C
.
A
tanBtan 6'=
cot
(j \
cot
(j \
cot
^'
J
.(xxxii).
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102
MR.
TUCKER,
THE
COSINE
ORTHOCENTRES
OF
A
If
c7,5C=Vr^,cr^C5=V'\,
then
8in(g^'ifr^)
^
acoag
sill
^j
i
cos
C
'
whence
cot
-^j
=
2 cot
5
+
cot'JB
tan
(7,
]
and
cot
^/
=
2
cot
(7
+
cot'C
tan
5,
[...
(xxxiii).
therefore
2
cot
^
.
cot
^'
=
4
cot'o)
3.
'
From
(viii)
e
see
that
AE,AM=:b^
cos'
A
==JN\
therefore
circle
round
-ft/j^ilf
ouches
AB
at
N]
similar
results
hold
good
for
the
other
sides
(xxxiv).
The
equations
o
the
circles
LE'M^
LFN
are
respectively
a?'
+
y*
=
cy
cos
Cj
sin
vl,
and
a:'
+
y*=5iy
C08i5/3in4...
(xxxv),
where
CL,
LA
are
the
axes
;
whence,
if
D,
U
are
their
diameters,
e
have
D
+
D'
=
B
cos
(B
-
C).
The
trilinear
equations
o
the
circles
in
(xxxiv)
re
FLNj
aC=Jj[b
cos
Ccosu4a
+
o
co8 5/8
+
6
cos'CV]]
DML^
hC=L\c
cosMa
-f
c
cos
-4
cos
B^
+
a
cos*
6V]
[
ENM^ cC=L[b
cosMa
+
a
cos'Bff
+
a
couB
cos
Cy]
and
E'MLy
a
(7
=
Z/
[c
cos
^
cos
-4a
+
c
cos'^/3
6
cos'*
C7]
(xxxvi),
J? -^i/,
(7
=
i
[c
cosMa
+
a
cos
C
cos
^/3
+
a
cos*
C7J,
D'LN^ cC=^L\h
cos^Aa
+
a
cos*jBi8
f
i
cos
4
cos
C7],
if
(7=
0^87
+
^qt
4-
ca/8
and
i
=
aa
+
JyS+
C7,
From
(xxxv)
we
see
that
if
/ ,,
^,
p, ;
p'^,
'
p\
are
the
two sets
of
radii,
hen
P
\PtPz
=
^*^ ^^*
-^
^^*
^
^^*
7/
=
PiP\p\^
\
ap^
+
1/)^
c/ jj
2jB*
8in4
sin^
sin
C=^ap\
4
Jp't
^pVJ
,.., , ,
(xxxvii).
Some
general
properties
f
the
cubics
given
by
the
equation
C;E(a +
J/9
+
C7)
(-+
A
+
-)
^k'
''
\a%
bti
cy)
we given
in
the
Messengerof
Mathematics
(vol,
l.,
1864,
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TKIANdLE
AND
A
CUBIC
THROUGH
THEM,
103
p.
116)
and
in the
Reprint
from
the
Educational
Times^
(vol.
.,
p.
38)
;
we
propose
here
to
consider the cubic
Cj
=
cot
o)
tan
A
tan
B
tan
C==^k\..
(xxxviii),
which
passes
through
the
points
of
reference,hrough
0-^,cr,,
and
through
the
orthocentre.
The
equation
an
be
put
into
the
form
(aa
+
J/8
+
C7)
(hc^y
+
ca'^a.
+
aJa^S)
A;'aJca^7...(xxxix),
from
whence
it is
seen
that
it touches
the
minimum
circum-
ellipse
87/a
7 /c
+
a)8/c=0
at
the
points
f
reference.
The
centroid
of
the
triangle
s the
centre
of this
ellipse,
and
HA^
HB^
HC^
which
are
normals
to
the
ellipse,
re
also
normals
to
the
cubic,
and,
from
above
statement,
are
drawn
from
a
point
on
the
cubic.
The
six
points
^
t^,
t
t,
(xxvii),
nd
T,
tan^/a,
tan
0/6,
tt^AJc (xl),*
T
,
tan
CJ
a
J
tan
-4
/J,
tan
jB/
,
are
manifestly
n
(7
and
they
also
lie
on
the
ellipse
abal3
+
bcl3y
cay
a
=
cot
cot
-4
cot
5
cot
0
=
(aa+J/3
+
(yy) (xli),
wliich
IS
concentric
and
similarlylaced
with
the
above
minimum
ellipse.
The
cubic
also
passes
through
the
six
points
^^,
cot^/a,
cotjB/J,cotC/c,
j8,,
cot
-4
/a, cotC/6, cotJB/c,
/Sj,
cot/?/
,
cot
A
I
by
cot(7/o,
;9^,
cot
5/
a,
cotC7/6,
cot-4/c,
ySg,
cotO/a,
cot
-4/
6,
cotjB/c,
)8g,
cotO/a, cotJ5/6,
cot^/c.
^
which
lie
also
on
the
ellipse
a6ai8
+
645^87
^7a
=
tan*
(aa
+
i/8
+
7)*
concentric
and
similarlylaced
with
the
minimum
ellipse*
March,
1887
.(xUi),
*
These
pc^nts
are
easily
oonstrncted
in
a
difEerent
way
from
that indicated
in
(xxTii)
for
o-,
(a,),
-
(oj),
(o),
t'
(a')
we
have
the relation
between
coor-inates
to
be
a^aui
=
a*
{aa^
=
^^Pi
=
'
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(
104
)
NOTE
ON
THE
MULTIPLICATION
OF
NONIONS,
By
G. O.
Morrie^.
The
object
of
the
present
note
is
to
present
the
multi-^
plication
able
of nonions in its
proper
form.
We
have
m
and
n
for
our
two
fundamental
ternary
matrices
with the
condition
nm
=
pmny
where
p
is
a
primitive
ube
root
of
unity,
i':=n'=l.
Now
in
any
? uch
multiplication
or
a
group
of
operators,
and in
particular
or
substitutions
(vide
Dyck's
Gruppen^
theorettsche
Studien
//.),
it is
important
to
consider
the
per-,
mutations of
the
elements
of
the
initial
ow
which
leads
to
any
one
of the
following
ows.
In the
present
ase
we
have
cle?^rly
non-primitive
roup,
the
substitutions
which
inter-,
change
the
triads
(1,
wi,
? *),
( ,
wn,
m*n)j
(n*,
wn*,
V)
being
cyclic,
nd also
the
substitutions
which
interchange
the letters
within the
cycles^
We
may
callthe
substitutions^
1
,
5
,
5
,
's%
8',
S*s,
8*s\
Let
us
form the
matrix
J/,
1
,
m
,
w*
,
n
,
mn
,
m*^
,
w*,
7WW*,
m*n*
5
8
will
effect
a
cyclicnterchange
f
columns,
S of
rows*
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MR.
CHREE,
VORTICES
IN A
COMPRESSIBLE
FLUID.
105
Moreover,
he
simplest
orm
of
m
is
m'=
(1,
0,
0
)
,
0
.
Pt
0,
0,
P*
m
(
h
0,
0,
P%
0),
0
0, 0,
p
and
it is
easy
to
verify
hat,
or
example,
the
elements of
the
second
row
are
those
of the
matrix
formed
by
multiplying
'
by
8
(M)^
the
third
row
w'*
by
s^
(M)^
and
so
on.
June
21,
1887.
VORTICES
IN
A
COMPRESSIBLE
FLUID.
By
CharUB
Chree,
M.A.y
Fellow
of
King's
College,
ambridge.
The
following
aper
contains
certain
applications
f
the
equations
f
vortex
motion in
two
dimensions
to
a
compressi-le
fluid.
The
equations
f
vortex
motion
in
an
infinite
fluid
are*
^^dP
dN_dM\
dx
dy
dz
^^^^dL^dN.
dy
dz
dx
^
'*
dP
.
dM
dL
If
dz
dx
dy
^
du
dv dw
^
.(2),
.(3)
hen
F^-^fjj^d^dy'dz'
gives
the
value
of
the function P at
a
point
(a?,
,
z)
at
a
distance
r
from the
point
{x\
y\
'),
here
6
has
the
value
^,
the
integration
xtending
through
all
space.
If
-j
denote
partial
ifferentiation,
nd
^
differentiation
following
he
fluid,
he
equation
f
continuity
s
dp dpu dpv
dpw
_
^
dt
dx
dy
dz
^
^
or
p
ot
gee
Lamb's
Maion
of
Fluids^
p.
160,
151,
for the
meaning
of
X,
Jtf,
V,. c.
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108
MR.
CHREE,
VORTICES
IN A
COMPRESSIBLE
FLUID.
be treated
exactly
as
in
an
incompressible
luid
by
the
aid
of
images.
Each
image
is
to
have
the
same
cross-section
at
every
instant
as
the real
vortex,
and
the
consequent
velocity
ue
to
variation
of
density
ill
in
every
case
be
parallel
o
the
boundaryplanes.
Consider
now
a
single
straight
ortex,
parallel
o
oz^
and
let
m
be
its
strength
and
a
its
cross-section at
time
t
;
then
the
components
of the
velocity
t
a
point
in the
surrounding
fluid,
hose
coordinates
are
x
and
y
relative
to
axes
through
the
centre
of
the
vortex
fixed
in
direction,
re
given
by
my
\ diT
X
Tir*
2ir
dt r^
1
diT
y
x
V
=
5
+
,
-
Trr
2ir
at
r
.(7).
For
our
specialbject
e
require
o
determine
the
motion
of
two
such
vortices,
hose
mutual
distance is
supposed
reat
compared
to
the
diameters
of
their
cross-sections.
Let
their
strengths
e
m^^ m^,
and
at
time
t
let
their
cross-sectionsbe
7j,
(7,,
and the
coordinates
of their
centres
(a?^,
J
and
(x,,
^)
;
then,
f
r*
=
(a?,x^)*
(y,
yj ,
e
have
dt
irr'
27r
dt
r*
dt
TTT*
^
27r
dt r'
^
^
^Jyi-y )
.
1.
^
^Sli5l
dt
TIT*
27r
dt
r*
dt
TTT* 2ir dt
f^
[S).
From
these
eqaations
e
get
Thus
if
when
s=0,r.
o?
\^ ^ '
2ir
rf
'
i
. 'i
and
r,
r
we
get
r
=
c'+-( r,+
r,-, r,-, r,).
.(9).
These
equations
lso
give
d
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MR.
CHREE,
VORTICES
IN
A
COMPRESSIBLE
FLUID.
109
But
if
8
denote
the
inclination
to
ox
of the
line
joining
the
centres
of
the
vortices,
,
y,
=
r
sin
e
and
x^-x^
=
r
cose,
and
the above
equation
ecomes
dt
Trr'
Thus,
if
e
=
0
when
=
0,
we
find
,^^,r
_
^
(10).
The
case
of
a
vortex before
an
infinite
plane
is
included
in
this
solution. We
have
only
to
take
^c
as
the distance
of the
vortex
from the
plane
at
the time
^
=
0,
and make
m,
=
ij,a^=a-^^
and
^a'^'=^ r^.
In
this
case
s
remains
zero.
The
stability
f the circular form
in
one
and
in
two
straight
ortices
has been
considered,
nder
the
title
of
Linked
VorticeSjy
Prof.
J.
J.
Thomson,
in his
Motion
of
Vortex
Rings. *
The main
object
of this
paper
is
to
extend
his
treatment to
a
compressible
luid. To
render
more
easy
comparison
with
Prof. Thomson's
results,
have
followed
his notation and
method
so
far
as
possible.
Consider first
a
single,pproximately
ircular,
ortex
whose
section
is the
same
for
all
values
of
2;,
and let
the
radius
of
the
cross
section
making
an
angle
6
with
a
fixed
direction be
given
at
time
t
by
jB
=
a
+
a,jC0Sw^
/8^sinn^
(11),
the last
two
terms
being
the
types
of
any
number
of
pairs
of
terms,
while
a^,
0^
are
supposed
small
compared
with
a.
Suppose
the
vorticity
nd the
density
o
be
at
any
instant
the
same
at
all
points
in the
cross
section. At
external
points
we
have
approximately,
s
for
a
vortex
of
truly
circular
cross
section,
he
functions
'27r
dt
logr.
Assume
for
the
fluid
outside the
vortex
i^=(7-^logr
{^ cosn^
+
5,sinn )f^y
...(12),
*
Part
in.,
p.
71.
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MR.
CHREE,
VORTICES
IN
A
COMPRESSIBLE
FLUID.
Ill
But
from
(11)
n0(a^Binnff4^^cosnff)
(15),
where
0 isthe
angular
velocity
ound
the
axis
of the
vortex
of
the
fluid
at
its
surface.
Thus
^
1
dif
I
dP
m
^
0
=
=-
+__--=
-
4.
terms
m a
and
p.
r
dr
ir
dd
ira
*
Thus
in
(15)
we
may
take
0 =
|
.
ira
We have also
a
second
value for the
radial
velocity
iven
by
dB_ldN
dP
dt
r
de ^
dr'
Noticing
that
57
==
-57
this
last
value of
-^
can
^
2ira
dt dt
^
dt
be reduced
to
i
(a
sinnff
pn
cosn^)
-j-
,
iTQ,
at
which
must
be
identical
with
(15).
Equating
the
coefficients
of
cosn^
and
of
sinrad,
e
find
^-
-/3,( -l)^.
rf/3.
.
,.
m
I
('^^
_.=
.(n-l)-.
whence
a/
+
^^'=
constant
(17).
Also,
eliminating3^
rom
(16),
e
get
,
d*a
.
da
rfa
.
/
^
\s
/^
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112
MR.
CHREE,
VORTICES
IN
A
COMPRESSIBLE
FLUID.
Multiplying
p
by
2
-^
,
and
integrating,
e
find
a (^y+wi'(n-l)'a '=
onstant
(18).
fin
If
then
a^
and
-^-f
vanish
initially
hey
always
do
so,
the
same
is
obviously
rue
of
)3^.
Suppose
initially
^
=
0,
and
so
from
(16)
^,
=
0,
while
a =oa,,
and
so
-^/=o h
^^.
i
where
a,
is
the
initial
value
of
a
;
then
from
(18)
=
oa cos
' ( -1)
-\'i
^'11
(1^).
similarly
/S^
^^a^
sin
j
(w
1)
1
\
Thus
the
form
of
the
cross-section
at
time t
is
given
by
^
=
a +
,a
cos
[nd
^m{n^
I)
T -I
(20).
The
section
thus remains
approximately
ircular,
nd
the
disturbance
in
shape
travels
round
the
cvlinder
in
the
grai-'
dually
arying
times
given
by
m
(n
1)
I
=i2tV,
where
i
is
an
integer.
If
we
suppose
^
= constant
=
^7,.
we
get
i?
=
a4.a.co,| ^-^i^=ill
The
period
of
the
ith
revolution
of
the
disturbance
is
given
by
Ti
=
-
\
e^ff-l)
1
U
fn{n-l)
,
and
so
increases
with
t
If
7
be
positive,
,e.
if
the
cros^-
section
of
the
vortex
be
increasing.
If
r
=
constant
=
Tra'*,
nd
m^coTra^
and
the
time
be
properly
chosen,
the result
(20)
is
identical
with
that
of
rrof.
Thomson
on
his
page
74,
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114
MB,
CHREE,
YORTICES
IN
A
COMPRESSIBLE
FLUID.
Since
we
suppose
afc
small
we
require
to
retain
to
our
present
egree
of
approximation
nly
Kj,
a
13^^
nd
0^.
Let
iR denote
the
radial
velocity
f
a
point
on
the
second
vortex,
and 60
the
velocity
erpendicular
o
the
radius
vector,
oth
being
taken relative
to
the
centre
of
that
vortex.
Then,
from
(21),
-w
(a'^
innff
-
P\
cosnO')
(25),
where
to
the
present
degree
of
approximation
^
Try-
But
we
have
also,
utting
r'
=
J
after
diflferentiation,
r
du
dr
r
dff
ttc
^
'
^
dP.
1
d(T.
,-,
,
The
terms
in
m^
and
in
-7^
are
introduced as
it
is
the
velocity
elative
to
the
centre
of
the second
vortex
that is
being
considered.
We
thus
get,
the
terms
in
n
being
of
course
merely
typical,
=
.|^.(a inn^./S;cosn^')+^j
-^ 8in2((?'-e)...
971.
a
+
*
-^i+j
{(a^cosns
iS^sinwe)
in(^' b)
TT
C
+
(/8^
OS
we
a^
sin
we)
cos(5'
)...|
-
-,
-^
h
cos2
{ff-
e).
*
2 ^
^
?^
^^ *
^^'^^
*
^*
^ ^^^
^
^^'
^^
-
()8^
os
ne-a
sin
we)
sin(^-
e)...}
(26).
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MB.
CHREE,
YOKTICBS
IN
A GOKPBESSIBLE
FLUID.
115
These
two
values
for
It
must
be
identical. The
terms
independent
f
ff
are
so
since
-j
=
27r6
-r*
and
the
identity
extends
to
the
coefficients
of
every
sine
and
cosine
of
mul-iples
of 0^
Equating
coefficients
f
cos^,
we
get,
after
reductioni
S S
^^^
^ 2b-
a,
sin2e)
g^.
J
(a,
cos2e+/3,
in2s)
but
to
our
present
degree
of
approximation
erms
of
orders
^
or
-J
-7-
are
negligible,
o
-^
=
0;
similarly
e
get
-^
=
0.
Thus
a'j,
8\
if
originally
ero
remain
so,
whether
the
fluid
be
compressible
r
not.
To the
same
degree
of
approximation
e
find,
rom
equat-ng
the
coefficients
of
cos2 '
and
of
sin
20'
in
(25j
and
(26),
rfa'
m,
^,
tn}
.
^ .
1
dfT.
^
,^^.
^_
.
a',
^co82e+J^
.
^'
8m2.=0
(28).
dt
irb
'
TTC
27r
c
ai^
^
'
It
is
scarcely
likely
that
these
equations
admit
of
a
complete
solution,
ut
general
ideas
of
the
motion
can
be
deduced.
Supposing
for
an
instant
there
were no
vorticity,
but
only
two
columns
of
fluid
of
varying
density,
e
should
have
nij
=
wij
=
8
=
0,
and
so
e?a',
_
\
h
d T^
_
ah
da
~dt
27rc''di
V
dt
^
dt
Thus
/8',
ould
be
wholly
unaflected,
hile
a'^
ould
alter
from
its
original
alue
^a
,
according
o
the
law
-/;? '*
')-
This
shews
that
a,
would
increase
or
decrease
according
as
a
were
decreasing
r
increasing.
hus,
if
both
columns
'Vere
diminishing
n
cross *3ection,
nd
so
approaching,
here
would be
a
decided
tendency
n
both
cross^'sections
to
assume
12
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116
MR.
CHREE,
VORTICES
IN
A
COMPRESSIBLE
FLUID.
an
elliptical
ort
of
outline,
he
major
axes
coinciding
ith
the
line
Joining
heir
centres.
It
is
pretty
obvious,
taking
into
consideration
the
existence of
vorticity,
hat the vibrations
will become
more
important
f
the vortices
are
approaching,
and
will
not
take
place
about
a
truly
ircular form.
If
the
density
f
the
vortices
vary
very
slowly
it
is
com-aratively
easy
to
trace
the
effect
on
the
vibrations.
Suppose
and
let
m^^
=
^a'o ,
w,
=
irb'^m.
It
what follows
we
suppose
7*
and
yt
to
remain small
during
the time
considered.
Now,
when the fluid is incom-
;res8ible,
,
and
/S',
experience,
s
is
shewn
by
Prof,
'homson,*
two
forms
of
vibration,
he
shorter
period
being
27r/ '
nd
the
longer
7r/n,
here
n={aya\+a 'b\)l ^^
(30).
If then
the
period
to
which
our
equations
re
applicable
be
supposed
to
be
at
least
several
times
greater
than
the
longerperiod
f
vibration,
e
must
have
70*^
nd
7'c'o
mall
compared
to
cDa*^
o '6'^
thus
terms
in
7
or 7'
must
be
neglected
hen
terms
in
cd
or
'
occur.
Terms
in
(7^)*
r
{y'tY
re
negligible,
nd
terms
in
n
are
small
compared
to
terms
in
a
or
a '.
Removing
the
terms
which
are
negligible
ccording
o
the above
hypothesis,
e
get
for the
equations
in
a,
and
/8',,
from
(27)
and
(28),
^'-Ko'(l-7'0i8',- a',V '(l
i70
8in2n^=ol
f'
)...(31).
?Ji
a,'(l
ry' )
',
a'^Vo'*
^
+i7'0
cos2n =0
These
give
^^
+
ft)
(1 27V)a', a\b^ioc;'ft)'
2w
-
^myt)
cos2nf,
^
+
(1
277)
/3\
a'Aa)c/(ft)'
2n
-
la Wt)
sin2nt.
*
Motion
0/
Vortex
Rings^
p.
77.
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MR.
CUBEE,
YOSTICES
IN
A
COMPRESSIBLE
FLUID.
117
Suppose
a'y
0
=
)8',
hen
e
=
0,
then
to
the
same
degree
of
approximation
he
solutions
of
these
equations
re
;=^T-^^j
(l
7'0cos2n -(l
i'/0cos '
+
i
^,
1
fn^e)
sin 'fl
(32),
'^' -oT^^
(^
+
'y''^
in2ne-(l
i'/0
in 'e
+
i 7Vcos ^}
(33).
Putting
/sO
we
obtain
Prof.
Thomson's
results
on
his
page
77.
The
case
m^s- 7n
a
=
J,
which
applies
o
a
single
ortex
Earallel
o
an
infinite
wall,
has
not
been
specially
reated
7
Prof.
Thomson.
It
presents
certain
peculiarities
hich
seem
worthy
of
notice.
First
neglecting
he
compressibility,
the
equations
30)
become
^- V.- 'AV j'
0.
dt
Here
o ',
,
e
are
constants; thos,
if
oC^ O ff^^
hen
=0,
the
solutions
are
a -6V(l-cosa,'0]
,o.^
/8;=
iVsIna '
j
^^*^-
The
form of
the
cross-sectionat time
t
is thus
given
bj
5'
=
6
-
iV*
cos2^
+
6V'
cos(2^-o)'0
(35),
the wall
being
perpendicular
o
the
line
^
=
0 at
the
distance
^c.
Thus the diameter of the
vortex
perpendicular
o
the
wall suffers
a
shortening
b^c^^
nd that
parallel
o
the
wall
an
equal
lengthening,
nd
the
vortex
has
a
single
isturbance,
of
period
2irjm\
about this
altered
position.
In
the
same
case,
considering
he
compressibility
lone,
we
have
^^
=
constant,
0
say,
and
from
(29)
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118
COL.
CUNNINGHAM,
DEPRESSION
OF
Thus
',
y._J^'
(36),
neglecting
igher
powers
of
(blc^Y.
The
diameter
perpendicular
o
the
wall
would thus
become
c:
'
and that
parallel
o
the wall
would
become
2(6-a',)=26-2/,
+
J^^.
Supposeoa'g=:0,
hen
the
diameter
perpendicular
o
the
wall is
obviously
he
greater
if b be
decreasing,
.e.
if the
vortex
be
approaching
he
wall
;
the
reverse
is
the
case*
if
the
vortex
be
receding
from
the
wall.
Considering
hen either the
vorticity
lone
or
the
com-ressibility
alone,
we
come
to
the
conclusion
that
a
single
vortex
in
presence
of
an
infinite
wall will
not
retain
a
truly
circular
cross-section,
nd that the deviation from
the
circular
form
varies
inversely
s
the
square
of
the
distance from
the
wall. When both
vorticity
nd
compressibility
re
considered,
the deviation
from
the
circular
form will
still
vary
inversely
as
the
square
of
the distance from
the
wall,
but the
exact
change
in the form
of
the
cross-section
could
only
be
deduced
from
a
complete
solution of
the
equations
(27)
and
(28)
for
the
case
6==0,
and
this
I
have
been
unable
to
obtain.
DEPRESSION
OF
DIFFEEENTIAL
EQUATIONS-
By
LU'CoL
Allan
Cunningham^
R,E,f
Fellow
of
King's
Coll.,
Lond.,
o.
[BeferenceB
o
Boole's
Treatise
on
Differentialquations^
nd.
Ed.]
1.
General Notation. As follows
:
a?,
y
are
always
the
variables
of
the
originalquation,
f,
u
the
variables
of
a
^'depressedequation.
(^11
ydy
( ,jyi)j
( .i
.))
(^4?J
^^e
the
variables
of
the
first,
econd,third,
nd fourth
depressed
qua-ions
formed
in
succession.
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DIFFERENTIAL
EQUATIONS.
119
Lagrangian
notation
is
used
for
differential
coefficients
where
compactness
is
required
thus
:^
y\
y ,...y^''^
enote
differentiation
ith
respect
o
a?,
u\u'\...u^r)
^^
^^ ^^
^^
t^
2.
Boole's
DepressioriB.
t
is shown
in
Boole'a
Differen-ial
Equations,
hap.
X.
that
a
differential
equation
dmits
of
depression
y
one
order in
each
of the
following
ases :
i.
When
not
involving
.
ii. When
not
involving
?.
iii.
When
homogeneous
in order
1,
and
when
homogeneous
in order
fit.
iv.
When
homogeneous
in
order
oo
.
But
it
is
not shown
whether
these
depressions
an
be
either
carried
out
in
succession^
r
repeated.
3.
Object
f
Paper.
It
is
proposed
ere
to
investigate
he
possibility
f
successive
and
of
repeated
depressions,
hus
greatly
extending
he
power
of the
process
of
aepression.
As
to
successive
depression
he
following
important
esult
will
be
shown
^'
A
differential
equation
dmits
of
successive
depressions
as
follows
:
i.
By
r
orders when the
r
quantities
,
y',y ,...y^*^''
are
all
absent.
ii.
By
one
order
when
x
is
absent.
iii.
By
one
order
when
homogeneous
in
any
one
order.
iv.
By
one
order
when
homogeneous
in
any
other
order.
Thus
an
equation
ot
Involving
?,y,
y',y , . y^' *\
nd
also
homogeneous
in
two
orders,
admits
of
depression
y
(r+3)
orders.
To
prove
the above
itwill suffice
to
show
that
the
success
sive
depressions
if
applied
n
suitable
order
do
not
affect
the
remainingsingularities,
.e.
produce
depressed
quation
still
possessing
hose
singularities.
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120
COL.
CUNNINGHAM,
DEPRESSION
OF
It
will also be
shown that in
certain
cases
one
of
the
depressions
an
be
repeated
ithout
affecting
ny
of
the
other
singularities;
nd
that
in
certain
cases
the
application
f the
depressionsi.,
ii.,
r
iv.
produces
an
equation
ossessing
ne
of
the
singularities
.,
ii.
not
present
in
the
original,
nd
therefore
susceptible
f
further
depression.
4,
Homogeneity,
t
is
convenient
first
to
investigate
ome
properties
f
homogeneity.
Supposing
a;,
^
to
be
quantities
f
degrees
X,
fi
respec*
tively,
hen
the
following
cale
of
degrees
obtains
:
*
Quantities...
?, y,
y',
j/\
f\
...y^
Degrees
X,
/*, /*
X,
/*
2X,
/x
3X,
../x
rX.
Also
the
degree
(N)
of
any
term
Q consisting
f
the
product
of several
of
these
quantities
s
clearly
qual
to
the
sum
of
the
degrees
f
its
component
factors
;
thus,
if
then
-AT^i^X
j/t
+
a
(/i X)
+
iS
(/A
2X)
+...p
(/A
rX)
s=2/.X
+
Jf./tt,
where,
or
shortness,
icr^
-
(a
+
2/8
+
37
+...rp),
if=j-|.(a+
iS
+
7
+..../)).
Now,
it
is obvious
from
first
principles
that
all
terms
with coefficientsof
equal
degree
connected
by
the
signs
+,
^,
=
in
any
sort
of
equation
must
necessarily
e
^
same
degree
N) throughout
he
equation
hence
the
theorem
:
Every
differential
equation
ith
coefficientsof
equal
degree
in
each
term
is
homogeneous
in
Xj
y,
y\
y ,
c.,
and
may
therefore be
depressed
ne
order.
A
homogeneous
differential
equation
being
then
a
sum
of
terms
of form
Q
of
equal
degree
N
may
be
written
s((2)=o,
and
the
homogeneity
is
expressed
y
the
following,
hich
may
be called the
equation
f
homogeneity,
L.\
+
M.fi
=
N(9,
constant
for
every
term).
Def.
The
ratio
v^fi;\
of
the
degrees
of
y
and
x
is
called the
order
of
homogeneity,
nd the
quantity
is
calledthe
degree
of
homogeneity,
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122
COL.
CUNNINGHAM,
DEPRESSION OF
But
these
give,
introdacing
n
arbitrary
ultiplier
j
L
(Xj
+
*X,)
3f
(/ij
kii^
(N^
+
kN^^
also
a
constant,
and
this
result
exjresses
that the
equation
as
homogeneity
of
order
v^
=
(/i^
f
ft/*,)
(X^
+
A\))
and
therefore
of
every
order,
ince
k
is
arbitrary.
.E.D.
As
to
the
possibility
f
such
multiple
omogeneity,
he
necessary
and
suflScientonditions
are
easily
een
to
be
L
and
My
both
constant
for
every
term,
or,
writing
ut,
;?
-
(a
+
2^
+
37
+...+
r/o)
ir,
(a
constant
for
every
term)^
2+
(a+
P
+
7
+. +
p)=Jf,(a
constant
for
every
term)^
and these
conditions
can
be satisfied
by
more
than
one
distinct
system
of
values of
^, j,
a,
/S,
...p
only
when
at
least
three
of
these
quantities
nter
into
every
pair
of
systems,,
r
(which
is
the
same
thing)
when
at
least three
of
the
quantities
a,
y,
y\ ...y^^
ppear
in
every
pair
of
terms:
hence
^^
Multiple
omogeneity
s
possible
nly
in
an
equation
containing
t
least three
of
the
quantities
;,
y,
y^^
. y^^
n
every
pair
of
terms,
The
fact
that
duplex
homogeneity
nvolves
multiple
omo-eneity
bears
the
following
onsequences
:
^'
A
differential
equation
hich has
multiple
omoge-
geneity
cannot
in
general
be
depressed
ore
than
two
orders in
virtue
solely
f
homogeneity.
^'The
depression-formulse
uited
to
any
order
of
homogeneity
that is convenient
may
be
applied
to
depress
a
differential
equation
which
has
homo-eneity
of
any
two
orders.
Examples
ofmultiple
omogeneity.
t
is worth
notice
that
multiple
omogeneity
occurs* in all
final
differential
quations
of
rational
algebraic
quations
(formedby
elimination
of
all
the
constants).
6.
Depeession
i.
Absence
of
y,
y',y , ..y^'^'^
r
quan-ities).
The
depression-formulae
re
(Boole,
h.
X.,
Art.
1)
*
ThiB
property
will be
proved
in another
paper.
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DIFFERENTIAL
EQUATIONS.
123
Thus
this
depression
ntroduces the
new
independent
variable
(t)only
in
place
of
the
old
one
(x)
when
present
in the
original.
As
to
the effect
on
homogeneity,
t is
clear
that
if
a?,
y
be reckoned
of
degrees
X,
/x,
then
*,
u
must
be
reckoned
like
iu,
y**^,
.e.
of
degrees
X,
/*
rX
(Art.4).
Thus
a;,
y^' \
y**\ y*^
re
replaced
y
the
quantities
,
w,
w',
...i*^ **'^f
degrees
X,
il/,ilf X),
(Jf-2X),
...{il/-
n- r)X},
writing
if=(/Lt
rX)
for
shortness
;
i.e.
by
quantities
f
like
degree
with themselves. Thus
the
degree
of
each
term
in the
de-ressed
equation
is
the
same
as
that
of the
corresponding
term
in
the
original.
Hence,
if the
original
ad
single
homogeneity
of
any
order
fi
:
X
and
degree
Nj
the
depressed
equation
has
singleomogeneity
f
order M
:
X
and
of
same
degree
N,
Similarly
f
the
original
as
multiple
omogeneity,
so
also
has
the
depressed
quation.
Thus this
depression
oes
not
destroy
he
remainingingu-arities
ii.,
iii.,
v.
;
but
it
introduces
the
new
dependent
variable
(w),
so
cannot
be
repeated
unless
*^,y *\
c.
be
also
wanting).
As
this
depression
s much
easier of
appli-ation
than
depression
i.,
ii.,v.,
it
would
usually
e
applied
fio
as
to
depressbi/
s
many
orders
as
possible
t
one
step,
and
therefore could
not
be
repeated,
ecause
y^*'^
s
hereby
supposed
present
and
would introduce
u.
7.
Depeession ii. Absence
of
X.
The
depression-formulae
are
(Boole^
h.
X.,
Art.
1),
y^t,
y'^Uy
y''
uu%
y'''uW'-^uu'%
y
=
m'w'
+
4m'm
V
+
wm ,
c.
;
and,
in
general,
^/*
V
=
(^
^^Y
**
As
th se
depression-formulae
hus
generally
ntroduce
the
new
dependent
ariable
(m),
with
one
exception
noted
here-fter
(Art.
21,
wherein
u
cancels
out)
depression
.
cannot
be
applied
fter
the
present
(except
in the
case
reserved).
Also this
depression
lways
introduces the
new
independent
variable
(t)
in
place
of
y,
and
therefore
cannot
be
repeated
when
y
was
either
originally
resent
or
when
introduced
by
depression
,
(see
remarks
at
end
of last
Article).
As
to
the
effect
on
homogeneity,
he
depression-formulae
fihew
that if
a;,
y
be
reckoned
of
degrees
X,
/a,
then
tj
u
must
be
reckoned
like
y,
y
of
degrees
fi^
fi-^X
respectively,
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124
COL.
CUNNINGHAM,
DEPRESSION
OF
80
that
the
quantities
ty tt,
t*',
tt ,
t*' ,
...
t* ^
are
of
degrees
fly
(/i
X),
-
\,
X
/A,
X
2/A,
...
X
(r
l)/ .
On
referring
o
the
depression-formulaB,
t
will
be
seen
that
y,
t/\
t/'\
y^'^
re
replaced
by
homogeneous
functions
of
t,
M,
w',
...w^* *^
f
same
degrees
s
their
own.
Hence,
if
the
originalquation
have
homogeneity
of
any
order
fjb
:
X
and
of
degree
N
(so
that
each
of
its
terms
are
of
degree
-AT),
the
depressed
equation
as
homogeneity
f order
(/liX)
:
/*
(the
ratio
of
the
degrees
of
m,
f),
and
of
the
same
degree
{N}
as
the
original.
Hence
also,
if the
original
quation
aa
multiple
omogeneity,
he
depressed
ill
also
have
multiple
homogeneity.
8.
Depression
iii.
Homogeneity
of
order
v.
Here
v
may
be
zero,
but
not
infinite.
The
depression-formulae
re
(Booky
Ch.
X.,
Art.
3,
Class
II)
X:=tJuj
y^x.ty
BO
that
D^^'y
(e~/?
uDtY
{s^ ^'^/?
m
+
vt)].
The
Quantity
?
or
eJ
will
be found
to
cancel
out
of
the
depressea
equation,
hich
may
be
formed
directlyy
re-lacing
a:
by
1,
y
by
f,
y'
by {u-\-vt\
f
by
{uw'
(2i/-l)M
+
v(v-l)f},
f
by
{uV
+
ttw
+
3
(v
1)
uu'
+
(3v'
6v
+
2)
m
+
y(y-.l)(v-2)f},
c.
As these
depression-formulae
hus
generally
nvolve
both
of
the
new
variables
(ty
u)
with
certain
exceptions
onsidered
hereafter
(Arts.
1,
22)
(wherein
f,
u
cancel
out)
depressions
i.,
i.
cannot
be
applied
fter
the
present
(except
n
the
cases
reserved).
As
to
the effect
on
homogeneity,
he
depression-formulas
give
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126
COL.
CUNNINGHAM,
DEPEESSION
OF
As
these
depression-formulsB
hus
generatlj
Dtrodace
tite
new
dependent
ariable
(w),
with
an
exception
noted
hereafter
(Art.
21)
(wherein
u
cancels
out),depression
.
cannot
be
applied
fter
the
present
(except
in the
ease
reserved).
Again,
as
they
introduce
the
new
independent
ariable
( )
only
in
place
of the old
one
(a;),
his
depression
oes
not
affect
the
applicability
f
depressionii.,
hich
can
therefore
be
applied
fter
the
present
if
x was
originally
bsent.
As
to
the
effect
on
homogeneity,
he
depression-formulaar
give
from
which itfollows
that,
if
x
be
reckoned
as
of
degree
f
y
and
y,
y\
y'\
c.
all
of
equal
infinite
degree
(see
Art.
4)^
then
t
must
be
reckoned
(likea?)
of
degree
I,
and
u
of
degree
1,
so
that
(by
Art.
4)
the
quantities
t^
w,
u',
w , ' ,
..
u^^
are
of
degrees
1, 1,
2,
3, 4,
...(r+1).
On
referring
o
the
depression-formulas
t will
be
seea
that
oj,
y,
y',yf\
e.
are
replacedby homogeneous
functions'
of
t w,
u\
v \
e.
of
degrees
s
below
:
Degree
of
function
-1, 0,
1,
2,
3,
.,.
r,
replacing
oj,
y,
y\
y ,
f\
...
y('*^ .
Hence
this
depression
estroys
he
homogeneity
of
a
equation
possessing
ingle
homogeneity
of
order
go
,
and
therefore
cannot
be
repeated
on
such
an
equation.
But
an
equationpossessingultiplehomogeneity
has
necessarily
homogeneity
of order
zero
(Art.5),
wherein
x
is
reckoned
of
degree
-
1,
and
y,
y',y ,
c. of
degrees
,
1,2,
c.
But
these
have
been
shewn
above
to
be
the
degrees
of
the
functions
(of
f,
tt,
u\
c.)
which
replace
hem.
Hence,
in
cases
of
original
ultiple
homogeneity,
his
depression
roduces
n
equation
ith
singh
homogeneity
f
same
degree
(N)
as
the
original
nd
of order
1
(since
f,
u
are
of
degrees
1,
1).
Hence
an
equationossessingultipleomogeneity
dmits
of
two
successive
depressions
and
no
more)
in
virtue
thereof,
viz.
Ist*
One
depressiony
formulas
for
homogeneity
of
order
go
:
thisleaves
single
homogeneity
f
order
1.
2nd* One
more
depression
y
formulse
for
homogeneity
of
order
-
1
:
this
destroys
he
homogeneity.
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DIFFERENTIAL
EQUATIONS.
127
10.
Depression^FormuhB.
able
II.
shews
the
functions
t)f
,
w,
w',m ,
'
that
are
to
be
substituted
for
a:,
y,
y',
y ,
y'
in
performing
any
one
depression,
nd also the
general
sub-titution
for
y^^K
The
substitutions
for
homogeneity
re.
not
ihe
actual
values of
a?,
y,
y',
c.,
the functions
of
a:,
y
which
would
necessarily
ancel
out
of
the
depressedquation
eing
t)mitted.
11.
Elevatton-FormulcB.
Table
III.
shews
the
functions
of
a?,y,
y',
y ,
c.
that
are
to
be
substituted for
t,
u,
',m ,
c.
in
raising
he order
of
any
differential
equation
in
t^
Uj
also
the
general
substitution
for
u^K
These substitutions
are
actual
equivalencies.
Depbession-Formul^.
Table
II.
\xf
the
original
ariables
;
t,
u
the
variables in
depressed
equation].
N.B.
In
these
Tables
the
sign
H
should
be read
as
becomes^
or
may
be
changed
nto.
Depression
i.
y,
/,
y ,
...y^ '^
anting.
x
=
t^
y^'-^
,
y'^
=
',
y^'^^w ,
c.,
y(' ^)
tt .
Depression
ii.
x
absent,
y
=
t,
y'=^u,
y ^uu\
y' ==uV'+ttM ,
y'-
uV
+
4ttVtt
+
' ,
Depression
iii.
Homogeneity
f
order
y
ss
1.
ajEl,
yE^,
y^'Eu-t^ y''
uu'-Su
+
2t,
y '
=
uV
+
ttw
-
6mw'
+
llw
-
6^,
y '
ttV4
4MVw''+ttM''-
10(uV'+um'')
35wm'-50m+24^,
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128
COL.
CUNNINGHAM,
DEPRESSION
OP
Depression
iii.
Homogeneityf
order
f
= O,
y '
u
V
+
uu'^ Sue/
+
2tt,
Depression
iii.
Homogeneity
f
order
v
=
I.
y'
t M' +4wVm +
tttt -
2
(mV'+
mO
-
Mtt'-f
tr,
Depression
iii.
Homogeneityf
order
y
=
2.
OJEI,
y
=
f,
y'
=
w
+
2^,
/'
=
wi*'
+
3m
+
2 ,
y'
m'm
+
wu'
+
3wu'
+
2m,
3^
=
uW+
4mVm + wm' +
2
(mV'+
ttO
-
ttw'-
2i^^
y(r)
(g-/?
^J3
J'
{s/f
m
+
20).
Depression
iii.
Homogeneityf
order
v,
xEl,
y
=
tj
y'Eu
+
vty
y
=
wM'
+
(2v-
l)w+
v(K-l)f,
y' EwV'+WM +3(v-l)t*u'+(3v'-6v+2)ti+v(v-l)(v-2)f,
+
(4v'-18v
22v-6)m
+
f(v-1)(f-2)(v-3) ,
y(-)
(e-/?
DX'
{e '^/-
+
vf)}.
Depression iv.
Homogeneityf
order
oo
.
a;=f,
y
=
l,
y'Ew,
y
=
w'-fM*,
y -
w'
+
4mm
+
3m'
+
6mV
+
u\
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CONTENTS.
PAGB
Vortices
in
a compressible
fluid
(continued).
By
C.
Chreb
- -
-
113
Depression
of
differential
equations.
By
Lt.-Ck)l. Allan
Cunningham
-
118
The
following
papers
have
been
received
:
Mr.
Bnchheim,
Note
on
Matrices
in
Involution.
Prof.
Oayley,
Analytical
formulae
in
regard
to
an
octad
of
points,
A
correspondence
of
confocal Cartesians with
the
right
lines
of
a
hyperboloid,*
Note
on
the
relation
between the
distances
of
five
points
in
space.
Prof.
Mathews,
Geometry
on
a
quadric
surface.
Articles
for
insertion
will
be
received
by
the
Editor,
or
by
Messrs.
Metcalfe
and
Son,
Printing
Office,
Trinity
Street,
Cambridge.
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ISVS'
jr;
7cfA
No.
CO.]
NEW SERIES.
[December,
887
THE
MESSENGER
OP
MATHEMATICS.
EDITED
BY
J.
W.
L.
GLAISHER,
So.D.,
F.R.S.,
FELLOW
OP TRINITY
COLLEGE,
CAMBRIDaE.
VOL.
XVII NO.
8.
MACMILLAN
AND
CO,
^I'iHonOon
ani
ambrilige*
1887.
Price
One
Shilling.
METCALFE
AND
SON,
CAMBEIDGB,
-AAfl'
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diff iiektial
equations.
129
Elevation-Foemul^.
Table III,
[t,
the
original
ariAbles
; a;,y the
variables
of
the raised
equation].
Elevation i.
y,
y\ /',
...y^'
anting,
t^x,
u:=y^'\
'
=
y(^'\
m'=3^''%
c.,
u( =y(' ** .
Elevation
ii.
x
absent
^^.._y^y-^yyy +3y
3^ y
Elevation iii.
Homogeneityf
order
vsO
. .
aw .
' '=ft^.) ^'.
Elevation
Iii.
Homogeneity
f
order
v
1.
a;'
^
a' xy -y
'
Xxy-y
V
VOL,
XVII.
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130
COL.
CUNNINQHAM,
DEPRESSION
OF
Elevation iii.
Homogeneity
f
order
v.
t
p
.
U^
p
y
U
7
Kj
oj
'
a;
^y
-^yy
\xy
-
yy
V
a
Elevation
iv.
Homogeneity
f
order
oo
,
y
y
=
-3
u(^)=Dp(yj
12.
Successive
Depression.
umming
up^
it
is
seen
that
to
apply
the four
Depressions
n
succession;
he
following
must
be attended
to
:
Depression
1.
This
should
always
be
applied
first,
s
it
does
not
affect
the
remaining
Depressions,
hereas
they
if
applied
first
will
generally
prevent
its
applicationbyintroducing ).
'Depression
i.
This
should
precede
Depression
iii.
(because
the
latter
in
general
introduces
()
;
it
changes
the order of
homogeneity (when single)
from
i/
=
/Li:\toi/
=
fi
\;/Li.
Depression
iii.
This in
general
introduces
,
and reduces
multiple
homo-eneity
to
single
of
order
v
=
1,
so
cannot
precedeDepressions
ii.
or
iv.
Hence
in
cases
of
multiple
homogeneity
the
procedure
is
;
(1)
When
not
preceded
by
Depression
iv. it
may
be
applied
once
with
any
convenient
value
of
Vy
and
again
with
the
value
v
=
1.
(2)
When
preceded
immediatelyby
Depression
iv.
it
must
be
applied
with
value
i;
=
1.
(3)
When
preceded
by Depressions
iv.
and
ii.
in
turn
it
must
be
applied
with value
v=2.
Depression
iv.
This
may
precede
either
Depressions
ii.
or
iii.,
ut
cannot
follow
Depression
iii.
(because
the
latter
leaves
homogeneity only
of
order
v
=
+
1)
;
it reduces
multiple
homogeneity
to
single
of
order
v
=
-l.
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DIFFERENTIAL
EQUATIONS.
181
The
following
able
shews
the order
in
which
the
Depressions
hould
be
applied
o
equations
ontaining
wo
or
more
of
the
singularities
etailed.
Thus
in
certain
cases
there
are
two
or
more
Courses
open,
viz.
2
Couwes
in
eqnations
with
two of
the
singnlarities
os.
ii.,
ii.,
r.
8
Conises
in
equation
with
sinffolaritieB
os.
ii.,
ii
,
ir.
K2
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132
COL.
CUNNINGHAM,
DEPRESSION
OP
The
particular
ourse
most
advantageous
ill
of
course
depend
on
the
nature
of
the
question
but it will
probably
generally
e
advantageous
to
use
Depression
ii.with the
general
alue of
Vy
whenever
available,
s
this
leaves
a
constant
to
be
hereafter
determined in such
away
as
to
simplify
he
integrations.
13.
Combined
Depresston-Formulm,
he
results
of
substitu-ion
by
the
several
Depressions
pplied
n
succession
may
be
readily
computed
nce
tor
all
so as
to
shew
the
result
after
any
number
of
successive
Depressions.
hese
are
exhibited
in
Tables
V,
VI,
in
which
a:,
y,
y,
y%
c.,
belong
to
the
originalquation,
^v Vv Vx'i
yd
^^'J
'
^^^^
depressed
quation,
^
J
y,
Vi^
yi\
second
3
^8^
y
\
yd
c-
y
yi
^^^^^
w
^vyA^yl^yd^^'t
fourth
By
the aid
of
such
a
Table the
first,econd,third,
r
fourth
depressed
quation
ay
be
formed
at
once
without
the
labour of
forming
the
intermediate
depressionsy
simply
substituting
he functions in the Table
belonging
o
the
required
epressedquation
or
the
original
;,
y,
y\
( c.
Successive
Depressions.
Table V.
Depression
.
Step
I
in
all
cases
.
oD
x^j y
y,j
y
y\^y
yi
y\
y\
jotu...
Two
Successive
Depressions,
Depressions
.
and
ii.,
ombined,
yi=^,i
y/=y,j
yi'=y,y2i
2^r=3^,(y,y, +y/'))
yr-y,Q/:yr^^y.y:y.''^yry
Depressions
.and iii.
(v
=
0),
combined.
x^=l,y,
=
a;,,
y,'
y
y
=
y,
(y/ 1),
3'. '2y;(y.i/. '
*:^.'y;'-. )+.y.Cy.'-)(y.'-2)(y;-
).
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DIFFERENTtAL
EQUATT02(S.
135
Four Successive
Depressions.
Depressions
.,
i.,
ii.
(v
=
0),
and iii.
(y
1),
combined,
y,
E
1,
y.'
1,
y,
x
y.'
x^
(y,
+
2x,
-
1),
Depressions
.,
ii.,
ii.
(v=
1),
am? iii.
(v
=
1),
combined,
Depressions
.,
i.,
ii.
( v),
an(?
iii.
(y=i
1),
^mbined.
y,=iiy,'^i,
y,
=' ' + ',
y. '=a;,y,
(a;,
v)
(2a:,
2^
-
1),
yi
=
'.y,
(^ y/
y
+
'^r*
7
v
-
3)
+
(a'
')
.^^4
3f -
2)
(2x^
+2v-
1).
Depressions.,i.,
v.
awrf
iii.
(y
=
1),
combined.
y,
=
l,
y/=l, y.
x
y/ =y,-ir,
+
2x/,
y.
=y.
(y/
+
'^r,
S)
+
x,
(2a;,
1)
(Zx,
2).
Depressions
.,v.,
i.
anc?
iii.
(y
=
2),
combined.
y.=i,
y.'^i,y, =^.+i,
y. '='^4y.+(- 4+i)(2a;,+i),
y. '=a'/(y y;+7y.+i7a;,+6)
1.
14.
Repeated
Depression.
The several
depressions
on-idered
above
may
in
some
cases
be
repeated.
15.
Depression
.
repeated.
he substitution
=
f,
y^''^=Uj
applicable
hen
the
r
terms
y,
y',
y,
...,
y* *^
re
all
wanting,
and
which
depresses
orders
at
one
step,
is
evidently
quiva-ent
to
r
repetitions
f
this
depression,epressing
ne
order
at
each
step,
viz.
by
the
substitutions
but,
as
the
same
final result
may
also
be obtained
by
the
single
ubstitution
(x
=
tj
^^'^^=w),
here is
no
advantage
in
this
repetition.
16.
Depression
i.
repeated.
n
referring
o
the
depression-
formulae
(Art.7),
it
is
seen
that
(in
the
absence
of
x)
y,
y^
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136
COL.
CUNNINaHAM,
DEPRESSION
OF
become
t,
m,
and
t
does not
recur
in
the substitutions
for
y,
y' ,
c.
Hence,
if
y
be absent from
the
original,
will
be
absent
from
the
depressedquation,
o
that
depression
l.
may
be
applied
second
time.
The
h
priori
ecognition
f the further
applicability
f
this
depression
s
more
diflScult.
Thus,
if
u
be
absent
from
the
first
depressed
quation
as
well
as
^),
this
depression
ould
be
appliedgain,
inasmuch
as
y'
=
w,
and,
as
further,
enters
into the substitutions for
y'^
and
all
higher
differential
coefficients
(Art.
7),
it is
clear
that
the
absence of
u
is
secured
only
when
y',
y'',
y,
c.
occur
in
the forms
(given
in
Table
III)
equivalent
o
u\
u^\
c.,
(and
in
no
other).
The
results
of
the double
application
f
this
depression,
applicable
hen
a?,
y
are
both
absent
from
the
original
equation,
re
shown in
Table
VII,
with
the
new
variables
of
Art.
13.
It is
probable
that
this
double
application
ill
not
often
be
advantageous,
s
the
depressiony
two
orders
might
in
the
same
case
(a?,
both
absent)
be
generally
ore
simply
efi^ected
by
the
use
of
depressions
.,
i.
applied
n
succession.
The
results of
further
application
f this
depression
cannot
be
shown
in
an
equally
compendious
way
with
the last.
As
by
Art.
7,
a
single
application
f
depression
i.
does
not
destroy
homogeneity
(merely
changing
the
order,
hen
single),
t
follows
easily
that
repeated
application
ill
not
destroyhomogeneity,merelychanging
its
order
if
single.
17.
Depression
ii.
repeated.
he
a
priori
recognition
f
repeated
pplicability
with
same
value
of
v)
is
by
no
means
simple.
The
original
quation
must
of
course
possess
homo-eneity
of
some
finite
order
(v)
in
a:, y.
Besides
which
the
depressed
quation
must
also have
homogeneity
f
the
same
finite
order
(y)
in
tyU]
this
involves that the
original
quation
should
contain
a?, y,
y',y ,
c.,
only
in
the
forms
given
in
Table
III
as
the
equivalents
f
t,
w,
u\
w ,
c.,
and
should
possess
homogeneity
f
order
v
in those functions
;
a
condition
so
complex
as
to
make the h
priori
recognition
f
the
repeated
applicability
f
this
depression
ifficult.There
is,however,
one
case
of
easy
recognition,
iz. that of
multiple
omogeneity,
in which
it
has
been
shown
(Art. 8)
that
this
depression
may
be efi'ected
twice
with
the value
v
=
1
in each
instance.
By
reasoning
similar
to
that
in
Art.
8,
it
is
seen
that
the
repeated
application
f
this
depression
enerally
estroys
the
applicability
f
depression
i.
(by
introducing
he inde-endent
variable)
also,
that
in
an
equation
possessing
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DIFFERENTIAL
EQUATIONS.
137
inultipleomogeneity
s
well
as
the
property
above
mentioned,
one
application
f
this
depression
ill still
leave
multiple
homogeneity.
18.
Depression
v.
repeated.
he
h
priori
ecognition
f
repeatedapplicability
s
by
no means
simple.
The
original
equation
must
of
course
have
homogeneity
of order
oo
in
a;,
y
;
and
the
depressed
quation
ust
have the
same
in
t^
u.
This
involves
that
the
original
equation
should
involve
y,
y,
y,
c.,
only
in the forms
given
In
Table
III
as
equivalent
o
w,
u\
v \
c.,
and should further be
homogeneous
in
those functions reckoned of
equaldegree.
Ex.
/
r^
-
^]
0
is of this
kind.
As this
depression
ntroduces
the
independent
ariable
only
in
place
of
the
original
;,
if
present
in
the
originalquation,
it
IS
clear that
its
repeated
application
oes
not
affect the
applicability
f
depression
i.,
hich
can
therefore be
applied
after
repeated
pplication
f
depression
v.
if
a;
were
originally
absent.
Similarly
n
an
equationossessingultiple
omogeneity,
the
repeatedapplication
f this
depression
eaves
finally
(as
in
Art.
9)
single
homogeneity
f
order
1.
19.
RepeatedDepression
ormulae.
Table
Vll
gives
the results
of
a
double
application
f
each
of
the
depressions
i.,
ii.,
v.
Repeated DEPRESSiON-FoRMULiE.
Table
VII.
a;,
y
the
original
ariables
;
^1)
Vx
5
^j?
y%
^^
variables
of
the
first
and second
depressions,
Repeated
Depression
ii.
Absence
ofx^
y.
Repeated Depression
Iii.
Homogeneity
of
order
v \
in
aj,
y
and
in
ar^
y^.
dj=l, yEl,
y=a;,+
l;
/'
=
a;,
(y,
+
a^,
+
1),
y''= y,K(y,y/'+y -i)
+
3(y,+ar,)(y;+i)}
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138
COL.
CUNNINGHAM,
DEPRESSION
OP
Repeated
Depression
iv.
Homogeneity
f
order
oo
in
Xj
y
and
in
x^^y^.
x=:x,,
y=l,
/
=
1,
y
=
y,+
l,
y ' y/'
+
3y^/
+
4y;
+
y/
+
7y/
+
6y,+
l.
20.
Other
depressible
ases.
Inasmuch
as
Depressions
ii.,
ii.,
v.
introduce
in
general
both
the
independent
nd
dependent
variables
(f,
u)
in
place
of
a?,y,
y\
c
,
it
may
happen
that
certain
functions of
a;,
y,
y\
( c.
in
the
original
give
rise
on
application
f those
depressions
o
a
depressed
equation
ree from
one or
other of the
new
variables
(m, ),
and
therefore
depressible
y
Depressions
.
or ii.,
although
the
original
quation
ay
have contained
y
or
a?,
and
was
not
therefore
depressible
y Depressions
.
or
ii.
21.
Absence
of
w,
c.
(Depression
.).
On
examining
Table
III.
it
is
at
once
seen
that
**
A
differential
equation
wanting
x,
or
homogeneous
in
any
order,
and
also
involving
,
y,
^',
,
c.
only
in
the forms
equivalent
o
,
t*'* ^,* ' ^* ,
c.
of
the
Depression
li,iii,
r
iv,
(as
the
case
may
be)
will
after
that
depres-ion be
found free from
the
r
quantities
,
t*', ,...tt^* - ,
nd
be therefore
further
depressible
y Depression
1.
22.'
Absence
of
t
(Depression
ii.).
On
examining
the
depression-formulae
f
Depression
iii.
(Table
II.),
it
is
seen
that
t enters
in
the
first
degree
only
into several of
the
substi-utions
for
y,
y',
y^\
c.,
and will
therefore
disappear
from
certain
simple
functions
thereof,
epending
n
the
value
of
Vj
as
follows
:
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DIFFERENTIAL
EQUATIONS.
139
Hence
it
follows
that
''A
differential
equationhaving homogeneity
of
any
order
v
{v
not
infinite),
nd
either
not
containing
the
{y
+
1)
quantities
^
y'j
\f\
' t^^^t
or
else
containing
them
only
in the forms shewn
above
for
the
value of
v,
will after
Depression
iii.leave
an
equation
free
from
ty
and
therefore
further
depressibley Depression
ii.'*
In
the
case
of
an
equation
with
multipleomogeneity,
there
is
thus
a
large
range
of
applicabilitj,
s
the
value
of
v
maj
then
be
chosen
at
will.
23.
Preparationf
equations.
An
equation
ot
possessing
the
singularities
ere considered
may
sometimes
be
trans-ormed
into
one
possessing
hem
by
a
suitable
change
of
variables.
Removal
of
x.
The
well-known results.
When
X
e*,
xD^^D,,
f(xD:)=f(D.\
a;'.i ;
A(A-l)(A-2)...{A-(r-l)}.
When
a
+
bx
=
* ,
f[(a
+
bx)DJi^f(D.\
(aH-6^X2 ;
A(A-5)(^.-2 )...{Z),-(r-l)J},
enable
the variable
x
to
be
removed
from
equations
which
contain
x
and the
diflferentials
only
in
forms
xD^^
f(xDJ,
aT.D;^
or
only
in forms
(a+ix)Z ^,/{(a+ia:)i)J,
a-\-bxyD;.
24.
Integration,
he
use
of
these
depressions
s
chiefly
as
a
help
in
the
integration
f
high-order
on-linear
difi^eren-
tial
equations.
The result of the
depressions
s either
a
diff^erential
equation
f lower
order
or
an
equation
free
of
difi^erentials
(when
the
order
of the
original
s
equal
to the
number
of its
singularities)
in the
former
case
the
depressed
difi^erential
equation
ust be
integrated.
In either
case
this
equation,
free
of
differentials,
s
the
starting
oint
for
a
series
of
ascending
steps
in
which
the
successive
depressions
re
reversed
one
by
one.
Each
reversal
of
a
depression
ives
rise
to
a
first-order
differential
equation
which
is
to
be
integrated
efore
passing
n
to
the
next
Step
:
these
correspond
o
the
depressions
or
homogeneity
and
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140
COL.
CUNNINGHAM,
DEPRESSION
OP
absence
of
x.
Lastly
there will be
a
series
of
r
simple
iiitegratious,
orrespoodiug
o
the
depression
or
absence
of
if'^y^y
y
25. Conditions
for
success.
The
conditions
for the
practical
solution
of
a
high-order
ifferential
equation
by
this
process
are
1 .
The
final
depressed
equation
musl
be
either free
of
difEerentials,
r
else
a
solvible
difEei-ential
equation.
'
2 .
The first-order
differential
equations
arising
must
be
separately
Bolyible.
8 .
The final
simple
integrations(r
in
number)
must
be
separately
possible.
26.
Example.
As
an
example
of
these
principles,
he
equation
9yY-45yyV
+
4 '
=
0,
which
is
the
well known
differential
equation
f
a
conic
may
be taken.
It
wants
a:,y^
y\
and
has
multiple
omogeneity,
so
may
be
depressed
five
orders
in
all,
he result
being
an
algebraic
quation.
As shewn in
Art.
12,
there
are
three Courses open, viz.
the
Depressions
ay
be
taken
in
any
of
three
orders
as
below,
(beginning
lways
with
Depression
.
applied
o
the
utmost
extent),
The
results
are
shewn,
step
by
step,
in
Table
VIII.
:
the
final
result
(Step
IV.)
of
each
Course
might
have
been
written
down
at
once
by
using
the formulae
of
Table
V.,
without
shewing
the
intermediate
steps.
The
integration
f
the three
final
depressed
quations
f
each
Course
is
shewn
in
Table
IX.
It
has
been
thought
sufficient
to
indicate
the
leading
steps
without
shewing
the
actual
details
of the
integrations
which
would
cover
several
pages).
It will be
seen
that
in this
particular
xample
it
has
been
possible
t
each
step
to
solve the first-oider
differential
equationslgebraically^
o as
to
exhibit the differential
coeflS-
cients,
s
explicit
unctions,
pon
which
the
solution
can
be
effected
by
separation
f variables.
In
particular
n Course
l *,
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142
COL.
CUNNINGHAM,
DEPRESSION OF
Course
V.
Step
IV.
Depression
ii.
(v=
1).
(the
final
depressed
quation,
herein
v
is
still
arbitrary).
Course
2*. Step
II.
Depression
i.
9^;
(M,
+
yn
-
45:r,y,y/
40y/
=
0,
(an
equationossessingultipleomogeneity).
Step
III.
Depression
v.
9^/
(y/
+
2y3 )
450^3^,
40
=
0,
^
(an
equationomogeneous
in order
v
=
-
I).
Step
IV.
Depression
ii.
y4
+
2(a?,-t)(a^,-|)
0,
(the
final
depressed
quation).
Course 3^ Step
II.
Depression
v.
9y;'-i8y,y/
4y,'=o,
(an
equationanting
or,,
and
homogeneous
in
order
v
=
1).
Step
III.
Depression
i.
(an
equationomogeneous
n order
v
=
2).
Step
IV.
Depression
ii.
(i' 2).
'r y.+2.(x,-|)(x,-i)=o,
(the
final
depressed
quation).
Differential
Equation of
Conic
Table
IX.
Integrationffinal
depressed
quationf
Course
1 ,
*
or
3*.
Course
1*.
Depressions
.,
i.,
ii.
(v
=
v\
and iii.
(v=
1).
Final
depressed
quation,
^4y4
+
2(a:,
v-|)(a.,H-v-i)
0.
Digiti
zed
by
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CONTENTS.
PAGH
Depression
of
differential
equations
(continued).
By
Lt.-Gol.
Allan
Cunningham
-
-
-
-
-
- -
-
129
The
following
papers
have been received
:
Mr.
Bnchheim,
Note
on
Matrices in
Inyolution.
Prof.
Cayley, Analytical
formulae
in
regard
to
an
octad
of
points,
A
correspondence
of
confocal Cartesians
with
the
right
lines
of
a
hyperboloid/'
'^Note
on
the
relation
between
the
distances of five
points
in
space.
Prof.
Mathews,
Geometry
on
a
quadric
surface.
Mr.
L. J.
Bogers,
An
extension
of the
A.
M.
and
G-.M. theorem in
inequalities.
Articles for
insertion
will
be
received
by
the
Editor,
or
by
Messrs.
Metcalfe
and
Son,
PrintingOffice,
Trinity
Street,
Cambridge.
.
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'XPn'
No.
CCIL]
NEW
SERIES.
[Februaiy,
888.
^
ir,-,...v -i.
y
tSE-
MESSENGER
OP
MATHEMATICS.
EDITED
BY
J.
W.
L.
GLAISHER, So.D.,
F.R.S.,
FBLLOW
OT
TRINITY
OOLLBGB,
CAMBRIDGB.
VOL.
XVIL
NO.
10.
MAOMILLAN
AND
00.
^l-i
ontton
anS
Cambriftfie.
1888.
Price
One
Shilling.
Digiti
MRTCALFB
Ain
BOK.
CAKBRIDQB
'm
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146
MB.
BOGEKS)
AN
EXTENSION
OF
A
Firstly,
et
a,,a,, ...o^
be
integers.
Then
we
merely
have
a
particular
ase
of the well-known
theorem,
wherein
we
have
a,
quantities,
ach
equal
o
J^,
c.,
the
whole number
of them
being
a^
+
a,
+...
a^.
Secondly,
et
a,,a,,
...
be
fractional.
Let
N be the
least
common
measure
of
their
denominators
and
let
Na^^A^^ ^a,
=
-4 c.,
then
we
get
by
what
is
proved
above
\
^,
+
^,+...
J
'
*
Taking
the
real
positive
^^
root
of each
side
we
get
after
reducing
he
bracketted
fraction,
he
inequality
1).
Thirdly,
etthe a's be incommensurable.
Then
we
may
substitute for
each of
these
quantities
fractions,
hich
may
differ from
them
by
less
than
any
assigned
quantities,
nd since the
theorem
ia
true
for
the
substituted
fractions,
e
may
assume
it
also
true
for
the
given
incommensurables.
Hence
we
may
consider
(1)
as
established.
It
willbe
found
conveniently
rief
to
write
s^
for
d^+a^-^ ..j
as we
shall do henceforth.
Let
J,=
for all values of
r
from
1
to
n,
then
from
(1)
/5A*i
1
1
( '
'+
) '^ ' a .'
2),
a
well known
result.
Write
flj'
or
a,,
/
for
a,,
c c,
and
let
5i
i ''
J,
a^'^j
where
m r.
Then
(1
)
gives
Again,
let
b^
a*''
where
t
r.
Then
Q'^CO^^'^'-'-r^^
Combining
hese results
we
get
fer a)
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148
MB.
ROGERS,
AN
EXTENSION
OF
A
Let
u
=
0
ia
(5),
o
that
^^f^^
8^8^J
we
see
then
that
in
the
same
way
and
so
on^
or
as we
may
better
write
it,
J^ J :I
y.
(6),
a
result
which
admits
of
easy
extension
to
n
suffixes
a,
^,
7,
... .
We
shall
now
pass
on
to
applications
f the
above
results
to
Integral
alculus.
2.
In
the
inequality
1
(3)
let
where
^j
+
wA
=
^,.
We
then
get,
after
multipyling
,
5,,
s^
by
A,
and
putting
f{ai)^yy
and
making
h
decrease
indefinitely,
{jy-dxr
wdxr'
wdxr*
o),
where
m r t^
and
the
limits
are
such
that
y**,
f^
and
y*
remain
finite
and
positive
or
allvalues
between
these
limits.
As
an
example
of
this
we
may
put
y
E
,
whence
after
changing
+
1
to
97Z,
c.,
it
follows
that
er
( )'(?)'
.
where
m r t
Here
we
take
fbr
limits
1
and
0.
From
(1)
we
may
observe
that
it
is
impossible
hat
ra
[a*
I
u'^dxy,
I
v'dx^l
(3),
where
the
limits
are
independent
f
m
and
taken
so
that
the
functions
w,
t;
should
remain
positive
etween their
re-pective
limits.
For
let
rwV^
=
^(7w).
Then,
by
(1),
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CEKTAIN
THEOREM IN
INEQUALITIES.
149
But
if
(3)
were
true
we
should
have the
reverse
inequality
also
true,
which
is
impossible.
Hence
(3)
cannot
hold
good.
As
an
example
we
have
/,
0
w
+
1'
80
that
no
function
u
can
be found
such
that
/;
h
where
a,
h
are
subject
o
the afore-stated
conditions.
As
we
deduced
(1)
from
1
(3),
so
may
we
draw
from
1
(4)
that
where
m T^
and
the
limits
are
under
conditions
as
before.
If
y
=
0?,
we
get
(r+ir (7n+iy (5),
where
m r.
3.
If
we
treat
the
inequality
1
(3)
in
the
same
way
as we
deduced
1
(0
fi*^ the
well-known A.M. and
G.M.
relation,
e
shall
get
(SaJT'
(2ay) ^
(SaJT'
(1).
From
1
(5)
we
get
2a6^2a5 2ay.Say
(2),
and
from
1
(6)
^aV^^^'^'
2aJ
^af
,.,
2^^
^ 2^
26
W-
These
give
results
similar
to
2
(1),
iz.
Uyv''dxn yv'dxr- {jyv^dxT-
(4),
jyv^dx
jyv'dx
jyv'^dx
yv*dx
(5),
jyv'^^'dx
^
jyv'dx
jy^dx
^
.
Jydx
Jydx
Jydx
^
^^
where
y,
v
are
functions
of
Xy
and
the limits
are
under the
same
conditions
aa
before.
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150
provided
he sines
are
all
positive.
^e
may
also
get
similar
inequalities
rom
(5)
and
(6).
As
in
2
(3),
e
may
also
show
that if
I
yv^dx
=
^
(rn\
where
o,
i,
y,
v
are
subject
o
the
same
conditions,
hen
we
cannot
have
with
conditions
as
before.
The
following
quations
re
therefore
absurd
:
I
M
dr
=
-p-
,
or
-^ r^
-.
if
w * is
always
positive
etween
a:
=
Xj
and
a:
=
x
and
ar^,
r,
are
independent
f
m.
4.
We
may
also obtain
a
few
inequalities
rom
taking
logarithms
n
1
(1),
whence
,
SaJ
SaloffJ
Fron;i
this
may
be
deduced
.
Jvydx
fvlogydx
^ ^
Jvdx
^
Jvdx
with
restrictions
as
before
as
to
limits.
These
last
inequalities
o
not
appear
to
lead
to
very
interesting
esults.
-
Ozfcod,
b9.
1,
1887.
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(
151
)
GEOMETRY ON A
QUADRIC
SURFACE.
By
Prof.
Mathews.
From
a
fixed
point
P of
a
quadric
urface
project
lane
sections
of
the
surface
upon
a
plane
which is
parallel
o
the
tangent
plane
at
P;
then*
the
projections
ill
all be
similar
and
similarly
ituated.
This
is
easily
provedanalytically;
r
geometrically,
y
straining
he
stereographic
rojection
f
a
sphere-
In
particular
uppose
tnat
P is
an
umbilic
Uj
then the
projections
ecome
circles,
educing
to
straight
ines
for
sections
passing
through
17,
and
conversely
o
every
straight
line
or
circle
in the
plane
of
projectionorresponds
plane
section
of
the
surface.
To
a
coaxal
system
of
circles
corresponds
he
system
of
conies in
which
the
quadric
surface
8
is
intersected
by
an
axial
pencil
f
planes.
If the
axis
of
the
pencil
eets
8 in
two
real
points
and B
every
conic
of
the
coaxal
system
will
go
through
A and
P; if,
n
the other
hand,
A and P
are
imaginary,
no
two
conies
of
the
coaxal
system
will
intersect
in
real
points.
In
this
latter
case
there
are
two
point-
conics
of the
system,
viz.
the
points
f
contact
of
the
two
planes
f
the
pencil
hich touch
8,
The
plane
of
the
pencil
which
passes
through
U
meets
8
in
a
conic,
which
may
be
called
the
radical
axis
of
the
system.
The
axis of
the
pencil
of
planescorresponding
o
a
coaxal
system
may
be
called the
polar
axis
of
the
system.
Let
a
be the
tangent
plane
to
o
at
the
point
17,
apd
let
g
be
any
straight
ine in this
plane.
Then
the
coaxal
system
of
which
g
is
the
polar
axis
projects
nto
a
system
of
circles
having
their
radical axis
at
mfinity;
hat
is,
concentric
system.
Hence
we
get
a
construction
for
what
may
be
called
the
centroid
of
any
conic drawn
upon
8.
Namely,produce
the
plane
of
the
conic
to
meet
cd
in
a
straight
ine
g
;
then
the
point
f
contact
of the other
tangent
plane
to
8^
which
can
be
drawn
through
^,
is
the
centroid in
question.
r
again,
join
U
to
the
pole
of
the
section
;
this line
will
meet
8
in
the
required
entroid.
The
centroids
of
a
coaxal
system
lie
on
a
conic
passing
through
U,
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152
PROF.
MATHEWS,
GEOMETBY
ON
A
QUADRIC
SURFACE.
Let
g
be
the
polar
xis of
a
coaxal
system
;
then,
f
g'
be
conjugate
o
g
with
regard
o
/S,
the coaxal
system
of which
g'
IS
the
polar
axis
may
be
called
conjugate
o
the
other
system.
The centroids
of each
system
he
on
the
radical
axis
of the
other.
Any
two
conies
upon
8 determine
two
points
orresponding
to
the
centres
of
similitude
of
the circles
into
which the conies
project.
They
may
be
found
as
follows.
Find
the centroids
of
the
two
conies and
through
thfem
draw
two
planes
nter-ecting
0)
(the
tangent
plane
to
8
at
TJ)
in
the
same
straight
line.
Suppose
these
planes
eet
the
given
conies
in
A^
B
and
Cy
D
respectively
then,
if the
conies
UA
(7,
UBD
meet
in
0
and
VAD^
UBC
m
0\
0,
0'
are
the
points
equired.
It
is needless
to
go
into further
detail,
or
it
isevident
that
every
known
theorem
in the
geometry
of
the
straight
ine
and
circle
gives
a
corresponding
heorem in
the
geometry
f
conies
upon
8.
For
example,
two
conies
P
and
Q
upon
8
may
be
such
that
a
finite
number
of other
conies
may
be
drawn
upon
8,
each
touching
and
Q
and
two
adjacent
conies
of the
series;
nd
this
can
be
done,
if
at
all,
n
an
infinite
number
of
ways,
c.
Or
again,
s an
example
of metrical
theorems,
et
Aj
B be
any
two
fixed
points
pon
/S,
and
U
an
umbilic.
Draw
two
conies
upon
8
passingthrough
17,
A
and
U^
B
respectively,
and
intersecting
t
a
given
angle
in
U]
then
the
locus of
their
other
point
of
intersection
is
a
conic
passingthrough
A
and
B.
Again,
to
a
conic
in the
plane
of
projection
orresponds
upon
8
a
quarticaving
a
conjugateoint
at
Z7;
and
we
have upon
8
a
kind
of
projective
eometry
by
which any
such
curve
may
be
derived
from
a
plane
section
of
8^
and
its
properties
nvestigated.
EXPKESSIONS
FOR
0(a:)
AS
A DEFINITE
INTEGRAL.
By
J. W. L.
Olaisher.
In Vol.
V.
(1876),
.
173
of
the
Messenger^
gave
without
proof
an
expression
or
0
(x)
as a
definite
integral.
This
expression
ontained several
errors
which
were
corrected
in
a
paper
in
Vol.
ill.
(1887),
p.
61-66
of the
Proceedings
f
the
i3ambridge
hilosophicalociety.
This
paper
contained
six
expressions
orthe function
0
(a:),
ith
an
explanation
f
the
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(
154
)
HOMOGRAPHIO
INVARIANTS AND
QUOTIENT
DERIVATIVES.
By
A. J2.
Forayth.
1.
The
investigations
ontained in
the
present
paper
were
originally
egun
with the
purpose
of
finding
the
relation,
n
which
a
class
of
functions
called
quotient
erivatives
stand
to
reciprocants.
hese
quotient
erivatives
have,
among
other
properties,
hat
of
being
covariantive
for
homographic
rans-ormations
of the
dependent
and
the
independent
ariables,
when
applied
imultaneously
nd therefore also when
applied
separately.
But
it is
evident from
their
forms that
their
aggregate
does
not
constitute
the
complete
series of
such
functions
;
and
my
firstaim has been
to
obtain these
complete
series
for each
of
the
combinations of
homographic
trans-ormation.
The
character
of
the
invariance
is
less
general
han
that
of
M.
Halphen's
ifferential
Invariants,
hich
reproduce
hem-elves,
save
as
to
a
factor,y
the
substitutions
ax
+
by
+
c
a'x
+
b'y-^c'
a aj
+
6 y
+
c *
Functions of the kinds
herein considered
have
been
previously
suggested
y
Mr. L. J.
Rogers,*
ho
has,
except
in the
case
of
the
first
kind,
limited
his
investigations
o
the
deduction
of
the
partial
ifferential
equations
hich
are
satisfied
by
the
functions.
His aim
was
the derivation
of
homographic
reciprocants.
It
is
by
a
comparison
f
the
quotient
erivatives
with
these
homographic
reciprocants
hat
the
desired relation
has
been
obtained.
The
cubic
derivative
has been
expressed
n
terms
of
them,
but the
combination
is
not
legitimatef
or
the
preservation
f
reciprocal
nvariance.
The
relation thus
suggested
s
proved
to
be
general.
2.
Perhaps
the
simplest
ay
of
obtaining
he
quotient
derivativea
is
as
follows. Without
reproducing
he
general
investigation
n
which
they
arise,
onsider for
example
a
Homographic
and
Circular
Reciprocants
(first
aper),
roc.
Land,
MaJth.
Soe.,
vol.
XVII.
(1886),
p.
^20 231,
t
Sylvester,
Lectures
on
the
Theory
of
Bedprocants,''
mer. Joum,
MaHh,^
vol.
viii.
(1886),
.
212.
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MB.
FOESTTH,
HOMOGBAPHIC
INVABIANT8.
155
cubic
equation
of
wbich
the
primitive
s
and let
y
be
the
quotient
of
two
differentsolutions
w,
and
w,
of
this
equation,
o
that
Since the
third and
every
higher
derivative
of
u
vanish,
we
have
y +3/v+
3y '
=0,
and
therefore
eliminating
^,
m^',
,
which
are
linearly
independent
of
one
another,
we
nave
(now
indicating
differentiation
by subscriptntegers)
\j/i^l^
=
0.
/sy
3y
By,
This
is
a* differential
equation
of
the
fifth
order;
its
primitive
s
_Aj^Bx+^
y'^D-\-Ex
Fx''
The
function
on
the
left-hand
side
is
called the
cubic
quotient
erivative.
The
quadratic
quation
--p^
=
0 leads
similarly
o
the
well-known
Schwarzian
the
qnartic
equation
^4
=
0
leads
to
a
similarly
ormed
quartlc
uotient
erivatire
[y ]i'
Vi,
4y. %,.
^Vi
Va
5^4*
10y
lOy,
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156
MR.
FORSYTH,
HOMOORAPHIC INVARIANTS
and
60 on.
And
the
property
of
homographic
invarlance
already
eferred
to
is
constituted
by
the
equation
Some
other
properties
ill
be
obtained
later,
ne
of
them
in
particular
45)shewing
why
in
these
derivatives
the
highest
ifferential
coefficients
of
y
which
occur
are
only
those defined
by
alternate
integers.
Invariants when
the
independent
ariable
is
subject
o
hom H
graphicransformation.
3.
Let
f
y,
x)
be
a
function
which,
hen
the
independent
variable
x
is
transformed,
eproducestself,
ave
as
to
a
power
of
-T
,
BO
that
we
may
write
The
exponent
m
iscalledthe
index of
the
Invariant.
4.
Then
the
followingroperties
f the
general
forms
of
such
functions
are
easily
erived
if
(i).
The
independent
ariable
does
not
eocplicitly
ccur.
For
a:
=
-f
c,
where
c
is
an
arbitrary
onstant,
is
a
possible
transformation
;
if
x
occurs on
the
left-hand
side
of the
above
invariant
equation,
here
will,
after
the
substitution
for
a?,
arise
a
term
or
terms
in
c,
which
do
not
occur
on
the
right-
hand
side.
(ii).
he
irreducible
invariantsdo
not
explicitly
ontain
the
dependent
ariable.
For,
if
a
given
invariant
^
contain
y,
it
can
be
arranged
n
powers
of
y
in
the
form
If
the
transformed
value of
(f
e
4 ,
then
we
may
write
*
Proc.
Royal 8oe^
12th
Jan.,
1888.
t
These
oorrespond
to
the
propositionBiren
by
Halphen,
Thdse
^
Sur
Ic*
inTariantft
diff6rentielB,
.
21.
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AND
QUOTIENT
DERIVATIVES..
157
and
since
^
is
an
invariant,
e
have
Henoe
*.-(|) t.+y{*,-|) *,}
....0.
Now
f ^
nd
I ^
o
not
explicitly
ontain
y,
which
is
subject
to
no
variation
;
hence
Similarly
*t-(^)
i
=
^
and
so
on;
from which
it
follows
that
^^,
^^j
^^
invariants
of
index
m.
And
they
are
all
explicitly
ree
from
the
dependent
ariable.
(iii)
nvariants
are
of
uniform
grade^
that
is,
he
sum
of
the
orders of
diflferentiation
f the
dependent
ariable in
the
factors
of
any
term
of
an
invariant
is
the
same
for all the
terms
of that
invariant.
For
a
possible
ransformation
is
z =
axj
where
a
is
a
constant
;
and
then
dx
dz^
so
that
the
effect
of
the
transformation
on
any
term
is
to
multiply
t
by
a
power
of
a
equal
to
the
grade
of
the
term.
But
the
effect
of
this
transformation
on
the
invariant
is
to
multiply
t
by
a *,
hence
the result.
From
this
it
at
once
follows
that
the
index
of
an
invariant
of
the
class
at
'present
considered
is
equal
o
its
grade,
(iv).
Irreducible
invariants
are
homogeneous
in
the
diffe-ential
coefficients
f
the
dependent
ariable.
For
let
an
invariant
^
of
index
m
be
arranged
n
the
form
where
l)p
s
the
aggregate
of
terms
in
yfr
of the
degree
p
;
then
we
have
Now
a
change
of
the
independent
variable
in
any
differential
oefficient
f
y
gives quantity
hich is
linearin
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158
MB.
FOBSTTH,
HOMOQBAPHIC
INYABIANTS
the
new
differential
coefficients
;
hence
the
degree
of
a
term
in
the
invariant
is unaltered.
The last
equation
herefore
shews
that
the
aggregate
of
terms
of
any
degree
transforms
into
a
corresponding
ggregate
of
the
same
degree,
nd
that
this
aggregate
is
an
invariant.
It
IS
evident that allthese
results hold for
any
transforma-ion
of the
independent
ariable.
5.
Now
for
any
transformation
the
general
law
of
differentiation
s
where
Cr,
=
coefficient
f
p'
in
]p^i
^
p\
+
q-j
p\
+
[
J
dsi
d'z
and
z,
=
-^,z,
=
^,....
One method
of
obtaining
he
characteristic
differential
equations,
atisfied
by
the
invariants,
s
to
consider the
effect
on
the
invariant
equation
consequent
on
an
arbitrary
infinitesimal
change
in the
independent
ariable.
Such
a
change
may
be taken
in the form
where
8
is
an
infinitesimal
and
constant^
nd
/i
is
an
arbitrary
function
of
a;
;
we
then
have
and,
for
r
1,
Hence
we
deduce
Cr.r=l
+
re;A
and,
for
5
r,
n
^
and
therefore
d' d'
_
f r d-' r
.
(T*
rl
d
- ^'^^OlVld.
r
rl d^'
d
when
quantities
f
only
the
firstorder
are
retained.
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AND
QUOTIENT
DERIVATIVES.
159
6.
In the
case
of
a
homographic
ransformatioD,
ay
z
=
so
that
i2?,
^05+
A'
^
eh-fg
{gx^hj
2 9
it
is
sufficient,
or
reduction
to
what
precedes,
o
take
e
=
A,
y
=
0,
(jr
^e
J
and then
^
=
1
+
so;,
so
that
fi^
=
x^
and
higher
differentialcoefficientsof
ii
vanish. The
pre-eding
general
ormula
now
becomes
d d d
^
.
^
d^'
Applying
this
to
the
equation
=
(1+W2sa;)0(y,
),
we
have
on
substituting
or
the
differential
coefficients
of
y
on
the
left-hand
side
an
equation
=
(l
+
iwsaj) ^(y,),
where
in
terms
multipliedy
s
we
may
take
or
a?
indifferently
as
the
variable.
Since this
equation
s
to
be
identically
true,
we
have
an
index-equation,
r
say
the
grade-equation
and
2r(r-l)y..g 0,
a
form-equaiion.
When
the
form
of the function
isdetermined
by
the
latter
and
the index is
inferred
from
the
form,
then
the
grade
equation
s
identically
atisfied.
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160
MR.
FORSYTH,
HOMOGRAPHIC
INVARIANTS
7.
These
equations
oincide with
the
equations
therwise
obtained
by
Mr.
Rogers
(I.e.);
nd,
as
he
has
given
a
succession
of functions
satisfying
hem,
it
is
not
proposed
here
to
do
more
than
merely
to
state
results.
An
important
roposition,
hich admits
of
easy
proof
and
will
be
subsequently
sed,
is
the
following
If
^
he
an
invariant
of
grade-index
n,
then
y^
^
fny/f
is
an
invariant
ofgrade-index
+
2.
For
tM_thi$^
\dx)
\dz)
so
that
dx\y:)-^^dz\/d^^\'
or
-J-
[4^y^)
s
an
invariant
of
grade-
dex
unity,
nd
there-ore
y,
-7
wiy, ^
s
an
invariant
of
grade-index
+
2.
It
is
by
the
use
of this
proposition
hat
Mr.
Bogers
obtains
his
succession
of
educed
functions,
eginning
ith
the
customary
Schwarzian
y,y,
|y,*;
from
and
after
the second
educt,
owever,
simpler
orms
can
be
given
as
follows.
8.
The
function
of
the first
degree
is
The
functions
of
the
second
degree
re
:
f.^Vxy.
^^y.y,
losy.y.
^^y:^
and,
generally,
-
J^[
2jg
+
l 2jg )
^''^ ^r'^'^ 2^ '-'^ 22?--5-fll2
where
e,
=
i(-l)'j
and
for
*s=0,
1,
2,
...,p-
1,
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CONTENTS.
PAGE
Homographic
invariants
and
quotient
derivatives
(continned). By
A.
R.
Forsyth
-
-
-
-
- - -
-
161
The
following
papers
have been
received
:
Hr.
Buchhoim,
Note
on
matrices
in involution.
Prof.
Cayley,
Analytical
formulse in
regard
to
an
octad
of
points/
A
correspondence
of
confocal Cartesians
with the
right
lines
of
a
hyperboloid/'
*'
Note
on
the
relation
between
the distances
of five
points
in
space.
Articles
for
insertion
will
be
received
by
the
Editor,
or
by
Messrs.
Metcalfe
and
Son,
Printing
Office,
Trinity
Street,
Cambridge.
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