Annals of Mathematics

Determination of a Sphere Which Cuts Five Given Spheres at the Same AngleAuthor(s): Thomas CraigSource: The Analyst, Vol. 7, No. 1 (Jan., 1880), pp. 13-16Published by: Annals of MathematicsStable URL: http://www.jstor.org/stable/2635949 .

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?13?

universal; but the expression is the result of very different operations in

the two systems. Modern mathematics has given new meanings to old

symbols, or, more properly speaking, it has used old symbols to represent new ideas, and thus the same symbol has a double meaning ? and in some

cases several meanings. Thus, in algebra, dx implies that x is multiplied

by d, but in calculus d implies an operation; in the expression d2x, the fig? ure 2 may imply a square or a second operation; ?1 in the expression a-1

implies a reciprocal, but in the expression sin-1^ an inverse function; the

period (.) is used to denote a decimal fraction, multiplication, or a fluxion;

ft = 0 in algebra means that /? is zero, but in modern geometry it may rep? resent the equation of a line; &c. Conversely, the same symbol may be put under different forms; thus we have

a-J-6 = ?; a"1 = -; dx = x; j/?1 = (?1)*; &c. o a

Hamilton's system of quaternions and the imaginaries of algebra have a

relation to each other similar to that between Chemistry and Alchemy. The false science of alchemy gave birth to the true science of chemistry,* but the latter, instead of solving the problem ofthe former?the transmu-

tation of metals into gold?shows it to be all the more impossible, and has

entered new fields of inquiry; so the imaginaries of algebra gave rise to the

real system of Quaternions, but instead of geometrizing the former leaves

them as imaginary as before and enters its own field of inquiry by develop- ing new processes of analysis.

As soon as chemistry was established there was a deluge of pretenders in

science who attempted to find the philosopher's stone; so after the establish-

ment of the science of quaternions, there were, and still are, some who hold

that all imaginaries are so only in name, but are in fact real quantities. (To be continued.)

DETERMINATION OF A SPHERE WHICH CUTS FIVE

GIVEN SPHERESAT THE SAME ANGLE.

BY DR. THOMAS CRAIG, WASHINGTON, D. C.

General Benjamin Alvord a few days ago attracted my attention to this

problem as being one which had, at one time, possessed a peculiar interest

* J. Andrew, in the ninth edition of the Encyclopaedia Britanica, says, "Alchemy was, we may say, the sickly but imaginative infancy through which modern chemistry had to pass before it attained its majority,"

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?14?

for him on account of its connection with the large class of problems on

intersections and tangencies to which he has given so much attention, and

from which he has obtained so many valuable results by the elegant proces- ses of synthetic geometry. Gen. Alvord at the same time informed me that

he had communicated the problem to the Analyst, through the agency of

Marcus Baker, Esq., and that Mr. Adcock, in the Sept. number ofthe An?

alyst for 1877, had furnished the equations necessary for its solution, but

did not give the solution.

Having given the necessary equations, I presume that Mr. Adcock thought it mere journeyman's work to obtain from them the required forms of the

unknown quantities. The following solution is extremely simple but I

have thought that it might not be destitute of interest to the readers of the

Analyst on account of the symmetry of the expressions and the ease with

which the eliminations can be performed. Call 0 the angle of intersection and (xl9 yx, zx), (x2, y2, z2) &c, the co?

ordinates ot the centers of the given spheres, ru r2, r3, r4, r5 their radii,

p the radius of the required sphere and (a, /?, y) the coordinates of its center, then we have by equating the expressions for the distances between the cen?

ters of any one sphere and the required sphere, in polar and rectangular coordinates

cos e = (a?< ~ "P+fo-ft'+tsr-rP-ri'-/^ (1)

where i has the values 1, 2, 3, 4, 5.

These are the expressions given by Mr. Adcock for the determination of

the four unknown quantities (a, fi, y, p); after these are determined simple substitution will give a value of cos 0 for any given system of fiye spheres.

For the determination of (a, /?, y, p) we have the equations

^[(^-?)2+(3/r/5)H(?rr)2-^-^] = ^i[(^-?)2+(2/3-/?)2+(23-r)2-^-/'2],

rst(xr?Y+(yrPT+(zrrY-r: -y] = r, {.(xr?Y+(yrP?+(\-rT-ri-P2i

(2)

Expand these expressions and for brevity write

r^rz-r^+r^xl+yl + z^-r^xl+yl+zl) = Py

rxr5(r5-r1)+r5(x\+y\^^r1(x\+yi+^) = P5,

Px, P2 &c, being of course determinate constants. We find now

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-15?

(r2-r1)(a2+/?2+rV)-2[(?'2a;1-riiC2)?+^22/1-^i2/2)^+(?-221-^122)r]=-Pi

(r3-r1)(ai+l?+f-p2)-2l(rsx1-r1xs)a^-(rsy1-r1ys)l3 + (r^-r^^y] = P2

(r4-r1)(a2+/32+r!!-/'2)-2[(r4a!i-ria;4)?+(''42/1-',i2/4)^+('*^i-',124)r]=-P3

(r6^1)(^+^,+r,-/o')-2[(?,6a?i-?,ia!?)a+(r5yi-riy?)^+(?-5?1-r1z5)r]=P4

(3)

, r4?rA, r5?rx respectively Divide these through by r2?rl9 r3- and place

= ?i, = ?2 &c-;

then subtract the second equation from the first, the third from the second

and so on : by this means we eliminate the quantity a2 + /?2 + f ?

p2

and in fact eliminate p altogether and have left three equations containing three unknown quantities a, /3, y.

The result of the first substitution is of the form

\r"(x^~x2)+r^r2(x1-xs)+r1rs(x2?x1)']a Q _Q (rB-riKr,-^)

+ ""' Vl ^

the terms in j3 and y are obtained by merely changing x into y and z respec?

tively. The next equations have their first terms in the forms

\r\(x?? x^+r.r^? x^+r^^x^? xx)~\a , _ Q _Q -_ -j- . . . . ? v?2 *?3j {**?*!) (^4? ri)

[rl(x5?x^+r^r^? x^+r^^x^?xt)]a + . _ = Q Q .

These equations can all be given in the determinant form as follows:

'19 ' 3> xl9 #3,

rl> r4>

\a +

? +

? + ?/l>2/4> 2/5 /3 +

1> ' 2> !1> 22> ' 1> ' 1) ' 1) ' &> S1J 2S>

!1! *4> "5 ri, ri> i\

r^ft-GiXViXv.).

r = (&-&)(Vi)(V.)-

(4) The coefficients of a, /?, 7- are constants and of course can be readily com-

puted, for any particular case, by merely expanding these determinants; the

same is true of the quantities on the right hand side of the equations, they are constants and readily computed. Call the values of the determinants

in the first second and third equations, respectively,

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?16?

A,,

A2,

As 5,

2; 2>

L3> ^3> ^3' and denote the second members of the equations by

respectively, then these are Dx, Dit D 3>

Ata

A2a

+ BJ + Cir = Du + B2p + C2r = D 2> (5)

A3a + ?3/9 + C3r = D3;

writing for brevity J = the determinants of the coefficients A, B, G, i. e.,

J =

we have by direct solution

Ax, A.?

L3>

D,

?Bi,

-S?,

jDo, jBn 'l>

'2> D? .B

/? = -

'3?

Ci,

3>

C2, i>2 Co, -Z/q y3>

^1, *1,

L3>

A, ^3

(6)

Nothing in the second members of these equations but known quantities, therefore the center of the sphere is determined. In order to find the radi?

us p, it is only necessary to substitute these values of a, /?, y in any one of

equations (2), and solve the result for p2. In fact we have

2 = r^-af+^.-py+^-y)^

r2~-r1 The position of the center and the radius of the required sphere are now

found. If it is desired to find at what angle the sphere intersects the five

given ones, the quantities a, /?, y and p must be substituted in equation (1) and the result will be the cosine of the required angle.

It is worth while to observe that the coefficients of a, /?, y in the preced? ing linear equations are all derived from the same determinant, are in fact second minors of

V x1}

zi>

'2> x2, V2>

'3> x3>

Z 3>

'4) X4>

2/5

This consideration would enable us to replace A, B, C by certain second differential coefficients of V, but it is not necessary to go into that.

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