L. VANHECKE and L. VERMEIRE

ON ASSOCIATE RECTILINEAR CONGRUENCES

In the study of rectilinear congruences we meet several pro­perties with a similar character; they are particular cases of the following property.

Suppose S and S' are two surfaces with position vectors x{u,v)

and x{u,v). Points corresponding to the same values of the para­meters u and v are called homologous points. The homologous points of the surfaces Sa, represented by

xa=xcos a + x' sina , (a is any constant) , (1)

belong to °°2 concentric ellipses with center at the origin. The homo­logous points of the surfaces S' a, given by

xa' = x0{u,v)+xa (2)

also belong to °°2 ellipses. But now the centers coincide with the

homologous points of the surface Xo(u,v). Any four surfaces Sa

or S'a determine on the associate ellipses sets of four points with a constant cross-ratio.

Now we treat some examples. We use the same method as in [ 1 ] , [ 2 ] , [ 3 ] , [4] (x), without further explicit reference.

0) See References.

— 128 —

Central points and middle points on homologous lines.

1. Rotation with a sheaf as congruence of axes of rotation. Starting from a point-envelope, taken as origin, a congruence

Co is defined by

X0 = u + tv3 (3)

where

The congruences

with

u=uivi + u%VcL . (4)

Xa=ua + tvd (5)

(6) uia~ui cos a —u% sin a ,

u2a = ui sin a + u% cos a ,

are called associate congruences Ca. Their middle surfaces are given by

^ m « = ^ « + - 2 (Qia + a2a)V3 (7)

or, by a result in [1] :

^ « = ^ m O C O S a + ^ * s i n a • ( 8 ) m 2

The central point of a line for a ruled surface in C a is given by

XCa=Ua + laV3 • (9)

If we consider the central points of the homologous lines for the corresponding ruled surfaces in C and Cn_, i.e. for ruled surfaces

2

with the same image curve on the spherical representation of the congruences Ca, then (9) can be written as

Xca={u + lvz) cosa + ( ^ 4 - ^ f 3 ) sina 2 2

or

X ra = X„0 cos a + X n_ sin a . (10) " 2

LCci y i c 0 c

— 129 —

Thus (8) and (10) enhance:

PROPERTY I.

Consider associate congruences. The homologous points of the middle surfaces belong to °°2 concentric ellipses with center at the origin. Any four middle surfaces cut each ellips in four points ivith the same cross-ratio.

An analogous proposition holds for the central points of the ruled surfaces with the same image curve on the spherical repre­sentation of the associate congruences.

2. Rotation with any parallel congruence as congruence of axes of rotation.

Let us generalize the above considerations. Instead of a sheaf with center at the origin, we take any parallel congruence Cr as congruence of axes of rotation. Then the middle surface are given by

where xr represents the surface of reference for C, . Equation (11) becomes:

Xma = Xmr + (^m0 ~ ^mr) C 0 S « + (%m ± ~^mr) s i n « ' (1 2) 2

For the central points of the homologous lines of the correspon­ding ruled surfaces we find:

XCa = XCr + (^cO - %cr) C 0 S a+(Xr± — %cr) s i n « • (1 3) 2

Consequently:

PROPERTY II.

The homologous points of the middle surfaces of associate con­gruences belong to °°2 ellipses. The center of an ellips coincides with the homologous point of the middle surface of the congruence Cr. / / we consider the ruled surfaces with the same image curve on the spherical representation of the associate congruences, then the central points of homologous lines belong to °°2 ellipses with

9

— 130 —

center at the corresponding central point of the homologous line in Cr.

3. Associate normal congruences.

Assume that all the associate congruences Ca have the same middle envelope and the same anomaly, or, equivalently that C0

and the congruence of axes of rotation Cr have this property. Then (12) becomes

Xma=Xmri u cosa + u^ sin a (14) 2

and we obtain °°2 circles. Let us consider the special case of normal congruences. One

easily computes the equation of the orthogonal trajectories:

Xna = Xnr-\- (u — fv3 - du ) cosa + ( Un_ — f v3 m dun) sin a. (15)

2 2

These surfaces are called associate surfaces ^ a . Thus we have:

PROPERTY III.

Suppose the associate congruences are normal. Then the homo­logous points of the associate surfaces belong to °°2 ellipses. The centers of the ellipses coincide with the homologous points of an orthogonal trajectory of the congruence Cr. The four points in which each of the °°2 associate ellipses is cut by any four surfaces 2a are in constant cross-ratio.

Similarly, if we suppose that all the associate congruences (5) are normal, then the orthogonal trajectories ^ a are given by

Xna^Ua—hi'dUa • ( 1 6 )

Under a suitable choice of the constants of integration we become

Xna==Xn0 cos a-f X n_ sin a . (17) n 2

If these surfaces are called associate surfaces %a, the normal con­gruences (5) satisfy the next result.

— 131 —

PROPERTY IV.

The homologous points of suitable orthogonal trajectories %a

belong to °°2 concentric ellipses. Any four associate surfaces %a

determine on the ellipses four points with constant cross-ratio.

In [5] P. VINCENSINI investigates a special case of the above situation. He starts from the close relation between the normal congruences of Appell and the real minimal surfaces. The con­gruence of axes of rotation is a sheaf with center at the point of Appell. Further the author proves several properties, which we mention here.

Consider associate normal congruences of Appell, then we have:

(i) the tangent planes in homologous points of the middle sur­faces are parallel;

(ii) homologous elements of area of the middle surfaces are equal; (iii) at homologous points of two associate surfaces of Appell, the

difference of the principal radii is the same; (iv) consider two associate surfaces of Appell % and 2a, then the

angle between the lines of curvature on %a and the lines of curvature on X equals a /2 .

It is one of our main goals to generalize these results of VIN­

CENSINI to congruences with associate minimal surfaces as enve­lopes. This is done in the next three paragraphs.

4. Associate congruences, defined by associate minimal en­velopes.

We consider the congruence C0 represented by

X0=x + u + tvs , (18)

where x is the coordinate vector of the minimal surface S. The congruences

with

xa—x cosa + Xn_ sin a (20)

— 132 —

and x«_—fv3xdx , (21)

2

are called associate congruences; Xn_ represents the adjoint mini-_ 2

mal surface and xa represents the associate minimal surfaces.

The middle surfaces of the associate congruences are given by

Xma=xa-\-ua+- (Qia-i-o2a)v3 (22)

or, with [1] :

Xma=[x + u+ - fer^) v9]cosa + [xn_+Un_+- (Q „+o n)vd]sina. £ 2 2 ^ 2 2

This enhances again relation (8):

^ m « = ^ m O C O S a + - ^ - s i n a •' (8) 2

For the central points we find:

Xca=xa + ua + lav3

— (x-\-U+lv9) COSa + (Xn_+U3t_ + ln_VB) Sill a 2 2 2

=Xc0 cos a + X n sin a . (10) c 2

We find again equation (10). Consequently, also the associate congruences (19) satisfy the

property I.

5. Now we consider the case where the congruence C0 is nor­mal and has the minimal surface S as middle envelope. We know from [1] that all the associate congruences are normal and that they have the corresponding associate minimal surface as middle envelope.

The equation of the middle surface of Ca is

Xma=xa+ua=xQa . (23)

— 133 —

From [1] it follows easily that the normals to these middle sur­faces are given by

a = (Q%° — <>i!ei)Vi + (?iQ — Qi°)v% + (Qi<*% — °iQ%) v3 • (24)

It follows:

The tangent planes iri homologous points to the middle sur­faces of the associate congruences (19) are parallel.

As proved in [1] the distance between the limit points for homologous lines in the associate congruences is the same. This enhances:

The difference of the principal radii at homologous points for two associate orthogonal trajectories of associate congruences is the same.

One easily calculates that this orthogonal trajectories are re­presented by

Xna=x0a+tvs , t=-fv3. dxQa (25)

or

X n a = X n 0 c o s a + X ^ s i n a . (17) 2

We again arrive at property IV. The lines of curvature of the orthogonal trajectories are deter­

mined by

with

d^na = a i V i + a^V% • (27)

From [1] it follows easily that

Thus the equation (26) becomes

PlcPt* +ft«0>81 = 0 » ( 2 9 )

— 134 —

or

°ia (W23 — W13) + 201«w13tt>23 = 0 ( 3 0 )

where

a,„=Oi cos a + o, sin a , (31)

@ia=£i C 0 S a — ai sin a .

Hence the equation of the lines of curvature reduces to

°i (w'23 ~ w ' i i ) + 2^iw/i3ft>,23 = ° (32)

with

a . a co,

13 = a ) J 3 c o s - + o)23 s i n - ,

, . a a co 23 = — a>i3 sin — + a>23 cos —

(33)

Thus it follows:

Consider two associate orthogonal trajectories % and 2a of two associate congruences. The angle between the lines of curvature on 2a and the lines of curvature on % equals a /2 .

6. Associate congruences defined by any pair of two envelopes.

We generalize the previous theory even more. Let S and S' be two surfaces with a mapping between them and such that the tan­gent planes at homologous points are parallel. The congruences C a

defined by

Xa=(x + u) cosa-f(#' + Un) sin a +ta3 (34) 2

are also called associate congruences. We first deduce the formulae for the adjoint congruence C with

surface of reference

X ^ ' + H * . (35) 2

— 135

Let us put

anc

wi ith

dx/ = ai/vi-j-a2

fv2 , (36)

dX=/V«i+/V«»+/V*« (37)

£l' = 0l 'o >13 + a l ' C ( ) 23 »

^ / = p3/o)13 + a2

,(o23 , (38)

It follows easily from [1] :

W13 ~~\]£' + B )• ° J 2S '

P'3=—oa)iS + QO)2fi ;

consequently:

(40)

, _ r 2 ' r 2 ' 7i

" ' H W a. r i n

n n' r2 Q2 — " £ 7 +A=— — — j ^ + ^i »

Hence the double of the absciss of the middle point is equal to

ei' + < = £ - fri' + rt) + °i — Qz * (41)

— 136 —

while the anomaly is equal to

1 Ot'—Qz-g ( ri + r2) - - (Qi + 0%) • (42)

For the associate congruences (34) these quantities are

Qia+<J2a=(Qi+<Jz) cosot + tei'+V) sin a , (43)

ai« — Q2a=(0i- Q%) cosa+(ait—e2')sina .

Once more we obtain property I. From (41) and (42) we further deduce that all the associate

congruences are normal and are referred to the middle surface if

and only if x and x both represent minimal surfaces. Suppose this to be the case in the sequal of this paragraph. We first investigate whether the tangent planes at homologous

points of the middle surfaces are still parallel. Therefore we cal­culate expression (24) for the middle surface of the adjoint con­gruence. If no other conditions are imposed on the congruences, then the necessary and sufficient condition for parallellism is

(44)

It means that x is the adjoint minimal surface of x. Hence:

The tangent planes at homologous points of the middle surfaces of associate congruences are parallel if and only if the correspond­ing middle envelopes are associate minimal surfaces.

Now we suppose again that x and x are minimal surfaces, but not necessarily adjoint. We investigate the orthogonal trajectories of the associate congruences. Their equation is

Xna= {x+u). cos a + (#'+Un) sin a \ tvs (45) 2

K' • —

K' n

n' *\

~K7~ K •

where

— 137 —

t = —[ f v3 • d(x + u)] cosa — [fv3 • d{x' + Un)] sin a . (46)

We obtain again

X„„=X„n cos a 4-X .-r sin a . *-na -^wO (17)

Hence property IV holds here too. If we calculate the distance between the limit points for the

associate congruences, we find for its square

4 (£ia+aia) = 4 [(£i cos a-\-Qx' sin a ) 2 * ^ cos a+o^ sin a)2]

= 4 (QI+0*) ^ c ^ t e ^ + a / 2 ) s in2a+ (47)

+ 8 (QiQi+ai°i) cos a sin a .

By (40) this equals four times

2

(e2 + af) cos2a +

+ 2 e i

rc £' £ — a,

71'

+ ~ F ; + -F + ei

71 n

K' K sirr a +

JT' c+Tr' + x, (48)

sin a cos a

If no other conditions are imposed on the considered congruences, it follows:

The difference of the principal radii at homologous points of two associate orthogonal trajectories is the same if and only if the corresponding envelopes are minimal surfaces.

Just as easily one arrives at the next property:

The angle of the lines of curvature on the orthogonal tra­jectories 2a with the lines of curvature on X equals a / 2 if and only if the considered minimal envelopes are associate minimal surfaces.

REMARK

One of the above properties can still be generalized. From (42)

— 138 —

and (43) we see that all the associate congruences are normal under the following conditions:

(i) C0 is a normal congruence;

(ii) C0 and the congruence of the normals to the surface x, which needs not be a minimal surface, have the same middle envelope.

Again the orthogonal trajectories are represented by (17), and the corresponding property IV is satisfied. We state then that each of the associate congruences has the same middle envelope as the congruence of the normals to the corresponding envelope x cos a + x sin a.

7. General cases.

We investigate the general case of two surfaces x0 and x\ with a mapping between them, but without the condition of parallel tan­gent planes at homologous points. The congruences

Xa=x0 cos a + x0' sin a + tvs (49

are still called associate congruences Ca. We put

dx0=piv1 + p2v2 + psvs (50)

with

#j = e2ft>13+a2co23 , (51)

h = QcoiS + 0(023 »

and denote the corresponding quantities for x\ with accents. Then one easily proves property I.

If all the associate congruences are normal, we find property IV. Remark that the essential assumptions in the properties I and

IV are :

(i) homologous points of the chosen surfaces of reference belong to concentric ellipses;

(ii) homologous lines of the congruences are parallel.

— 139 —

In a similar manner the basic assumption for the parallel con­gruences in properties II and III is: homologous points of the chosen surfaces of reference belong to °°2 ellipses. The middle

points of the ellipses constitute a surface xr which is surface of reference of the congruence Cr. Cr is built up by lines parallel to

the corresponding lines of Ca. Indeed, let three surfaces x{u, v),

x(u,v) and xr(u, v) be given. Consider the associate congruences Ca denned by

xa=xr+(x — xr)cosa + (x'—#r)sina .

If we denote the coordinate vector of a middle surface by the in­dex m, it is easily calculated that

Xma = Xmr+(Xm — ^mr) C 0 S a + l^m' ~ %$*) s i n a ' I53)

In a similar way we find the central point of the ruled surfaces with the same image curve on the spherical representation of the congruences:

Xc a = X c r + ( X c - X c r ) c o s a + ( X c ' - X c r ) s i n a . (54)

Finally, assuming that all the associate congruences are normal, i.e. C, C, Cr are normal, the orthogonal trajectories are given by

XJW = X I l r + ( X n - X l i r ) c o 8 a + ( X f l ' - X l i r ) 8 i n a . (55)

Homologous focal points of associate congruences.

8. We now consider the associate congruences (5) starting again from a point-envelope. The focal points of these congruences are determined by

* 2 — (Qia + a2a) t + Qi02 — 0 ^ 8 = 0 . (56)

We refer to a local (x, y, z) coordinate system with origin at the

point-envelope and axes defined by the unit vectors vl9 v2, v3. The coordinates of the focal points are given by

— 140 —

X = Q cos a — a sin a ,

y=o cosa-f Q sin a ,

z2+Qi°2 — aiQ2 = zl(Qi + °2) c o s a \-(ol — Q2) sin a] .

(57)

Thus the locus of the focal points is the line of intersection of the cylinder of revolution

x2+yz=Q2+o2 (58)

and the cylinder

+ (Q2 + O2)(Z2 + Q1O2-Q2O1) = 0 . ( }

Hence the locus of the homologous points of the focal surfaces is generally a non-degenerated curve of the fourth degree.

The locus is a degenerated curve if and only if (59) degene­rates into two planes. This happens only in two cases:

1) cr1 — p2 and pi + cr2 are both zero. We have associate nor­mal congruences of Appell. The locus degenerates into two circles symmetrically placed with respect to the (v±, v2)-plane.

2) p\(T2 —Oi/D2'= 0. All the associate congruences have the point-envelope as focal envelope. The corresponding focal points

belong to a circle in the -(vl9 #2)-plane. The other focal points be­long to the line of intersection of the cylinder (58) and the plane

x

y

Q

o

— o

Q

ai ~ 92

0 (60)

which turns out to be an ellips.

9. The previous setting can be generalized if we start from associate minimal surfaces. To study the focal points of the asso­ciate congruences we take the trihedral (v1,v2,v3) with vertex at the origin as local coordinate system. We denote the coordinates of x and ^^respectively by xl9 yl9 zx and x2, y*, z2. Then the coordi-

2

— 141 —

nates of the focal points are given by

X=(X1+Q) cos a + (x2— a) sin a ,

y=(yi + o)cosa+(y2 + Q) sin a ,

2z—2 (zt cos a + z2 sin o)+Qla + o2a\-

The last equation reduces to

z2-\-(zl cos a+z2 sin a)2 + ^^2 — O±Q2 — 2z (z1 cos a + z2 sin a) —

— Z[(QX-\-O2) cos a + (aj — a2)sina]-f (62)

+ (0! cos a+,z2sina) [(Q1-\-o2)cosa-{-(o1 — ̂ 2) sina] = 0 .

Thus the locus belongs to the cylinder

1(72+9) x—{x2-°)y?Jrl(xiJrQ)y—(yiJr°)x? =

= [(%+^) (y% + Q) — (yx+°)(x2 — °)r

and to the quadric obtained as follows: calculate sin a and cos a from the first two equations in (61) and substitute in the last one. Hence locus of the homologous points of the focal surfaces is again a curve of the fourth degree.

This curve certainly degenerates into two ellipses in the following cases:

— all the associate congruences are normal, and they have the respective associate minimal surfaces as middle envelope;

— all the associate minimal surfaces are focal envelopes (/0io-2 — — 0-1P2—0). Both ellipses have their center at the origin.

10. For the general associate congruences (49) we also find a curve of the fourth degree as locus of homologous focal points. Indeed it is represented by

x=x0 cos a-\-x0' sin a ,

y = y 0 c o s a + y 0 ' s i n a , (64)

(z — za)2-(Qla + o2a) {z—za) + Qlao2a~ olag2a=0

(61)

— 142 —

or

(yo'^--Vr)2+(yo^-^or)2=Kro'—y0V)2 > 65

(2 - zaf— (Qla+o2a) (z — za) +glao2a - olag2a=0

where za, gla, ola, g2a, o2a are homogeneous linear functions of x and y, since they are linear and homogeneous in sin a and cos a.

Moreover if these curves degenerate into two systems of conies and if one system consists of concentric ellipses, to does the other and they have the same center. Indeed, this is easily proved by putting Qiao2a — oiag2a = 0 in the last equation (64).

In an analogous way we obtain curves of the fourth degree as loci of homologous focal points of the congruences (52). We refer them to a local coordinate system with center at the origin and with

axes defined by vu v2, v3. Then their equations are

X=xr + (x — xr) cos a + (xf — xr) sin a ,

Y=yr + (y— yr) cos a+ (y' — yv) sin a ,

(Z-za)2— (gla+o2a) {Z—za)+Qiao2a — a l a e 2 a =0 ,

where za . gia, ala, g2a, o2a are linear in cos a and sin a. If these curves of the fourth degree degenerate into two systems

of conies then the two centers of the ellipses belong to the homolo­gous line of Cr and are symmetrically placed with respect to the middle point. This is easily proved when we consider the con­gruences (52) and suppose that the points xa are focal points of the congruence Ca. The other focal points are given by

Xa = xa + tva > t=Qia + o2a .

Homologous l-planes.

11. Now we treat another problem which leads also to con­centric ellipses as loci of focal points.

Through each line / of a congruence

X=xQ + tv. (66)

— 143 —

we bring a plane, called /-plane. We investigate the envelope of these two-parameter family of /-planes. A point of an /-plane is represented by

X=x0 + X(vl + rv2)+pv8 . (67)

To facilitate the study we choose the trihedral (v1,v2ii's) such that r = 0. Then the /-plane is represented by

X = # 0 + kvl + /uivz . (68)

To find the characteristic point F± we require that

v2' d {x0 + kv{ + ju.v3) = 0 , (69)

i.e.

/52 + Aa>I2 + ywa)32 = 0 (70)

for all values of o)]3, w23. Hence

o2- p2l —^==0 .

The position vector of the characteristic point F1 is then

— — Q2 — 1 — X = X0 VL + — {p!02+p2Q2)V3 .

Pi ^

Now we consider the general case of the associate congruen­ces (49). We take an /-plane through each line and assume that homologous /-planes are parallel. Then the homologous characte­ristic points Fla are given by

•Xa = * o « - — « 1 + — (Pl°2a+P2Q?a) ^3 ( 7 3 ^ Pi Pi

or

Xa=-XQ cos a + Xn_ sin a . (74) 2

(71)

(72)

— 144 —

This enhances:

The homologous characteristic points Fla of l-planes belong to 00 2 concentric ellipses.

Through each point Fla we bring a line in the corresponding /-plane. Suppose that homologous lines are parallel. The lines constitute c ° 1 associate congruences Ca\ for which the points Fla

are focal points. From § 7 and § 10 it follows that the other focal points F2a and also the homologous middle points belong to °°2

concentric ellipses. If the associate congruences Ca' are normal we have an ana­

logous property for suitable orthogonal trajectories.

12. Let us investigate whether analogous properties hold for the associate congruences (52). The characteristic point Fla of the /-plane (68) is again given by equation (73), where

Q2a = Q2r+(Q2—Q2r) C 0 S a + (Q2' —Q2r) S i n a > , „ c %

(75) a2a = a2r + (a2 — °2r) C 0 S a + (°2 ~ °2r) s i n a •

So the coordinate vector of Fia becomes

Xa=Xr+(X-Xr)cosa + (X'~Xr)sma , (76)

and we have:

The homologous characteristic points of the l-planes belong to 00 2 ellipses. Each ellips has its middle point at the characteristic point of the corresponding {parallel) l-plane in Cr.

Consider again through each point Fla a line in the correspon­ding /-plane, so that homologous lines are parallel. We become c°1 associate congruences C0 ' with similar properties as found under §11 . The ellipses are not concentric any more but have their center at the corresponding focal point of C/.

Application for H-congruences.

13. Now we study a congruence given by

X=Aa-rjub + t(xa+yb + c) , (77)

— 145 —

where a, b, c are fixed orthogonal unit vectors, x and y are the free variables and

X = X (x, y) , v ' • " ' (78)

/A=/A(X, y) .

It is easily computed that the equation of the focal points is

t% + (** + A*y) * + hf*y - V x = ° • (?9)

Consequently the congruence has a plane middle surface" if

^ + ft,=0 . (80)

It is a //-congruence [6] if furthermore

i , - A ^ - 0 • (81)

Under these conditions the surfaces

X=to+fj) + ic (82)

and

X= — jua + Xb + tc , (83)

with dt=— (xdX + y ^ ) ,

are harmonic surfaces [ 6 ] . Their respective normals are given by

xa -t- yb + c and — ya + xb + c . (84)

The congruences

X = ^« a +A*a^+*(*°+ ^ + C) (85)

with (

Xn = A c o s a — u sin a , (86)

/,6a = X sin a + /,* cos a ,

10

— 146 —

are called associate congruences. All the associate congruences are simultaneously //-congruences. The harmonic surfaces

Xa=Xaa + juab + tac (87)

where

dta = — (xdka + ydpa) (88)

are called associate harmonic surfaces. From the above formulae or from § 7 it follows that:

The homologous points of the middle surfaces of the associate congruences belong to °°2 concentric ellipses. The ellipses reduce to circles in case of harmonic congruences.

The homologous points of the associate harmonic surfaces be­long to °°2 concentric ellipses.

14. The previous theory can be generalized as follows. Let us consider the associate congruences

Xa=k'a + p'b + laa + JLIJ)+t(xa + yb + c) . (89)

The equation of the focal points is

t2 + ( V + Mv' + Kx + f*au) t +

+ ( V + KX) (My + May) ~ ( V + X*J ^* + ^ = 0 '

It follows immediately from this equation or § 7 :

The homologous points of the middle surfaces of the associate congruences (89) belong to °°2 ellipses. Each ellips has its center at the homologous point of the middle surface of the congruence of axes of rotation.

Suppose now that all the associate congruences are //-con­gruences.

The homologous points of the associate harmonic surfaces which are defined by the associate harmonic congruences belong to cc"

— 147 —

ellipses. The centers of the ellipses coincide with the homologous points of the harmonic surfaces which are defined by the congruence of axes of rotation.

15. As a result from § 10 homologous focal points of the asso­ciate congruences (89) belong to curves of the fourth degree. In

the coordinate system defined by the vectors a, b,xa-\-yb-\-c the curves are given by

X = A'+Acosa— fJL sin a ,

F = / / + A sin a + ywcos a ,

2Z= - ( V + fxy' + lax + vay) ± L ,

where

L* = ( V - /V + K, - A , / + 4 ( y + Xay) (ft/ + ft,,) . (92)

If X' •= fju •= 0, then each of these curves degenerates into two ellipses when all the associate congruences are harmonic.

16. For the /-planes

X=l'a + ju'b + Xaa + [iab +p(a + rb) + q(xa + yb + c) (93)

where r is a constant with respect to a, we have analogous proper­ties as is seen from § 12.

(91)

— 148 —

REFERENCES

[1] VANHECKE L., Enkele wentelingsproblemen bij stralencongruenties bestudeerd met de methode van Car tan, Mededelingen van de Koninklijke Vlaamse Academie voor Wetenschappen, Letteren en Schone Kunsten van Belgie, Klasse der Wetenschappen, XXVII, 6, 1965.

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[3] VANHECKE L., Minimaalvlakken en Stralencongruenties, Mededelingen van de Koninklijke Vlaamse Academie voor Wetenschappen, Letteren en Schone Kunsten van Belgie, Klasse der Wetenschappen, XXXI, 8, 1969.

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