DE. J. CASEY ON CYCLIDES AND SPHEEO-QTT AETICS . 657 
Hence the coordinates of any point on the cuspidal edge must satisfy the system of 
equations : 
x 1 ?/ 2 z 2 w 2 n 
(« + /t) 3_1 (6 + /t) 3_i (c + £) 3 1 (d + kf u ’ , . . . • 
. (125) 
a A by 2 cz 2 dw~ „ 
(a + /fc) 3 1 (b + kp 1 (e + /c) 3 ‘ (rf+A) 3 
. (126) 
aV by cA 2 dll 
(a + k) 3 ' (6 + Ar) 3_ ' _ (c + A') 3_ * + ' 
. (127) 
Hence 
a? 2 y 2 z 2 m> 2 1 1 1 1 
(a + /:) 3 ‘ (b + k) 3 ' (c + k) 3 ' (d+k) s ‘ ' \J/(«) ' F(^) ' F(c) ' F(d) ’ 
• (128) 
and substituting the values of ( a-\-Jc ), ( b-\-k ), See. from these equations in 
/y.2 nji 
«Ti+4TI+ &c '’ 
we get the equation (124). Hence &c. 
Cor. 1. By giving Jc any particular value, we see from the equations (125), (126), (127),. 
that, the points on the cuspidal edge of 2 are , eight by eight , the points of intersection of 
three quadrics. 
Cor. 2. From equations (126), (127) we see that the cuspidal edge is a curve on 
Clebsch’s surface of centres; and from equation (109) it follows that the sphero-quartic 
WU is a double line on the surface which has Clebsch’s surface of centres for a deferente. 
182. By eliminating k from any three of the four equations (128), we get the equa- 
tions of four cones standing on the cuspidal edge. Thus one of the cones is 
(»-c){4/^K}H(c-a){4/W}H(«-»){^(^}*=0^ • • ■ (129) 
The vertex of this cone is one of the vertices of the tetrahedron ; it possesses several 
properties. The following are some of the most important: — 
1°. It intersects the opposite face of the tetrahedron in Clebsch’s evolute of a conic 
(see art. 164). 
2°. It is the reciprocal of the corresponding double line of A — that is, of the developable 
formed by the tangent lines of WU. 
3°. Every edge of it is a line through two points of the cuspidal edge of 2. 
4°. Every tangent plane to it is a plane through two lines of % , and it is therefore one 
of the four cones which are the envelopes of all the planes through two lines of that deve- 
lopable. 
5°. The equation, cleared of radicals, is of the form 
|A 2 +B 2 +C 2 p=27A 2 B 2 C 2 (130) 
Hence it has six cuspidal edges lying on the cone of the second degree , A 2 +B 2 +C 2 =0. 
6°. Any tangent plane to it will intersect it in a pencil of four lines whose anharmonic 
ratio is constant. 
7°. The tangent planes, touching it along the lines of intersection of any tangent plane, 
