596 
DR. J. CASEY ON CYCLIDES AND SPHERO-QUARTICS. 
Hence the required envelope is 
a 
> 
n 
5 
m 
> P 
9 
S*-A, 
n 
b 
9 
l 
> 2 
9 
S^-B, 
m 
? 
l 
9 
G 
, r 
9 
s*-c, 
V 
5 
<1 
9 
r 
, cl 
9 
S*-D, 
s*- 
-A, 
S J - 
-B, 
S*- 
■c, s*. 
-D, 
0 , 
The expansion of this determinant gives evidently a result of the form U 2 + U 1 S^=0, 
where U 2 represents a function of the second degree, and Uj a function of the first degree 
in the variables ; and clearing off radicals we get U 2 — XJ 2 S=:0 ; and this is the equation of 
a surface of the fourth degree, having the conic of intersection of U 2 and U, as a double 
line. Hence the proposition is proved. 
26. The quantities x, y, z, w of the last article are evidently proportional to the tetra- 
hedral coordinates of the point «, /3, y, b, referred to the tetrahedron whose vertices are 
the poles of the planes A, B, C, D of the quadrics S — A 2 , S — B 2 , S — C 2 , S — D 2 , so that 
the equation of condition in x, y, z, w is only the equation of the surface F referred to 
this tetrahedron. Hence the method of generation of surfaces of the fourth degree 
having a conic for a nodal line is exactly the same as the method of generating cyclides 
given in art. 5 ; and in fact the two surfaces are identical, since the cyclide has the 
imaginary circle at infinity for a nodal line, so that by linear transformation we could 
get one surface from the other ; and to every property of a cyclide there is a corresponding 
property of the more general surface here considered : but I thought that it would be 
useful to show that their equations, the equations of the surface cutting them ortho- 
gonally &c., are identical in form ; so that for every theorem which I shall prove to hold 
for a cyclide the reader may if he chooses put in the more general surface here con- 
sidered *. 
* [Professor Cayley has remarked to me that, instead of the method of Chapter II., the immediate general- 
ization. would be to consider, instead of spheres, quadric surfaces of the form S + LM, S + LN, &c., and that it is 
a further generalization, or rather an extension, of S— A 2 , S— B 2 , &c. Professor Cayley remarks that it is a pity 
to omit the intermediate step. Before Professor Cayley had drawn my attention to it, the intermediate step had 
not occurred to me ; however, any person who reads Chapter II. will find it easy to supply, by the assistance of 
the two following propositions, the omissions referred to: — 
I. S + LM= 0, S LN = 0 are two quadrics ; it is required to find the condition that the pole of L with respect 
to S + LM will be the pole of M — N with respect to S + LN. Let 
S + LM vr -\-2(by + cz + dw )x, 
S + LN = x 1 -f y- + z" + w 2 + 2(1' y c'z + d'w)x ; 
and let A, p, v, g be the coordinates of the pole of the plane x with respect to the quadric S + LM=0 ; then we 
get the four equations: 
A -f- by. -j- Cy -j- dg — 1, \h-\-p — — 0, 
A c-\-v = 0, Ac?+£> =0. 
