610 
DR. J. CASEY ON CYCLIDES AND SPIIERO- QUARTICS. 
these are both the same thing. Hence it follows, when we represent a circle on U by 
the equation S* — L— 0, where ~L=ax-\-foj-\-cz=Q, that a, b , c are equal to the direction 
cosines of L multiplied respectively by the secant of the spherical radius of the circle. 
46. Now let us take, as in conics, the small circles 
St_L = 0, S J =M=0, 
and form the invariants of this system ; thus 
S*-L+£(S*-M)=0 
is a small circle coaxal with S= — L and S 1 — M, L — M = 0 being the great circle through 
and the absolute term in equation (B) is r H . Hence we get 
r 2 +r f2_ AP l Oi T ... ^ 
(« 2 + /q) 2 (P+^V+fO*' 
That is the sum of the squares of the radii of J and J'= square of the distance between their centres, and 
hence J and J' cut orthogonally. 
YI. The cyclide got from J and E is the envelope of the quadric S + /xC -f/r, where 
S zrr cdx 1 -f- bSf + c 2 z 2 , C = x 2 + y 2 + ~ 2 + 2 fx -j- 2 gy -f 27tz + r 2 . 
The Same cyclide, got from J' and E f , is the envelope of the quadric S' + XCf + X 2 , where 
and 
V^+f+^+^+jgL 
2a 2 f , 26V , 2c 2 7i n 
• ^+ 7-, . -y+-r-, — - z + r 
c +lh 
Now, to show that S + juC+ju, 2 and S' + xC' + X 2 represent different quadrics, we are to observe that the first is 
referred to the centre of J as origin, and the second to the centre of J' as origin. Now let us transform the 
first to the same origin as the second ; we must change x into x y into y— ~ , zin to z — — and, 
a+Hi ® +/q « +Pi 
in order that they may he identical, we must have p=p 1 + \ ; this will make the coefficients of x 2 , y 2 , z 2 the 
same in both, hut the coefficients of x, y, z will be different. Hence S+,uC + /i 2 and S' + A.C' + A 2 cannot repre- 
sent the same quadric. Hence we have the following theorem : — A cyclide which has no node may he generated 
in jive different ways as the envelope of a variable quadric. 
VII. If it be required to find how many double tangents can be drawn from a given point to a cyclide, let 
us substitute the coordinates of the given point in the quadric S + yG + y 2 , and we shall have a quadratic in y ; 
hence two quadrics of each of the five systems of generating quadrics pass through the given point, and two 
rectilinear generators of each quadric pass through the given point ; now each rectilinear generator of the gene- 
rating quadric is a double tangent of the cyclide. Hence we have the following theorem : — The tangent cone 
from an arbitrary point to a cyclide which has no node has twenty double edges. 
VIII. If E = (a, b, c, d, l, to, n,p, q, rdjx, y, z, 1) 2 = 0, 
J ^a"+y + z 2 -r = 0, 
the cyclide is given by the determinant 
a, 
«> 
m, 
lb 
-2,i’, 
n, 
b, 
i , 
9 . 
TO, 
l. 
C , 
r, 
— 2z 
lb 
9’ 
r , 
d, 
(x 2 + y- + 
— 2x 
-% 
— 2z, 
(_x : + y- + z- + rj, 
o, 
=0. — January 1872.] 
