DR. J. CASEY ON CYCLIDES AND SPIIERO-QUARTICS. 
607 
on the sphere, and to show that sphero-quartics may be generated in the same way as I 
have given in the Fifth Chapter of my ‘ Bicirculars’ for generating plane quartics having 
two finite double points. 
44. The equation of a right cone whose semivertical angle is § is 
Ft — cos - (j(x~ y 2 = 0, (46) 
where L is a plane through the vertex of the cone : now this cone intersects a sphere of 
radius unity whose centre is at its vertex in two small circles ; and I say that the two 
factors of the equation (46), namely L+ cos §(^ 2 -j-,y 2 + may be taken to represent 
these two circles; for the equation (38), which represents a small circle on the sphere, 
will become by transformation to three rectangular planes L= cos £(# 2 +3/ 2 -{-2 2 F, and its 
twin circle will be the other factor, L+ cos^+^+s 2 /. 
other four focal quadrics ; and hence we have the following system of Cartesian equations of these focal 
quadrics : — 
+ 
+ - 
-=1, 
(r + p-2 V 2 + jj. 2 c 2 + /x 2 
" +*£r+:£r=l. 
« S + f* 3 b ' + H C “ + i a 3 
- + y + - 
r + ( a 4 b 2 + jj. 4 c 2 + jm . 4 
"" +T g-+*=l, 
« 2 + /* 5 6 " + fb-> C ' 2 + F-5 
so that the five focal quadrics are con focal, as we know otherwise. 
III. Since the equation 
may be written in the form 
u~f- u-q - u-h- „ „ 
- 5 — — + -r-, + 4- /r + wr 
rr + ju, U- + [X, C“-j- [Af 
-&+i£r+~&= 1+- 
cr + [A 0 * + p cr + [). 
F. 
and this is the discriminant of /xF + J, where 
E=5 + ^+A-l = 0, J = ( X -fy+(y-rjY + (z-h) 2 -r~0 
(see Salmon’s ‘ Geometry of Three Dimensions,’ p. 146), we infer that the same values which will make jU,F + J 
a cone will also make S+/xC + p (see II.) a cone. The two cones will have a common vertex, their equations 
referred to that vertex as origin being 
a-^cr + fi.) if-jb- + fx) ~(c~ + /a) _ q 
cr b~ <? 
ar(cr + [a) + y 2 ( b ~ + M-) + z\c- + /a) — 0. 
Hence we have the following remarkable theorem : — If F and J be a corresponding focal quadric and sphere of 
inversion of a cyclide, and if y. v p. 9 , /x 3 , >x 4 be the four roots of the biquadratic which is the discriminant of 
/xF -J- J, then if F be given in its canonical form, 
-.+£+i-i=o, 
cr b- c~ 
the equations of the four other focal quadrics are got from this by changing a 2 , b 2 , c 2 respectively into (cr-f /xj, 
(b' + ^i), (c 2 + ^ifi &C. 
