602 
DR. J. CASEY ON CYCLIDES AND SPHERO-QUARTICS. 
The double lines of % are plane sections of F , ~F" , F", F w , 
57 
ii ** ii 
F' , F'" , F w , F , 
V' 
ii ii 
F", F w , F , F , 
sni 
ii — * ii 
F'", F , F , F" , 
•s?im 
ii ‘-i ii 
F , F , F" , F". 
35. If we take the first of the equations (34) to represent W, the corresponding sphere 
of inversion is a 2 ; but this, in virtue of the identical relation a 2 +/3 2 +y 2 + ^ 2 d-s 2 =0, is 
also given hy the equation i3 2 +y 2 +<$ 2 -f-s 2 =0 ; and eliminating /3 2 , y 2 , ci 2 , s 2 in succession 
between W and a 2 , we see that each of the four binodal cyclides are inscribed in W, 
(b-cW+(b-d)l*+(b-ey= 0, j 
(c-b)p 2 +(c-d)tf + (c-ey=0, | q7 
(d-b)p 2 +(d-c)y 2 +(d-ey=0, ' 
(e-b)(3 2 + (e-c)y 2 +(e-d)h 2 =0, j 
and these cyclides have the double lines of "$ as focal quadrics. 
It is plain that we get corresponding results for each of the five forms (34) in which 
the equation of W may be written, so that the cyclide W is circumscribed about ten 
binodal cyclides. The focal quadrics of these binodals are plane conics , and through each 
conic two developcibles pass. 
Section II. — Sphero-quartics. 
36. Let P, a point on the surface of the sphere U, be the centre of the small circle S on 
the surface of the same sphere, O a fixed point, also on the surface of U, which we shall 
take as origin, OX a great circle of U corresponding to the initial line in plane geometry, 
and let O P—m and the angle POX=m ; then m and n are what I shall call the spherical 
coordinates of the point P ; and whenever I shall use the term spherical coordinates it is 
in the sense here explained. Now let $ and be the spherical coordinates of any point 
Q of the circle S, then (the reader can easily construct the figure) we have from the 
spherical triangle OPQ, r being the radius of S, 
cosr=cosw cos g+sinw s i n ^ cos (0 — (38) 
This equation may be taken as the equation of the small circle S. Now if in the equation 
(38) we substitute the spherical coordinates of any point Q' whose distance from P is the 
arc 7 J , we plainly get cos r— cos r'; but cos r— cos r l is equal to the perpendicular let fall 
from the point Q! on the plane of the small circle S, hence we have the following 
theorem: — The result of substituting the spherical coordinates of any point on the surf ace 
of a sphere radius unity in the equation of any small circle on the sphere is equal to the 
perpendicular distance of the point from the plane of the small circle. 
37. If any sphere Q intersect a cyclide W, the curve of intersection is a sphero-quartic. 
Demonstration. Let W=«a 2 +fy3 2 +cy 2 -f <7!5 2 , and let perpendiculars from any point P 
