6G0 
DR. J. CASEY ON CYCLIDES AND SPIIERO-QITARTICS. 
infinity lies in the plane touching the cyclicle along a tangent line to the circle at infinity ; 
hence the tangent plane to the cyclicle is also a tangent plane to the surface of centres. 
Again, the sphere of curvature at any point P of a cyclicle is the quadric through the 
imaginary circle at infinity and through four consecutive points at P ; if P be any point 
on the circle at infinity, this quadric is indeterminate, and the pole of the circle at 
infinity is any point on the tangent plane at P. Hence any point on the tangent plane 
may be regarded as a point of intersection with a consecutive tangent plane ; in other 
words, the tangent plane to the cyclide at any point along the imaginary circle at infinity 
is a stationary tangent plane to the surface of centres. 
Cor. If the imaginary circle at infinity be a cuspidal curve on the cyclicle, it will he 
a cuspidal curve on the surface of centres of the cyclicle. 
191. The points of contact of tangent lines from any point to a cyclide of the fourth 
degree W are the points of intersection of W with the polar cubic of the point ; but this 
polar cubic is evidently a cubic cyclicle. Hence the tangent cone which circumscribes a 
cyclide and has any point for vertex reduces to the eighth degree by rejecting the square of 
the cone to the imaginary circle at infinity. Or thus : — Draw any plane through the vertex 
of the cone ; this plane will cut the cyclide in a bicircular quartic ; and this quartic being 
of the eighth class, eight tangents can be drawn to it from the vertex of the cone. 
192. Class of Tangent Cone. — Let V be the vertex of the tangent cone and V' any 
other point, then the class of the tangent cone is plainly equal to the number of points 
common to W and the polar cubics of the points V, V'. Here we have three cyclides 
to consider, viz. W and the polar cubics. Let F, G, H be their focal quadrics ; then F, G, 
H have eight common tangent planes; and corresponding to each common tangent plane 
there will be a pair of inverse points common to the cubics ; therefore through the line 
V V' sixteen tangent planes can be drawn to the cone ; the class of the tangent cone is 
therefore sixteen. 
193. The equation of the tangent cone from any point to a cyclide may be found as 
follows. Taking the point which is to be the vertex of the cone as origin, let the 
equation of the cyclide in Cartesian coordinates be written in the form 
A(o; 2 +^+^) 2 -1-4B(^+/+2 2 ) + 6C+4D+E=0, .... (132) 
where A, E are constants, 
T>=lx-\-my -\-nz, 
D =px +qy-\-rz, 
fe(«, b , c,f g, hfx, y , z)\ 
In polar coordinates this becomes, by putting §cos u=x, §cos fi=y, §cos y=z, and 
putting for shortness 
B'hsZ cos a-\-m cos/3+w cos y, 
D'=p cos a-\-g cos /3+r cos y, 
C'=(«, b , c,f, g, hfcosa, cos/3, cosy 2 ), 
A § 4 +4By+GCV+4D'?+E=0; 
