DR. J. CASEY ON CYCLIDES AND SPHERO-QTJARTICS. 
G49 
four circles will belong one to each of the four systems of generating circles of the 
sphero-quartic WU ; but if the sphero-quartic be inverted from any arbitrary point of U, 
it becomes a bicircular quartic. Hence the anharmonic ratio is constant of the four 
generating circles of a hicircular quartic which touch each other at any point of the quartic. 
See ‘ Bicircular Quartics,’ art. 99. 
2°. If four points, 1, m, n, p, he taken on a hicircular quartic and normals he drawn to 
the quartic at these points , the normals divide the focal conics of the quartic homogra- 
phically. This follows from art. 149. 
163. Conversely, properties of spliero-quartics may be inferred from those of bicir- 
culars. 
If we take any line through two points E, F of the sphero-quartic WU, and through 
EF draw four planes each tocching WU in another point, these planes intersect U in 
four circles, which will become, if U be inverted into a plane, four circles intersecting 
a bicircular quartic in two common points and touching it, each in another point, but 
the anharmonic ratio of such a pencil of circles is constant (see ‘ Bicirculars, ’ art. 99). 
Hence is constant the anharmonic ratio of the four planes through EF. 
Cor. 1. If A, B, C, D he the four points where the planes through EF touch the sphero- 
quartic, the tangent lines to the quartic at A, B, C, D ( that is, the lines of A through A, 
B, C, D) meet EF in four points whose anharmonic ratio is constant. 
Cor. 2. The four lines of A of Cor. 1 are generators of a ruled quadric. 
Cor. 3. If through the lines of A at A, B, C, 13 (that is, the four lines of Cor. 1) he drawn 
four planes intersecting the sphero-quartic in a common chord, if the common chord 
varies it will generate a ruled quadric. 
CHAPTER IX. 
Osculating Circles of Sphero-quartics. 
164. If we consider the cy elide 
W = ad + h/3' 2 -j-cf-l-dh 2 =0, 
and the sphere U given by the equation 
u 2 = a 2 +/3 2 -br-f£ 2 =o, 
then the quadric 
ax 2 -}- hip + cz 2 -f- dw 2 , 
which will be the reciprocal of the focal quadric of W, will pass through the sphero- 
quartic WU, and x 2j r y 2J r z 2 -\-w 2 =Q will be the equation of U in the same system of tetra- 
hedral coordinates. 
The section of these quadrics by the plane w will be the conic ax 2J r by 2 -\-cz 2 =0 and 
the circle x 2 -\-if -j- s 2 = 0. N ow, following Clebsch, let us generalize the method of finding 
the evolute of ax 2 -\-hy 2 -f cz 2 (see Salmon’s ‘Geometry of Three Dimensions,’ art. 472). 
We have the following problem to solve, which will be the generalization of drawing a 
normal to a conic. Let it be required to find a point x, y, z on the conic ax 2 J r hy 2 cz 2 , 
such that the pole with respect to the circle x 2J ry 2 + z 2 of the tangent to the conic at 
