632 
DR. J. CASEY ON CYCLIDE8 AND SPHERO-QUARTICS. 
Hence the equation of the projection written in full, after replacing 2 by x, is 
0/.2 
r , 2/> 2 /3 
x- + a -i _ x + _ b * y 
a 2 — 7 2 , 2 , £ 2 /3 
To c? 
cV 
« 2 — fj 2 ' a 2 — c‘ 
ih */ // 2 - 
c 2 + y a </ b 2 — < 
p (a 2 — £ 2 ) (a 2 — c 2 ) 
X 
a 2 + c 2 a- + b ~ 2 2c 2 7 2& 2 /3 (/3 2 — a 2 — 7 2 
P-r 2 ^ + a 2 - // y ~ a 2 — c 2 ^ ” a 2 — A 2 ^ ' a 2 - Z > 2 
, 9 , ,2 
ir — c' 
\ jab V 7 a 2 — c 2 -t-ya */b~— c 2 ) 4 q 
( ^/(a 2 — 5 2 )(a 2 — c 2 ) j 
• ( 91 ) 
97. The elliptic projection of a sphero-quartic is a bicirculcir quartic. 
Definition. If through any point P on a quadric we describe two confocals, and if P' 
he the point where the line of curvature common to the two confocals drawn through P 
intersects the plane of xy, P' is what 1 call the elliptic projection of P. 
If X, Y be the coordinates of the elliptic projection of a point on the sphero-quartic, 
x,y the coordinates of the projection of the same on either plane of circular section by 
lines parallel to the greatest axis, then, by Salmon’s ‘ Geometry of Three Dimensions,’ 
art. 180, 
x 2 :X 2 ::f:Y 2 :: b 2 -c 2 : b 2 . 
Hence the locus of the point whose coordinates are X, Y is similar to the locus of the 
point whose coordinates are x, y. Hence the proposition is proved. 
98. The four spheres a, (3, 7 , h intersect respectively the four cones through the sphero- 
quartic (that is, each sphere intersects the cone whose vertex is at its own centre) in four 
sphero-conics, and the projection of these sphero-conics on the planes of circular sections 
by lines parallel to the greatest or least axis will be four circles, and these will be the 
circles of inversion of the bicircular which results from projecting the sphero-quartic. 
For if the sphere a intersect the line KPP' in Q (see art. 95), KQ 2 = KP KP'. We 
can therefore account for the four circles of inversion of the bicircular. 
99. The projecting lines of the four sphero-conics of the last article intersect the 
quadric in four curves, whose elliptic projections will be the circles of inversion of the 
bicircular which results from the elliptic projection of a sphero-quartic. 
This is proved in the same way exactly as art. 97. 
100. Lemma. If a sphere concentric with a quadric intersect it in a sphero-conic, 
and tangent planes to the quadric be parallel to the tangent planes to the cone whose 
vertex is at the centre of the sphere, and which stands on the sphero-conic, the locus of 
their points of contact is a line of curvature of the quadric. 
This proposition is plainly the converse of art. 158 of Salmon’s 4 Geometry of Three 
Dimensions but we can give a direct proof of it as follows. Let the sphere and quadric 
be given by the equations 
x 2 +y 2 +z 2 =r 2 , 
