DE. J. CASEY ON CYCLIDES AND SPHEEO-QUAETICS. 
629 
itself, then W and U invert respectively into a cubic cyclide and a plane, and hence 
the sphero-quartic inverts into a circular cubic. Hence if a sphero-quartic be inverted 
from any point on itself it inverts into a circular cubic ; and conversely , if a circular cubic 
be inverted from any arbitrary point in space , we get a sphero-quartic passing through the 
centre of inversion . 
91. If W^«« 2 + bft 2 -f- cy 1 -J- = 0, and U 2 =a 2 +/3 2 + y 3 + £ 2 = 0, and we eliminate car 
between Wand U 2 , we get the cyclide (b — a)ft 2 -\-(c— a)y--\-{d— afr = § passing through 
the sphero-quartic WU. But the cyclide (b — a)ft' 2j t~(c— «)y’ 2 + (^ — c/)b 2 —0 is the enve- 
lope of a variable sphere whose centre moves on a conic ; and inverting the curve WU 
from any arbitrary point on U, we get a bicircular quartic, whose generating circles will 
be the inverses of the circles in which the generating spheres of 
(b — a)f 3 2 + (c— a)y 2 + (d — af 2 = 0 
intersect U, and the centres of the generating circles of the bicircular will be the points 
in which the lines from the origin to the centres of the generating spheres of 
(b — a) ft 2 + (c — «)y 2 -f (d — af) 1 — 0 
pierce the plane into which the sphere U inverts. Now the locus of the centres of the 
generating spheres of (b — a)ft 2 -\-(c — a)f -\-{d — «)<P— 0 is one of the double lines of the 
developable formed by tangent planes to U along the sphero-quartic WU (see art. 42). 
Hence we infer the following theorem : — 
If a sphero-quartic WU be inverted into a bicircular quartic , the four cones having the 
point we invert from as a common vertex , and whose bases are the double lines of the deve- 
lopable "S formed by tangent planes to U along the sphero-quartic WU, will pierce the 
plane of the bicircular in four conics , ivliich will be the focal conics of the bicircular. 
92. If Z be a circle on IT which osculates WU, then it is evident that the pole of the 
plane of Z with respect to U is a point on the cuspidal edge of 2 (see last article). 
Again, when we invert (WU) into a bicircular, Z will invert into an osculating circle of 
the bicircular. Hence we have the following theorem : — 
If we invert a sphero-quartic into a bicircular , the evolute of the bicircular is the curve in 
which its plane is intersected by the cone ivhose vertex is the origin of inversion, and whose 
base is the cuspidal edge of the developable formed by tangent planes to U along WU. 
93. We give in this and the following article some important properties of bicircular 
quartics, which follow at once from the properties we have demonstrated for sphero- 
quartics. 
Since a sphero-quartic WU is the intersection of W and U, and WU is given by the 
general equation 
WU=(«, b, c, d, l, m, n,p, eq, rfa, ft, y, £) 2 =0, 
Avhere a, ft, y, h are circles on U, and U 2 is given by the equation (29), art. 28, hence, 
when we invert U into a plane, the curve WU will be inverted into a bicircular whose 
equation is (a, b, c, d, l, m, n, p, q, rfa, ft, y, d) 2 =0, and the following relation will be 
an identical one on the plane into which U inverts : — 
4 e 2 
