674 
DE. J. CASEY ON CYCLIDES AND SPHEEO-QUABf ICS. 
III. When U and V osculate. 
1°. Cone whose vertex is a point of A : 
o 
[* = o. 
II 
M 
Nr 
1! 
03 
U 
II 
o 
rH 
II 
o’ 
II 
CO 
lose vertex is ; 
II 
a 
II 
03 
co 
II 
2* 
u 
II 
1— l 
o' 
II 
oy 
II 
o 
Circular cubic of third class. 
i>=3, r=l, > A Cartesian oval of third class — that is, a cardioide. 
). J 
3°. Cone whose vertex is a point on a line of ^ and which stands on the cuspidal 
edge of % : 
(M=i, * = 2, 
*= 4 , 7 = 1 , 
i=2, 5=1. 
| Evolute of circular cubic of third class. 
4°. Cone whose vertex is a point on two lines of and which stands on the cuspidal 
edge of % : 
i^=4, z =? j , "j 
f=3, t = 1, > Evolute of a cardioide. 
;=0, 5=0. J 
230. If a plane curve whose degree is N be inverted from any point out of the plane 
of the curve, it will invert into a twisted curve, whose characteristics are easily found. 
1°. Let us suppose that the plane curve does not pass through the circular points at 
infinity. Then, since the curve passes through N points at infinity, the inverse curve will 
have this order of multiplicity at the origin of inversion ; and since the plane of the curve 
will invert into a sphere, the inverse curve will be the intersection of two surfaces of the 
degrees N and 2 respectively, having a multiple contact equivalent to 1 — -= points 
of ordinary contact. Hence we have m=2N, /3 = 0, 2A=3N 2 — 3N (see Salmon, p. 273). 
Hence, by Cayley’s equations, 
r= N 2 +N, ^ = 3N 2 -3N, 2y=N 4 +2N 3 -9N 2 +6N, 
«=6N 2 — -ION, 2x = N 4 -f 2N 3 — 3 N 2 — 4N, 2y=9N 4 -18N 3 -13N 2 +32N. 3 
231. Next, let us suppose that the plane curve passes Jc times through each of the 
circular points at infinity ; then it is easy to see that the inverse curve will be the inter- 
section of a sphere and a surface of the degree N —Jc. Hence we have in this case for 
determining the singularities, m=2(N — Jc), ,3 = 0, and 2£=3N 2 — SNC-f-GC 2 — 3N + 4L 
