676 
DR. J. CASEY ON CYCLIDES AND SPHERO-QTTARTICS. 
cone whose vertex is the origin of inversion, and which stands on the cuspidal edge of 
the developable formed by the tangent planes of the sphere along the inverse curve — that 
is, the sphere into which the plane inverts. These singularities will he those of the 
evolute of the original plane curve. 
The singularities of the developable will be got by the changing of letters as in the 
last article. Since the origin is a multiple point of the degree N — 2k, therefore the 
class of the cone will be r — (N — 2£) = N 2 — 2k 2 , since it is evident that, in finding the 
number of lines of the system which meets an arbitrary line through the vertex of the 
cone, we must subtract from the rank of the system the number denoting the multiplicity 
of the vertex. 
Since any arbitrary plane meets the cuspidal edge in a number of points equal to the 
degree of the system (that is, 3N 2 — Qk 2 — 3N), it is evident the degree of the cone is equal 
to this number diminished by N — 21' ; therefore the degree of the cone is 
3N 2 — 6/r 2 — 4N 2k. 
Again, the cuspidal edges of the cone will be equal to the number of stationary points 
of the system — that is, equal to 
6N 2 — 12& 2 — 10N+4£. 
In order to find the corresponding singularities for the evolute, we must plainly add 
N — 2 k to the last two singularities; for, N — 2 k branches of the inverse curve passing 
through the origin of inversion, each branch will add one to the number of cusps, and 
one to the degree, and we shall have for the evolute of a curve of degree N which passes 
k times through each of the circular points at infinity, but which has no finite double 
point or cusp, the following singularities, 
Class = N 2 - 2k 2 , 
Degree = 3N 2 — Qk 2 — 3N, 
Cusps =6N 2 -12£ 2 — 9N+2&; 
and by Plucker’s equations the other singularities can be determined. 
By putting N = 4 and k= 2, we find class, degree, and cusps of a bicircular quartic to 
be 8, 12, 16, and our formula is verified for the bicircular. 
Again, putting N = 3 and k= 1, we find the numbers for the circular cubic to be 7, 
12, 17, which we know otherwise to be the characteristics for the evolute of a circular 
cubic. If we put k — 0 in the above formulse, the numbers coincide with those in 
Salmon’s higher curves. 
The foregoing numbers are to be modified when the curve of the Nth degree has cusps 
at the circular points at infinity. In that case for each cusp at a circular point at 
infinity the class of the evolute will be diminished by unity, and the number of its cusps 
increased by unity, the degree remaining the same. If the original curve had finite double 
points or cusps, the surfaces, viz. the sphere and the surface of the Nth degree, will have 
ordinary contact for each double point on the curve, and stationary contact for each 
