DR. J. CASEY ON CYCLIDES AND SPHEKO-QUAETICS. 
675 
Hence by Cayley’s equations we get the following results for the other singularities : 
r= N 2 - 2& 2 + N— 2 k, 
w=3N 2 - 6£ 2 — 3N, 
a=6N 2 — 12& 2 — 10N+4&, 
2 y= N 4 + 2N 3 — 9N 2 + 6N-4(N 2 +N-l)£-4(N 2 +N-6)& 2 +8& 3 -f4£ 4 , 
2x= N 4 + 2N 3 — 3N 2 — 4N-4(N 2 +N-2)&-4(N 2 fi-N-3)£ 2 +8£ 3 -j-4£ 4 , 
2 g = 9N 4 - 1 8N 3 - 1 3N 2 + 32N - 1 0/c- 4(9N 2 - 9N - ll)/c 2 + 36L‘. 
We can easily verify these results in the case of N = 4 and & = 2, wlrich is that of a 
bicircular quartic; they give the results previously obtained (see art. 224). 
232. Let us now find the singularities of the cone whose vertex is the point we invert 
from, and which stands on the inverse curve. The vertex of the cone is a multiple point 
on the curve, the degree of multiplicity being of the order N — 27c; but since the twisted 
curve is of the degree 2N — 2k, and the multiplicity of the vertex is N — 2Jc, it follows 
that the degree of the cone is N, .\ N. Again, the class of the cone is the same as 
the number of tangent planes which pass through an arbitrary line through the vertex ; 
.-. *=/•- 2(N-2£); 
and using the value of r in the last article we get 
v = N 2 — N — 2£(& — 1 ) , and £=/3=0. 
Hence by Plucker’s equations the other singularities of the cone are determined. It is 
evident we could get all these results at once, since evidently the singularities of the 
cone are the same as those of the original plane curve ; but getting them as done here 
verifies the equations of the last article. 
233. Let us find the singularities of the section of the developable circumscribed to 
the sphere into which the plane inverts along the inverse curve made by the tangent 
plane to the sphere at the origin of inversion. The characteristics of the developable 
considered here are got from those of art. 231, by leaving r unaltered and by changing 
rn, a , g, x into n, (3, h, y, and vice versa; and the plane in question will be a plane of 
multiple contact of the degree N — 2k ; that is, it will touch the developable along N — 2Jc 
lines, and will intersect it besides in a curve of the degree 
r— 2(N-2/r)=N 2 -N— 2/<£— 1). 
We have therefore 
/a=N(N-l)-2£(£-l). 
The class of the curve is determined by the number of planes of the system which can 
be drawn through any point of the section ; and since in this case N — 2Jc planes coincide 
with the plane of the section itself, the number of remaining planes y = N, and we have 
i=zu= 0, and by Plucker’s equation the remaining characteristics can be found. These 
results are the reciprocals of those found in the last article, as they evidently ought to be. 
234. The most important problem in this inquiry is to find the singularities of the 
MDCCCLXXI. 4 Z 
