78 
of any symmetrical intersection of a sphere with a cone of the 
second order (i. e. when the centre of the sphere lies upon one 
of the principal axes of the cone) may be expressed by elliptic 
functions. This is easily deduced from the following formule 
for the rectification of the class of spheric curves represented 
by the polar equation, 
sin? pF(w) + sin 2X cosp = 1 (a) 
where ) is aconstant. Let s,,s., denote the two arcs which 
correspond to the same value of the polar angle w, and we will 
have 
2 r2 29 ] 
fey oo ssf ER a 
F—cos’A F 
/2 7 29 d 
s, — 5 = sind | — siete a Se 
F—sin?A F 
r’ denoting the derived oat of F(w). 
Now in the case of the intersection of a cone and sphere, 
such as we have described, F(w) is a linear function of cos 2, 
and each of the foregoing integrals is reducible to an elliptic 
function of the third kind. 
In the case of the spherical ellipse, the two ares s; s, are 
equal; and when the vertex of the cone is upon the surface, the 
second arc s, vanishes altogether. 
The same relation between the principal angles of the 
cone, and the distance of its vertex from the centre of the 
sphere, will cause the parameters in both the functions II to 
vanish. And it is remarkable that, in this case, when the sum 
and difference of the arcs s,, s:, are each expressible by a trans- 
cendant of the first order, the curve will coincide with the 
locus of the vertex of a spherical triangle, the base of which is 
given, and of which the product of the sines of the semi-sides 
is constant, and less than the square of the sine of the fourth 
part of the base. 
This result is strikingly analogous to M. Serret’s theorem 
on the rectification of the Cassinian curve, published in M. 
Liouville’s Journal, vol. viii. p. 145. 
