DR. BOOTH ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 317 
which they meet the surface of the sphere are termed the foci of the spherical 
ellipse. 
IV. Every umbilical surface of the second order has two concentric circular sections, 
whose planes, in the case of cones, pass through the greater of the external axes. Per- 
pendiculars drawn to the planes of those sections, passing through the vertex, — they may 
he called the cyclic axes of the cone — make with the internal axis of the cone in the 
plane of 2^ — the plane passing through the internal and the lesser external axis — equal 
angles n, such that 
cos 71 = 
sin/S 
sin*' 
(3.) 
Let a series of planes be drawn through the vertex, and perpendicular to the suc- 
cessive sides of the cone. This series of planes will envelope a second cone, which 
usually is called the supplemental cone to the former. The cones are so related, that 
the planes of the circular sections of the one are perpendicular to the focals of the 
other, and conversely. 
V, The equation of the spherical ellipse may be found as follows, from simple 
geometrical considerations. 
Let 2 c 4 and 2(3 be the greatest and least vertical angles of the cone ; the origin of 
coordinates being placed at the common centre of the sphere and cone. Let the in- 
ternal axis of the cone meet the surface of the sphere in the point Z, which may be 
taken as the pole. Let p be an arc of a great circle drawn from the point Z to any 
point Q on the eurve. %// being the angle which the plane of this circle makes with 
the plane of 2a, we shall have for the polar equation of the spherical ellipse, 
1 cos'-f/ sin^vf/ 
tan'^p tan^a"^ tan^/3 
To show this, through the point Z let a tangent plane be drawn to the sphere. This 
plane will intersect the cone in an ellipse. This ellipse may be called the plane base 
of the cone, while the portion of the surface of the sphere within the cone may be 
termed the spherical base of the cone. The plane of the great circle passing through 
Z and Q will cut the plane base of the cone in the radius vector R ; and if we write 
A and B for the semiaxes of this ellipse, whose plane touches the sphere, we shall 
have, for the common polar equation of this ellipse, the centre being the pole, 
1 cos®\|/ , sin^vj/ 
^=-^+-32-- 
Now the radius of the sphere being h, and p, a, j3, the angles subtended at the centre 
by R, A, B, we shall clearly have 
R=^tanjO, A=^tana, 
whence 
B = A tan/3 ; 
