38 Mr. W. Lupton on Spherical Geometry. 
of the sphere with a cone or central surface of the second degree, 
the vertex of the cone or centre of the surface being at the centre 
of the sphere.” 
6. We shall now apply the principle contained in arts. 3 and 
4 to deducing the spherical equation of the great circle, and to 
the establishment of certain formule which will be found useful 
in investigating the properties of spherical curves. In what fol- 
lows, the axes of spherical coordinates are supposed to be rect- 
angular, and the radius of the sphere equal to unity. 
A great circle is formed by the intersection of a sphere with a 
plane passing through its centre. Its equation is therefore 
found from 
LX+MY+NZ=0...1, 
by substituting for X, Y, Z, their values as given in (B), to be 
Lz+Ny+M=0, 
where L, M and N are the cosines of the angles between the 
normal to the plane whose equation is (1), and the axes of X, 
Y, Z respectively. Hence, if P be the pole of the great circle, 
and CA, CO and CB be the axes of rectilinear coordinates, we 
have 
L=cosPB, M=cosPO, N=cosPA. 
But 
cos PO=cos PM. cosOM 
= cos PN . cos ON, 
and 
tan PM= tan NO. cos MO 
tan PN = tan MO.cos NO; 
therefore 
M= cos PO=cos PM. cosOM 
tanPM sinPM 
maeriires tan NO tanMO 
Similarly, 
We also have 
L? + M?+ N?=1. 
Hence if a and 5 be the coordinates of P, the pole of the great 
circle, we have 
