Mr. W. Lupton on Spherical Geometry. 37 
Z 
yY ” 
R.2a 
X= + —_____—_____ abetinreieee 
(1+a2+y? + 2zy cosw)® f 4) 
yo ee ee ae 
(1+a?+ y?+ 2zy cos w)? 
Z= BY 
+-—__—4____ 
~ (l4+a2+y?2+42zxy cos o)* 
4. When the axes of spherical coordinates are rectangular, the 
equations (A) become 
> =n, 7 
Z 
yo! 
Sas dhe (B) 
~(1+a2+y%)? 
Vp ae 
(1+a?+y?)? 
Pa AB 
(l+a*+y?)? 
5. Whenever, therefore, it is required to find the spherical 
equation of the curve formed by the intersection of any given 
surface with the sphere, it is sufficient to substitute for X, Y, 
and Z in the equation of the surface their values as given above, 
aud the resulting equation in w and y will be the spherical equa- 
tion of the curve. 
Hence ‘a spherical equation of the nth degree represents a 
curve formed by the intersection of the sphere with a cone of the 
nth degree whose vertex is at the centre of the sphere.” 
Similarly, it may be seen that if m be an even number, “a 
spherical equation of the nth degree represents the curve formed 
by the intersection of the sphere with a central surface of the nth 
degree.” 
As a particular case of the above theorems, we learn that “a 
spherical curve of the second degree is formed by the intersection 
