36 Mr. W. Lupton on Spherical Geometry. 
v=y=tan 90°= a. Hence the great circle AB may be said 
to correspond with what in plane coordinate geometry is desig- 
nated the line at infinity. It is also evident that this circle 
divides the whole spherical surface into two hemispheres, in 
each of which there is a distinct origin. We shall, however, at 
present confine our attention to the consideration of one hemi- 
sphere. 
3. We shall denote by X, Y, Z the ordinary rectilinear coor- 
dinates of any point on the surface of a sphere whose spherical 
coordinates are v, y. Now if the axes of spherical coordinates 
are inclined at an angle , it is evident that the tri-coordinate 
planes are such that the axes of X and Z are inclined at anangle 
@, and are each perpendicular to the axis of Y. Hence the 
equation of the sphere, whose centre is at the origin of tri- 
coordinates, is 
X?+ Y?+ Z?42XZ cos o=R?. 
And if 
£=PH ji V=GH, i 0G, 
we have 
_p sin PM 
“sin BM’ 
cos PM 
Y=R.cos OM. sin BW’ 
4 
r= tan OM=z. 
Hence we have 
Z __ tan PM 
Y cosOM‘ 
But 
sin PM _sinPM _ sin NAO 
cosOM sinAM~ sin MPA’ 
and 
sin MPA _cosNO 
sin NAO” cosPM 
therefore sci 
cos sin NAO 
tan PM. cosMO~™ sinONA’ 
or 
tan PM _ sin NAO il 
cosMO~ cosON snONA 
We have, therefore, when the axes of spherical coordinates are 
oblique and inclined at an angle , the following equations :— 
= tanON=y. 
