Mr. W. Lupton on Spherical Geometry. 35 
or the appearance of any remarkable cloud without a long wordy 
description. Considering that clouds are the only indications 
of those lofty and otherwise invisible currents and other atmo- 
spheric changes, producing ultimately those results which we 
take so much trouble to record on the surface of the earth, it 
is impossible to understand why their study has been so much 
neglected. 
Sydney, New South Wales, 
January 22, 1857. 
III. On Spherical Geometry. 
By Wii11aM Lupton, Esq., M.A.* 
ae coordinate principle, which has been so successfully em- 
ployed in investigating the properties of surfaces and plane 
curves, may be used with advantage in discussing the properties 
of curves traced on the surface of a sphere. 
The system of coordinates which I propose to employ, has 
already been suggested by Professor Graves of Trinity College, 
Dublin, who, in the appendix to his translation of Chasles ‘On 
Cones,’ has shown how some of the fundamental relations may 
be deduced from principles of projection, as well as from the 
ordinary rules of trigonometry. There is, however, a peculiarity 
of the system of coordinates which he has adopted that seems to 
have escaped his notice, but which appears to be of considerable 
importance, as it enables us to deduce the spherical equations of 
curves from their common tri-coordinate definitions, and con- 
versely from the properties of spherical curves, to derive the 
corresponding properties of the surfaces by whose intersection 
with the sphere they are formed. 
1. Let A A’ and B B’ be two great 
circles cutting each other in point 
O. These great circles are the axes 
of coordinates, and their point of in- 
tersection is the origin. Nowif we 
set off on the axes of coordinates 
OA=OA!'=90°andOB=OB'=90°, 
the position of any point P may be 
determined by drawing through that 
point the axes APN and BPN, 
and taking the trigonometric tan- 
gents of the axes OM and ON as the coordinates of the point 
P. These coordinates may be expressed by the letters w and y. 
2. Ifa great circle be drawn through the points A and B, 
it is clear that the coordinates of any point on it will be 
* Communicated % the Author. 
2 
