Sbction hi., 1908. 



[2B5] 



Trans. R. S, C 



XXIII. — )An Outline of Analytical Spherical Geometry. 

 By S. Beatty, B.A., 



Fellow in Mathematics, University of Toronto. 



(Presented by Professor Alfred Baker, and read May 26th, 1908.) 



I. In analytical spherical geometry several systems of axes 

 present themselves. The first to be considered is that analogous to 

 the Cartesian in plane geometry. It will be subsequently noted 



that the great circle plays the same 

 ]; role in spherical geometry as the 



straight line in analytical plane geo- 

 metry. Consequently, we choose as 

 axes any two great circles passing 

 through a fixed point O called the 

 p origin; much labour will be avoided if 



the axes are taken as making an angle 

 <1> \ ^■"^N.s^^^ of 90° at the origin. Furthermore, the 



^ ^^ lines OX and OY are each taken a 



X quadrant in angular distance. The 

 co-ordinates 6 and of a point P are 

 defined to be the angular intercepts on OX and OY made by the 

 great circles FP and XP respectively. The following diagram makes 

 clear the meaning intended. As in 

 plane geometry the directions OK and 

 OY are by convention positive. The 

 reverse are negative. 



II. As has been intimated, the 

 simplest curve on the surface of the 

 sphere is the great circle. The equation M 

 in the case where the angular inter- 

 cepts are a and ^ is readily found. From 

 triangles JVFK and XPX we have sin 

 (a — 6) tan V = tan PX = cos 6 tan 0. r^'' 

 But sin a tan V = tan /?. On eliminat- 



TT^L X- rxu X • . tan 6 , tan QÎ) 



ing K the equation of the great circle, , + 7, = 1, results. 



tan a ' tan /? 



III. This may become the equation of any great circle on the 



surface of the sphere by giving to a and /? particular values, as long 



H 



N K 



