traced upon the Surface of the Sphere. 



297 



Let M be the projecting point, A, B the poles, C the centre of the 

 sphere, ADB the meridian in which the point is situated, and RST its 

 intersection with the plane of projection. Then S is the point into which 

 D is projected, and RS is the line or distance sought. 



PutCM = i; BC = a; CR = c; AD = (j) ; RS = w,; and CN = v. Then, 



EM = b + a cos 0, and 

 EM : MC : : ED : CN, or 



. , ab sin d> 



b + a cos : o : : a sin : v = * r 



b + a cos 



b : b -f- c : : v : v, = 

 which is the distance sought. 



Again, CM : RM : : CN : RS, or 

 I v a(b + c) sin (f> 



h + a cos <p 



(I-) 



(2.) 



2. The angles made by the meridians A D, &c. are obviously equal to 

 their projections on the plane II S. 



3. Dropping the subscribed accent from v, in (2), and resolving the 

 equation for sin and cos <p, we obtain 



sin = - 



(b 



, 6w 2 -f-(6 + c) /(a* i 

 COS< P = ~ ^rA , /A 



, 



