THEORY OF POLYGON IC PROJECTIONS. 103 



We have for the square of the distance OM to the origin 



, 1 —cos X' cos if 

 ^ 1 +COS X cos (f 



We should note that the general equation of the circles 

 traced upon the sphere and that of circles traced upon 

 the map have exactly the same form when we take for 

 coordinates (p and X on the sphere and .<^' and X' upon the 

 plane. On the unit sphere we have 



X = cos X cos <p 



y = sin X cos (p 



2; = sin (p. , 



If we substitute these values in the equation of a plane 



Ax + By+Cz + D = 0, 

 we obtain 



{A cos \-\-B sin X) cos <p-\- sin (p + D = 0. 



This is the equation of a circle determined by the inter- 

 section of the plane with the sphere. 



The general equation of a circle in the plane is given by 



(x-ay-h(y-hy = c\ 



or on substitution of the values of x and y in terms of 

 ^' and X' we obtain 



/ sin X' cos <,' _ V +/ ^nf _ jV ^ 



\l+cosX cos<^ / \l+cosX cos<^ / 



or on development 



1-cos X' cos (p' 2a sin X' cos cp^ 2h sin <^' 



1+cos X' cos (p' 1+cos X' cos (p' 1+cos X' cos <p' 



1— cosX' cos (p' — 2a sin X' cos (p^ — 2h sin <p' = c^ — a^ — ¥ 



+ {c^ — a^ — ¥) cos X' cos <^' 



{a^-\-h^ — (^—l) cos X' cos (^' — 2a sin X' cos v?' — 26 sin <p^ 



+ a^ + ¥-c^ + l==0' 

 or 



U' COS y + B' sin XO cos (p' + C sin (p' + D' = 0, 



