146 



If we project stereographically upon the sphere 



52 _^ ,^2 ^ (^ _ 1'2)2 Z= 1 4 



whose south pole is the point (0, 0, 0) and whose north pole is the point 

 (0, 0, 1), the circles given by the transformation v ^ x, Ce— " cz^y we shall 

 have the upper part of the xy — plane represented conformally upon the 

 hemisphere Lbd — O. The x — axis goes into the great circle Lbd and the 



I 



Fig. 1. 



circles at right angles to o— x go into circles at right angles to Lbd. 



If now we project orthogonally upon the plane Lbd we shall have the 

 representation in question as chords of Lbd. Since i ?; C are the co-ordin- 

 ates of the sphere we get as the equations of transformation from the plane 

 to the sphere, 



x = (n/(i-c) 



y= ( ri)l(\ -: ) 



This gives for the circle 



x= + y- — V'x -t- ii^ — 0= oc 2 = 

 the plane 



(1 — r' + 02 cc ■') : — 2 3 s f ,32 — o=/oc 2 = 



which is independent of // . It therefore represents the trace of the plane 

 on tlie ijlane // = and hence the required straight line in the i C — plane. 



