146 



If we project stereographically upon the sphere 



£2 -f ^2 -f (C — 1 2)2 = 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— " c=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 ^' n C are the co-ordin- 

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



to the sphere, 



x=( ? )/(l -C ) 

 y= ( ,/)(!_: ) 



This gives for tlie circle 



x= + y- — Vx -h p — C- Qc 2 = 

 the plane 



(1 _ p -j_ c^ a 2) : — 2 3 = + ,32 _ 0= QC 2 = 



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

 on the plane // = and hence the required straight line in the 5 C — plane. 



