newson: continuous groups. 97 



duced b}' the two degenerate line conies AA , and BB,. Let A 

 decrease still further and become negative; the point of contact of 

 K approaches C^ from one side and the point of contact of K' ap- 

 proaches the same point from the other side. For some value of A 

 (usually an imaginar}' root of unity ) the two conies coincide with 

 the degenerate line conic CC,. Let A approach — x : the conic K 

 then approaches its limiting form BB,, while K' approaches its 

 limiting form AA^. Thus we reach the second pseudo-transforma- 

 tion of the group which is produced by the same two degenerate 

 conies BB^ and AAj; but now taken in the reverse order, showing 

 that the two pseudo-transformations form an inverse pair. If ii be 

 taken not between o and i, another combination of line conies will 

 produce the pseudo-transformations. 



The real group contains two real infinitesimal transformations 

 which are inverse to one another. The conies K and K' which 

 determine these infinitesimal transformations differ by an infinitesi- 

 mal amount from the coincident conies which produce the identical 

 transformation of the group. (The case where the transformations 

 of the group are not real will be discussed elsewhere.) 



The analytical expressions for a one-termed group G., can readil}' 

 be written dow^n from the properties pointed out above. Let the 

 invariant triangle be ABC, and let the transformation T whose 

 equations we wish to find be that one which transforms the point 

 P (x, y, z) to the point P^ (x,, y^, Zj). The anharmonic ratio of 

 the pencil C(ABPP^) is A; in terms of the co-ordinates of the 



Yi y y, y 



points P and P^ this is seen to be A=^ — : — . Hence — =A — . In 



x, x X, x 



like manner the ratio of the pencil A(BCPP,) is A"": and its analy- 

 tic expression in the co-ordinates of P and P, is A"'''^ — : — ; hence 



Yi y 



z^ z 



we have — =A"'>' — . The anharmonic ratio of the pencil B(CAPP^) 



Yi y 



Xj X 



is similarl}' found to lead to — =r:A^-i — . 



z^ z 



These three equations express the transformation T which trans- 

 forms P to P^ : if we have a second transformation Tj of the same 

 group which transforms Pj to P.,, its equations will be 



y, ^y, z. ^ z^ Xj ^ Xj 

 Xg x, y^ yj Zg Zj 



