newson: continuous groups. g3 



infer that 



CY CBj 



I— k=(CAYBi) = : ; 



YA B , A 



CY CBj 



and I — k' = (CAY*Hj):= : . Dividing the first hv the 



YA B,A 

 second we get 



r— k CY CY 



=^ : ^(CAYY')=:Ay. 



i_k' YA Y'A 



Again taking the ranges along the line x, we have Ax=(BCXX). 

 k^^CXCBAj), and k' - .(X'CBAj): therefore 



k BX BA^ 



=(BCXAi)= : — - 



k— I XC AjC 



k'— I BA, BX' 



and ^ = (BCy\,X')=r : . Multiplying together these two 



k' AjC X'C 



results we have 



k(k'— I) BX BX' 



k'(k — I) XC X'C 

 By multiplying together these values of A;^- , Ay, A^ we can verify 

 the former theorem that the product of the three characteristic an- 

 harmonic ratios along the three invariant lines is unity: thus 



k(k'_i) k--i k' 



A,^ Ay Az = . . — =: I . 



k'(k— I) k'— I k 



Thcorrni 14. For any projcclive transformation of the kinJ T tlw 



tlircr charartrristic aniiarnionic ratios a/o/ij^ the three invariant lines x, 



1', z max he expressed in terms of the two tan<:;entiai aniiarinonie ratios 



of tlie tu>o eonies K and K' luliieh determine the transformation: thus 



k' k— I k(k'— I) 



Xj =: , Ay ^= , Ay r= . 



k ' k'— I k'(k— I) 



If we express these in terms of A and a, these relations are found. 



k' k — I k(k'— I) 



A^^ — : A-'-i^ : A"":^ . 



k k'— I k'(k~i) 



We wish to find out how to select the pairs of conies which pro- 

 duce transformations belonging to a one-termed group. We must 

 first express k and k', the tangential anharmonic ratios of the con- 

 ies K to K', in terms of A, the characteristic anharmonic ratio ot 

 the transformation. By theorem 14 we have 



