1279 



p{h) {P,S-PA) ^i'^) ^ Pi^) \iQ.P,- Q.PyiV^ + (P, 7',- P,7\)Q(.^.)^ 

 Putting now 



p{d) QA-Q A 

 p(h) PA-PA 



we get 



^ /Ho) SJ, -SJ, ^ ^ p{a) Q^P-Q^P^ ^ 



p{a)P,l\~P/l\ 

 p{b)P,S,-PA 



= (i 



P(.r)=zy7'(..) + rfQ(.;). 

 It is easy to verify the condition 



p{b){aö~fy)=p{a) . (17) 



We thus see, that S and P may be calculated from T and Qby 

 a linear substitution. Replacing these functions by what they mean, 

 we have 



K[a'. b) = a K(.v,a) f fl 



dE{A:,ri) 



= y K{x,a) + 6 



èK{xoi)\ 



(18) 



That, on the other hand, these equations satisfy (14) appears from 

 the fact that in (18) the determinant of the left hand quantities for 

 two values of .i', say x and //, is equal to {luS — iiy) times the deter- 



minant ot the quantities A {x,a) and ■- 



a' and y; this shews, in connection with (17), that (14, is satisfied. 

 The inverse substitution of (18) is 



for the two values 



K{^v.,a) = a' K{x,b) + /?' 



(18a) 



= y' K{xM + Ö' 



èK{x,i])\ 



with the condition 



p{a){a'(i'-i-i'y')=:p{b) (17a) 



If it be impossible to determine //, and //, so that .S\ /^, — *S\P, t^O, 

 S{x) and /-*(.(,') will be proportional and from (14^/) it is seen that 

 in that case also Q{,v) and 7'(./') will be proportional. This may also 

 be considered as a consequence of (18). 



In the case p{a)=iO, p{/)) ^ it follows from (14^?) that S{x) 

 and F(x) will be proportional and as then the proportionality of 

 <S(,?;) and P(.i') is a consequence of (18) and (17r/), it is allowed to 



83* 



