1885.] K', E', J', G' in ascending jmvers of the modulus. 237 



Substituting these values of K', E', &c. in the equations given 



in the last section we find that in every case the terms mul- 



4 

 tiplied by log ^ cancel each other, and we obtain the following 



21 equations involving K x , K 2 , E x , E 2 , &c: 



(i) 



(ii) 

 (iii) 



(iv) 



(v) 

 (vi) 



(vii) 

 (viii) 



(ix) 



00 



(xi) 



(xii) 



(xiii) 



(xiv) 

 (xv) 



(xvi) 

 (xvii) 



EJ\ 



KK* =1 



I 2 K X -IJ\ =1 



g 2 k x - a t K 2 = i 



UJS X - U X K 2 = 1 

 ^ - V X K 2 = 1 



w 2 k x -w x k 2 = i 



EJ 2 -I X E 2 -1 

 KU a - ZZ Fl =4 



G x w 2 -a 7 w x =\(k i -'y 2 ), 



LG, 



LG„ 



v i w 2 -r 2 w 1 = ik\ 



U 2 W X -TJ X W 2 = W\ 

 E W -EW=i 



1 " 2 ^2 " 1 2' 



F^, - V 2 G X =U!\ 



(xviii) e x v 2 -e 2 v x =ir, 



(xix) IJJ X -IJJ 2 =U", 



(xx) 



2 1 



E,n 



e 2 u x =i(i+n 



(xxi) I 2 V X -T X V 2 =K1+^ 2 )- 



§ 37. It will be seen that by the substitution for K', E', &c. of 

 their values in terms of K x and K 2 , I x and I 2 , &c. all the 21 

 equations have been rendered uniform and symmetrical, viz. the 

 sign of the second term is negative in the equations involving K t 

 and K 2 as well as in the others, and each expression is symmetrical, 

 vol. v. pt. iv. 17 



