1884.] K',E',J', G' in poivers of the modidus. 191 



I found it advantageous in working with the quantities E. J, G 

 to use, instead of J, the quantity — J. Putting therefore /= — J 

 the three quantities considered are E, I, G. 



It was found to be convenient also to denote by separate letters 

 the three quantities |(/+ G), ^(G + E), §(E + I). Denoting 

 them by U, V, W we consider therefore the six quantities 

 E, I, G, U, V, W, denned by the equations 



I=E-K, 



G = E-k'*K, . . 



U = ±(I+G), 

 V=^G + E), 

 W = -l(E+I). 



The six quantities E, 7, G, U, V, W, § 8. 



§ 8. Expressing the six quantities in terms of E and K, and 

 of / and K, we have 



E = E =I+K, 



I = E-K =1, 



G = E-k' 2 K =I+k 2 K, 



U = E-±{l+k' 2 )K=I+±k 2 K, 



V=E-lk' 2 K =I + l-{l+k 2 )K, 



W = E-\K =I + ±K, 



the six quantities E ', I', G', V, V, W denoting the same func- 

 tions of k' that E, I, G, U, V, W are of k. 



Systems of formulas for K, E, I, G, <&c. and K', E', T, G', &c, 



1 9— 11- 

 § 9. The systems of formulae given in §§ 2 and 3 may be 

 expressed uniformly in the following manner. 



Let 



