1884.] K' , E' , J' , G' in powers of the modulus. 201 



We thus obtain as a particular integral 



/ l 2 l 2 3 2 l 2 3 2 5 2 



w = ^( /i + 2 T4 A2 + 2 T6 A4 + 27i 2 T6 2 78^ + &c - 



that is, in the notation of § 9, 



u = A .2G l . 



If we take the root ra = we obtain infinite values for A x , 

 A 2 , &c. and we conclude therefore that the general integral of the 

 equation is of the form 



u = aG t log bh + T, 



the coefficients in the series T being determined by substituting 

 this expression for u in the differential equation and equating co- 

 efficients. 



Since G' satisfies the differential equation it must be of this 

 form, and comparing its value, viz. 



l ^M^h+G„ 



with aG t log bh + T, 



we see that it is included as the particular case a = £, 0== te> the 

 series T being equal to 6r 2 . 



In the solution of each of the seven equations we obtain 

 directly by the ordinary process of integration one series proceed- 

 ing by ascending powers of h. These are the series K v E x , I v 

 &c. The series K 2 , E 2 , T 2 , &c. are the values of T in the different 

 cases. 



Series for K, G, W> involving sines of multiples of the modular 



angle, § 23. 



§ 23. In Vol. XIX. (pp. 51, 52) of Crelles Journal, Guder- 

 mann has given the following remarkable formula in which 

 denotes the modular angle : 



/ l 2 l 2 3 2 l 2 3 2 5 Z \ 



K = it ( sin# -f ^,sin 50 + ^-. , sin 90 + ^' ..,',., sin 130 + &c. . 

 V Z 2" . 4 2 2 2 . 4' . b" / 



The chief interest of this formula consists in its elegance, as 

 it is of course not so suitable for the calculation of K as the series 

 proceeding by powers of k 2 . The method by which Gudermann 

 obtained the above series for K is in effect as follows : 



1 — k 



By the change of k' into = j , K' is changed into ^(1 + k)K', 



vol. v. PT. III. 14 



