( 450 ) 



of the first kind is reducible, and it is [)Ossibli' to eoiistruet rational 

 functions on the Riemanx surfticc Ï', which are doubly periodic 

 functions of this integral. 



Let us suppose that in the (^-function of p variables (9(«;7-), 

 where u denotes the p normal inteo^rals and t denotes the given 

 period matrix, we make a substitution of oi'der >• associated with 

 the Abelian matrix 



aft' 



raft\ 

 Wft') 



of 2/' rows and 2/> columns of integers, in such a way that the 

 separate matrices «, ft, «', ft' satisfy the equations 



a' = , aft' = ft'a = a'ft' = ft'a = r , ft ft' — ft ft = 0. 



According to this substitution the integrals ?« are replaced by 

 other integrals w determined by the equation 



and 0{iijT) of the hrst order becomes a function (?,(«•;/■') of 

 order r with a period matrix t' which can be derived from the 

 equation 



«r' = /? + Tft'. 



From the above relations we can immediately calculate the incre- 

 ments £2, taken by the integrals to, when by describing some closed 

 curve on T the normal integrals ti are increased by 



CO = k -\- T k' , 



where k and k' denote two columnletters. 

 In the first place we find 



a S2 = 0) =z k -\- T k', 

 and also by multiplying by the matrix ft' 



ft' a 11 = r Si =ft'k-\-ft'T k'. 



*) For the notation compare ; ]?akek, Abc-Ps Iheoreui and the <tUkd Ikeunj. Cam- 

 bridge 1897. 



