562 PROCEEDINGS OF THE AMERICAN ACADEMY. 



belong to the periods w = 1, w' — 27rV — 1/ log 5, and let ABCD 

 be a parallelogram with vertices 



27rV— 1 27rV— 1 



logg log 5 



respectively. At homologous points of BC and AD, P(i) has the same 



value, since P(t + 1) =P(/). 



To obtain the relation between P(t) at homologous points of AB 

 and DC we consider first the matrix 



(59) P{x)= y^-H.v)Fo(.T). 



It is apparent from the form of the elements of Yo(.v) near .r = as 

 given by (58) that, if a positive circuit of x = be made, Yo(x) 

 will change to Yq{.v)K, where 



likewise, upon a similar circuit, 1 oo(.^") will change to • 



(-i)M.--^^'.-Slr^(.i-)L, 



where 



L = (e-2-iV=i5..). 



If these modified matrices be substituted in (59) we obtain the form 

 which P(.t) assumes after x has made a positive circuit of the origin. 

 This is 



(_ i)Me-2-^ V=i '^^2.-iP (.r) 7v. 



If therefore j^tj (-^O denotes the element in the /th row and jth 

 column of P{x), such a circuit modifies 'Pij{x) to 



(_ 1) Me-2-'' V^ «^g^ g2.(.i+pj) V-T^j.. (.^.) , 



But this circuit in the .v-plane corresponds to a passage from a point 

 of AB in the i-plane to the homologous point of DC; in this way, let- 

 ting pij (/) stand for the element of P{f) in the ?th row and jth column, 

 we find 



(60) Pis{t+~-^\^) = (- iyc-'-'^^-''JS',,^-^''i+^? V=i p,.^t) 



