(451 ) 

 From 



« T.' = /:? + r ft' 



we have 



jf T — T a — ft, 



so tluit we get for the system of the increments il of the integrals w 



rn = ft'k — ftk' + r'ak'. 



By supposing tliat in the first row of the matrix t' the constituents 

 ^'is, ^13 • • • • '■':/) ïii't' ïi'l equal to zero we shall find that every 

 increment i2i of the integral wi is expressed by 



riii = ftj\k — ftyk' -\- r'li a^k', 



where the first columns of the matrices ft\ ft, a are denoted by 



ft\, ft\, «1- 



Hence the moduli of periodicity of the integral noi corresponding 

 to any closed circuit are always multiples of 1 and t\i and therefore 

 this integral must be an elliptic integral. 



It may be noticed that in the case /> = 2, the same conclusion 

 holds for the integral rwo, so that for /j = 2 there exist two redu- 

 cible integrals or there is none. 



Assuming rw^ to be an elliptic integral we can easily find how 

 many zeros the function 0(rwi] r'n), of the single variable noi 

 and the period r'n, possesses on the surface T. We have only to 

 calculate the value of the integral 





taken round the boundary of the simply connected surface T\ into 

 which T is resolved by the customary p pairs of cross-cuts Ah and 

 Bh . On opposite edges of a cross-cut A^ the variable «o, has values 

 the difference of which amounts to ft'h\, so that on both edges 

 dlug {rwi ; r^i) has the same value and the integrals taken in op- 

 posite directions round these edges, destroy one anotlicr. 



31* 



