388 Mr Baker, On Abelian Functions in connexion [Jan. 24, 



where /a is a variable real quantity ; again, since —i(v% a — K) is 

 real on the arc a x c 2 , it has conjugate complex values on the arcs 

 (hPi, tti-D/ j so that we have on these arcs respectively 



where /x is a variable real quantity ; therefore 



while, on the other hand, 



v a lt ^ — yd', Cj 1 JJ 1 ^qt!, C,' 

 ?■ r »• ' 



where II represents the value obtained in passing from one side to 

 the other of the barrier through (7/, = — 1 when r = 1 and = 

 when r =}= 1, it being supposed that, as in the figure, the arc c^ is 

 in the counter-clockwise direction ; hence we have by addition 



2i>«i. c > = n, 



giving the result enunciated. This equation is also easily seen to 

 be the condition that the function vi> a be continuous at _D/. 



For these results cf. loc. cit. (6) ; the proof here given is de- 

 signed to bring out the identity with some corresponding results 

 in the theory of a hyper-elliptic Riemann surface ; it is upon this 

 identity that an important result deduced below is made to rely. 



Further, since the real part of v*= r is constant upon the separate 

 arcs of the circle we obtain 



K 3 ' a * = ^r, 2 - fa, 3, K' ^ = fa, 3' 



of which for instance the value of v c f tti can be obtained by a 

 method similar to that employed above, by adding the values 

 obtained by passing from a 1 to c 2 , first directly along the circle 0, 

 and then along the path a 1 G 1 'G 1 a 1 / C2G 2 G 2 'c 2 , where a-[, c 2 ' are the 

 images of a 1} c 2 on the circle C . 



These equations shew that, for the definition of the half 

 periods, the points a, c 1} a lt c 2 , a 2 , ..., are exactly analogous to 

 the branch places denoted by the same letters when the period 

 loops of the hyperelliptic Riemann surface are drawn as in the 

 annexed figure (2). 



Hence from the theory of hyperelliptic theta functions we infer 

 that the theta functions © (v^> a ), © {v& c ) vanish respectively in the 

 places («!, ..., a p ) and (c 1; ..., c p ). 



