38,2 Mr Baker, On Abelian Functions in connexion [Jan. 24, 



As we shall require the result, and to illustrate the point of view, 

 we give a proof. 



Denote the real axis by C ; the points of the circles which are 

 nearest to the real axis by G-[, . . . , G p ', and these circles themselves 

 by the same letters, reckoning from right to left ; denote the region 

 bounded by C , G/, ... , G v ' by I2 ; denote the circles which are 

 the images of G-l, ... , C p ' in the real axis by C\, ... , G p , the points 

 of them which are nearest to the real axis being also denoted by 

 G 1} ... , G p ; denote the region bounded by the 2p circles (7/, . . . , C p , 

 C ly ... , C p by ft, the axis G being then disregarded. Then it is 

 known (loc. cit. (/3)) that there exists a function, G, single valued 

 in fl save for real periods, real on G , G-[, ... , C p ', and infinite at 

 an arbitrary point £= t like (P + iQ)j(t— t), where P, Q are arbitrary 

 real constants. Being real on G , this function will (Schwarz, Ges. 

 Abh., ii. p. 66), have the same value at any point of the circum- 

 ference C r as at the point of the circumference C r ' which is its 

 image in G , and will be infinite in XI, beside at t, at £ = t', where 

 t' is the conjugate complex of t, namely like (P — iQ)/(£— t'). Now 

 the region 12 when considered as the locus for functions whose 

 values at any point of one of the boundaries C/, ... , G p are equal 

 to the values at the conjugate complex point of the boundaries 

 G 1 , ..., G p , is a £>-ply connected Biemann surface, for which the 

 fundamental functions are well known (loc. cit. («)). It follows 

 therefore from the theory of functions on such a surface that for 

 proper values of the constants A lt ..., A p , A, the function desired 

 is given by 



G = -(P + iQ)Tf a -(P -iQ)Tf, a + A lV f a + ... + A/ p a + A; 



here A is an arbitrary real constant, and A 1} ..., A p are determined 



by 



A lTl , r +...+ ApT PiV = 2-JTl [(P + iQ) <f)r (t) + (P - iQ) <f> r («')], 



(r=l, ... ,p), 



where <j> n (£) denotes dv^ a /d%; the point £ = ais supposed taken 

 on G Q . 



By applying similar reasoning we find for the function giving 

 the general circulatory motion in fl , 



cf> + if=Brf a +...+B/; a + B, 



where B 1} ..., B p are determined by the values of ^ on G , 

 Ci, .... G p ; and infer, by the way, that v a is real when £ and a 

 are real, and that r r>s is a pure imaginary, so that the constants 

 A 1} ... , Ap above are real (loc. cit. (6), (A,)). 



