1898.] with two-dimensional fluid motion. 383 



Therefore when f, a are real 



(P + iQ)Tf a + (P-iQ)Vf l a , 

 is real ; as can be shewn from the explicit formulae. 



This function is single valued in H ; it has its imaginary part 

 constant on each of the circumferences (7/, ..., G' 



2. The problem of Riemann's paper (loc. cit. (a)), in which 

 however the boundaries are not necessarily circular, is the deter- 

 mination of the solution, for the region I2 , of the potential 

 equation 



WW 



which is finite within I2 and takes on the boundary an assigned 

 continuously varying value u. This is given, at t = \ + iy, by 



U: 



■H- 



where Pf is the function real on the boundary of Q 0) with only real 

 periods, which is infinite at the interior point t like - — . log (£— t). 



This function is known (loc. cit. (A.)), or follows as before from the 

 theory of a Riemann surface, being given by 



where t' is the conjugate complex of t, a is real, A is a real constant, 

 A 1} ... , A p are real constants determined by the equations 



/ * + A lThr +...+ A P T P> r = 0, (r = 1, 2, . . . , p). 



3. Among the single valued functions for the region I2 , with 

 rational infinities, there is one of particular interest — that namely 

 which is real on the boundary and has a single pole of the first 

 order at an assigned point of each of the (p + 1 ) separate boundary 

 curves C , 0/,..., G p '. This function effects a conformal repre- 

 sentation of the region fl upon an infinite half plane (p + 1) times 

 covered, and is thus the direct generalisation of the function giving 

 the conformal representation of a simply connected region upon a 

 simple half plane. 



In what follows we put down this function for the case when 

 the circles 0/, . . . , C p ' are unrestricted (save by the inequalities 

 which are necessary for the convergence of the series involved) ; 

 and then give the very simple form taken by the function when 

 the Abelian functions involved are those of hyperelliptic character. 



32—2 



