1898-.] with two-dimensional fluid motion. 385 



which, in turn, shew that, a being real, the function 



/, (S) = (P, - *&) i# + . .. + <p, - »&> rf;f + (p - »Q) rj •, 



is a single valued function unaltered by the fundamental substitu- 

 tions. The poles of /i (£) in the £ plane are however the same as 

 those of/ (£), since c/, c r are analogues of one another for the r-th 

 of the fundamental substitutions — and the functions /i(£)> f(0 

 vanish at the same place a. Wherefore, considering the residues 

 at the pole z, we infer 



(P + iQ)-if(0 = (P-iQ)-if 1 (0; 



as we have already shewn that for conjugate complex values of £ 

 the quantities 



(Pr+*0r)I* B , (P,-tQr)lf* 



have conjugate complex values, it follows that the function 

 (P + iQ)' 1 /(£) has the same property. 



This proves the theorem in question. 



4. The same argument can be used to establish the further 

 result :• Take p + 1 points entirely within the region O, so situated 

 that those which are not upon the real axis G consist of pairs 

 occupying conjugate positions in regard to this axis. The uniform 

 function unaltered by the p fundamental substitutions which has 

 these points for poles of the first order is necessarily real upon 

 C , C/, . . . , Gp (and C\, ... , G p ), or can be made so by multiplication 

 by a proper constant factor. For the region D, this gives the 

 theorem : take any m points within H or upon one or more of the 

 circumferences G-[, ..., G p ', and any k points upon G Q , so that 

 2m + k=p + l; there exists a function uniform in Q , real on 

 0/, ... , G p ', G 0> and infinite to the first order at each of the m + k 

 points taken in H . 



It may be of interest to give the functions of §§ 3, 4 for p = 1. 

 Let the fundamental substitution be 



and v — | \f/JL [ . Then if co be any real quantity, 



C.'B C B I G^B , a), to. (1%-Bs 



cr = 



g;b g b g;b , « , », /i£-/A 



AG?' v = ACjACr "^V 10 ^' "-S^UfTZj- 



(forms which hold when C becomes a circle, C being the point 

 where the straight line C'/Ci cuts the circle G ), the general and a 

 particular form of the function of § 3 are given by 



tf (u) + V' (v) _ ff>' (u) + $>' (w + co') p' (u) 

 $ (u) - p (v) p (u) - p {w + co') ' $ («) - e 3 ' 



