in certain Problems of Viscous Fluid Motion. 



623 



It is shown that <f>(V) = K(.i') and /(#)=#(#) satisfy (5) ; 

 hence by substituting and equating coefficients of similar 

 exponentials, it is found that a, 0,ry, ... v are the roots of the 

 algebraic equation 



P 



+ - 



x— p x 



Q v 



—■+.... + — 



-q x- 



4-1 = 0, 



(») 



while the coefficients in K(a?) satisfy the equations 



A 



+ .7T- 



B 



N 



-p £-/> 



1 = 



(10) 



A B N , 

 — - + -o — + ....+ +1 = 



OL—V p — V V—V J 



The solution of (10) leads to 



t>) 



(*-^)(rf-7)...(«-v) 



(11) 



Before proceeding, we may note alternative forms of these 

 results which are of use later. If we write 



F(«) = (*-jO*C*-9)*...t*-tO T , • • ( 12 > 

 the equation for the new exponents a, f3, <y ... is 



F»+F(*)=0. . . . . (13) 



Further, if we put 



and <j> [a) = (a —p) {x — q)... {x_ — v\ 



the coefficients in (11) are — 0(a)//'(a), where « is a root 

 of (13). 



Whittaker remarks that if the number of exponential 

 terms in (6) is supposed to increase indefinitely, a theorem 

 appears to be indicated, namely, that in the solution of a 

 Poisson's integral equation whose nucleus is expressible as 

 a Dirichlet series, the solving function is also expressible 

 as a Dirichlet series, but with a different set of exponents 

 for the exponentials. 



4. Returning now to equation (4), we see that it is an 



