in certain Problems of Viscous Fluid Motion. 623 
It is shown that ¢(«)=K(x) and f(7)=«(x) satisty (5) ; 
hence by substituting and Paes coefficients of similar 
exponentials, it is found that a, B,y,--.v are the roots of the 
algebraic equation 
12 
ap Q tb aoo0 te = (9) 
w—p x—-q vv 
while the coefficients in K(z) satisfy the equations 
A B 
Se ot oon +1=0 | 
a—p B—p Yires?? 
Se ee ge ea) 
Agia kB N | 
maa Y Ban ean oqe == aril | 
The solution of (10) leads to 
py a= P@=9) (20) 5 
G=2)C=) =C=n 
_ C= NOH DiaclV=o) oa 
(=a)v—A)...v—m)?” OP 
Before proceeding, we may note alternative forms of these 
results which are of use later. If we write 
F(z) =(a—p)?(@—g)®...(7—v)¥, . . (2) 
the equation for the new exponents @, 6, y... is 
Hy (eee) =O) an) Se G2) 
Further, if we put 
fle) =(e—2) (eB) ... (e—») 
and $ (x) =(«—p)(@—q) ... (v—v), 
the coefficients in (11) are — ¢(@)//’(«), where & isa root 
of (13). 
Whtitaker 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 eq teuon 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 
179 
