in certain Problems of Viscous Fluid Motion. 627 
where the summation extends over the positive roots of 
1 2 
alo: Vr ceaam ails pe. (C2) 
The equation for the exponents of the solving function is 
iia 1 1 ae ae 
ies careoe amen oe . +1=0. (30) 
Writing «= —v)2/a?, equation (30) reduces to 
AJo(X)+ATA)=O,. . . . . (81) 
where k=2zpat/I. The equation can be deduced from (29), 
by logarithmic differentiation, as indicated in (12) and (13). 
The equations for the coefticients of the solving function 
become 
Ay As As Vv eas 
AP~—p? a Agi — pi 3? — pp? F xa2aap v 
Ay ee Ay or Bawions —-+=0 5 (ez) 
a 
where p,, pz, ... are the roots of (29), and X,, A», ... the roots 
of (31). 
To solve these equations, we may adopt the same plan as 
before. Assuming that a function f() can be expanded, in 
the range 0 <r<1, in the series 
f(r) ==BJ(Ar), 
the summation extending over the positive roots of (31) 
we have 
DNS ( AGM, 2G 
B= {PEACE Epp TaF(n) qf IO7)rar. o (a3) 
Now take f(r)=J.( pr), where p is a positive root of (29) ; 
after obtaining the expansion and putting r=1, we arrive at 
the result 
2k? 
= ; mo 0 0 (ol UAL 
Th DEE Op) Se 
p being any one root of (29) and the summation being with 
respect to the roots of (31). 
183 
