of a Solid Body in a Viscous Fluid. 632 
In the previous notation, we have 
ee bas, By 
(2) = o( cosh + 3 “sinh =), 
% (20) 
By has\ . b Qui ha? 
(2) =(f4 + ose e* cosh ——? 
ANE oph. VE 
the formation of the latter being clearly indicated in the 
reduction from (18) to (19). The coefficients of the solving 
function can now be formed by (11). Finally, substituting 
in (6) and carrying out the integration, we arrive at the 
result 
_ 4p a = aot 
I~ ch ~at—(2yloh) (1+ 2ph/o)a* + pra? + 6up aloh + p” 
Se ESE) 
the summation extending over the roots of (19). 
3. We may verify the result by other methods which are 
available in this case. We choose Bromwich’s method of 
complex integration*, referring to his paper for the general 
principles, and writirig down the results briefly for the 
present problem. 
Suppose the fluid velocity and the displacement of the 
plane to be given by 
1 
v= 55 uetda; y= se retda: 0 o (x) 
where u and 7 are functions of «, and the paths of integration 
are in the plane of a complex variable « and enclose all the 
poles of these functions. The differential equation of fluid 
motion, 07/Ot=vd7r/d2”, with the conditions w=0 for «=h 
and u=dn/dt for e=0, gives the solution, for x positive, 
_ dnsinh{a3(h—z)/v?} 
‘dé sinh (ath[vt) —~ 2) 
From the boundary condition (1), after introducing terms 
due to the initial conditions y=a and dy/dt=0 for t=0, and 
using (23), we obtain 
Quast eh i 3 
= n coth = +op'n= ( + = coth — i 
oan + a (24) 
v oy 
Hence we have 
_ 1 (afat(2u23/cv*) coth (hai/vt) ter da 
Y= dni) a+ Qyallon) cath (hati) +p? | OO) 
Forming the residues of the integrand at the zeros of the 
* T. J. PA. Bromwich, Proc. Lond. Math. Soc. xv. p. 401 (1916). 
189 
