of a Solid Body in a Viscous Fluid. 630 
Consider the integral equation 
$0 +) $)x—n)ar=/0, 
where the nucleus is the sum of n exponentials 
x(t)= SP ri oes Bap baad be) 
Whittaker’s solution * is given as 
t 
s0=/O-[AnKe—nd, . . . ©) 
0 
the solving function being also the sum of n exponentials 
KO) =p Abe 2sin Wau neelay aa) 
il 
The indices « are the roots of the equation 
P P Pn 
“+7 4.,,4—* 41=0. . . (8) 
U—p, U—py L—Pn 
Further, if we form the functions 
eee Cage ZN . (9) 
a(0)=(1-= )(1- 2) (1-2 | 
it may be shown that the cocflicients of the solving function 
are given by 
(Nea eal Maris -1)" (a) /0"( 10 
al “Abe OD PAE GM) 
where « is a root of (8). It should be noted that if p, is 
zero, and we write W(v)=2(1—2/p.) ... (1—2/pn), then 
A= rr (c)IPIO(a) 2). GD) 
We shall assume that these results hold in the limit when 
the number of exponential terms becomesinfinite. Hquation 
(3) then comes under this form, except that it is an integro- 
differential equation. Hquation (8) for the exponents of the 
solving function gives, on summation, 
2h 
Vex 
(a) = (v3a2/h) sinh (hz2/v?), 
6() =cosh (ha?/p*) + (ov2x?/2p) sinh (hi?/v?) 
* [1 T. Whittaker, Proc. Roy. Soc. A, xciv. p. 867 (1918) 
eoth® 41=0. oe oo GY) 
Also we have 
187 
