630 Prof. T. H. Havelock on the Decay of Oscillation 

 Consider the integral equation 



*(*)+f+W*(«--T)dT*/(«), 



J o 

 where the nucleus is the sum of n exponentials 



K(t) = %F r e p r t (5) 



i 



Whittaker's solution* is given as 



*(0=/(0-f,/(T)K(t-T)rfT, ... (6) 



Jo 

 the solving function being also the sum of n exponentials 



K(t)=XA r e art . (7) 



The indices a are the roots of the equation 



+'-^-+...+-^-+1-0. . . (8) 



(9) 



X—p 1 iC~p 2 X—p n 



Further, if we form the functions 



it may be shown that the coefficients of the solving function 

 are given by 



A= (^ + " -+ £ -i r f(a)/ *' (a) ' • (io) 



where a is a root of (8) . It should be noted that if pi is 

 zero, and w T e write ^r(#)=#(l— #/jo 2 ) . . . (l—x/p n ), then 



A=-+(-)/P l fl'(«) (11) 



We shall assume that these results hold in the limit when 

 the number of exponential terms becomes infinite. Equation 

 (3) then comes under this form, except that it is an integro- 

 difFerential equation. Equation (8) for the exponents of the 

 solving function gives, on summation, ' 



2^ 



COth^r- + l = (12) 



Also w r e have 



yjr(ai) = (yha^/Ji) sinh (hxijvi), 



0(#) = cosh (hxi/pi) + (o-V2X*/2/jl) sinh QiaFJv?). 



* E. T. Whittalier, Proc. Roy. Soc. A, xciv. p. 367 (1918). 



