52 Proceedings of the Royal Society of Edinburgh. [Sess. 
the given equation admits of solutions in terms of a finite number (^n) of 
the functions gfi(x). 
In particular, consider the integral equation (4) of § 2, 
f/3 ^ 
u{x) = X I K(i«, s)u{s)ds, 
supposing, for simplicity, the nucleus symmetrical and decomposable into 
n products as follows : 
s) = "'^Cigi(x)gi{s) ..... (26) 
i=l 
It follows that u{x) admits of the development 
i—n 
(21) 
7=1 
where the constants are to be determined. We thus have 
r^i= 
■I— a rt' %=znj = n 
'^Cigi(x) = igj{s)ds 
7-=l a i=l j=\ 
i=n j—n 
i=i .'ft 
and hence the n coefficients are to be determined by the solution of the 
n simultaneous linear equations 
j=n 
s, = a2A«c, (28) 
j=^l 
where 
= Ci j gt{s)gj(s)ds (28a) 
The constants can therefore be determined, apart from a constant 
multiplier, by solving the set of n equations between the n-\-l variables 
Cj, ... Cn and X. 
As an example, let us consider the equation 
dH , 
g,g> + [a — 4/ COS 2x + |-Z2 cos 4,r]?/ = 0 . . . ( 29) 
with which is associated the integral equation * 
r)iTr 
ii{x) = Xjcos^ .... (30) 
The nucleus of this integral equation may be expanded in the form 
pH (sm2 »+siii“ s [1 (.Qg 3^, ^.^^g 3g 3 (3Qg pQg g ^ 3^- gj^ 3s + f sin X sill ,s'] 
* Whittaker, loc. cit. (4), p. 22. 
