134 Mr. Bootr on the Analysis of Discontinuous Functions. 
CONNEXION OF FOURIER’S THEOREM WITH THE THEORY OF 
EQUATIONS. 
1. Theorem.—The value, v, of the definite double integral, 
5 NS. dadvesW/=1 f(a, 0/1), (1) 
is symbolically expressed by the equation 
ane 
— (- AG e’), (2) 
provided that, after expansion and differentiation, we assume @ = 0. 
For if 6=0, 
J (a vf =1) = f(a, v/—1¢’) 
= f(a, ef Fle ty/=1) 
== AU: Wain f (a, e’), by Taylor’s theorem, 
Hence =1(8 JZI)# fase). (3) 
1 2 © 
1: x) SJ. dadv &"/=i(yx/ 1) 4 f (a, EE (4) 
Now 
x = 
WN (y4/ 1) = (- _ e(a—z)eXV 1 
when A is a constant quantity ; and if, in the right hand member of (4), we sup- 
pose f(a, €’) to be developed in descending powers of ¢’, then in each term of the 
Oe , : 
expansion, — will be represented by a constant quantity, by virtue of the known 
Pp rT Pp Yi q ys by 
theorem, 
d 
p ay — p(n ye". 
Hence 
a nS = 
ero ZT) f(@e) = (— REV flase 
and substituting in (4), 
