of Definite Integrals. 19 



method of investigation, it appears that the functions <p and \p 

 may be any whatever, consistent with the required relation 

 between them. But if we are obliged to integrate by series, 

 they will in general be subject to the restrictions mentioned in 

 (1.) and (2.); I say, in general, for infinite quantities may 



vanish by the operations ( ^ ) • 



To give an example in each of the theorems : in (1.) let 



We find 



w=2>P=2»4'(^)='/^. 



{K0}V)=|(f). 



and , V 1 



then 4/(a)='/a, 



as it should be. 

 In (2.) let 



We find 



In the last example n and p are not conformed to the re- 

 strictions, but the infinite quantity goes out by differentiation. 

 The theorems (3.) and (4.) are likewise satisfied by these ex- 

 amples. It must not be supposed that the values of <p (a), 

 given in (1.) and (3.), or in (2.) and (4.), are necessarily equal; 

 for they will not reduce the one to the other. Yet we may 

 have 



^ ^ 



in both cases ; since we know from examples that the inte- 

 grals of different functions may be of the same form. 



Gunthwaite Hall, near Barnsley, 

 May 24, 1847. 



C2 



