542 THIRD REPORT — 1833. 



sistent with general values of the symbols : m the second we 

 should assume the existence of all the terms which may cor- 

 respond to values of the symbols, whether general or specific, 

 ancl then prescribe the form which they must possess, con- 

 sistently with the conditions which they are required to satisfy. 

 If we adopt this second course, and assuming ti — f {x) and 

 u' = <p (x + h), if we make 



u' = M + A /«" + B /^* + C h' + &c., 



the inquiry will then be, if there be such a term as A A", where 

 A is a function of a: or a constant quantity, and a is any quantity 

 whatsoever, what are the properties of A by which it may be 

 determined? For this purpose we shall proceed as follows. 

 It is very easy to show, from general considerations, that 

 if uf be considered successively as a function of x and of h, 



-i — r = -TTT > for all values of r, whether whole or fractional, 



positive or negative : it will follow, therefore, (adopting the 

 principles of differentiation to general indices which have been 

 laid down in the note, p. 211,) that 



dh''~ r(i) •^ + r(i + 6-«) + *^^-' 



omitting the arbitrary complementary functions, which will in- 

 volve powers of /<. In a similar manner we shall get 



d'tc' d^u d" A ,„ fZ-'B ,. „ 



If these results be identical with each other, we shall find 

 r{\ + a) ,. _ d^n 



~r~i\)~' dx"' 



1 d" u 



and, therefore, A = ^,^ ^ . . ^^, since r(l) = 1. It is easy 



to extend the same principle to the determination of the other 

 coefficients, and we shall thus find 



or, in other words, it follows that the coefficient of any power 

 of h whose index is r will be 



1 d u 



r(l + r)' doif' 



