242 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, 

 and 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 u ■= (^ {x) and 

 2^' = (p (a; + li), if we make 



u' = « + A /i" + B /^* + C h" + &c., 



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

 A is a function of x 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 iH be considered successively as a function of x and of h, 



-7—7 = -rrr. , 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 



d'^u' _ r(i + a) ^ r(i + &) B,,_„ . . 



omitting the arbitrary complementary functions, which will in- 

 volve powers of h. In a similar manner we shall get 



d'u' d^u d" K j„ d^B ,, 



dx" dx" dx" dx"' 



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



r (1 + a) y _ d^ 



rji) ' dx"' 



and, therefore, A = -=-?! — ; — n • -i-~^, since r(l) = 1. It is easy 



r{l + a) dx"' ^ ^ •' 



to extend the same principle to the determination of the other 

 coefficients, and we shall thus find 



d" u h" # u //* o ,, . 



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)'dx'^' 



