844 THIRD REPORT — 1833. 



and in the second we get, 



— — (t'u 2 ^ ' ■ 



(I) 



3 



\a — a) 



= >/•=!; 



dx^ Ar(0) X X (a -a) 



since r(l) =s 1 = r(0), and the symbol in the denominator 



3 3 



= — ^, is a simple zero. The corresponding developement 



of m' under such circumstances is 



^/-«2.A^ + -/-I. A^ 

 a result which is very easily verified. 



If we pay a proper regard to the hypotheses which deter- 

 mine the existence of terms in the series for u' for specific 

 values of the independent variable, we shall be enabled without 

 difficulty to select the indices of the differential coefficients 

 which can present themselves amongst the coefficients of the 

 different powers of h in the developement. For, in the first 



place, /*», and the differential coefficient whose index is — , will 



possess the same number of values, and the same signs of affec- 



tion. If there be a term in u which = P (ar — «)», where P 

 neither becomes zero nor infinity, when x — a, and where the 

 multiple values of P, if any, are independent of those contained 



- .„ ,1 j.(J^ .V .{x — «)» 



in (a: — «)», then it will appear that the term ot — 



m 



dx» 



m 



which is independent of {x — «)» is P . — ^ > and that 



d X" 



m 

 fill • fl# 



all the other terms of —, being either ssero or infinity when 



d x^ 

 X ^= <t, or, if finite, introducing, through the medium of the 

 factorial function by which they are multiplied, multiple values 

 which are greater in number than those contained in u , must 

 be rejected, as forming no part of the developement. It will of 

 course follow, that the function P will become, under such cir- 

 cumstances, a function of h, and if we represent it by P', and 

 denote its values, and those of its successive differential coeffi- 

 iiients, when h = 0, by p, p\ p", p"', &c., we shall find 



