REPORT ON CERTAIN BRANCHES OF ANALYSIS. 243 



The next step is to adapt the series (1) to the different cases 

 which an examination of the constitution of the function ?/ will 

 present to us. 



If we suppose x to possess a general value, then u' and u 

 will possess the same number of values, and no fractional 

 power of h can present itself in the developement. In this case 

 T(l + a) = I . 2 . . . a, and it may be readily proved that 

 the successive indices a, b, c, &c., are the successive numbers 

 1, 2, 3, &c., and that consequently, 



, d u J cV^ u h^ cPu h^ o 



du dx^ 1.2 dx^ 1.2.3 



It will also follow that the series for m' can involve no negative 

 and integral power of h ; for in that case the factorial T (1 + «), 

 which appears in its denominator, would become oo, and the 

 term would disappear. If it should appear, also, that for spe- 

 cific values of x any differential coefficient and its successive 

 values should become infinite, they must be rejected from the 

 developement, in as much as in that case the equation 



would no longer exist, which is the only condition of the intro- 

 duction of the corresponding terms. In other words, those 

 terms in the developement of u' must be equally obliterated, 

 which, under such circumstances, become either or oo. 



If the general differential coefficient of u could be assigned, 

 its examination would, generally speaking, enable us to point 

 out its finite values wherever they exist, for those specific va- 

 lues of the symbols which make the integral differential coeffi- 

 cients zero or infinity. For all such values there will be a cor- 

 responding term in the developement of u under those circum- 

 stances. Thus, if we suppose u ■= x t/a — x, we shall find 



1 .J-u (-')"'^(l) [ K«-.)-(|-r).r ] 

 r{\ + r)r[^-rj 1^ {^--rya-xy-i J 

 if we make x = a, this expression will be neither zero nor 



infinit?/ in two cases only, which are when r =i—, and when 



3 



»' = -g- : in the first case we get, 



rf* u , — - 



(1)-'-^ 



R 'Z 



