REPORT ON CERTAIN BRANCHES OF ANALYSIS. 241 



riable part of ^" d x is no longer expressible by a function 



under the form r' but by one which must be determined 



n + i 



by independent considerations. A knowledge, however, of the 



nature of its form in this particular case has enabled algebraists 



to bring it under a general form, by which the sign of failure 



or impossibility is replaced by the sign of indetermination 



A /r" + ' a;" "*" ' a" "*" ' 



jr-; for if we put — -— r + C = — — ^ + C, (borrowing 



from the arbitrary constant,) we shall get an expression 



-«" + ' 



n + 1 



which becomes -^ when n = — 1, and whose value, determined 



according to the rules which are founded upon the analytical 

 properties of 0, will be log a; + C. 



A more general example of the same kind, including the one 

 which we have just considered, is given in the note to page 

 211, where it is required to determine the general form of 



— — — and of -, — . — (where n is a positive number) for all 

 dx"" X dx'' x" ^ 



values of r : a formula is there constructed, from our knowledge 



of the form in the excepted case, which is capable of correctly 



expressing its value in all cases whatever. 



The cases in which the series of Taylor is said to fail are of 



a similar nature. Thus, if u = <f (x) = x + \f x — a, then 



du , , d^u W , d^u h^ ^ 

 u' = <p (:c + h) = u + j-^h + j^, Y72 + J^' T7273 ^^- ' 



ft U 



and if we suppose x =i a, all the differential coefficients ^, 

 -^—2, &c., become infinite, which is an indication that no terms 



of such a form exist in its developement, which becomes, under 

 such circumstances, a + V'/'- The reasons of this failure in 

 such cases have been very completely explained by Lagrange 

 and other writers ; but it is possible, by presenting the deve- 

 lopement which constitutes Taylor's series under a somewhat 

 different and a somewhat more general form, that the series may 

 be so constructed as to include all the excepted cases. 



There are two modes in which the developement of <p (.r + h) 

 according to powers of h may be supposed to be effected. In 

 the first and common mode we begin by excluding all those 

 terms in the developement whose existence would be incon- 



1833. R 



