g02 PROFESSOR KELLAND ON GENERAL DIFFERENTIATION. 



(n + m)(n + m + l) 1 . 



+ 172 •^^?~^"- 



(^_1)»_ '^ + '^ 1 1 



'»^ 





]^ \ « + nt 



^(_1)». ^^ + ^ '^' 



= a" 





1 



which is the proposition to be proved. 



28. The example given in the last article will furnish us with a ready means 

 of exemplifying a theorem analogous to that of Maclaurin ; for the coefficient 



of <2?~" in the expansion of — in terms of negative powers of x, is, accord- 



ing to that theorem, supposing it extended to the case before us : {oc being = 0) 



(i+^^ 1 which is equal to 



, ., l—n + m a~" 1 , ^ 



f — n-« ^ — . . —, — - when x=0 



f — 1)"~" 1^ ir = 



^ ^ P^ ■ (1 + a ^)-" + '« (-!)-» /O ' 



-» /: 



/ —n + m a ■■ n 



~ I'm /F 



For instance, ifii-^m; since /0^= oo , the coefficient is zero. 

 Ifn—m, the coefficient is 



/O cr^lm 

 Im /O 



If ?i=m + r, it is 



r 



= a-^'^ + ''' m{m + V) . . . {m + r-V) x -^^ 



Im /O /O 



Now, /T = — 1 l^-i = (-l)"^ fV f^r if r be a whole number : in this case 

 the coefficient is 



, ^y m(n + V) ... (ni + r — l) 1 

 ^ ^ 1.2 ... r ^« + '-' 



But if r be a fraction, /^ is finite and / oT infinite, therefore the coefficient 

 is zero ; results which are all obviously correct. With one more example we 

 shall conclude the present memoir. 



