582 . PROFESSOR KELLAND ON GENERAL DIFFERENTIATION. 



We cannot help remarking, that the process adopted here is most satisfac- 

 tory, as we do not take for granted even that the function /n + l satisfies the 

 condition /« + l = n/n except in those cases in which 7^ is a positive quantity, 

 where, of course, we have no difficulty. It, therefore, confirms our h3^othesis, if 

 indeed confirmation is requisite. 



13. We have still another mode of verifying our formulae from the theorem 

 before us. We have seen that 



rf'*^" /(n + l) ._, 



V- . 



dzi^ /(W-/X + 1) ■ 



where /(^ + 1) = — — ■ 



But the theorem gives, by writing n for ^t, 



^" ^" = illlA", [ /m + n-n .r /m + n — l + . .. + (—iyn...(n-r + l) /m + n — r\ 

 dx" Im \ ^ 



but n=r—m, therefore n—r + m=0, so that the last term of this expansion is /o 

 X a quantity. Also, /Tx = /l ; therefore /O = cc , consequently the last term 

 is infinite ; we may therefore neglect all the other terms compared with it, and 

 we get 



(-i)".(-i)'-/^rn /o 



f{n + 1) = — . === 



jr — n Jn—r-\-\ 



TT 



+ 1 sm W TT 



= (—1) /w + l./O, 



Hence the constant factor which we omitted in /(w + 1) is , and we have 



TT 



now the complete value of that factor. The result completely verifies the for- 

 mula (2). 



14. But its effect is not confined to this particular formula. 



We have generally 



-^^ = ^^i£^ w(/^-l) . . . (w-r + 1) /O" 

 ^^» /r-n 



a .jc^ _A_U ^(n-ii)...(n-ix-s + l) /{) 



dx^i" /s-(n-iJL) 



if n-/Libe positive; 



I 



«/"-'* 



and 



of-'' _ (-1)"-^ /O 



from the fundamental formula, if w-^ be negative. 



