gg2 PROFESSOR KELLAND ON GENERAL DIFFERENTIATION. 



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

 tory, as we do not take for granted even that the function /m + 1 satisfies the 

 condition fn + l = n Jn except in those cases in which w is a positive quantity, 

 where, of course, we have no diflaculty. It, therefore, confirms our hypothesis, if 

 indeed confirmation is requisite. 



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

 before us. We have seen that 



rf'^a^" ^ /(w + 1) ^n-^ . 

 dxi' /(«-/x + l) 



where /(" + !) ^TT • 



dz 



But the theorem gives, by writing n for ii, 



d" X" _ (^^)" I i„i + „-n .r lm + n — l + . . . + {-\y n. ..{n — r + \) lm + n — r\ 

 dx" Im \ ' 



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

 X a quantity. Also, /"Cx = /I ; therefore /O = ce , consequently the last term 

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

 we get _ 



{-ir.{-Vf ln + 1 /O 

 •^^ ' /r—n /n—r + l 



TT 



I + 1 Sin n TT 



^(_l)» + ^fi^ii:iiL. /n + l. /o. 



TT 



Hence the constant factor which we omitted in f(n + l) is ^^. and we have 

 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 



d_^_ = L=AL^ n(n-l)...{n-r + l) /O 

 dx" /r-n 



if n-/Lt be positive; 



1 



.'r^~" ( — 1)""'' /O 



and '■ = -'^ — == — 



dx"-'' /M-« 



from the fundamental fonnula, if re- ju be negative. 



