578 PROFESSOR KELLAND ON GENERAL DIFFERENTIATION. 



use it without any restriction as to the value or sign of the symbols which it in^ 

 eludes. That our formulae are correct, we shall give abundant evidence as we 

 proceed. At present we shall be occupied in the demonstration of two or three 

 propositions which will be useful hereafter, and will also serve to verify our re- 

 sults. 



Mr Peacock suggests the adoption of the following form of the differential 

 coefficient. 



rf'^a;" /« + l 



dx^ /n — p. + l 



Jn fact, if /i be a positive integer, we get 



X 



,« — A» 



dxi^ 



=n{n-V) ... (n-fi + l)x'^'^ 



-^±1-.-'^ 



/n — n + 1 



and are thus in possession of a form very convenient to be adopted as a defini- 

 tion. We can easily shew that it is a correct /orw by the following process : 



Since = A a;"^'" where A is a function of n and u, if we take the 



dxi^ ^ 



(«-/x)th differential coefficient of each side, we obtain 



:A 



^ a;" d'^i' ar' 



A = 



rfar** dx"^^ 



ifi n 



if therefore — be abbreviated by /(n + l), we obtain 



/(w-yu+l) 

 dx'' /(n-fX + l) 



the form required. 



10. Mr Peacock is, however, not justified in assuming that/(« + l) coincides 

 with /(n + l). If we examine equation (2), we shall find that, as far as that for- 

 mula is concerned, we may write 



(-ly sin nTT /n + l for f(n + l) 



Taking this as true, we obtain 



( _ 1)»— 1 sin (« - 1) TT /rC=f{n) 

 or (— l)"sinw7r/»»=/n 



/(n + 1) = (— 1)" sin w TT /w + 1 

 = ( — 1)" sin « TT » /» 

 = «/(n). 



