PROFESSOR KELLAND ON GENERAL DIFFERENTIATION. 



Therefore, in the latter case, 



rf''a;" . . , n(n-l) ... (n-r + 1) (-l)'" 



583 



dxf^ 



= (_l)^//x_» 



/r—n 



(-1)" 



/fx-n In + l (-!)'■ 



I r—n ln — r + 1 «''■""" 



_ /_-iNi^ + l sin w TT I fx — n ln + \ 



which coincides with formula (3). 



lin and to + a« are both negative, we can obtain the results by means of the 

 fundamental formula alone, without having recourse to the theorem before us. 



15. We subjoin the fundamental formula, with its three modifications, in 

 order to apply it to a few examples. 



.u. 1 



the fundamental formula. 



(2.) 

 (3.) 



(4.) 



dx'^ 



dxi" 



dT^ 

 dx'' 



'^'^ 



x" 



In 



/(n + fl) 



^n + f. 



= (-1)" 



/n + 1 



/n-fl + 1 sm{n-fi)'7r 



.^n-y. 



(-1)' 



:(-!)" 



^ + 1 siawTT /n + 1 . //j.- 



■w 



it- 



+ 1 



nz 



— (»+rt 



di^ ' sin(— w + |U7r) In ll-(n + fM) 



Ex. 1. Find the differential coefiftcient of ,3f to index 1. 

 Here, if we adopt the fundamental formula, we obtain 



and by supposing m = —n, this gives 



But 



and 



Ex. 2. Find the f^*-^ differential coefficient of of where fJ- is an integer. 

 1. If M be less than n, we may use formula (2), and we obtain 



VOL. XIV. PART II. 5 M 



