PROFESSOR KELLAND ON GENERAL DIFFERENTIATION. 533 



Therefore, in the latter case, 



dxi^ ^ Ir—n a^—^ 



Ir—n /n — r + 1 x^ ^ 



_ / ,N;a + l sinwTr /jx — nln + l 



TT a;'^~'* 



which coincides with formula (3). 



If n and n + iJ- 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. 



(!•) ^=-4(-ir. /Jll±]^ the fundamental formula. 



' dxi^ In-fx + l sm(n-fx)rr 



, d'^x'' _._-,N,>* + l sinwTT /w + 1 ■ lix-n 



^^•^ dz^ ~^ ^ ' ^ ■ ^-^ 



dl^ — 



(4.) ^" ^(- 1)^ + 1 ■ ^ . _ 1 a:~^'^ + '^) 



<?«'' sin ( — n + /x tt) /w /1_(/«+ju) 



Ex. 1. Find the differential coefficient of x" to index 1. 

 Here, if we adopt the fundamental formula, we obtain 



dx-"^ _ 1 ,_^' lm + 1 

 dx Im ' x^'^™' 



and by supposing m = -n, this gives 



dx"" —1 II — n 



dx l—n ' X 



1 — n 



But ll — n——nl—n. 



ll-n _ 



l—n 

 dx . .. , 



and --7— = —r^ = ^ ^ 





Ex. 2. Find the M^^ differential coefficient of os" where M is an integer. 

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



VOL. XIV. PART II. 5 M 



