PROFESSOR KELLAND ON GENERAL DIFFERENTIATION. 



where n + ^ is a negative quantity, 



dx'^ In 



and since w + m is negative, let it equal —m ; 



577 



m{vi — V) . . . (m.— t+T) 



(t being the integer next greater than m) 



/t—m 



"'^~^'^' /mTl ■ /m-t+1 



1 v 



= (~V)' - 



^ ^ ' /m + 1 sm(l-m)'rr 



■n- 1 



sin m IT /m + 1 



» + /« 



sin(-M + ^.7r) /n /l-'^^ 



If « + /x=0, the result is infinite; but it is constant ; consequently we may 

 suppose some arbitrary constant to have been omitted in the differentiation. In 

 other words, when // is negative, the fundamental formula does not give the com- 

 plete result. It must therefore be rectified, as in the case of ordinary integration, 

 by the introduction of an arbitrary constant of the form of the integral. The com- 

 plete result, then, we shaU assume to be 





dx'' sin(-w + /x'7r) /n /l-(n + fx) 



Now . c •• 



—T~ — : = -7r when ri + u = 0; 



sm(— w + yii7r) " 



hence we must find its value by the usual method of differentiations, and we ob- 

 tain 



-'°S-J 1. . 



— ("TT) cos {n + jJ^TT TT ° 



d-^xr-' 1 1 X 



~j^^^ = {-V) . —log.-. 



We shall recur to this process in the sequel. 



9. We have thus deduced from a single formula results which are applicable 

 to any case, and we may consequently adopt this formula as our standard, and 



