PROFESSOR KELLAND ON GENERAL DIFFERENTIATION. 57'J 



where w + yu is a negative quantity, 



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



/_^=(_])' 



/^ 



-w 



m (m — 1) ... (m — i* + 1) 



{t being the integer next greater than m) 



jt—m 

 ^^~^^' '/tt^Tl ■ /m-t + \ 



1 TT 



^ ^ ' /m + 1 sin(?— m)7r 



TT 1 



or 



sin ?« TT /m + 1 



= (_l)^ + i_ 



TT 



a;-^ + ^ 



sin(-w + //.'7r) /w /l-^fi^ 



If w + /z=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 /jl 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 0/ the form of the integral. The com- 

 plete result, then, we shall assume to be 



"^'''^~" =(-1)^ + 1- '^^ . _ ^ - . (a;-(» + ^)_a-(» + ^)). 



dx'^ sm(—n + iJL'7r) /n fl — (n + iJi) 



Now = jr when w + /x=0; 



sin ( — w + yU tt) " 



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

 tain 



X 



a 



log - 1 X 



= — losr . — 



rrr O /-> 



■ (TT) cos (w + yU) TT TT °' a 





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 



