576 PROFESSOR KELLAND ON GENERAL DIFFERENTIATION. 



Ijl^n in-)l-{n-il) /» + l 



hence -7=^ 1=^^- \-^) -_^ - 



l—n In — n /« — |U+l 



rf'' ir» /« + l __n— ^ 



aiul 



/^ /w-^u + l 



and the above formula is not correct. 



We can, however, render it applicaljle by adopting the following mode of 



proceeding, viz. by treating the formula (-1)'' • ^ 'T^' \~ as p'emera/ with respect 



to n, but not general with respect to fi ; and consequently, writing sin n ttcos/u tt 

 for sinra— /x.TT, when /x is an integer as well as n. It must be borne in mind that 

 our present transformations are not intended for the purpose of obtaining formulae 

 general in their nature and form, but ai-e merely formulfe of calculation. We 

 wish, in fact, to shew that the fundamental formula itself includes all others, 

 and consequently that, in all cases of general operation, it can be adopted without 

 error. 



Bearing in mind the restriction imposed by the last case, we are able to 

 make use of the following formula in our calculations whether n and «-/x are 

 integers or fractions. 



d'^x" ,,^ /n + 1 shiwTT „_„ 



= (-1)' ._=-^.^^,^_,__ __,,_.,. 



dx'' ' /n-fx + l' f^in (n-fj.) IT 



7. III. Next let n be negative, but less than /j. : then shall we obtain from 

 the fundamental formula 



dx'' ~ /- 'I 7''-" ■ 



Now we have already shewn that 



/-n = 



re(« — 1) ... (n — s + 1) 



Is — n l\ + n — s 



.(-1/ . -^-^- . -..i=^ 



sin (« — «)7r /n + 1 



sin w TT /n + 1 



= (-1)" 



'^''=^_,' 1^^^-l -^'""'^ /w + 1 /fX-t 



dx'' '^ 



8. IV. Lastly, if the form be 



dx'' 



