PROFESSOR KELLAND ON GENERAL DIFFERENTIATION. 5^7 



M. LiouviLLE, on the other hand, makes ^ =A + Bx + . . . + Cx"' ; by add- 



dx 



ing- any rational integral function of x, Avhich he calls the complementary func- 

 tion. In applying the analysis to differential formula, this result, if admissible, 

 would totally stultify all our processes. We should prefer writing the result in 

 the form which Euler has given ; for then we could proceed to differentiate a 

 second time, and obtain the differential coefficient of x, without reference to a com- 

 plementary function, which would be desirable, otherwise the complementary 

 function could not depend in any determinate way on n and jjl. We conceive 

 that, whatever may be the value of the differential coefficient, its form ought to 

 be such as to resolve itself into a known function when subjected to known ope- 

 rations. On this account, we should think it advisable to write 



aa;^ ^ ^ /3. sm -| TT 

 Now, differentiating this function to the index \ , the result is 



^^ /| /I ■ sinOTT sin-lTT 



sin 77 



x" =x'' =1. 



smTT 



X 



2 



sinOTT 



The logarithmic form of Ex. 4. cannot be here introduced, on account of the 

 process in the first part not being ajlnal process ; the introduction of a constant 

 at all going on the supposition that the differential shall vanish for some value 

 of a;, which, in the case before us, it cannot do. 



d^ 

 17. Ex. 9. To find -1-* • (a const.) 



dx^ ^ ' 



By form (3) 4^ =(-1) t '^^"^ f^^ • li^~n 



Let »=0 



d^ a , ^ , 3 sin V TT 



= a(-l)s- 



rfar* TT x^ 



a sin , -, . > 1 



VOL. XIV. PART II. 5 N 



