PROFESSOR KELLAND ON GENERAL DIFFERENTIATION. 53^ 



rf* X 



M. LiouviLLE, on the other hand, makes 7- =:A + Ba; + . . . + Cj-'" ; by add- 



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

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

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

 the form which Euleb 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 //. We conceive 

 that, whatever may be the vjalue of the differential coefladeut, its fm^m 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 wi-ite 



rfa;* ^ ^ /I" sin -^ T 

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



'^'^ l\ /I sinOTT sinlTT 



sinT 



x" ^x' =1. 



sinOTT 



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

 process in the first part not being 2, final process ; the introduction of a constant 

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

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



17. Ex. 9. To find 



By form (3) 

 Let 



VOL. XIV. PART II. 5 N 



