PROFESSOR KELLAND ON GENERAL DIFFERENTIATION. 579 



This function will consequently satisfy the requisite condition, as well as 



/n + 1. 



In order to assure ourselves whether our induction is true Oi wot, we must 

 inquire whether it will satisfy the fundamental formula. 

 By adopting it, we obtain 



dx^ /(-n-/x + l) 



^— «+^ 



(-!)-« sin (-»» TT; /-w + 1 _^— ;^ 



. ^_— . ^ nfi ~ 



"(—!)—('*+'') sm(-?»-/z)7r/-»-yu+l 



X 



~,i_Xf smwTT 



sin(/i + /x)7r /_^_^ + l 

 Now we have shewn under formula (2) that 





X 



j—n' /n' — fi + l sin (n' — fx) TT 



/-(n'-fx) jn' + l ■ sm(n''7r) 



Let n'=n-l, and write —fi for fx ; 



/— w + 1 M + fx smn + fx—l'rr 



/—n—fx + l /^ sinw — Itt 



/n + fx sin {n + }x)'rr 

 jn sin » TT 



therefore, by substitution, 



This value of /(?^ + l), therefore, completely verifies the general formula. 

 We must not at the same time conclude that it is complete, although we may safe- 

 ly trust it as the variahle factor of the complete form. It wiU be seen in the 

 sequel that the other factor is infinite ; but, as each function has the same factor, 

 this produces no effect on the result. 



We see in this circumstance a remarkable instance of the failure of the prin- 

 ciple of the permanence of equivalent forms, as it is called. According to that 

 principle, f{n + l) should have been equal to /n + \, and not to ( - 1)" sin w tt /t^ + 1. 

 There can be no doubt that such a principle has no real existence, sanctioned as 

 it is by the names of the greatest analysts. But we forbear discussion of this 

 matter. 



11 . Our next proposition is the following, analogous to that in the theory of 

 whole differentials. 



VOL. XIV. PART II. 5 L " 



