PROFESSOR KELLAND ON GENERAL DIFFERENTIATION. 579 



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



In order to assure ourselves whether our induction is true Oi not, we must 

 inquire whether it will satisfy the fundamental formula. 

 By adopting it, we obtain 



« -^ /(_„ + !) _— _ 



Jl_= -LS 1 i J. — n + ft 



dxi" /(-«-M + l) 



( — 1)~" sin( — MTT; i~n + \ 

 "(-1)— ("+'') sm(-ra-|U)7r/-w-/x + l 



!+|lt 



/ is„ sLnnTT / — w + 1 



= ( — '■J — : 7 ^ • '^=^=^^z^^=^- . n + w 



8111 («+;/) TT /_„_^ + l a; 



Now we have shewn under formula (2) that 



j—n' /n'—fx + l sin (n' — fx) v 



/—{n'-jx) ~ /n' + l sin (n' tt) 



Let n'=n-l, and write — /u for fj, ; 



/—n + 1 M + H sinw + yu— Itt 



/—n — fjL + 'i M sinn — l'Tr 



/n + fx sin (n + fx)7r 

 Jn sin M TT 



therefore, by substitution, 



1 



d^ 



*'L=(_l)f > + M 



This value of /(w + 1), 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 variable 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 priro- 

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

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

 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 n. 6 L " 



