PROFESSOR KELLAND ON GENERAL DIFFERENTIATION. 297 



log(i + -) 

 Prop. 6. «.+„=(! + a,) ^ ^% 



For u,+n^e "" •%, = {l + nA+ y^^ + &c.) u, 



dx i- --i 



log (\ 4 -) 



It must be observed that x is considered constant with respect to a< in the 

 formula (l + -) ' ■ Had we supposed it otherwise, we must have taken 



account of the diflerential coefiicients of - itself. This would have given the fol- 

 lowing theorem. 



Prop. 7. A«, = {(l-e-'')-(D + i)_i}„^ 



For .uMe^^-l)u^= (^4 + ^ iiy^^^)-^ 



= ((D + l)e-'' + ^(D + 2)(D + l)e-2''+&c.)M(, (B) 



= {(l + A,)-l°g(l-^-^)(l-e-0-^-l}2^<, 



Prop. 8. ..,„=.-..= (I'^'^i+oG^)' ^^-)"^ 



= (l + »i (0 + 1)6-" +j^(D + 2)(D + l)e-2'' + &c.)M^ 



= (l-«e-'')-(D + l)„^ 



= (l + A,)-"'g(l-"^"Vl-»«-0~'«. 

 It is manifest that these formulae do not foUow the distributive law. They 

 cannot, consequently, be applied with any great advantage to the solution of equa- 

 tions of differences. We shaU exhibit their application only in one instance. 



VOL. XVI. PART III. 4 F 



