•290 



PROFESSOR KELLAND ON GENERAL DIFFERENTIATION. 



(12.) 



■ dx e — 1 . 



Id X 

 — e 



± n± (n + l)£. (n + 2)± 



,_(e<i-'=-l)(e<i-^ + e <<^ + e ''•^ + &c.)i 



= -{^^x + ,i+'^^x + n+l + ^ «^ + „ + 2+&C.) 

 Cob. If« = 0, 2'M,,= -(tt,, + M^ + i + %+2 + &C.), 



which coincides with Cor. 1, formula (2.) 



(13.) 



(n-l)— — (n-1) JL 



d 



e^-1 



I 



-,d X 



(n-l) 





d_ 



e^-1 [e'^-lV „ \ 



which coincides with the Cor. to formula (8). 



Thus formulae (1), (2), and (8), include formulae (11), (12), and (13). 



It is evident that by the same process aU the ordinary formulae in finite dif- 

 ferences, which are usually obtained by the aid of generating fiinctions, may be 

 easily obtained. 



For example the following : 



(n + l)g-P. (» + l)g-2g 



/UN / IN f (W + 1)- — i- „ ( 



(14.) »A. + „ = (w + l)|e<^ + ' 1.2.3 ^ ^^'-1+ 1.2.3.4.5 



»r-l» o 



A* M._<> + &C. 



..{ 



«.r - 1 + ■ 



12 3^" "■t-2 + &c. 



) 



We have 



1 + A 



'--"<'"'" {i-a+-)l {'-if.) 



1 + A 



Now 



(Y-af-az 



iy-ay-%^ (i-«)^-f^ 



*■ ' 1 + A ^ ' 1 + A 



when expanded in terms of a, gives as the coefficient of a", 



(»+i){i+^^f:^% + &c.} 



