PROFESSOR KELLAND ON GENERAL DIFFERENTIATION. 289 



(8.) u = (^\ -V= l\- -^\ -"« 



ii(n + l) „ . 



Cob. If «=!, m.c+i = «,, + am^_i + a^ «i;-2+ <^c. 

 (9.) u =(-!)-" f-^_l^-»^, 



^ ' x + n ^ ' \1 + A / X 



= (-l)-"(A-" (l + A)" + »A-('' + l)(l + A)" + l+&C.)«^. 



^ ' |_ x + n x+n+l 1,2 ^ + 11 + 2 J 



In strictness we ought to write a-" for 2", but the latter notation is more 

 familiar to the eye. 



(10.) "-«=-"C-^)"«^=-''(i-i^)'"^^ 



= A«{l + U (l + A)-l + ^-^(l + A)-%&C.]«. 



= A«M^ +nA''u^_i+ 1^+^ A" «,,._ 2 + &e. 



Formula (10) is a particular form of formula (1), for by formula (1), 

 A ~ "'".r + m = M^ + »* % _ 1 + &c., which is reduced to ( 1 0) by multiplying by a™. In the 

 same manner we may reproduce formula (2.) 



The last class of relations which we shall produce are such as do not depend 

 on the general expansion of the binomial. 



n—- ji__ Jix_-i 



dx ^_ d X ^ -•• 



(11.) t(^ + n = e 'iix=e 



d 

 e^-1 



d X , , rf d ^ d 



-J- («-!)-;- (.1-2)-— (n_3)-- 



/ d X i\ / ' d X , ' d X ^ ^ ' d X t > 



= (e — 1; (e +e +e + &c.) u^ 



d 



jl X 



= AM^ + „_l + AM^ + „_2-t AM^ + „_3 + &C. 

 COE. 1. IfM = 0; M„ = AM^_i+Am^_2 + Am^_3 + &C., 



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

 Cor. 2. If n be a positive integer 



Ux + n = ^^x+n-\+^'U'x + n-2 + ^'i- + ^ ».r + l + ^ % + '^ «.r- 1 + &C- 



=a«^+„_i+am^^.„_2 + &c. +^u^+i + ^u^ + u^; byCor. 1. 



VOL. XVI. PART III. 4 D 



