288 



d 

 n 



or 



w- 



PROFESSOR KELLAND ON GENERAL DIFFERENTIATION. 



-1 , n (n + 1) 



.,j^ . +'1^1)2^,, _&c. 





= tt.v + n-»^ 'in + n + '^^j~2~' ^ "^^.t + n-^C- 



A « ,. : 



= M , — n ^ «. , r , -„- 



2d d 



(5). 



-.if »('» + l) , ^ 



= (-1) |M., + n2M.„ + i+ 12 22% + 2 + &c. j 



d 



edx_i 



-{n + \)— — n + 1 



= fe '«-»(erf^_l)" + »e d-'^ (e'^-'^.l) +&c. j «,,, 



These formuke are all quite independent of the value of n, and serve to con- 

 nect the wth difference of a function of x with differences of functions oix + n, &c., 

 x + \, &c. 



We shall now obtain the converse series of connections, those of «,„ + „ with 



Ux, &c. 



(6). 



(7). 



= n,. +n Au,. + n ^, — ^ a'^ tc. + &c. 

 .1 X 1 _ 2 



M^ + „ =(A-|-1) ?<,. = (A -l-nA +n ^ ^' A +&C.)tl., 



n 11-1 n(n~l) n-2 



= A M^ + »A "■' "*" 1 2 "■"' 



&c. 



If n were a positive integer, formula (1) would coincide with formula (2) ; 

 and formula (6) Avith formula (7) ; but in our present calculus they are by no 



means the same thing. 



