288 PROFESSOR KELLAND ON GENERAL DIFFERENTIATION. 
d d ad 
—g de {1—n (et eDiets (e4 2 aay tsice, } 
d 

=f 7 {aga es Un, + ——— saw U »—ke, | 
=Up ind ees A” tty pn —&e. 
‘ = n (n+ 1) 
or Meant gen Ti a Be tn 7 We: 
a 
n —n ett Sl 
(4). gal) ce : ) Un 
Atte | 
d d d 
ae 
2d 
—n — —1 a —2 
=(—1) {l+nel (e* *—1) +2 Qe) fe (*—1) + de. } we 
= n(n+1 
=(-1) { te 3 tty a+ Mee eh te 2+ ke. | 

n et —n 
(5). A U,=\—; —1 Uy 
= 
apo tod " eee * EIS 7 
={e el (ee 1) +ne A aes 1) +é&e Yue 
1 n(n+1 2 
=A"u, ,tna”* Meee ee Un m9 + Wie. 
These formule are all quite independent of the value of n, and serve to con- 
nect the nth difference of a function of # with differences of functions of x +n, &., 
z+1, &e. 
We shall now obtain the converse series of connections, those of u,,,, with 
Un, &C. 
(6). Unyn=A1+A)"u,=1+nAa+ 
n(n—1) ,» - 
“To & + &.) wu, 
=n, +n Au, es 5) A? ng + &e. 

(7). tig an =(A+1)"a, = rbecritene Te 
+ Ge.) u, 
n(n—1) )y 
TD unt be. 
=i 
=A Up tMA’ thy + 
If n were a positive integer, formula (1) would coincide with formula (2) ; 
and formula (6) with formula (7); but in our present calculus they are by no 
means the same thing. 
