29() PROFESSOR KELLAND ON GENERAL DIFFERENTIATION. 
(12.) Ue og ee ee, 

a al. (Moretti 42) 2. 
= —(e%*—1) (e 7 * +e dO @ 
= —(A Ug i ntA Ugengt1 tA Uganso+ Xe.) 
Cor. If 2=0, 3 a, = —(uypt+ Up 41 +494 &e.), 
which coincides with Cor. 1, formula (2.) 
d \d d 
(n—1) — Ease (m —1)— 1 
(13.) Wa, =e da oda Un =e = 5 


“=e 5 + &e. ) Up 
=Up4n—-1tA Ue 4m — Qt A? Ue -n—3 + Ge. 
which coincides with the Cor. to formula (8). 
Thus formule (1), (2), and (8), include formulze (11), (12), and (13). 
It is evident that by the same process all the ordinary formule in finite dif- 
ferences, which are usually obtained by the aid of generating functions, may be 
easily obtained. 
For example the following : 



1)?-1 1yP—1?. ~+1?—2 
(14.) tear = (0) faye GEESE at, + StU at u,_2+&e. } 
2_ 42 
—n {te +7 534? t-9+ ke. 
a 
We have ane ‘Tra = 
Loo {1-a +a) } {1-;%. \ 
1+A 
a 
1 1l+a ¥ 
an 5 in? 
oN aia Da pace 
Now aaa when expanded in terms of a, gives as the coefficient of a”, 
(n+1)?—1? 
(n+) {1+ e353 
2 + &e. } 
