
PROFESSOR KELLAND ON GENERAL DIFFERENTIATION. 989 
1 —n A —n 
A n(nm+1) A? ) 
=(1+n,2-+ io (eae ye 
n(n+1 
=U, +n AU, 1+ = ue Uy, 2+ &e. 

Cor. If 2=1, wu, 41 =Uy +A Ug_1 +A? U,_9+ &e. 
9.) u=(-1)” (53-1) ms 
=(—1)-” (a-” (1+a)y"+na-@t) (14 a)"*! + &e.) u, 

7: —n fn n+1 n (n+1) n+2 
=(—1) {3", +72 Ue to ts ngat hee} 
In strictness we ought to write a-” for 3", but the latter notation is more 
familiar to the eye. 
PELL TIAN OTA Aa eine i 
(10.) Bama (—) Un, =A G-75) ta 
=A, + n A” ty A GODT Ke U, 9+ Me. 
Formula (10) is a particular form of formula (1), for by formula (1), 
mau, +mM U,_1+&e., which is reduced to (10) by multiplying by a”. Inthe 
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. 
NEO 

ae no ett] 
ax a“ 
(11.) Usps = 2 = u, 
et %@_] 
d d d 
(n—1)— (n—2)-— + (n—3)—— 
=(e*"—1) (e OS ae Cielo 7? 1 &e.) the 
d 
=" 1) (Me 4n—1 + Up 4n—-2Un+n—3 + &e.) 
=A Ua nH 1tA Upen—at A Uptn—3+ Ke. 
Cor. 1. Ifn=0; u,=Au,_1+44,_9t AU, 3+ &e., 
which coincides with Cor. 1., formula (1.) 
Cor. 2. If n be a positive integer 
Un n= Ug an LEA Upin—-gt Ge. +AU, 7 +AU, +A U,_1+ he. 
=A Upp nt Ugyn—gthe. +AU, 1+ 4%, +%~; by Cor. 1. 
VOL. XVI. PART III. 4D 

