292 PROFESSOR KELLAND ON GENERAL DIFFERENTIATION. 
When 1 is less than 1, therefore, it is necessary to seek some other method 
of obtaining the nth difference. The following method, analogous to that by 
which we obtained the nth differential coefficient of a logarithm in Art. 2 appears 
to be the most simple. 


Let x be represented by “ Sa where q is of a higher order than p, and 
both are 0. 
Then aC r= (e? —1)” Co (e4—1)” el ® 
ie 
(p+ Pat E+ be)" o? (gt be.)e” 
iL, 

5 2 
(p +77 5 + &e.)" (l+p2+ &e.)— (9+ 445+ &e.)" (1+9¢2+d&c.) 


Pp 
wd n prt? n(3n+1) n+2 es n_ ng t!_n(8n+1) wee ; 


n+1 ngrt? 
n+2 
a gL ah ts a a Sr 

Pp 
per? gut? 
5 + &e—= + &e. 
es = a 

+ &e. 
If n>0<1, every part vanishes except the constant, which is infinite: if 
n=1, Ax=1; if nl, every term is zero. 
If n is negative, there will still exist the infinite constant which may be re- 
garded as part of the arbitrary constant; there will also exist in some instances 
infinite functions of «, which, as will easily be seen, may be considered in those 
cases as part of the arbitrary functions. 
Let »=—1; then 
p+ &e. 
A7 a = const. eS aaa 
= const. + —_ 
Let n= —2) ‘and 
2 
pa 
A” “#=const. + const. 7— tae 
and so on. 
Ex. 4, NA aaa Hetn (nin Dae (~w+n—2)"—&e. 
by the first formula. 
