D44 PROFESSOR KELLAND ON GENERAL DIFFERENTIATION. 
Now C is altogether independent of n: if, therefore, we take n positive and 
greater than (—/), which can always be done, we shall have proved generally, 
that 
dt e& 
ae Oyen (a)s 

It will be observed that the properties on which the truth of equation (2) is 
based, are these,— : 



Mn n+ 
j, MAE pla, | 
dl xt gee whatever be n. 
2. -/n+1=nj/n j 
3. : = = “e~*"a"~1q a, when n is positive. 
d* log x 
2. To find “Ge 
1 : . . Pare a” log x 
n my previous Memoir, Art. 19, I obtained an expression for ae by as- 
suming that of fF log 2 ; an assumption which owes its correctness to the admit- 
ted possibility of the introduction of an arbitrary constant of integration. Con- 
sequently, the conclusions at which I arrived can only be correct generally, by the 
aid of an arbitrary function of differentiation. Now, it is our object to avoid the 
use of such functions, and to obtain expressions for the general differential coeffi- 
cient of all functions which shall be complete in themselves, so far as relates to 
the satisfaction of every law of combination to which they may be subjected. It 
; : ‘d : 
becomes necessary, therefore, to-reject the equation ie = = log x, and to substitute 
in its place some other function of 7. The following process appears to be per- 
fectly satisfactory. 
The value of 

eat 1+p log x+ &.—1—g log r—&e. 
Pp P 
=log «—T logz+Ap+t &e. 
If, therefore, g be of a higher order than p, such as p’, it is manifest that 
a? — a1 
pP 

will be a simple representation of log x, provided p=0 and i —O, 
By adopting this mode of representation we obtain, 
r= it | = 1 
(-1y"42 pee end 
P| —p (ie 
d“log a __ 
hee at g wee 
ad x” 
This expression comprehends every case, and appears to be the most simple 
