PAG PROFESSOR KELLAND ON GENERAL DIFFERENTIATION. 
——s 
similarly qa =—/u%(1+Bg+ &.); 
and from the expression in Art 2. 
Oee# (1H a4 Bp+ Be) (1 +p log # + &e.) 
=e (L+By+ &e.) (1+glog # &e.) 4 
=(— af 
3. The expression given above for the differential coefficient of a logarithm is, 
therefore, perfectly general, and is applicable to all cases. It is essentially ana- 
lytical in its nature, and does not appear to be reducible to a more arithmetical 
form so as to retain its general character. The expression which I previously 
gave exhibits very simply the mth differential coefficient of a logarithm as well as 
its nth integral, when n is a whole number, and may be, consequently, regarded as 
the most comprehensive arithmetical form of this function which we can at pre- 
sent obtain. 
It may not be considered out of place here to introduce the deduction of the 
d”" log x : se ' : 
value of ae , when is a positive or a negative whole number, from this form 

also. 
The equation is 
d” log x_|n(—1)"+1 1 a a 
ae = x {toe ge (az In * n(w—1) * 


1 it 1 i: 
2 (w—1) (m—2)* 3D ars &) } 
(Part I, Art. 21.) 
(1.) If be a positive whole number, the only terms in this expression which 
are not indefinitely small, are, 

a = 1 1 
; j=1a” +09 (gerne) * wa) (n—n) ares, 
_/n(—1yr** i eee 1 
Fie Cae, Ga ee) 
jn(—1yr*? __ fn(— 1)" *7Jn=n 
a a" (n—n) a" |n—n—1/n—n+1 
_ fa(—1"*2(m—n—1)_|n(—1)"*? _ Cee 12) aaa) 
a” a x 



the well known form. 
