PROFESSOR KELLAND ON GENERAL DIFFERENTIATION. 945 
form under which the pth differential coefficient of a logarithm can be repre- 













sented. 
We shall reduce it in the different cases : 
1. When p is a negative whole number =—=m. 
|u—p=|—(m+p); and [—p=(—p—1)/—p—1 
=(—p—1) (—p—2)...(—p—m) |—(m +p) 
nlL+m+p 
(aj) ee 
=P _ pr ilte 
—? =(-1) 
[-p /l-p+p 
dag’ ge pple 
lag [l—p+¢ 
Hence aenee =(—1 Peel Py Nemes hy peer g har tae! 
d xt eee P ean + qian 
jl+p Deu. 2 
—————— 1—pA+& eS yocaaet 
But anew: = mh pA+&c.) where A= Tat = 
and pit 
[l+q 1 
al SS == 5S SS (ld) Oi 
vi=1+¢ log x+&c. 
d* log x _#@ * G-pA+t&e.) (1+p log x + &e.) 
dx [i—u p 
pee (l—~gA+&c.) (1+glog a2 +&c.) 
jl—p P 
1 i* 
= — (log «— A— j log at + se.) 
= Be 
= = (log x—A), since p and - are both equal to 0. 
a lore rea 1 Ne Y cir n 
Hence is -aail log #— G +5 + &e. +— ) \ which is a well known 
(m) 
expression for da log x 
2. If w be not a negative whole number, / is finite; and 
(PaaP [E=p [7 
f_* = —*_*=—/n(1+Bp+ &e.) 
ERS ees: 
by supposing this function (which is finite) expanded in terms of p ; 
VOL. XVI. PART III. 5@ 
