PROFESSOR KELLAND ON GENERAL DIFFERENTIATION. QAT 
(2.) If n be a negative integer = —m ; 
gmtl ml ymt2 ii ~m+s 






Let y = a2 1) 2G@=)@-2 * 3@—2) GH ss 
gmt i} gm+2 1 gint3 x 
= m(m+ 1) * 2(m+1)(m+2) * 3(m+2)(m+8) 1” 
1 
Coals 2 + Be, 
whence, by eee 
2 ill Me a Sra one er 
= ala RIED oO Ga ae pe 
ik 1 ym gm 1 
3: Wee Ht ia DIG re ioe =| 
1 
Stee =, log —2)— te zs (L-2). 

Consequently, the value of y between the limits 0 and 1 is 
1 Le 1 
~ m(m — Ea dee nears ary) 


LNG ten 1 1 
fiat. aan eee aaT ~ m—1 
: (F+5+ke. + al ee 

~ m(m— m(m—1) \I m m—1 
1 naa 1 
Fee awa 15 = 3 t &e. + =; 
—m a3 iF) m+1 
and 3 WEP =! Sr Cen { log @—1—m(m—Ly } 
Cpe | Sj 

Mee ae Nii, (C2 
ama). (2) 7 NI e+) 
pole? rf 1} plies eabtyh 
~ m(m—1)...2 og a (7+5 Bea a 
es : (m) 
which is the expression for d x” log x. 
4. In my previous memoirs, I have obtained the general differential coefti- 
cients of several functions, and have applied the results to the solution of analy- 
tical and mechanical problems. It will be my object at present, to extend the 
science itself by exhibiting the solution of differential equations, and by investi- 
gating some of the properties of finite differences. In every instance I shall select 
the most simple problems which will serve to illustrate the process employed. 
Of the process itself, consisting entirely of the application of the calculus of opera- 
