YAP PROFESSOR KELLAND ON GENERAL DIFFERENTIATION. 
dt x eg : 
— =(—1)" a a”, whatever ben and yw. This form can be proved to be 
Lx i 
the correct one in every interpretable case, and can be deduced from the gene- 


; : d¥ e&” 
ralization of : 
3 when n is negative.* We shall at present assume it as the 
defining property or definition of — 
When, from this definition, we can deduce the differential coefficients of ¢* 
and of log a, that is, of the ascending and descending index-function, we are in 
possession of the three fundamental forms from which all others may be de- 
rived. The following mode of arriving at those differential coefficients is differ- 
ent from that which has hitherto been given, and appears to leave nothing to be 
desired. 




dé e* 
1. To find 2 
f d x 
aa Cm eo Weer 
e sltexty gt pa 3t &e. goes 
ey fee cf gt 
d xt [=o oi free 

=(Sicyr iM { ( a) 4 CP ad eee” +&e.} 

[0 1-p = d-»)@-}» 
UL = il! —2 
=f & {= " + aueaes 1 pee 2" + be, } 3 where z=cz; 
* See Part I., and the excellent Memoir of M. Liovuvit.e, referred to in that Treatise. 
Another formula has been proposed, viz. 
dt eet ie 
da /T+n—w 
I have lately received from Mr W. Center, of Langside, some judicious remarks on these formule, 
contrasting the results arrived at by them respectively. He shews that (without continual introduction 
of an infinite arbitrary constant) the latter formula is inapplicable in many of the most simple cases : 

for example, in d“ of expanded positively, it gives, when applied, infinity on one side and not on 

1 
l+a 
L Sih} ID Ce d” a 
the other, and when expanded negatively, infinity on both sides; and again, it gives for ‘asf or 
d 
0 
qa” 
edie the value 
d x” me 


a ™, which is a function of « when wu is a positive proper fraction. 
