187 
disincumbered from all signs of differentiation or integration 
to be effected upon them. In this latter respect my solution 
seems to possess an advantage over those which have been 
given by Drs. Hargreave and Boole. 
«I will now proceed briefly to describe the manner in 
which I investigate the effect of 7 upon yz", when m and 2 
are negative or fractional. 
«© As before, I commence with particular and simple cases, 
expecting that the results will guide us, by the observation of 
analogies, to a general conclusion. 
«Let us first calculate r y%z'. Putting p in place of 
jD.+kD;, we have 
. Db ay) = —jy*z" ats ky iz, 
pty st=- 9! foe = 91 Pz, 
D3 ytzt=3! jytet4 2) hye? 4 Ql jyrets 3 lhyizt, 
piytzt=4lyizt+2.212lytzt+4l yz, 
pi ytet=-5! jytzt—4lhytz?- 2.312! jyte® 
—~2.2!3! hkytet—4ljy?z-5!l hy'z*, 
De yizt=_6! yz 3.412! y%29-3.214 1 y%2*-6 lye, 
_ Hence, 
2 
2 T mee =yz) —2 Gy?z* me ky 27) ue = (2 ! yz Lig 2 lye) 
: 
1 ai (3! jytet + 2! ky sz? 42! jy2z? + 3! ky te) 
x 
+g Alytet+2.21alytzts 4tyz”) 
ee f ‘ 
= (6 liytet + A lhytet 42.3121 jy te 
+2.2'3!kyiz*+4 lyytze +5 | kytz-*) 
6 
— 7 (6!yte" +3 412 Raye 84.05 2 LAL yates G !ytz?) 
. 
(A) 
