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. 
‘<T will now proceed briefly to describe the manner in 
which I investigate the effect of 7 upon y™z", 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 7 yz. Putting bin place of 
jD,+kD;, we have 
> ye = —jy?z" ky z, 
Db? = Dp yet — 9 |: en 
D3 ytzt=3! jyt2142! kytz?+ 2! jy?23+ 3! kyle, 
Bigeye +2. 212 by sehad bytes, 
Be ytzt=-5! jy’z1-4! hyez*- 2.312! jytes 
—~2.213!hkytzt- 4) jy%2z5-5 | ky iz, 
BS yizt=-6! yi -3.4!2!y%23-3.2!4! y8z*-6! yz, 
a yiztaylet— a (jy2 + hye) - a! lasz eae) 
3 
By _ (3! jytz3 + 2) ky 22+ 2) jy?2? +3! ky te) 
4 
a a (4 Dopben tee DD I yze+ 4! yz?) 
5 
= 55 (5! jyte + 4 hy te? + 2.3! 2) jy tz 
EO) | 3) ky?zt+4! ljy?ee+ 5! hy iz) 
Fy Glytet+ 3.41 21244 3,214 ye + 6 lytzt) 
fe 
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