$86) -'Fractional and Negative Exponents. ‘B47 
fluxions is established ; and however ingenious these demonstrations 
may be, they cannot be admitted as the proper demonstrations by 
which the ‘truth. is first tobe established. ‘To the common demon- 
strations given in elementary books, we object that they are not 
eneral. The law of the coefficients is shown for the first coh 
cients, and this induction is generalized without, any solid demon- 
stration. (Nov. Comm. Petr. vol. xix. p. 103 ; Phil. Trans. 1806, 
p> 318.) “A demonstration for those cases, which ‘is at once satis- 
factory and elementary, seems, therefore, not to be generally 
known, and may be very desirable. We intend to submit to the 
judgment of the reader one which appears to us to have these cha- 
acteristics, and is, as far as we know, new.* We Shall suppose 
that it has been proved that 7, being an entire number, 
1 
(a+ lpea@tn@bt —_ OP 2b? Bevis oc 
“S wat - = +? gh" ty, &c.; and shall prove that, n being 
any, number, the same expansion is true. i ail pvavile scum 
or the sake of abridgment, we shall denote the binomial coeffi- 
—1l.n—2...n—r41 es Seine Dae iaiae ha 
* : See ees I", where the number on 
the right denotes the number of factors both in the pumergtor and 
denominator, and the number on the left expresses the first factor, 
m, or the exponent of the binomial quantity. We have, therefore, 
generally, the following relation between two such coeflicients :-— 
(4) a = tip orl n —7 =r + 1* +11" (8) whatever 
“number 7 may be, and the theoréiti for 7 afi enitiré number is thus 
expressed :— s) 
(a + by" =qt fn qr=- 1 +. J" gr—2 Zz a and 7[* a -7 33 
&e. + L”. il . 
We shall now demonstrate that, whatever p or g may be; we have 
always 
ES RE ad I Ce ede ao) COR a) a Cr 
75]? *]a + &e. .. Ae. satis 5 
For r = 1, it will be easily seen that 'I@+? = "I? + 'Iy, 
. n. 
cients 
‘ \ ~ Cn) 
* We say that tle demonstration is new, because we, believe the demonstration 
in that general form in which we liave given it to be new. Euler’s demonstration 
in Novi Comment, Acad, Petrop, 19, is indeed much tlie same, as far as it goes ; 
but Euler shows only the form of the first two coefiicients, and says, Quemadmodum 
hie duos primos coeflicientes per literas m et n determinare licebat, ita mani- 
festum est, si superior multiplicatid wlferius continuaretur inde etiam sequentes 
coefficientes C, D; E, per easdem literas m et n definiri posse guamvis calculus 
_mox ita foret molestus ut maximum laborem requireret,, [t,is evident that the agree- 
Of the first two coefficients with thavaae coeflicients for entire numbers, which 
is Euler's démonsration, exnuot be dutisfuctory and strict. The same objection 
applies with equal force to Dr. Robertson's demonstration, which seems nearly to 
agree with that of Euler in the paper above referred to, 
Ya 
