1816.] Fractional and Negative Exponenis. 349" 
= p=—stt+g—t) . fori —tF ype , 
Gm ay (PASE) = @s— 241 
"E #+1 2 and s—t—l]p t+ if¢ $ 
—s+tt+lgqg-—t-l t+1]q r—tIp ok 
ee a a 
two parts give together +"? . ¢7 TP, 
And it follows that the whole product will be 
s+ fo 4 s]e ft 4 ste Ye 4 sf? 3]e, Re, s+IYe, 
which is of the same form. If, therefore, the proportion be true 
for 7 = s, it willalso be true fory = s + 1. But as we proved it 
to be true for s = 1, s = 2, s = 8, it follows hence that it will 
be true for all succeeding values of r, or that it will be generally 
true. 
Let us next assume two quantities of this form :— 
a? 4 MTP gt" h + YP gh—2 h? + SIP gh 3 05, &c. and 
a+ Wa-'b + a&—2 l? + It at—-3 b, &e. 
Where p and 7 may be any positive numbers, and multiply them 
together, the product will be as follows :— 
s—t—1Jp t+ 2qy 
Sige L's) 
atta +4 PQ geta—) b Jey ghta—2]2 sf) aete—s Y3 &e, + 
+ eg + a} + 9 Eg 19 
ye, Ie iJ sf 
9 kd 
TW) a 1-7 Uy, 
r—lp ral 
r—2]p i 
r—S]p sJa 
&e. 
a 
It will now be readily seen that the coefficient of a? *+¢~" Ur n = 
*l’ + %, and that the whole product will be 
ata4 eta gto) 4 Bore a +92 O°, &e, + YP+¢ 
@ +=" }°, &c. which, being still of the same form as the factors 
which produced it, gives, if multiplied by another quantity of the 
same form, 
a+ 'la-'h+ w-* Ll’, &c. the product 
a? +947 + 1J? + QO +r atqt+r—1 b ss 2 +QO+7 aetitr2 oe, &e. 
which is still of the same form. This being multiplied again by a _ 
quantity of the same form, would produce a quantity of ‘a similar 
form. Let us now suppose that all the p, g, 7, &c. are equal, let 
that number = 7, and the sum 7p = m an entire number, so that 
= = , and all those quantities which are multiplied will be equal, : 
and we shall have for their product 
fe + amb + Wem? &e.]” = gh 4 re gremt yy + 
