148 

 ha\e a Fractional Index, for since f'=p •'• ' p^r— ' p 



m — w 



that is j5 1 =p'~^ =ox—p . Thus although we cannot con- 



m 

 r" 



sider the Dividend p, whose Index is the Fraction ", to have 



been produced by a number of multiplications as in the case 



where the Index is an Integer, yet the same Law of Division 



through the subduction of the Indices is shewn to apply in 



the one case as well as in the other, by considering the Divi- 



th 

 dend and Divisor under the form of the r Roots of powers 



of p wliose exponents are Integers ; we effect this by the 

 known method of reducing the Radicals to the same Deno- 

 mination which is the necessary step for their Multiplication 

 or Division. 



Cor. II. From the Cases where ^ is affirmative, we shall by 

 algebraical Division be able to prove that the Law of the 

 Terms Avill be the same when the Index of the power is 



m 



negative. In this case we have the negative power/* + a- '' — 

 1 1 



(/)+.r) f-ir';.rp-\-cxp + kQ. 



7" — r — ^ 1 r — 2 



p--'^ap+(^l—<^)xp + &c. 



Cor. 



