139 



Where the Index of the power of a Binomial is a positive 



integer, this Law Avill appear from the arrangements of 



common multiplication: That the Law is the same where 



the Index is an affirmative or negative Fraction, will follow 



th 

 from the rule which I have given for extracting the r Root 



i/t 

 of the m power of a Binomial, viz. by shewing that if from 



l+x or from l + mx + Cx + Dx+Ex + &c. + Zx you subduct l,in 

 the remainder, the simple dimension of >r is the lowest, and 

 the second member of the Root is '-^x. That by subducting 



l + y'x from the given power, in the remainder the second 

 dimension of x is the lowest and the next member of the 



Root is of two dimensions. That by subducting (1 + "x) + ex, 

 in the remainder all the dimensions before the third will be 

 destro^'ed, and the new term is of three dimensions. That by 



subducting {l+yX + cx) + dx from the given power, in the 

 remainder all the dimensions below the fourth will be de- 

 stroyed and so on, ad libitum. This will now be shewn by de- 

 monstrating that such Coefficients in the Subtrahends are equal 

 to the corresponding Coefficients in the given power, since r 

 times the member added to the polynomial, is their common 

 excess above equals. 



VOL. XI. V Demonstration. 



V 



