"138 

 first terms in both the powers are the same. For n being the 



ih 



Index of the term added, it will enter no term of the r power 

 before the n+l, and there it is multiphed (from the nature 



— Ih 



of' Involution) by the product of r multiplied by the r — 1 

 power of the first member of the Root, or by the product of r 

 into unity. 



EXAMPLE. 



, n — I n 



l+ix+cx + Scc.+sx -i-tx = 



n— I I n— I „ 



1+6X + &.C.+SX =l + 'Bx+Cx + kc.+Sx + Tx-\-8cc. 

 J— 1 



n n — 1 n 



+ rxtxxl+bx + 6cc. + sx=. +rtx + &c. 



+ &C. = +&C. 



1 n — 1 n 



=:l-i-'Bx+Cx+8iC.+Sx+{T-^rt)x+&.c. 



th ih 



PROP. In the r Root of the m power of the quantity 



l+AT, that is in 1+;^, there will be found no Fractional 

 powers of the second member of the Binomial quantity, and 

 the Indices of the powers of x in the successive terms will be 

 the series of natural numbers. 



"Where 



