member of the Root, divide the remainder aS before, theh 



conceive the first, second, third, and fourth members found, 



to be a Quadrinomial, &c. 



This Rule follows from the nature of Involution, for, if you 

 ' th 



subduct the r power of the first member of the Root from ther 



th — tk 



r pmvei" of a Binomialj the remainder will be r times the r — ! 

 power of the first member of the Root multiplied simply by 

 the second member, + plus certain multiples of powers of the 

 first member, having a lower number than r — 1 for their 

 Index, and having certain powers of the second member for 

 their Cofactors. 



Although this method of extracting Roots supposes that from 

 the power of 1 + ^ whose Index is the Integer ;«, there are suc- 

 cessively taken the powers of Polynomials having the integer 

 r for the Index of the powers, yet to demonstrate the law of thte 

 Indices of the terms in the Root, it will not be necessary to 

 suppose the actual Coefficients of integral powers of polyno- 

 nrials to be determined, tyat merfely the relation of indetermi- 

 nate Coefficients to be known, wliich Avill appear from the 

 following Law. 



If to a multinomial of h terms,ortol + hx + cx+dx + Scc. + sx, 

 there' be ^dded a term containing the next higher dimension 



n- —ik tk 



of X as tx, then the n+l term of the ».-,pp\yer of the new mul- 



tiiiomial exceeds the »H- 1 term of tli^f p6wer of the multi- 

 noinial of ii terms, by r times the member added, and the n 



first 



