191 



the roots are successively diminished by quantities in a de- 

 creasing arithmetical series, they are successively diminished 

 by lesser numbers, and therefore in each transformed equation 

 the roots will be greater than in the preceding, by the com- 

 mon difference of the substituted numbers. But where the 

 hio-hest term of the equation has a co-efficient, there are frac- 

 tional roots, whose denominators, if rational, are some oi the 

 numeral divisors of m. For, the product of the roots with 



the sio-ns changed is — ; and since those fractional roots must 



differ in the successive transformed equations by the common 

 difference of the substituted numbers, if the difference of the 

 substituted numbers be reduced to the same denominators 

 ,\vith the fractions, the numerators of the fractional roots will 

 differ by the numerators of the fractional expressions for the 

 difference, that is by the. common difference, multipHed by 

 the root's denominator, which denominator, in the case of 

 rational roots, is an integral divisor of the highest term. 



Amongst the rational and irrational divisors of those quan- 

 tities, (i'28;h + 64P+32Q + 1611+8S+4T+2V+W), (?h+P+Q 

 . + R+S+T+V + W), (W), (— wi+P— Q+R— S+T— Fi-W), 

 &c. &c. there should be as many arithmetical series as is the 

 number of.dimensions of the equation; for, so many roots are 

 successivtilyi ■diminished i from the nature of the operation. 

 , But only the terms of the: rational series, and the. rational 

 ]Troducts or contents of the irrationals, can be found among 

 .. ., the 



