20^ 



55, 5, — 1, 1, — 1, 5, we have a trinomial divisor a:— 3i'+ 1 =0, 

 ■whose roots can be discovered by resolving a quadratic equa- 



3 2 



tion. Also the divisor lar— 15.r+42.r — 35=0, which is de- 

 duced from the numbers 442, 133, 10, — 35, &c. is the only 

 quadrinomial derived by the particular rules. But, by the 

 general rules, from the numbers 1870, ^5, — 10, 7, —10, ^^, 



6 4 z 



wc find the quadrinomial ^x — 2x — l8jf + 7=0, which is re- 

 ducible to a cubic. 



A divisor of five dimensions is found from the numbers 

 715, ^o, — 1, — 5, &c. and although, in this example, from 

 the two first divisors we may investigate the others by help of 

 division, yet the rule discovers them immediately, and iu an 

 equation having no divisors of such low dimensions, the usual 

 rules would be inadequate to the discovery of the others. 



Sometimes no divisor of a given number of dimensions can 

 be found, which shall succeed in the division, viz. when the 

 content of so many roots is not found among the rational di- 

 visors of the absolute term. However, the content of yet a 

 greater number of roots may be ratioaal, and among all the 

 divisors, rational and irrational, there will be always as many- 

 arithmetical series as there are roots, for all the roots are ne- 

 cessarily diminished, as we have shewn, from the nature of 

 transformation. 



When some of the divisors of the absolute terms of the 

 given equation are odd, and others even, we can, frima facie, 



reduce 



