Resolution of Algebraical Equations. 143 



of the m roots precisely the same result as before obtained in 

 p. 141. 



We see likewise from this, that if there is an equation of n 

 dimensions, and we transpose the highest power, so as to put 

 the equation under the form x = J/P, then applying Lagrange's 

 Theorem, to find x, and multiplying the first group of n terms by 



1-K 47T / . 4? 



H-7T / " . 37T *7T / . -tTT 



1, cos h *./ — 1 sin — , cos 1- d — I sin — , &c. 



n ^ n n n 



and the 2 nd , 3 rd , &c. groups of n terms respectively by the same, 

 we shall get a 2 nd root of the equation, and multiplying them by 



4 IT / . 4>7T 



1, cos hJ-lsin — , &c. 



n n 



we shall get a 3 rd root, and so on for all the roots. 



Suppose a series is to be reverted, or an equation solved, such 

 as a =x — a i x° — a„x s , &c, we can easily express the law of the 

 general term in the value of x, thus by Section I. 



x = coefficient of - in— 1. \l — (— + c^x + a^x", kc.ji; 



therefore the general term of x = coefficient of - in 



° x 



(— + a^ + a^, &c.j . 



1/ 



C^+a^ + a^, &c.V 



Now _ n i- = coefficient of a" in e '. e"' x . e" a ^\ &c. 



1 . 2 . . . n 



Hence the general term of # = coefficient of 1.2...B — 1 - - in the fol- 

 lowing product, viz. 



(l + a S + ^L + &c .\( 1+a0ia . + ^^ j &c.)(l+a«^ + &C.), 



