OPERATIVE DIVISION 231 



the far-from-elementary method attributed to Lagrange and 

 developed by Murphy and also by way of Lagrange's and Bur- 

 mann's differential expansions. But I can find in the books 

 no information as to how to use it for calculating all the real 

 roots of an equation, nor as to when it is convergent or not ; 

 nor indeed any note of its importance. It is, however, only one 

 of the family of series given by operative division — which I 

 believe will ultimately prove to be a process of elementary 

 importance in algebra. 



(4) These results may be written somewhat more formally 

 as follows : 



Let o = f(x) be the proposed equation, and let x\, x 2) x 3 . . . 

 denote its real roots. Let 



xi = pi + gi x 2 = p 2 +z 2 x 3 = p 3 + z 3 etc.; 



and let a, b, c, d, . . . be now written respectively for f{p\), 



-f'(Pi), T\ f " {pl) > ~ y\J'"^> etc ' Then 



o = f(p x -f z x ) = a — bzi + C0i 2 — dz* + . . . 

 Putting z\ = by and dividing by b 2 , 



o = ~ t — y -f cy s — dby 3 + eb*y k — etc. 

 Inverting this and writing s for -=-, we have 



Xl = p 1 +"{ I + cs+ (2c 2 -db)s*+{$c* - $cdb + eb*)s*+ etc.} 



*2 = P2 + similar terms in f(p 2 ), f'(p2)} etc. 

 *3 = p3+ similar terms in f(p3), /'(ps), etc. 



But the quantities p u p 2 , p 3 , . . . must be so near the re- 

 spective roots that s x , s 2) s 3 , . . . shall be small ; and the ultimate 

 condition of convergence is that the slope of the curve 



o = £- 2 — y+cy* — etc., at its root is certainly greater than 



— 2 ; as will be shown in the next part of this paper. 



All these propositions may have been given entirely in 

 operative notation. They are merely instances of the general 

 theorem that 



(5) Examples. 



Numerical equations may be solved directly from the various 



