228 SCIENCE PROGRESS 



where the indices in brackets denote factorials — e.g. 

 q {m) = q (q — i) (q — 2) • ■ • {q—m+i). This operation applied to 

 the absolute term gives a numerical series which is divergent if 

 the two possible positive roots are unreal, but which always 

 converges to the value of the lesser positive root if it is real. 

 The convergence is slow if the two roots are equal or nearly 

 equal ; rapid if they are far apart. 



If c is negative in the original equation write o = 1 — x — cx n , 

 and the series obtained for [o-\-co n ]~ l will have alternately 

 positive and negative signs. In this case there must be one, 

 and only one, positive root, and this is given by the series if c 

 is numerically small. But if c is numerically too large the series 

 becomes divergent though the root is real — as for example in 

 the equation o = 1 — x — x*. The series will certainly not 

 be convergent if the slope of the curve 1 — — co n as it crosses 

 the axis of x is algebraically less than — 2 ; and the convergence 

 is quickest when the slope of the curve at that point is nearly — 1 ; 

 facts which the ingenious amateur will doubtless be able to 

 explain even before he reads the rest of this paper. But the 

 slope of the curve can always be modified by suitable transfor- 

 mations as will now be described. 



(3) We proceed to inquire how all the roots of an integral 

 rational equation may be obtained in succession by simple 

 ascending operative division. 



In Horner's process we must first find by trial the integers 

 between which a required root lies. The lesser of these integers 

 is called the first figure of the root ; and the origin is transferred 

 to this point. The next figure of the root is obtained, again by 

 trial, from the transformed equation, and the origin is again 

 transferred to this second point ; and so on, until the required 

 degree of approximation is attained. In solution by simple 

 ascending division we must often ascertain the first figure 

 of the root by trial (or other means l ) ; but a single division 

 usually suffices for the rest of the approximation. 



Write the general equation in the form 



= a' - b'y+ c'y 1 - d'y 1 + . . . 



Now for simple division it is absolutely necessary that the 

 terms a' and b'y and at least one of the other terms should 

 exist ; to ensure convergence of the quotient it is generally 



1 Which will be indicated in a later Part. 



