OPERATIVE DIVISION 229 



necessary or advisable that a' and b' should have different 

 signs and that the absolute term, a', should be as small as 

 possible ; and it is convenient for the process of division that 

 V should be unity, and that as many of the other coefficients as 

 possible should be integers. 



In order that a f should be small and that a' and b' should 

 have different signs, we must, if necessary, transfer the origin 

 to a point less than and close to the required root (the nearest 

 integer generally suffices). The nearer this point is to the root 

 the smaller will be the absolute term of the transformed equation 

 and the more certainly will its next term {b'y) have a different 

 sign. 



This being done, and supposing o = a' — b'y + c'y* — etc., to 

 be the transformed equation (c', d', e' , etc., having either sign), we 

 have next to reduce the coefficient of the second term to unity. 

 If we merely divide throughout by b' ', the new coefficients may 

 all be fractions — which is not desirable. It is, therefore, 

 usually better (A) to make the substitution y — b'x and divide 



by b'\ which will give o = tt^ — x + c'x z — a 1 ' b'x 1 + etc. ; in 



which, if the original coefficients were integers, the new ones, 

 except the new absolute term, will be integers also. Or (B) 

 we may make the substitution already suggested in (2) above, 



namely y = —x, and divide by a' ; in which case the first two 



new coefficients will be unity, but the other coefficients will 

 generally be fractions. These small changes do not affect the 

 result but only facilitate the working ; they are not absolutely 

 necessary, and, for numerical work, the selection of the most 

 convenient method will depend upon the original coefficients. 

 By method (A), the absolute term should be considerably less 

 than unity, to speak roughly ; and by method (B) all the other 

 coefficients should be less than unity — the coefficient of x 

 being of course unity in both. 



The final condition for convergence is, as stated at the end 

 of the previous sub-section, that the slope of the curve at the 

 required root must at least be algebraically greater than — 2 ; 

 and, for rapid convergence, is that it be about — 1. Where 

 the required root is multiple the slope of the curve will tend 

 towards zero the nearer we approach the root, and the con- 



