59 8 SCIENCE PROGRESS 



(10) x* + y?-2x- 5=0 (Part II, p. 395). 



By p. 582 the two leading-term critical points \/z and ^5 are both > X, and 

 f'X is therefore </'( ^2) = I2"485. Hence the former critical point is not suitable ; 

 and the latter one gives a very slowly approximating series {X= 1 '330058 . . .). It 

 it better therefore to centre the equation and proceed as in Example P (11). 



P. Two-change Equations, A - B + C = o. 



Here there are two possible positive roots, but they are often unreal. In 

 Note II such equations are called Plenary Quadratics, and it is shown that if the 

 positive root (plenary critical point) of A — B = o is not considerably greater than 

 the root of B-C=o, the positive roots of fx cannot be real. For example, in 

 yc 6 + ior 5 -f- 19** - 30.iT 3 — 105^' + 645.2: + 119 = 0, the root of 3-tr 6 + io.r 5 + 19* 4 — 

 3ar 3 - 105.tr* is less than the root of 19^ — 30X 3 - 105.iT 2 = o, which is 3*23 . . . ; and 

 the rootof 30.r 3 + 105.*" 2 -645.tr- 119 = is greater than the root of 3o.v 3 + 105^-645^ 

 which is 3*25 ; and as 3'25>3'23, the whole equation can have no positive roots. 

 In fact the roots of fx lie between the roots of the inner partial function 

 19-r 4 . . . 645.tr, and must be unreal if these are unreal. A descending iteration 

 based upon the superior plenary critical point must reach the superior root ; and 

 an ascending one based upon the inferior critical point, the inferior root. The 

 latter will also be the reciprocal of the superior root of the x = z~ x equation, given 

 by descending iteration. If the roots are not real, both iterations will pass through 

 the gap between the curve and the axis of x — rapidly if Newton's iterand be 

 employed ; so that the reality of the roots is easily ascertained. For the series it 

 is generally sufficient to select the minor critical point which is nearest to the 

 appropriate plenary critical point. Part II, Example C, p. 395, is a case, and the 

 reader may show that the critical points there selected yield convergent series, 

 though both series are slow. 



(11) x z - 5.r+ 1 =0 (Part II, p. 395). 



Ex. O (10) centred. The point ± «/5 gives the greater positive root 2*1284 . . . 

 and the negative root, —2*3300 . . . ; and the reciprocal equation gives the middle 

 root x = o'2oi6 . . . , all the series being valid. 



Q. Three-change Equations, A — B+ C— D = o. 



Dealt with in the same way. If all three possible positive roots are real, the 

 root of A -B = o refers to the greatest, and that of A —D either to the greatest or 

 the least. Similarly with the reciprocal equation ; and the minor critical points 

 refer to the same — there being no leading-term positive critical points which refer 

 to the middle root, and therefore, apparently, no series for it (without trans- 

 formations). 



(12) x 3 - \bx* + 65* -50 = (p. 582). 



The series q=i, g = 16 converges slowly to the greatest root, 10. The series 

 ? = 3,g= &S° converges to the least root, 1, though it may not at first appear to 

 do so, and the same root is given by simple ascending division. Put x= Soz -1 , 

 divide by 50, and compare. For the middle root, 5, centre the equation after 

 putting x=yJ3 and multiplying by 27. 



R. Four-change Equations, A-B+C-D + E = o. 



AB refers to the greatest root, AD generally to the third, ED to the least 

 and EB to the second. By multiplying a plenary cubic by a factor x-a we can 

 raise it to a plenary biquadric and thus obtain a series for its middle root. 



