OPERATIVE DIVISION 595 



IV. Analysis by Partial Roots. — The following is a useful method of localising 

 roots by means of the partial roots of a single setting. Let fr — fx = £r where fx 

 is the given equation— of an even degree. Let (x be an arbitrary function of the 

 same degree and with its first two or more terms the same as those of fx, and let 

 its roots be all double roots. Then (x is always positive ; and £r is two degrees 

 less than fx and its roots bear definite relations to those of fx. For example, let 

 (x* - 4X + 5) {x 1 - 4X + 3) = x* - 8x* + 24_i' s - 32^ +15=0 bea test equation. Take 

 for (x the arbitrary standard equation, x* - 8-r 3 4- 22.r' — 24^ + 9 = o, of which the 

 roots are 1, 1, 3, 3. Then £x = -2x i + 8.r — 4 ; and while £r is always positive, t~x 

 is always negative except between its two roots 2± J2 — so that the roots of fx lie 

 between those of 1-x ; which is evidently the case. Or let fx = (x 2 — 4X + 5) 

 (x s -4x + 7) = x*-8x 3 + 2&r I -48.r+ 35 = 0. Take the same standard equation as 

 before. Then £r= — 6x* + 24^ — 26 ; and as this is always negative, fx has no 

 roots. If fx is of an odd degree it may be treated in a similar manner; or may 

 be raised one degree by multiplication by an arbitrary factor x ± a. In all 

 cases the actual roots of fx can occur only when fr and £x have the same sign — 

 Section VII (1). 



V. Critical Coefficients. — Many propositions in the Theory of Equations may 

 be given more elegantly by using these. For example, if x i —flx' + gx — r = o, we 

 write x* - bx* + bcx — bed = o, where b = p, c = qlp, d=r\q. The same thing can be 

 done if the original coefficients are affected by the binomial (Newtonian) co- 

 efficients. 



VI. Remarks on the Tables. — On inspection of Tables I and II it will be seen 

 that the series for yf?- 1 and (^n)" 1 consist of the same combinations of the original 

 coefficients b, c, d, . . . grouped in the same way, but that the numerical sub- 

 coefficients are different; in fact the series V^'y is the same in form as the 

 algebraic quotient of y n + l divided by yfr n y (y is used for^- in the Tables as the latter 

 symbol is required for one of the coefficients). Now if in place of b, c, d, . . . we 

 use weighted coefficients p\,pi,p* . . . , we shall easily see that the general minor 

 terms of ^ l y is (if a, b, c . . . are any integers) 



w-\\ n ) e - a\b\c\. . . - p i p < p »- "'I ' 



where e = a + b + c . . . , and iv = aq + br + cs . . . and is the combined weight of 

 the whole group pqp\p c s - . . . Hence we can always find the numerical sub- 

 coefficient to be attached to every one of these groups, and also the power of y to 

 which the group belongs. Conversely if the power of y be given, we can find the 

 groups of original coefficients which will belong to it in the series and their 

 numerical sub-coefficients — by the same process as the one employed for finding co- 

 efficients in the multinomial theorem. Indeed, the sub-coefficient {—\) e .e\ja\b\c\ 

 is simply the sub-coefficient of the same group in the ordinary expansion of {ty n )~ l \ 

 while the sub-coefficient — {{w— \)\n)J{w- 1) is the special one given by opera- 

 tive division. On further inspection of the series we see that the middle 

 coefficients of y (in small brackets) consist of groups which are all of the same 

 order, e ; and that the major coefficients of y (in large brackets) consist of groups 

 which are all of the same weight w. The sum of the numerical sub-coefficients of 

 all the terms within small brackets is the binomial coefficient (w— i) e -i; and when 

 the products of both the numerical sub-coefficients are fractions, the denominators 

 of these fractions consist only of powers of n. 



In Section VIII (6), p. 590, it was shown that the series are ultimately con- 

 vergent when the corresponding iteration is so — which usually can be easily 



