594 SCIENCE PROGRESS 



because the powers of x in these sets are of increasing order of magnitude. Hence 

 the equations A - B = o, B -C = o, A-C — o must each have a root and, by the 

 previous proposition, only one root — which we may denote by AB, BC, AC. . . . 

 The curve A begins at the absolute term and, if it contains powers of x, constantly 

 increases, reaches positive infinity, and has no root. The curve A — B also begins 

 at the absolute term ; is always less than A, which it follows closely at first ; but 

 then bends downward, crosses the axis of x at AB, and finally becomes negative 

 infinity. The curve A -B + C begins at the absolute term ; never again touches 

 A — B ; at first lies between A and A — B ; but crosses A when A = A - B + C, 

 that is, at the point BC ; and finally becomes positive infinity. Obviously if C is 

 small enough, A - B + C may cross the axis of x at two points ; the first of which 

 lies near AB and the second of which lies near BC, both the points of crossing 

 (which are the roots of A — B + C = o) lying between AB and BC : but if C is too 

 large both these roots will disappear. Similarly the curve A — B + C-D begins 

 at the absolute term ; never again touches A — B + C ; at first lies between A - B 

 and A-B + C; but crosses A-B when A - B = A - B + C-D, that is, at the 

 point CD ; and finally becomes negative infinity. It must therefore have one 

 root and may have three. Similarly the curve A —B + C— D + E lies between 

 ^4-^+Cand A -B + C-D until it crosses the former at the point DE. And 

 so on. 



This construction makes it obvious that if all the possible roots exist we must 

 have roughly not only that AB < BC < CD, . . . but that there must be sufficient 

 intervals between these points to give room for the roots, so to speak. Moreover, 

 all the roots must lie between the least and the greatest of these points ; and each 

 of the former will lie somewhere near each of the latter in order. Thus A-B+C 

 cannot possibly have a real pair of roots unless AB<BC by a considerable 

 interval ; and A - B + C— D can have only one root unless AB < CD. It may be 

 thought that if CD<AB, the curve A-B + C-D may still be able to cross the 

 jr-axis thrice between them ; but then this curve might possibly cut the curve 

 A—B thrice also — that is, C—D would have three roots; which is impossible. 

 Further study of this subject must be left to the reader. He should also con- 

 sider the position of AD, AF, AH . . . and other secondary critical points, which 

 may often be the bases of iterations, and examine Example P. 



I propose to call one-change, two-change, three-change . . . equations plenary 

 linear, quadratic, cubic . . . equations ; and the points AB, BC, AD, . . . plenary 

 critical points. 



These propositions are best studied under the Arithmetical Variable — consisting 

 only of signless numbers. Much confusion is frequently caused by the pro- 

 miscuous use of the One-Dimensional and the Two-Dimensional Variables. Thus 

 Cauchy's theorem that an equation of the nth. degree has »-roots has been hailed 

 as " the fundamental theorem of algebra," but with both the Arithmetical and the 

 One-Dimensional Variables it is simply untrue, and Descartes' Rule should take 

 its place. If we go outside these variables, why not use the Three-Dimensional 

 Variable at once and put everything upon a Quaternion basis ? 



III. Sterile Tracts. — The function Q — R + S can have no bend-points unless 

 its tangential Q' — R' + S' has roots ; and operation upon them with any number of 

 p + xo factors will not give them bend-points or roots. Hence if fx contains such 

 a group of terms it will have at least two roots less than the possible complement. 

 This is the simple general theorem which, when Q, R, and S are consecutive 

 single terms, becomes Newton's Rule (p. 583). This rule was proved but not 

 explained by Sylvester in 1864— after many mathematicians had failed. 



