Febeuaey 24, 1922] 



SCIENCE 



191 



f{x) or else separates them, j. e., has roots on 

 each side of it. This form is immediately sug- 

 gested by the corresponding mechanical sys- 

 tem; it is evident that a point of equilibrium 

 must either be collinear with all the repelling 

 particles, or else the latter must be seen under 

 an angle of more than 180° from the former. 



This result is only one of many concerning 

 the relative distribution of roots of f{x) and 

 f (x) that may be inferred from the conditions 

 of equilibrium of our mechanical system; we 

 have deduced it by taking account only of the 

 directions of the repelling forces. By con- 

 sidering their magnitudes as well J. Nagy 

 ( Jahresbericht der Deutschen Mathematiker 

 Vereinigung, Vol. 27 (1918), page 44) has 

 obtained a number of interesting theorems of 

 which the following is one of the most striking: 

 // a is a root of the polynomial f(x) of degree 

 n, and /3 is a root of f'(x), every circle through 

 the points /3 and y = j8 + (n — l)(j8 — a) 

 contains at least one root of f (x). The proofs 

 given do not, however, make explicit use of 

 the mechanical analogy. In a paper read 

 before the International Congress of Mathe- 

 maticians at Strasbourg J. L. Walsh has util- 

 ized Gauss's theorem in discussing the case 

 where the roots of f{x) lie in two circles. 



If the repelling particles exert a force 

 inversely proportional to the square of the 

 distance we obtain theorems of relative dis- 

 tribution of roots in which f'{x) is replaced 

 '^y fMf"(^) — [/'(^)]"; from a root of the 

 latter function the roots of f(x) must be seen 

 under an angle of at least 90°, and the polygon 

 of Lucas is replaced by one bounded by ares 

 of circles. Other extensions of this sort sug- 

 gest themselves, but nothing, so far as I am 

 aware, has been published along this line. 



An immediate corollary of the polygon the- 

 orem states that all the roots of all the derived 

 functions lie within the polygon of Lucas. It 

 is well known that the centroid of the roots of 

 f{x) coincides with that of the roots of its 

 derivative of any order. An often discovered 

 theorem places the roots of f'(x) at the foci 

 of a curve determined by the roots of f{x). 



In 1912 Jensen, in a very suggestive memoir 

 on the theory of equations (Acta Mathematica, 

 Vol. 36), stated without proof a theorem for 



equations all of whose coefiieients are real 

 which may be regarded as an improvement on 

 the polygon theorem. If f{x) is a real poly- 

 nomial its complex roots form conjugate pairs. 

 The resultant force of repulsion due to parti- 

 cles at such a pair of points is directed away 

 from the real axis at a point not on this axis 

 and which lies outside the circle whose diam- 

 eter is the line segment joining the pair; we 

 designate this circle the Jensen circle of the 

 pair. At a point within the Jensen circle and 

 not on the real axis the resultant force due to 

 the pair is directed toward the real axis, while 

 on the real axis and on the circumference of 

 the circle it is parallel to the real axis. Thus 

 at a point which is neither on the axis of reals 

 nor within or on the circumfei'ence of any of 

 the Jensen circles corresponding to the com- 

 plex roots, the resultant force of repulsion due 

 to the whole system of particles at the roots 

 of f(x) cannot vanish, for the force due to 

 each particle on the real axis is directed away 

 from that axis, and the same is true of the 

 forces due to pairs of particles at the complex 

 roots. We thus have Jensen's theorem: The 

 roots of f'(x) which are not real must lie 

 within or on the Jensen circles of f(x). To 

 be more precise, a root of f'(x) cannot lie on 

 a Jensen circle unless it is real, or unless it is 

 a multiple root of f(x), or unless it is also 

 within or on another Jensen circle. 



Since the addition of a constant force par- 

 allel to the real axis does not change the above 

 argument, Jensen's theorem remains valid 

 when we substitute for f'{x) the function 

 af{x) + f'{x) where a is any real number. 

 Another extension indicated by Jensen con- 

 cerns the regions within which roots of the 

 successive derived equations lie, these regions 

 being defined in terms of the roots of f{x). 

 Thus the complex roots of f"{x) are in the 

 Jensen circles of f'(x), whose centers are on 

 the axis of reals and whose vertical diameters 

 are within the Jensen circles of f{x). The 

 solution of a simple problem in envelopes 

 shows that all the complex roots of f"{x) lie 

 within or on ellipses each of which has a pair 

 of complex roots of f{x) at the ends of its 

 minor axis and has a major axis whose length 

 is \/2 times that of its minor axis. For the 



