190 



SCIENCE 



[Vol. liV, No. 1417 



of the roots of other polynomials; we are then 

 concerned with relative distributions of the 

 roots of two or more polynomials. 



Theorems of separation for real roots of 

 real equations are numerous, and are among 

 the most familiar results in elementary mathe- 

 matics. I need only mention Descartes' rule, 

 which gives a superior limit for the number of 

 roots on the positive real axis, or Sturm's 

 method for obtaining the exact number in any 

 real interval. Eolle's theorem, in the form 

 which states that between each consecutive pair 

 of real roots of a real polynomial f(x) there 

 lies an odd number of real roots of the derived 

 function f'(x), is perhaps the most important 

 proposition concerning relative distributions of 

 real roots of two real polynomials. 



No such progress has been made with sim- 

 ilar propositions for complex roots, although 

 the widening of the field of observation from 

 the real axis to the complex plane vastly 

 increases the range of possibilities. To be 

 sure, we have extensions of Stuiin's theorem, 

 and other methods, both algebraic and tran- 

 scendental, which give criteria for the exact 

 number of roots within a region, but in prac- 

 tice these prove so cumbersome as to be of 

 little use. The great desideratum is a body of 

 results whose simplicity and range of appli- 

 cations would make them comparable with 

 Eolle's theorem, or the Budan-Fourier theorem 

 in the real case. As Jensen has remarked, the 

 solution of important problems regarding the 

 zeroes of transcendental functions may be de- 

 pendent upon progress in this direction. 



The significance of RoUe's theorem naturally 

 led to attempts to extend it to the complex 

 plane almost as soon as the now familiar geo- 

 metric representation of complex numbers had 

 been adopted. A line of attack is clearly indi- 

 cated by the identity of the logarithmic deriva- 

 tive 



fix) _ _J. 1 1 



fix) 



+ 



+ 



+ 



where /(a;) is a polynomial of degree n, whose 

 roots are dj, a^, . . , a^, and f'{x) is the first 

 derivative of f{x). Gauss was probably the 

 first to give this a mechanical interpretation 

 which depends on the representation of a com- 

 plex number ;c — a as a free vector whose 



length, la; — a I , and direction are those of the 

 directed line segment from the point which 

 corresponds to a, or, more briefly, from the 

 point a, to the point x. The conjugate of the 

 reciprocal of x — a, which may be denoted by 



the symbol K , corresponds to a vector 



*^ X — a 



having the same direction as the vector x — a 

 but with a length equal to the reciprocal of 

 U_a|. This is precisely the vector which 

 represents the force at x due to a particle of 

 unit mass at a which repels with a force whose 

 magnitude is equal to its mass divided by the 

 distance. If, then, we take the conjugate of 

 both sides of the identity of the logarithmic 

 derivative, we have the theorem of Gauss: 

 The roots of f'(x) which are not also roots of 

 f(x) are the points of equilibrium in the field 

 of force due to particles of unit mass at the 

 roots of f (x), each of which exerts a repulsion 

 equal to its mass divided by the distance. 



From this result it is but a step, though one 

 not taken for many years, to the polygon 

 theorem of Lucas, now sufficiently well known 

 to have a place in Osgood's "Lehrbuch der 

 Funktionentheorie," but discovered and redis- 

 covered, proved and reproved in most of the 

 languages of Europe— and all the proofs are 

 substantially the same! This ignorance of the 

 work of others characterizes even some of the 

 most important contributions in this field. 

 Lucas, for example, seems to have considered 

 himself the discoverer of the theorem of Gauss, 

 which really antedates his work by many years. 

 The polygon theorem, in its usual form, is a 

 theorem of relative distribution which states 

 that the roots of the derived function f'{x) lie 

 within or on the perimeter of the smallest 

 convex polygon (or line segment) which 

 includes within itself or on its boundary all 

 the roots ot f(x). This statement implies that 

 there is but one such polygon (or line seg- 

 ment), which reduces to a point if /(a;) has all 

 its roots coincident. In ease the polygon of 

 Lucas does not reduce to a line or a point, the 

 only roots of f'(x) on its perimeter are mul- 

 tiple roots oif(x). An equivalent form giving 

 a separation theorem for the roots of f{x) 

 states that every straight line through a root 

 of /'(.-b) either passes through all the roots of 



