192 



SCIENCE 



[Vol. LV, No. 1417 



rth. derived equation the result is the same 

 except that the ratio of lengths of axes is y/r. 

 Jensen states that this is also true of the func- 

 tion g{D) . f{x), where g(D) is a linear diifer- 

 ential operator of order r with constant coeffi.- 

 cients whose factors are all real, and that f{x) 

 may be an integral transcendental function of 

 genus zero or one. 



In a recent paper (Annals of Mathematics, 

 Vol. 22 (1920) p. 128), J. L. Walsh notes 

 some results for non-real polynomials which 

 follow from considerations that led to Jensen's 

 theorem. He also gives an answer to the ques- 

 tion which at once suggests itself as to how 

 many roots of f'{x) lie within a Jensen circle 

 when f{x) is real by a method of interest in 

 itself, doubtless suggested by Bocher's treat- 

 ment of a similar problem which we shall note 

 later. By allowing all the roots of f{x) out- 

 side a Jensen circle to move out to infinity, 

 noting what roots of f'{x) may enter or leave 

 the circle, and counting those within the circle 

 at the end of the process, Walsh concludes that 

 if a Jensen circle has on or within it k roots 

 of f (x) and is not interior to nor has a point 

 in common with any exterior Jensen circle, 

 then it has on or within it not more than k -{- 1 

 nor less than k — 1 roots o/ f'(x). In a paper 

 not yet published I have obtained a result a 

 little more precise than this in which, for the 

 sake of simpler statement, I will suppose 

 neither f{x) nor f'{x) has multiple roots. By 

 the term "root of even index" I designate a 

 real root of f {x) between which and the next 

 real root of. f{x) to the right or left there lies 

 an odd number of real roots of f'{x) ; if f(x) 

 has no real roots this term denotes every other 

 real root of f'(x), starting with the least. All 

 the real roots of even index of f'(x) can be 

 shown to lie in or on Jensen circles, and every 

 such circle that has no point in or on it within 

 or on any other Jensen circle has within it 

 either just one real root of even index of 

 fix), or just one pair of complex roots of 

 f'{x). The region covered by a system of 

 Jensen circles each of which overlaps or 

 touches some other of the system has within it 

 the total number of real roots of even index 

 and of pairs of complex roots of the derived 

 equation which the circles would have if they 



were separated, but there may be circles of the 

 system containing no such points. General 

 criteria to determine whether even an isolated 

 Jensen circle contains a pair of complex roots 

 or a real root of even index of f {x) are lack- 

 ing, though Walsh discusses special cases, in 

 some of which we may use a circle smaller 

 than Jensen's. 



Relative distributions of the roots of a real 

 polynomial f{x) and of its derivative in vari- 

 ous special cases have been discussed by H. B. 

 Mitchell (Transactions of the American Math- 

 ematical Society, Vol. 19 (1918), p. 43). The 

 identity of the logarithmic derivative is used, 

 but the mechanical analogy and Jensen's the- 

 orem are not cited. 



So far we have been concerned only with 

 theorems of relative distribution for the roots 

 of a polynomial and of its derivative. In a 

 most suggestive paper by Bocher ("A Problem 

 in Statics and its Relation to Certain Alge- 

 braic Invariants," Proceedings of the American 

 Academy of Arts and Sciences, Vol. 40 (1904), 

 p. 469) our mechanical system is generalized 

 by assigning to particles at points e^, e^, . . e 

 masses to^, m^, . . m respectively, with the 

 same law of repulsion as before. Negative 

 values for the masses are admitted, the repul- 

 sion becoming an attraction in the case of the 

 corresponding particles. The field of force is 

 then given in both magnitude and direction by 



(TO, «i, m„ \ 



—^ + -^ + + — f-;- 



The eases of greatest interest are those in 

 which the sum of the masses is zero. By pro- 

 jecting such a system stereographically upon 

 a sphere (the same result could be established 

 by inversion on a circle about x), Bocher 

 proves that a point cannot be a position of 

 equilibrium if it is possible to draw a circle 

 through it upon which not all the particles lie 

 and which completely separates the attractive 

 particles which do not lie on it from the repul- 

 sive particles which do not lie on it. 



A remarkable property of these systems 

 whose total mass is zero is now developed by 

 introducing homogeneous variables 



