194 



SCIENCE 



[Vol. LV, No. 1417 



all of whose roots are real, and to polynomial 

 solutions of linear differential equations. 



Before leaving this phase of our subject we 

 may note, with Laguerre, that similar the- 

 orems hold for each of the successive polars of 

 a binary form with respect to a point. An 

 interesting field hardly touched as yet is that 

 of separation theorems for the successive polars 

 of a form with respect to a sequence of points 

 defined as the roots of another form. By 

 taking the two forms in a special case where 

 they are apolar Grace has proved (Proceedings 

 of the Cambridge Philosophical Society, Vol. 

 11 (1901), p. 35) a result equivalent to this: 

 If the distance apart of two roots a^, a, of a 

 polynomial f{x) of degree n is 2a, there is at 

 least one root of f'(x) on or in the circle 



ivhose radius is a cot -, and ivhose center is 

 n 



^(ai + as)- I" tl"s paper lack of references 

 indicates ignorance of Laguerre's work. The 

 same result was proved later by Heawood 

 {Quarterly Journal of Mathematics, Vol. 38 

 (1907), p. 84) by allowing all the other roots 

 of /(a;) to vary suitably. Here, again, there 

 is no reference to any other work in this field. 

 To return to more recent work on the van- 

 ishing of the Jacobian of two forms /j and f„, 

 we note two very interesting papers by Walsh 

 in the Transactions of the American Mathe- 

 matical Society, in which are discussed cases 

 where the roots of the ground-forms are in 

 three circles, instead of two. An added inter- 

 est is shown to attach to the Jacobian because 

 the numerator of the derivative of a rational 

 function 



u{x) _ f^jX^jX^) 



v(x) f^_(x^,x,) 

 is xl multiplied by the Jacobian of f^ and /,. 

 Separation theorems for the Jacobian are then 

 interpretable in terms of this derivative. The 

 results of these papers are, of course, only a 

 first step to the consideration of still more 

 general separation theorems. The field is the 

 more interesting in that its investigation 

 involves a combination of mechanical, alge- 

 braical, and geometrical considerations. 



I must close with only a mention of certain 

 extensions of the problem we have so far con- 



sidered. Thus Bocher', generalizing a method 

 due to Stieltjes, considers the positions of 

 equilibrium of a system of free particles of 

 equal mass in a field of force due not only to 

 a number of fixed repelling particles, but also 

 to their own mutual repulsions according to 

 the same law. If the total mass of fixed and 

 moving particles is 1, the positions of equi- 

 librium of the free particles are determined 

 by the vanishing of covariants, of which some 

 examples are given by Boeher. These results, 

 as well as some obtained by adding a force 

 function K[f(x)], are useful in the study of 

 polynomial solutions of differential equations. 

 "We must regret that Boeher was never able to 

 fulfill the hope twice expressed in this paper 

 that he might be able to return in detail to 

 these problems which he had merely sketched. 

 Their investigation requires considerable skill, 

 but, if successful, would add a new and im- 

 portant chapter to algebra, with a striking 

 application of invariant theory. 



D. R. CUETISS 

 NOETHWESTERN UNIVERSITY 



WILLIAM BATESON ON DARWINISM 



Aside from the fine impression created by 

 the admirable series of papers and addresses 

 in biology, zoology and genetics in Toronto at 

 the Naturalists' meeting, a very regrettable 

 impression was made by a number of passages 

 in the addresses of Professor William Bateson, 

 the distinguished representative of Cambridge 

 University and British biology. On the morn- 

 ing following his principal address the Toronto 

 Globe (December 29, 1921) published, in large 

 letters: "Bateson Holds That Former Beliefs 

 Must Be Abandoned. Theory of Darwin Still 

 Remains Unproved and Missing Link Between 

 Monkey and Man Has Not Yet Been Discov- 

 ered by Science. Claims Science Has Out- 

 grown Theory of Origin of Species." In inter- 

 mediate type it announced: "Distinguished 

 Biologist from Britain Delivers Outstanding 

 Address on Failure of Science to Support 

 Theory That Man Arrived on Earth Through 

 Process of Natural Selection and Evolution of 

 Species. Have Traced Man Far Back but 

 Still He Remains Man," and, in smaller type: 



The missing link is still missing, and the Dar- 



