Febeuaet 24, 1922] 



SCIENCE 



193 



If the above expression for the field of force is 

 reduced to a common denominator within the 

 parenthesis, the numerator is the product of 

 xl and a covariant <p of iveight 1 of the n 

 linear forms e'.'x^ — e'.x„. The points of equi- 

 librium are roots of the covariant <f>, and 

 (j) vanishes at no other points unless two of the 

 particles coincide. If the points e. are defined 

 as the roots of a system of binary forms / ^, 

 the masses of all the particles corresponding to 

 each / being equal, (ji is an integral rational 

 covariant of the forms / , and we are thus led 

 to theorems of relative distribution for the 

 roots of a system of forms and those of a 

 covariant of the system. In particular, if the 

 system consists of but two fonns, the covariant 

 <j) is their Jacobian; in all cases <p can be ex- 

 pressed as a polynomial in the ground-forms 

 and Jacobians of pairs of the ground-forms. 



The conditions of equilibrium of the cor- 

 responding mechanical system can now be 

 interpreted as theorems of separation for the 

 roots of the forms. Thus if /^ and /., are two 

 binary forms whose roots are all in circles 

 C^ and Cj respectively, and these circles do not 

 touch or overlap, then all the roots of the 

 Jacobian of /j and /, are in Cj and C^- The 

 actual number of roots in each circle is ob- 

 tained by allowing the roots of /^ to coalesce 

 at a point a^^ and shrinking C^ to this point; 

 during this process C^ is always to include aU 

 the roots of /j. At the end of this process the 

 Jacobian has Pj — 1 roots at a^, where p^ is 

 the degree of /^. We conclude that the Jaco- 

 bian originally had this number of roots in C^, 

 and a correspondingly deteimined number in 

 Cj. The circles C-^ and C, may be replaced by 

 circle-are polygons. 



The polygon theorem of Lucas corresponds 

 to the special case where one of the ground- 

 forms reduces to x„. 



A ease of especial interest is that where one 

 of the two ground-forms is linear; we have 

 just noted a particular instance. The Jacobian 

 of y^x^ — y-^Xr, and f{x^,x^) is the first polar 

 of iy-^ty^) with respect to /. In a series of 

 papers dating from 1874, to be found in his 

 collected works, Laguerre had developed sep- 

 aration theorems for a binary form and its 



y = x- 



polars, without the use of our mechanical 

 analogy. Bocher seems to have been unac- 

 quainted with these results, which, however, 

 are directly obtainable from his own. If the 

 circle C^ of the preceding paragraph is re- 

 placed by the point (j/i, «/,), we have La- 

 guerre's theorem which states that if this point 

 is outside a circle C^ that contains all the roots 

 of /(aij, aij), then all the roots of the polar 

 yj' -\- yj' lie within C^. Laguerre gives 



1 2 



this a more striking form by supposing 

 (xj, x^) taken arbitrarily and determining the 

 "derived point" {y^,y^) as the point which 

 makes the polar vanish. Every circle through 

 a point and its derived point either has all 

 the roots of f (x^, x,) on it, or else there is at 

 least one root within and at least one root 

 without the circle. In non-homogeneous vari- 

 ables the derived point y oi a. point x with 

 respect to f{x) is 



V(*)' 



where n is the degree of f{x). The first ap- 

 proximation to a root of f(x) being x, the next 

 approximation by Newton's method is 



X . Thus we have a most interesting 



fix) 



light upon Newton's method in the complex 



plane; it replaces a; by a point within a circle 



on which x lies, and which surely contains a 



root of f{x). 



A point coincides with its derived point 



when and only when the point is a root of f{x). 



Let a be such a simple root, and let (3 be its 



derived point with respect to F{x), where 



f{x) = {x — a)F{x), and the degree of f{x) 



is at least two. Since F{a) — /'(a), and 



F'ia) = V2f"{a), we have 



F{cl) /'(a) 



^ = a— (w — 1)— ^ = a — 2(m — l)-^!-^- 



F'(a) /"(a) 



Each circle through a and ^ either has all the 

 roots of f{x) upon it or else at least one is 

 within it and at least one is without. There 

 is thus at least one root whose distance from 

 if («) I 



a is not greater than 2(n — 1) 



I /"(a) 



Laguerre and others have made interesting 

 applications of these results to polynomials 



