478 PROCEEDINGS OP THE AMERICAN ACADEMY. 



and/2 which do not lie upon it. Some of the roots of f^ will form the 

 vertices of Ti, while the others lie within it or on its boundary. It fol- 

 lows by the principle we have already made use of, that the roots of the 

 Jacobian must lie in the polygons 7\ and T,^. 



By a slight additional consideration we can even determine how the 

 roots of the Jacobian are distributed between the two regions. For this 

 purpose let us allow the roots of f^ and f,^ to change, always remaining 

 in /S'l and S^ respectively, in such a way that the roots of /< all approach 

 a point a, within S^. If this change is a continuous one, the roots of the 

 Jacobian will also change continuously on the complex sphere, and there- 

 fore such of these roots as originally lay in *S', will remain there. We 

 may, however, during this process allow the regions S^ and S.^ to shrink 

 down towards the points a^ and a.^^ respectively, while they always include 

 the roots oi fx and/o. Accordingly all the roots of the Jacobian which 

 originally lay in S^ must be approaching a, as their limit. But we have 

 seen above that when h particles whose total mass is not zero fall to- 

 gether at a point, h — 1 positions of true equilibrium coalesce into a 

 position of pseudo-equilibrium. The Jacobian must therefore have had 

 just pi — 1 roots in S^^ and p, — 1 roots iu S.^. Hence the theorem : 



If the roots of a binary form fi of degree pi lie within or on the boun- 

 dary of a region T^, and if the roots of a second binary form fa of degree 

 P2 lie within or on the boundary of a second region Tj which does not 

 overlap or touch the first, and if these two regions are bounded by arcs of 

 circles each one of which circles separates the roots of fi tvhich do not 

 lie on it from the roots of fj ichich do not lie on it; then the Jacobian of 

 fi and i.2 has just p^ — 1 roots in Ti and pj — 1 roots in To.* 



The method of proof here used can be immediately extended to the gen- 

 eral case of k ground-forms y*!, . . . ff.. If all the roots of such of these 

 forms as correspond to the positive constants of the set m^, m^, . • . m^. 

 lie in Ti and all the roots of the other forms lie in T^, we see in this 

 way that all the roots of the covariant ^ lie in Ti or T^, and that </> has 

 in 7' one less root than the ground-forms have there. 



The principle we have used so far is not the only one which helps us 



* Tlie speciiil case in which one of the ground-forms reduces to x„ gives the fol- 

 lowing theorem, wliich is an immediate consequence of Gauss's tlieorem quoted 

 above, and was first explicitly stated by F. Lucas, Journal de I'Ecole Poly technique, 

 Cahier 4G (lb79), p. 8. 



The roots of tiie derivative of any polynomial in x lie in any convex rectilinear 

 polygon in the complex plane which includes within itself or on its perimeter all 

 tiie roots of the original polynomial. 



