BOCHER. — A PROBLEM IN STATICS. 481 



numerical factors are included in A and // so as to make them have 

 integral coefficients without common factors. 



This problem of Stieltjes admits of generalization in still another di- 

 rection by considering not the field of force due to a number of fixed 

 particles wliich repel according to the law of the inverse distance, but 

 the field of force given in the plane by the formula 



K[f{x)l 



where f{x) is any analytic function of the complex variable x* We 

 should here seek the positions of equilibrium of a group of k unit particles 

 which repel one another with a force equal to the reciprocal of the dis- 

 tance. Denoting the complex quantities which determine the positions 

 of these movable particles by Xj, Xj, . . . x^, and letting 



</> {x) = (x — Xi) (X — Xo) . . . (x — x„), 



the equations of equilibrium are 



Let us consider here in more detail the special case in which 

 f(x) = — cx, 



where c is a positive constant. The force here may be described as 

 attractiou towards the axis of imaginaries and repulsion from the axis of 

 reals, the intensity of the force being in both cases directly proportional 

 to the distance from the axis in question, and the factor of proportion- 

 ality being the same in the two cases. It is clear that the particles can- 

 not be in equilibrium unless they all lie on the axis of reals, and it is 

 equally clear that there is at least one position of equilibrium of this 

 latter sort. In order to find it write the equations of equilibrium iu 

 the form 



c}>" (x,) - 2cx,c^' (x,) = 0, (i=l,2, . . . k). 



These equations show that the polynomials (f>(x) and (f>"{x) — 2 cxc^i'ix), 



* My colleague, Professor B. 0. Peirce, calls my attention to the fact that this 

 amounts to considering the field of force which has the real part of [/(t) dx as 

 a force function. We are thus dealing with the most general field, which has a 

 force function with continuous first and second partial derivatives and satisfies 

 Laplace's equation. 



VOL. XL. — 31 



