480 PROCEEDINGS OF THE AMERICAN ACADEMY. 



equal mass which are all free to move in the plane or on the sphere, and 

 are acted upon not merely by the field of force due to the fixed jjarticles, 

 but also by one another, the action here again being repulsion of intensity 

 equal to the product of the masses divided by the distances. We may 

 without loss of generality take the mass of each of the moving particles as 

 unity, in which case the total mass of the fixed and moving particles must 

 evidently be +1 in order that the force acting on each of the moving par- 

 ticles should be tangential to the sphere.* The problem of determining 

 the positions of equilibrium of the system of moving particles is readily 

 seen, by refereoce to the results already obtained, to have the same 

 invariant character as in the special case already considered. Neither 

 Stieltjes's original treatment of the problem nor my own earlier methods 

 brought out this fact in the analytic work, the positions- of equilibrium 

 being obtained as the roots of the polynora^ial solutions of certain homo- 

 geneous linear differential equations of the second order, — equations 

 whose invariant character was in no way evident. I hope to take up 

 this whole subject before long from the point of view here indicated. 

 I content myself here with pointing out, by a simple example, how 

 Stieltjes's problem can be brought into connection with the elementary 

 theory of algebraic invariants. 



If i is a binary cubic with distinct roots, and if three fixed particles, 

 each of mass \ (\ — k) are situated at the roots off, then the position of 

 equilibrium, if any exists, of k movable particles of mass + 1 is given by 

 the vanishing of a certain covariant </> q/" f, which in the simplest cases 

 k ^ 7 is: 



k = S (fi = J, 



k = 4: there is no position of equilibrium, 



k = 5 (f> = JI-J, 



k = 6 <i> = ^f^-l H\ 



k =^1 there is no position of equilibrium. 



Here A and H denote the discriminant and Hessian of /, and J the 

 Jacobian of/ and H, where, however, in order that the formula for ^- = G 

 be correct, we must suppose that, / being written : 



/ = Qq ^1* + 3 Oi Xi^ a-2 + 3 Oj Xi x^- + a, Xo', 



* This restriction is not necessary wlien, as has always been the case hereto- 

 fore, the Tplane. problem only is considered, and no attempt is made to bring out its 

 invariant character. 



