A PROBLEM IN STATICS AND ITS RELATION TO 

 CERTAIN ALGEBRAIC INVARIANTS. 



By Maxime B6cher. 



Presented May 11, 1904. Received November 16, 1904. 



It is the object of the present paper to show how a certain problem in 

 the equilibrium of particles, when treated by the use of complex quanti- 

 ties, can, by the introduction of homogeneous variables, be brought into 

 connection with the theory of algebraic invariants. A very special case 

 of this problem was considered by Gauss and F. Lucas, while another 

 case, much more general in some respects, was made use of by Stieltjes 

 in his discussion of polynomial solutions of certain homogeneous linear 

 differential equations of the second order. Nowhere, however, so far as 

 I know, has the mechanical interpretation been carried so far as I carry 

 it here, nor has the connection with the subject of algebraic invariants 

 been pointed out. The following pages are intended to be suggestive 

 rather than exhaustive, only a few of the simpler applications of the 

 method being taken up. 



§ 1. The Fip:ld of Force. 



Given a number of fixed particles Pj, A, . . . P„ in a plane, with 

 masses mi, m^, . . . ni„', let us suppose each of them repels with a force 

 equal to its mass divided by the distance. We do not exclude the pos- 

 sibility of some of the particles having negative masses, in which case 

 these particles will attract instead of repelling. 



To obtain the field of force in the plane due to these centres of force, 

 let us take the plane as the plane of complex numbers, and determine 

 the positions of the particles Pj, . . . P„ in the ordinary way by means 

 of the complex numbers e^, . . . e„. Then, if any point Pin the plane 

 is determined by the complex number x, the force at P, due to the cen- 

 tres of force at Pj, . . . P„, is given both in magnitude and in direction 

 by the complex quantity : 



(1) 



^, / nil rrio m„ \ 



A — ^ + — + . . . + "— ], 



\x — ei X — e., X — e„y 



