BOCHER. — A PROBLEM IN STATICS. 475 



We have so far regarded (f> as an integral rational covariant of weight 

 1 of the system of linear forms 



(3) ej'xi - ejx., (i = 1, 2, . . . n) 



which involves n real parameters ?Mi, . . . w„ connected by the relation 



mi + Wo + ■ • ' + i^n = ^•* 



Instead, however, of determining the positions of the particles by the 

 vanishing of the linear forms (3), as we have practically been doing, 

 we may, if we prefer, build up forms of higher degree yi,yo, . . . f^ by 

 multiplying the linear forms (3) together in groups, and then regard the 

 positions of the particles as determined by the vanishing of these forms 

 f. The form <^ will still be a covariant of the system of forms /, but it 

 will in general be an irrational covariant of this system. If, however, 

 the particles corrresponding to the form f^ all have the same mass, is 

 rational and symmetric in the roots of /, and therefore rational in the 

 coefficients of/,. Thus we obtain the result: 



Given a system of hinary forms fj , fj, . • . f^ of degrees Pi, po, • • • Pt 

 respectively^ and k real quantities m^, m2, . . . m^ subject to the condi- 

 tion pi mi 4- . . . + Pk m^ = ; and suppose that at each of the p, points 

 determined in the complex plane or on the complex sphere by the equation 

 f i = are placed equal particles each of mass m; which repel with a force 

 which varies directly as the mass and inversely as the distance ; then the 

 positions of equilibrium in this plane or spherical field of force are de- 

 termined as the roots of a certain integral rational covariant cj> of the 

 system of forms f. 



Besides vanishing at the points of equilibrium, </> vanishes only at the 

 points of pseudo-equilibrium described above. Such points can occur 

 only at the multiple roots of a form f ; or at a common root of two of these 

 forms. 



The simplest case is when ^ =: 2. Here pi mi + p., m« = 0, and 

 therefore, since only the ratio of the masses mi and m^ is important, we 

 may without real loss of generality assume that mi = p^, m^ = — pi. 

 If we let : 



* It would be possible to consider the more general case in which the quantities 

 m,- are complex, the force having then not merely an attractive or repulsive com- 

 ponent, but also a component at right angles to this. Thus if the quantities /«, are 

 pure imaginaries, we have, in the plane, the electromagnetic lield due to the steady 

 flow of electric currents through long straight wires which cut the plane at right 

 angles. 



