470 PROCEEDINGS OF THE AMERICAN ACADEMY. 



where the symbol ^is used to indicate that we are to take the conjugate 

 of the complex quantity which follows it.* 



This problem may be extended in a useful manner by the following 

 considerations. 



Let us take first the special case of two particles whose masses are the 

 negatives of each other. Let the positions of the particles be determined 

 by the complex quantities Ci and e.^, and let their masses be respectively 

 m and — m. Then the force at any point x in the plane is by (1) : 



mK-. — - . 



{X — Ci) {X — e^) 



By considering first the angle and then the absolute value of this complex 

 quantity we obtain at once the two results : f 



L The lines of force of two particles Pi and P^ whose masses are the 

 negatives of each other are the circles through Pi and P^, the force at 

 any point being directed away from the particle of positive and towards 

 that of negative mass. 



II. The intensity of the force at a point P is given by the formula : 



Pi A 



PiP- P^P' 



The first of these statements shows that if through Pi and Po we pass 

 any spherical surface, the force is tangential to the spherical surface at 

 all its points. The same will, therefore, be true if on any spherical sur- 

 face we have any number of pairs of particles Pj, Pg ; Pi', PJ ; • • •, pro- 

 vided that the masses of the two particles of every pair are the negatives 

 of each other. Now it is evident that on a spherical surface any system 

 of particles Pi, Po, . . . P„ the sum of whose masses is zero is equivalent 

 to such a system of pairs of particles. For let the masses of Pi, Pg, . . . 

 P„ be mj, ^2, . . . /w„ respectively (wi + m^ -\- . . . + ?w„ = 0), and let us 

 at an arbitrarily chosen point Pq of the spherical surface place n coincident 



* If, in particular, jhj = m, = . . . = m„ = 1, and if we let 

 f(x) = (x-ei)(x -e^) . . . (x-e„), 



the field of force becomes A' ( -~ \ I > 



V /(■'■)/ 



and the points of no force are given by the roots of y (,r) = 0. This theorem was 



stated by Gauss (Werke, 3, 112) and afterwards rediscovered by F. Lucas 



(C. R.. 1868). 



t These results are very well known, and can easily be proved by other methods. 



