BOCHER. — A PROBLEM IN STATICS. 471 



particles of masses — m^ , — Wj , . . . — m,, . Since the sura of the masses of 

 these new particles is zero, they do not affect the field of force ; and now 

 we have our pairs of particles Pi,Po, with masses ?ni, — ?«i ; P^j fo, 

 with masses m.2, — mo, etc. We have thus proved the theorem : 



If a number of particles lie on a spherical surface, and if the sum of 

 their masses is zero, the force j^roduced hy them at any point on the spher- 

 ical swface is tangential to that surface.* 



It is this spherical field of force we now wish to study. For this pur- 

 pose let Pq be any point on the spherical surface, and project the sphere 

 8tereogra})hica]ly from P^ onto the diametral plane perpendicular to the 

 diameter through Pq. Call the projections oi Pi, . . . P„ on the plane 

 pi, . . . p„, and let us consider the plane field of force due to particles 

 of masses mi, . . . m„ situated at these points.f We will prove the 

 following two theorems : 



I. The direction of the force at any point P of the spherical field is the 

 stereographic projection of the direction of the force at the corresponding 

 point p of the plane f eld. 



Thus the lines of force on the sphere are the projections of the lines of 

 force in the plane. 



II. The intensity of the force at any point P of the spherical field is 

 equal to the intensity at the corresponding point p of the plane field multi- 

 plied hy \ {\ H- r-), where r is the distance from the centre of the sphere 

 to p, and the radius of the sphere is taken as the unit of length. 



We will first prove these two theorems for the special case in which 

 there are only two particles situated at P, and Pq with masses ?«, and 

 — m. respectively. Since the circle P,PPq is a line of force on the 

 sphere, the force acting away from P, or towards it according as /w, is 

 positive or negative, and since the projection of this circle is the straight 

 line pip, and this is a line of force in the plane, theorem I is obviously 

 true in this case. 



Since the intensity of the force at/> is 



fJ^' 



P.P 



* The same reasoning shows tliat if a number of particles lie on a circle, and if 

 the sum of their masses is zero, the force produced by them at any point on the 

 circle is tangential to the circle. It also admits of inmiediate application to spher- 

 ical multiplicities in space of n dimensions. 



t If one of the points P, lies at Pq, the corresponding point p,- will be at infinity, 

 and the particle in question in the plane is simply to be omitted. 



