472 PROCEEDINGS OF THE AMERICAN ACADEMY. 



and, by a formula obtained above, the intensity of the force at P is 



we see that 



P,P'P.3 ^' 



Now it is well known that the stereographic projection of the sphere 

 on the plane is equivalent to an inversion with regard to a sphere having 

 Pq as centre and radius V^. Accordingly : 



2 2 



p^p=4-= ■ 



PoP Vl + r' 



Moreover, by a familiar property of inversion, the triangles P^P.Pand 

 PqPP' ^^^ similar, so that : 



P.P ^ f^ 



p.p Pop: 



Substituting these values we find 



1 4- r^ 



as was to be proved. 



Having thus established theorems I and II in the special case of two 

 particles, one of which lies at Pq, the proof in the general case follows 

 by replacing the n particles Pi, . . . P„ as we did above, by the n pahs 

 of particles Pi, P^; P^, P^; . . . P„, Pq. We thus get « forces acting 

 at P and n corresponding forces acting at p. Moreover each force at 

 P acts in the direction which is the stereographic projection of the direc- 

 tion of the corresponding force at p ; and accordingly, since angles are 

 not changed by stereographic projection, the figure formed by the 

 directions of tlie forces at P is congruent with the figure formed 

 by the directions of the forces at p. Moreover every force at P 

 is }j (1 4- r-) times the corresponding forces at p. Accordingly the 

 forces at P are represented by n vectors, which form a figure similar (in 

 the geometrical sense of the word) to that formed by the n vectors which 

 represent the forces at p. If we now complete these figures by con- 

 structing the resultant in each case, the figures will clearly remain 

 similar. Accordingly the directions of these resultants, since they make 



