The Variation in Attraction Due to the Attracting Bodies. 231 



repulsive effect from the rotatiou measured by the system, of a circle- 

 Draw f b perpendicular to a C, and from the initial jjoint a, drtw diagonals 

 a b and a b . Under the hypothes's that no component parti -le excep ing 

 partic'e a, has freedom to ciiange position in the ellipsoid, it is evident that 

 this exceptional particle acted upon by the forces combined vs'ould be at 

 the end of the firat unit of time at b instead of at b, as it wa,i in Case 1st. 



Diagram 11. 



If during the second unit of time attraction did not act the particle 

 would continue to move in direction a b to c' making distance b c' equal 

 to a b. It is evi-lent under the action of the forces combined at the end 

 of the second unit of time, the particle would be found at some point c in 

 in line c c' drawn parallel to b C. The position of the particle for the 

 third or any unit of time in due order can likewise be determ ned. 



Tri ingles C b c and Cbc' are equal from having same base Cb and equal 

 alt tudes; triangles Cbc' and cab are equal from haviog equal bases be' 

 and ab and same altitude; triangle Cab and Cab,, are equal from having 

 same base Ca and equal altitudes. It is now evident that the triangle 

 evolved by the forces combined for any unity of time is equal not only to 

 the one for the first unity of time but also to the one that would be evolved 

 by the initial impulse of rotatiou combined with an attraction equal to the 

 versid sine due to repulsion from rotation. 



Measured, with reference to the arc ab,, of the circle attraction is to repul- 

 sion as af to fa, but measured with reference to the arc ab in th-i path of the 

 particle acted upon by the forces combined attraction and repulsion are 

 equal and in equilibrium in the production of arc ab or any other ar c in 

 the path of revolution of the particle. 



The following is the euuuciati )n of a pi'oposicion well kao vn to be true 

 from demonstration. "If a body -describes an ellipse, being continually 

 urged by a force directed towards the focus, that force must vary inversely 

 as the square of the distance." In the case, then, under consideration if 



