84 O* ACCELERATING FORCES. 



237. Theorem. If two forces act in the 

 same right Hne on a moveable body, varying 

 inversely as the square of its distance from two 

 given points, situated at the distance a from 

 each other, the magnitudes of the forces being 

 expressed by b and c at the distance d, the 

 square of the velocity generated in the passage 

 of the body, between any two points of which 

 the distances from the first centre are succes- 

 sive values of cl\ is the difference of the cor- 

 responding values oi 2cP (-+ -7-). 



The sum of the forces, acting on the body, is h — ±c 



XX 



and since vdvzz^ado?" (235), vdiV— — da; ± 



(a±j)2 XX 



cdd , , vv hdd cdd . 



ax, and—-— , consequently vv 



{a±xY 2 X a±x 



/2bdd 2cdd\ , .^ ^ j , 2b 



ir( 1 ) : and it czz 0, vz=.dsj — . 



V X a±.x^ X 



Scholium. This proposition is not altogether entitled 

 to a place among the elementary doctrines of motion, 

 having arisen from an inquiry into the origin of the me- 

 teoric stones : but it serves as a very good illustration of 

 the utihty of the 235th article. In the case of a body 

 projected from the moon towards the earth, dzz2(i 900 000 

 feet, az=LQOd, &=:32.2 feet, the velocity produced in a 

 second at the earth's surface; and c—-^h, nearly; then 

 taking a:=ffjct, at the moon's surface, and |^«, at the 



point wl: ire the force becomes neutral, we have (-; 



a '' 



