30 



OF CENTRAt FORCES. 



The sum of the forces actfng on the body is — ± — — --> 



ax (a±:x)' 



. ^ . M'i rd'r tit) W 

 and smce vvzzfi, vzz i;, — ; — -, and — = 



•' XX («±.ry 2 X 



ccP /2M' , 9rd'\ , ., /2f\ . 



— — -,wt;=::+:( K — — ) ; and if c=:o, v—^/ — )<«• 



a±x \ x a±.xl \^ I 



Scholium. In the case of a body projected from the 



moon towards the earth, dr:20,900,ooo feet, arieod, b'zz 



32.2 feet, the velocity produced in i" at the earth's surface; 



_ 1 . . ■. , • 2ig 



czz—b, nearly; then taking xzz — a, at the moon s sur- 



face, and — a, at the point where the force becomes neu- 

 94 



tral,we have ( — X220 and { — I ) 



a V219 70/ a ^84 roo/ 



X94, of which the I difference is , or .09646M, 



a 



and its square root about 8070 feet. Hence, if the velocity 

 of a projectile from the moon exceed 8070 feet, it may pass 

 the neutral point, and descend to the earth ; where its velo- 

 city will become more than 36000 feet in a second. 



SECTION III. OF CENTRAL FORCES. 



238. Definition. An accelerating force 

 tending to a point out of the line of direction 

 of a moving body, deflects it from that line, 

 and is then usually called a central force. 



239. Theorem. The force, by which a 

 body is deflected into any curve, is directly 

 as the square of the velocity, and inversely 

 as that chord of the circle of equal curvature, 

 which is in the direction of tl>e force; and 

 the velocity in the curve is equal to that 

 which would be generated by the same force, 

 during the description of one fourth of the 

 chord by its uniform action. 



For the force is as the space described by 

 its action, beginning from a state of rest, or 

 as the evanescent sagitta through which the 

 body is drawn from the tangent of the _curve 

 in a given instant of time : but the portion AB 

 of the tangent described in a given instant is 



as the velocity, and BC=: —=, or ultimately 



ABq 

 CD 



, which is as the square of the velocity directly, and 



inversely as the chord of the circle of curvature of the arc 

 AC. 



Now the velocity generated during the description of BC 

 is expressed by twice BC, since the force maybe considered 

 for an instant as constant : consequently it is to the orbital 

 velocity as twice BC to AB, or as twice AB to ED, or as 

 AB to half CD ; and if the time of the action of the force 

 were continued during the time that half CD would be 

 described with the orbital velocity, it would generate a 

 velocity equal to that velocity ; but in this time one fourth 

 of CD only would be described by its action. 



240. Theorem. When a body describes 



a circle by means of a force directed to its 



centre, its velocity is every where equal to 



that which it would acquire in falling by the 



same uniform force through half tlie radius; 



and the force is as the square of the velocity 



directly, and as the radius inversely. 



For in this case the chord, passing through the centre, 

 becomes a diameter. 



241. Theorem. In equal circles the forces 

 are as the squares of the times inversely. 



For the velocities are inversely as the times, and the de- 

 flective chords are equal. 



242. Theorem. If the times are equal, 

 the velocities are as the radii, and the forces 

 are also as the radii, and, in general, ihe 

 forces are as the distances directly, and the 

 squares of the times inversely ; and the 

 squares of the times are directly as the dis- 

 stances, and inversely as the forces. 



For the velocities are as the distances directly, and as the 

 times inversely ; and the squares of the velocities are as the 

 squares of the distances directly, and as the squares of the 

 times inversely ; consequently the forces are as the radii di- 

 rectly, andthesquarcs of the times inversely; and the squares 

 of the times are as the radii directly, and as the forces in- 

 versely. 



243. Theorem. If the forces are in- 

 versely as the squares of the distances, the 

 squares of the times are as the cubes of the 

 disl^ances. 



