56 Mr. J. J. Waterston on the Intensity of Gravity 



Let U be the velocity with which the agent of gravitation 

 acts, and w the velocity of the descending body in the direction 

 of its action • then, if by supposing U to be doubled the force of 

 gravitation is doubled, it is evident that U diminished to U — w 

 (which it is by the motion of the descending body) diminishes 



the force by jj part. Again, if U is supposed by doubling to 



quadruple the force of gravitation, U diminished to U— w dimi- 



nishes the force by -^ — -r^ ■ Thus upon whatever power of 



U the force of gravity may be supposed to depend, the decre- 

 ment of force is always in the simple ratio of w to U, so long as 

 w is small in comparison to U. 



An orbital body in descending from aphelion to perihelion ac- 

 quires centripetal velocity w, which attains its maximum value 

 when radius-vector is perpendicular to axis, thence diminishes 

 to zero at perihelion. From aphelion to perihelion the force of 

 gravity is thus to be viewed as having a decrement, the amount 

 of which is proportional to w, increasing with it to a maximum 

 and diminishing to zero at perihelion. On the ascending side 

 of the orbit the force of gravity is to be viewed as having an 

 increment which passes through the same changes as the decre- 

 ment, and has the same maximum value. Let us suppose that 

 maximum value to be equal to 100m. The decrement of gravi- 

 tation caused by iv increasing to w-\- co is yy. This is equal to 

 the decrement that would be caused bv the sun's mass losing a 

 proportionate part — = fr ' Hence the proportionate increase 



n . . . . Ba co 2a — r 



ot semi-maior axis is — = t? • 



J a U r 



In the same way the increment of gravitation caused by 

 w diminishing from w to w—co is jj; an( l the proportionate 



decrease of a is — = ^ • l - . This is negative and larger 



a U r x ° 



than the preceding, because the point in the orbit where w dimi- 

 nishes is nearer the perihelion than the point where the same 



, £ „ , . , w /2a— r x 2a— r\ . 



value ot w is lound increasing : nence ^ ( l I is 



D U \ r l r / 



the value of the ordinate to w as abscissa. The area resulting 

 from such ordinates is the integral value of the proportionate 

 decrement of semi- major axis made good in the descending semi- 

 revolution. 



In the ascending semirevolution gravitation is augmented. 



