128.] CENTRAL FORCES. 67 



tricity, with the sun at one of the foci. As a comet is within 

 range of observation only while in that portion of its path 

 which lies nearest to the sun, a portion of a parabola, with the 

 same focus and vertex, can be substituted for this portion of 

 the elliptic orbit, at least as a first approximation. 



It is also found from observation that the motions of the 

 moons or satellites around the planets follow very nearly 

 Kepler's law. A planet can therefore be regarded as attracting 

 each of its satellites with a force proportional to the mass of 

 the satellite and inversely proportional to the distance. 



128. All these facts led Newton to suspect that the force of 

 terrestrial gravitation, as observed in the case of falling bodies 

 on the earth's surface, might be the same as the force that 

 keeps the moon in its orbit around the earth. This inference 

 could easily be tested, since the acceleration g of falling bodies 

 as well as the moon's distance and time of revolution were 

 known. 



Let m be the mass of the moon, a the major semi-axis of its orbit, 

 T the time of revolution, r the distance between the centres of earth and 

 moon; then the earth's attraction on the moon is (Art. 126) 



or, since the eccentricity of the moon's orbit is so small that the orbit 

 can be regarded as nearly circular, 



On the other hand, the attraction exerted by the earth on a mass m on 

 its surface, i.e. at the distance R = 3963 miles from the centre, must be 



f" = mg. 



Now, if these forces are actually in the inverse ratio of the squares of 

 the distances, we must have 



