68 KINETICS OF A PARTICLE. [129. 



or, since the distance of the moon is nearly = 60 R, 



F' = 6o 2 F. 

 Substituting the above values of F and F\ we find 



^= 4 7T 2 X60 3 X^ 2 . 



With R 3963 miles, T= 27* f 43"*, this gives 



=32-0, 



a value which agrees sufficiently with the observed value of g, consider- 

 ing the rough degree of approximation used. 



129, In this way Newton was finally led to his law of universal 

 gravitation, which asserts that every particle of mass m attracts 

 every other particle of mass m' with a force 



r 2 ' 



where r is the distance of the particles and K a constant, viz. 

 the acceleration produced by a unit of mass in a unit of mass at 

 unit distance (see Part II., Art. 257, 261-262). 



The best proof of this hypothesis as an actual law of physical 

 nature is found in the close agreement of the results of theo- 

 retical astronomy based on this law with the observed celestial 

 phenomena. 



It may be noticed that, according to this law, -the path of a 

 projectile in vacuo is only approximately parabolic, the actual 

 path being a very elongated ellipse or hyperbola, one of whose 

 foci is at the earth's centre. 



130. Taking Newton's law as a basis, let us now turn to the 

 converse problem of determining the motion of a particle acted 

 upon by a single central force for which f(r) = p/r* (problem 

 of planetary motion). 



It has been shown in kinematics (Part I., Arts. 239-242) that 

 if the force be attractive, the particle will describe a conic section 



