20 I- KINEMATICS OF A POINT. LAWS OF MOTION. 



From which 



h* = ^(l-e*), 



1 



Now since |^ is by the third law constant for all the planets, the 

 factor by which the inverse square of the radius vector is multiplied 

 in order to obtain the central acceleration is the same for all the 

 planets and depends only on the sun. We have thus obtained a 

 complete kinematical statement of the law of gravitation for the 

 planets. 



Newton tested the law of the inverse square by applying it to 

 the motion of the moon about the earth, and comparing its accelera- 

 tion with that of a body at the surface of the earth as directly 

 observed. Supposing the moon's orbit to be circular, of radius a } 

 with period J, since the tangential acceleration is zero, its velocity 



is constant, and equal to ^r" Its acceleration, which is entirely 

 normal, will accordingly be by 30) 



If the acceleration varies inversely as the square . of the distance, the 

 acceleration experienced by a body at the earth's surface a s will be 

 given by 



where R is the earth's radius. Therefore 



Now we have T=27d. 7h. 43 m. = 39, 343 m., 2jrjR = 4-10 7 meters, 

 a = 60 R, from which 



2* 60 s 4 10 7 meters _ Q . meters 

 0/8 = (39,343 60 sec.) 2 sec. 2 



Now terrestrial observations give for the mean acceleration of bodies 



at the earth's surface 9.82 - 2 -> which by a more exact calculation 



sec. 



is in agreement with the predicted result. 



13. Physical Axioms. Laws of Motion. It is necessary in 

 order to pass from the kinematical specification of motion to the 

 dynamical one to make use of knowledge drawn from a consideration 

 of terrestrial phenomena. This knowledge is summed up by Newton 

 in his three Axiomata sive Leges Motus. An axiom is defined by 



