35 



will be of use hereafter, and which also proves the third law of 

 Kepler. 



Because t = ;-= ^^=-=; consequently, 



since by equat. (16) the orbit is an ellipse, 



the periodic time is as (««> w "?-)! ^7) 



and hence about the same centre of force, neglecting the masses 

 of the revolving bodies, the squares of the periodic times are as 

 the cubes of the greater axes. 



2. Supposing the moon also acted on by the sun. A small force 

 P exists, and also a small alteration of the force R takes place, and 

 then the integral of the equation -(b) is less easily obtained 

 than that of equation (c). Now as the values of P and jR de- 

 pend partly on the relation of u and v, we cannot exactly ascer- 

 tain the former without knowing the latter ; therefore previously to 

 the integration of equat. (c), we must use approximate values of 

 these quantities obtained by the integration of equation (b), when 

 P = oandR = ^- 

 By equation (16) 



I-f-e cos (v — ot) 

 "= a ( 1—e^ 



or if we regard only the first power of the excentricity, 



« = i (1 +e CO* (y—^y) (18) 



also h = ^ aM 



Therefore when P = o, equation (a) gives 



dt = ^ = ^'(1-2 e cos 0-0) 



g2 



