33 



ISTow'^=P« 4'+^" (IJ) 



dit du 



-3 



du Pu do 



J /du'^ \ dK _ Pu d u Pu I. 



and — (^_- + „.j^^- j-^ rf„ 



hence we easily deduce from equation (11) 



The equation a is equivalent to the first of the equations (L) of 

 Laplace,* and the equation (c) is equivalent to the second of the 

 equations (L) when s = o. 



(") 

 Application to the Lunar Orbit. 



1. If we suppose the moon only attracted towards the earth, 

 P = o, and R varies inversely as the square of the distance from the 

 earth. Let R = ^, M representing the sum of the masses of the 

 moon and earth. Let also a (1 + e) and a (I -e) represent the 

 greatest and least distances of the moon from the earth on this hy- 

 ^thesis : these distances must be invariable, because the centripetal 

 forces (the force P being = o on this hypothesis) are equal at equal 

 distances on each side of the apsids. We can obtain the values of the 

 constant quantities h and k of the equation (6) in the following 

 manner. Equation (9) becomes u' = —J -75 + 1^ - ^ f'^ 



therefore V and V representing the velocities at Perigee and Apo- 

 gee respectively 



When P=o equation (7) gives 



VOL. XIII. o 



* Mec. eel. p. 181. torn. 3. 



