relating to the Moon's Orbit. 279 



p being the projection of the moon's radius-vector on the plane 

 of the echptic. Multiplying the equations by 2dx, 2dy and 2dz 

 respectively, adding, and putting V for the moon's velocity, we 

 have 



, ,„ 2iJbdr m'rdr 3»i' , „ -7^ — -pc. 



- ~ {pW sin 2f^' + zdz) . 



It is proposed to integrate this equation so as to include all 



small quantities of the second order. That the approximation 



may apply to the case of the moon's motion, it will be assumed, 



in accordance with what is known by observation, that the true 



longitudes 6 and 6' always differ from mean longitudes nt + a. 



d& . 

 and 7i't + u' by small quantities. Hence -^ is a small quantity 



of the first order. Omitting, therefore, the terms in the above 

 equation which involve d6' and zdz, and putting for ?' the sun's 

 mean distance «', all small quantities of the second order are 

 still retained, and we get by integration, 



V' + C='ji + ^^V«^cos2«-39. . . (A) 



Again, small quantities of the second order being included, 

 d-u d^x Sm'p^ . n^ — ^ 



d'^z d'^x _ d'^z d^y _ 



The first of these equations is equivalent to 

 pHQ Sm'p^ . „ — ^, 



Za'-^ dt dU 



and 



sin 26^' ■^df= - sin {2n—n't + 2ai^u')dt nearly. 



Hence by integrating, and neglecting all small quantities of a 

 higher order than the second. 



p'^dd 3m' c, „ Pi — rz, 



—TT = c, + T-Fj-^ cos 26- 6'. 



Also we have 



^^~^i'^4«'V 



dz dx _ dz dy 



''Tt~^Tt-''^ ydi'-lt^"^- 



