152 ON THE SECULAR VARIATION OF THE MOON'S MEAN MOTION. [21 



Now JV 2 - E'^ ndt=- 1270" (^J nearly, 



where t is expressed in years ; therefore the numerical value of this term is 



This result will serve to give an idea of the numerical importance of the 

 new terms to be added to the received value of the secular acceleration, 

 and probably will not differ widely from the complete correction; though in 

 order to obtain a value sufficiently accurate to be definitely used in the 

 calculation of ancient eclipses, the approximation must be carried considerably 

 further. 



The new periodic terms added to the Moon's longitude are perfectly 

 insignificant, the coefficient of that involving cos c'mv, which is by far the 

 largest of them, only amounting to 0"'003. 



19. Transforming the expressions found above, so as to obtain the 

 Moon's longitude and radius vector in terms of the time, and writing for 

 convenience nt instead of $ndt + e, mnt instead of mnt + ^, and c'mnt instead 

 of c'mnt + e' w', we have 



11 / 5 \ 74 e'de' 



v = nt + m" 1 - C H ) sin (2 2m) nt ^- m 2 -j- cos (2 - 2m) nt 



da' 



3me' sin c'mnt 3 5- cos c'mnt 



ndt 



77 215 de' 



H -- mV sin (2 2m c'm) nt -\ --- m 2 T - cos (2 2m c'm) nt 

 16 48 ndt 



11 257 de' 



mV sin (2 2m + c'm) nt - - m 2 - ,- cos (2 2m + c'm) nt 



a 11 201 



- = au=l m 4 m 4 e 2 



K \ ono p'clp 1 



1 - e' 2 cos (2 -2m)nt+-.^ m 2 *. sin (2 - 2m) nt 



^ / 1 ndt 



- m?e' cos c'mnt 3m 3 r sin c'mnt 

 2 ndt 



+ - m*e' cos (2 2m c'm) nt m? r sin (2 2m c'm) nt 

 2 24 ndt 



1 91 de' 



- mV cos (2 2m + c'm) nt + - m 2 - j- sin (2 2m + c'm) nt. 



