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

 of Art. 9, similarly give 



15m 2 f, / 11 ,e'de'\ 21 m 2 (, / 77 ,e'de'\ ,3m 2 /", ill 'de'\ 

 -- \dvl--z-m* -j- --; <*"( 7Z m -jr)+7 " W--- 



2 aj \ 8 nrfy 4 aj \ 16 wctt/ 4 a, J 



T.--J- 

 16 ndt ] 



165 m 4 , 2 1617 m 4 /2 33 m 4 

 = -- -e' 2 + ~. -e 2 + r^ -e* nearly 

 32 a y 128 a, 128 a x ' 



495 w. 4 

 = -- e' 2 as their non-periodic part. 



12. Hence we obtain 



1 1 1 m 2 / 3 ,.A 9 m 4 495 m 4 27 m 4 /2 

 = - + 1 + e '- e - 1 - e + 



4 a 16 a 16 



_ _ 



2 a v 8 a x 8 a, 8 a 



or 



9m", ,, 441m 4 /2 9 m 4 , 



4 a, 16 a, 16 a, 



= --l--^ 2 -- mV 2 + 3m 4 + r m 4 e' 2 l 



a a, [24 64 J 



_ fyYi" 



Now m2 = 7T~~ \5 ' m Plana's notation, or (substituting the value of p given 

 in Plana, Vol. ii. p. 855), 



91 J 1 



m = m 2 1 1 - m- - me 2 1 nearly, 



13 3 



-m 4 - 



.. 



a a, [ 2 4 4 64 



Joo q q-i OQ 



___ -. l+m 2 --m 4 + mV 2 -- -m 



13. Again, by substitution in the equation for -r- we obtain 



