21] ON THE SECULAR VARIATION OF THE MOON'S MEAN MOTION. 151 

 17. In the expression for ; - just found, a, is absolutely constant, but 



itf 



e' is variable, consequently n will vary, and therefore m likewise, which is 



n' 

 connected with it by the equation m = . 



Taking the variation of the equation for n, and observing that 



Sm _ Sn 



m n ' 



8n /i 2\ , /3 3867 A s/ , 

 we have 0= (l-m 2 ) + -m- m 4 )d(e' 2 ), 



IV \ & O ^t / 



Therefore, if -ZV be the initial value of n, and E' the corresponding value 

 of e', 



g 



r /^ *377i \ r 



and hi^=m + -f|m s - -^ m 4 j I (e /2 - E'*) ndt. 



Hence the expression for the true longitude in terms of the mean, contains 

 the secular equation 



18. According to Plana, the corresponding terms in the expression for 

 the secular equation are 



/3 2187 



V2 128 



Hence we see that the terms now taken into consideration have the effect 

 of making the second term of the secular equation more than three times 

 as great as it would otherwise be. Of course, the succeeding terms will 

 also be materially changed. 



The principal term of the correction to be applied to Plana's value of 

 the secular acceleration is therefore 



5355 



