OF THE MOON’S MEAN MOTION. 
405 
17. In the expression for “just found, a, is absolutely constant, but e' is variable, 
consequently n will vary, and therefore m likewise, which is connected with it by 
rJ 
the equation ^ 1 =-- 
Taking the variation of the equation for w, and observing that — = — — , we have 
In 
3867 
Sra 
— = — I 
3771. 
64 
■m 
64 
e'^). 
m 
Therefore, if N be the initial value of w, and E' the corresponding value of e'. 
n=N- 
3771 
— E'^), 
and 
^ndt= 
3771 
64 
E'®)w<77. 
Hence the expression for the true longitude in terms of the mean, contains the secular 
equation 
2^ 
3771 
■“64 
m 
18. According to Plana, the corresponding terms in the expression for the secular 
equation are 
-(I 
nf — e’^ — E'*) ndt. 
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 
^{e'^ — '£I^)ndt. 
128 
Now 
E'")wrf^= — 1270"(^) nearly, 
where t is expressed in years ; therefore the numerical value of this term is 
- r '-66 
looy ' 
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 suffi- 
ciently accurate to be definitively 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. 
