172 SECULAR ACCELERATION OF THE MOON'S MEAN MOTION. [23 



The result which M. Plana obtains in this Memoir is 



dn d(e'*} f 3 351 4 } 



, = -\ - vnr + - m 4 > , 

 ndt dt { 2 64 J 



and the difference between this result and mine arises in the way I have 

 explained, viz., from his having neglected to take into account the term 



285 4 

 8 m< 



which is shewn in Art. 11 of my Memoir to constitute part of the non- 



j- f o f > dR / 



periodic term 01 2 Irj-av. 



J dv 



M. Hansen's value of the secular acceleration is not exhibited in an 

 analytical form, like those of MM. Plana and de Pontecoulant, and we can 

 therefore only compare his numerical result with theirs. These differ con- 

 siderably, and, in fact, much more than appears at first sight, on account 

 of a reason which I will explain. 



dn rr d(e'*} 



where K is the coefficient found from theory, the secular equation to be 

 applied to the mean longitude will be 



E' being the eccentricity of the Earth's orbit at the epoch from which t 

 is reckoned. 



Now I find that M. Hansen uses a smaller value of the integral 



than M. Plana does ; that is, he supposes a slower change in the eccentricity 

 of the Earth's orbit : and yet his resulting value of the secular equation is 

 larger than those of M. Plana. 



It may be inferred, either from the data in the Introduction to 

 M. Hansen's Solar Tables, or from other data in the Introduction to his Lunar 



Tables, that the value of the integral \(e'" E'^ndt which he employs is 

 5f, t being expressed, as usual, in centuries. 



