22] ECCENTRICITY AND VARIATION OF THE MOON'S ORBIT. 159 



I find the following expressions for the secular variation of the eccen- 

 tricity and inclination of the Moon's orbit, adopting Plana's definitions of e 

 and y : 



de 



dt = ''dt{64' 



dy = ,de'(22l 2 779 a _^ 199631 

 dt ~ ye dt 1 64 m 256 ^ 



I am engaged in carrying on the approximation to the value of -^ to 



Clt 



the same extent as I have done in the case of -^ , and in finding the 



part of the secular variation of the mean motion which depends on e 2 and 

 y. These terms, however, can only very slightly affect the numerical value 

 of the secular acceleration. 



Supplement to the foregoing. 



Since I sent my result respecting the secular variations of the eccen- 

 tricity and inclination of the Moon's orbit to the Society the other day, I 

 have found the leading terms of the secular acceleration of the mean motion 

 which depend on the eccentricity and inclination of the orbit. The result 

 is one of remarkable simplicity, considering the nature of the calculations 

 which have led to it ; and I should be glad if you would let it appear 

 in the Monthly Notices as soon as you conveniently can, as a supplement 

 or a note to my former communication. The result is, 



dn e'de' f 3771 27 , 27 ,] 



- --- - - 3m 2 + --- m 4 + &c. - we 2 + - 



-T. --Y- - - -5-7- . - - . 



ndt dt { 32 8 8 ' J 



[I have not written down the coefficients of higher powers of m, as 

 given in my former note.] 



It is curious that the coefficients of e 2 and y 2 , in this expression, are 

 equal and of contrary signs, although they are found by totally distinct 

 processes. The effect of the terms in e 2 and y 2 on the magnitude of the 

 secular acceleration is, as I anticipated, very insignificant. The term in e 2 

 increases the coefficient of the square of the number of centuries by 0"'036, 

 and that in y 2 diminishes the same coefficient by 0"'097 ; so that, on the 

 whole, the coefficient 5"'70, which I previously found, must be diminished 

 by 0"'06, or reduced to 5"'64. This value I believe to be within one-tenth 

 of a second of the true theoretical value of the coefficient of the secular 

 acceleration. Whether ancient observations admit of such a small value of 

 the acceleration is a different question. 



