27] OF THE MOON'S MEAN MOTION, ETC. 217 



which gives the relation between n and a. 



4. In the above, e' is considered constant throughout ; if now we 

 consider e' to be variable, we may choose n and a so that the constant 

 (or rather the non-periodic) parts of u and of H* may have the same 

 forms as before, and in this case we shall find the same relation between 

 n and a as that which has just been found, and n will continue to signify 

 the actual mean motion at the time to which 6 belongs, but n and a will 

 now become variable quantities, and, in order to satisfy our equations, it 

 will be necessary to add certain periodic terms to u and H* which would 

 not exist if e' were constant. 



Suppose then that 



w = - jl + Su - 1 mV cos n't + m* ( 1 - - e'' 2 } cos (28 - 2n't) + - rrie' cos (20 - 3rit) 

 ft ^ 2i \ Z / 2i 



-}-m?e' cos (2d-rit)\, 

 and 



H* = n*d* jl + 2Srj + -J-- m 1 + 2 4f m 4 e" + 1 m 2 f 1 - ~ e' 2 ) cos (20 - 2n'<) 



91 S 1 



Y mV 2 cos (26> - 3n') - - m?e' cos (20 - n't] Y . 



We will suppose e' to vary uniformly with the time, and very slowly, 

 or, in other words, we will suppose 



de' 4 . d'e' 



~r- to be constant, so that r-^ 0, 



and we will neglect \~7fi i ' 



A. 



28 



