208 



ON THE THEORETICAL VALUE OF THE [26 



and therefore that the new forces, the effects of which are to be found by 

 the present method, are Tbt and Pbt respectively. 



On consideration, however, it will appear that this is only true for the 

 non-periodic term in P, and that the periodic terms, whether in P or T, 

 will be changed by any given variation of E in very different ratios. 



For instance, the periodic terms in both T and P which depend on 



5 

 the angle 2D or F will vary nearly in the same ratio as 1 - - E 3 does, 



g 



instead of in the ratio in which 1 + - E 3 varies as in the above case. 



Zi 



Hence these terms will be changed by the above-mentioned variation 

 of E in the ratio of 1+b't to 1, where 



b = 5 - nearly. 



Again, the periodic terms in T and P which depend on the angles 

 S, FS and F+S will vary nearly in the same ratio as E does, so that 

 these terms will be changed in the ratio of l + b"t to 1, where 



<S ET 



b" = p nearly. 



Hence we see that the values of b' and b" are quite different from 

 that of b which belongs to the non-periodic term, and that b" is much 

 larger than the other two quantities. 



The correct way of finding 8T and SP, the changes of the disturbing 

 forces T and P due to change in the eccentricity of the Sun's orbit, is 

 to express T and P in terms of the Moon's coordinates v and r, the 

 Sun's mean longitude L and its mean anomaly S, and the eccentricity E. 



Hence 8T and SP may be at once expressed in terms of 8v, Sr, and 8E. 



Thus calling V the Sun's longitude, and employing the other symbols 

 in the sense before explained, we have 



