THE ORBIT OF URANUS. 



77 



The computation of these terms being extremely complex, a clieck upon their 

 accuracy is desirable. In the case of the secular variations of the coeificicuts, the 

 coefficients of the time are easily obtained by substituting in the integrated per- 

 turbations the variations of the eccentricity and perihelion of Saturn. Thus I 

 have found 



h = +0.0103 / sin (2^ — I') — 0.0094 t cos (2r/ — 

 +0.0027 t sin {Sj — l') — 0.0138 1 cos (3f/ — /') 



Tlie greatest discrepancy is found in the coefficient of sin (3^ — I'), and it amounts 

 to 0".003(S/, or about 0".4 in a century. But, owing to the great period of this 

 term, nearly 600 years, this difference, during any one century, will be nearly 

 eliminated through the mean longitude and mean motion. 



It may also be remarked that in this case the terms derived from the pertur- 

 bations of the elements are undoubtedly the correct ones, and will therefore be 

 employed. 



Tlie terms which the preceding integration fails to give, owing to the constant 

 terms introduced into ^<]Q and yjSQ, are found by (22). 

 We thus have 



nXpJc':^ = + 0".36 

 «2 q.k't' = + .27 

 ri^^p = 1 Mni' \ 0".36 sin g — 0".27 cos g \ 

 6v = ^ Mnf \ 0".36 cos g + 0".27 sin g \ 



= t' J0".0000038 cos r/ + 0".()000029 smg\. ' 



The greatest effect of these terms amounts to less than one-twentieth of a 

 second in a century. They may therefore be neglected in the present theory. 

 The other terms containing the square of the time are yet smaller. 



Applying the terms of the second order thus found to the terms of the first order 

 depending on the corresponding arguments, the perturbations of Uranus by Saturn 

 become 



T here represents the time counted in centuries from 1850.0. 

 The other terms remain the same as given on page 50. 



Perturhatloihs depending on the product of the masses of Jupiter and Sit urn. 



The values of (5i?'(fl, 5 — , and 3 -, depending on the products of the masses 



(?V ^p 



