OF ARTS AND SCIENCES. 61 



" Now, since all values of 7r= {w — v),w being the longitude on the 

 orbit, are d priori equally probable, and since the maximum value 

 of u is ± 90°, for e = 1, we have the a priori probabilities for e as 

 follows : — 



Limit of e. A priori probability. 



Limit e > 0.06 and < 1.00 |^ 



2-4 



¥Tr 



3:8. 

 50 



221 



e > 0.006474 < 0.01 -%%i 



" This d, priori probability, that e falls between 0.01 and 0.006474, of 

 f , is derived from a theory, which in a half-year's path of Neptune 

 presents throughout a probable discrepancy of 0".49 between theory 

 and observation. 



" The next inquiry is, how far this value of e is consistent with the 

 equations of condition between e and a, derived from the residual per- 

 turbations of Uranus. From the two full computations of Mr. Adams's 



Supplement to the Nautical Almanac for 1852, for values of -^ r= 



0.50 and 0.51, e varies from 0.16103 to 0.12062. Hence Mr. 

 Walker found the conditional equation, 



e = 0.16103 milU] (P^^^X 



\log. r.i^J 



Whence for a = 30.20058, e z= 0.0153883, which is the eccentricity 

 from Adams's computations, with this value of the mean distance. 

 The mean longitude of Neptune, according to Mr. Adams's remark, 

 also comes out right for this hypothesis. 



" It remains to consider Le Verrier's limits in his additions to the 

 Connaissance des Temps for 1849. In his first solution, he gives 

 for the minimum limit of the mean longitude of Neptune for 1800, 

 234°, whereas Elements II., with e<[ 0.0153883, would require at 

 that date a mean longitude of 226°. In his final solution, Le Verrier 

 finds the most probable value 240° nearly. The limit ± 5° gives for 

 the minimum 235°. If it be asked why Le Verrier and Adams differ 

 in their conclusions, it may be answered, that they differ in their resid- 

 ual perturbations required, from the more complete computations of 



