68 PROCEEDINGS OF THE AMERICAN ACADEMY 



in which p, q, and A are known functions of the masses and different 

 elements of the orbits. These equations give at once 



D2V = 2D^u' — D'w=(2?— p)sin. (V + A), 

 which, multiplied by 2 D V and integrated, becomes 

 'DY = ^ [R-'—{4q—2p)cos. (V-f-A)], 



in which li = 2n' — n ; 



if n = the mean motion of Uranus, 



and n' = that of Neptune. 



It follows from the value of DV, that if 



H^<4(? — 2p, 

 V-j- A cannot increase indefinitely, and that, therefore, the term 



(2 n' — n) t, 

 upon which its indefinite increase depends, must vanish, or in other 

 words 2 n' — n := 0, 



and V -{- A must oscillate in value either 

 about zero when 2p — 4 q is positive, 



or about 180° when 2p — 4 q is negative. 



The probability of the occurrence of this ratio depends, it will be 

 seen, upon the magnitudes of p and q, which are always of opposite 

 signs. It is evident, from inspeating the computations of Mr. Walker, 

 that Neptune's period of revolution is not less than in his second 

 hypothesis of 166 years ; and Professor Peirce infers from the investi- 

 gations which he has already made, that a period of 166| years, 

 which involves only a slight additional eccentricity, is already a suf- 

 ficiently near approximation to establish the exact permanency of 

 the period of 168 years. As soon, then, as there may be observa- 

 tions sufficient to prove that Neptune revolves in more than 166|, and 

 in less than 169| years, the conclusion is inevitable, that its year is 

 precisely twice as long as that of Uranus." 



Professor Peirce communicated, from Mr. Bond, of the 

 Cambridge Observatory, the following 



