290 PROCEEDINGS OF THE AMERICAN ACADEMY 



upon those integers whose ratios best represent those of the revolution- 

 periods. This character is probably most peculiar and beautiful, 

 because most distinct, when the period-ratio is 



1 [re an integer, p large,] 



n + 1 



i + ..i 



i + 1 

 p 



as in the actual solar system ; for were the period-ratio 



1 [m, . . a, 5, . . c, . . integers, & > 1,] 



M + . . 1 



a-\- 1 



^ + ..1 



c + .. 



the character due to the ratio ('«+-l) would be somewhat confused 



and overlaid by that of the lower ratio i» + -l). Since p is large, 

 the pattern nearly repeats itself after a few revolutions. 



Professor Peirce thought that different discussions would 

 require different criteria of excellence in the representation 

 of the constant-ratios involved ; and that it would be of inter- 

 est to determine that form of V which is smallest for those 

 approximate period-ratios which best represent the aggregate 

 of all " terms of long period " in the theory of two or more 

 planets. 



Five hundred and thirty-fifth Meetin;^. 



May 24, 1864. — Annual Meeting. 



The President in the chair. 



The Corresponding Secretary read letters relative to the ex- 

 changes of the Academy ; also a letter from Richard H. Dana, 

 Jr., in acknowledgment of his election into the Academy. 



The Report of the Council upon the changes which have 

 occurred in the Academy during the past year was presented, 

 and read in part by the Corresponding Secretary, as follows : — 



