of a Finite Conservative System. 559 



In the case of e having any real value between 1 and — 1, it 

 is convenient to put 



£ = cosa, -\ 



which gives p = cos a + ^sin a > (15) 



and p -1 = cos a — tsin a J 



30. Suppose now, for the first time of passing through 

 i/r = 0, the three coordinates and three corresponding mo- 

 menta, $j, xi, *u ?i> Vi, Si; to De a11 g iven 5 we find 

 cf> i+ , = Ajpx 1 + Ax V + A 2 /V + A//V 1 + A 3 p 3 * + ±Jpf* 

 % . + 1 = B^ + B/pr* + Brf + B a > 2 -' + B 3 ^ + B// 



S (16) 



5+ , = F lPl * + F/^r' + *W + F 2 > 2 - + F,p 8 ' + F 3 > 8 J 



where A la A/, A 2 , A 2 ' . . . F 1? F x ', F 2 , F/ are 36 coeffi- 

 cients which are determined by the six equations (16), with 

 2 = 0: and the six equations (8), modified by (9); with i 

 successively put=l, 2, 3, 4, 5 ; with the given values substi- 

 tuted for (/>!, xi, $i> f u Vi) ?u in ttem ; ^and with for 2 , ^ 3 , &c. 

 their values by (16). 



31. Our result proves that every path infinitely near to the 

 orbit is unstable nnless every root of the equation for e has a 

 real value between 1 and —1. It does not prove that the 

 motion is stable when this condition is fulfilled. Stability or 

 instability for this case cannot be tested without going to 

 higher orders of approximation in the consideration of paths 

 very nearly coincident with an orbit. 



[To be continued.] 



Postscript, November 10, 1891. 



The subject of periodic motion and its stability has been 

 treated with great power by M. Poincare in a paper, " Sur le 

 probleme des trois corps et les equations de la dynamique," 

 for which the prize of His Majesty the King of Sweden was 

 awarded on the 21st of January, 1889. This paper, which 

 has been published in Mittag-Leffler's Acta Mathematica, 13, 

 1 and 2 (270 4to pp.)? Stockholm 1890, only became known 

 to me twelve days ago through Prof. Cayley. I am greatly 

 interested to find in it much that bears upon the subject of my 

 communication of last June to the Royal Society u On some 

 test cases for the Maxwell-Boltzmann doctrine regarding 

 Distribution of Energy ; " particularly in p. 239, the following 

 paragraph : — " On peut demontrer que dans le voisinage 



