198 Sir William Thomson. [Nov. 26, 



for the elimination of the ratios 0|x[# is symmetrical with reference 

 to p and 1 p. Hence it is 



+/- 1 )+2C ......... (12), 



where , Cj, C 2 , C 3 are coefficients of which the values in terms of 

 11, 12, &c., are easily written out. The determinant equated to zero 

 gives an equation of the sixth degree for determining />, of which for 

 each root there is another equal to its reciprocal. We reduce it to an 

 equation of the third degree by putting 



.. .................. (13). 



Let ej, e 2 , e 3 be the roots of the equation thus found. The correspond- 

 ing values of p are 



In the case of e having any real value between 1 and 1, it is con- 

 venient to put 



e = cos , "i 



which gives p = cos a + 1 sin a V (15). 



and p~ l = cos a. i sin j 



8. Suppose now, for the first time of passing through ty = 0, the 

 three coordinates and three corresponding momentums, 0i, Xu ^i> 

 j ?7ij &> to be all given ; we find 



...(16), 



where A b A' 1} A 2 , A' 2 , . . . . , F 1? F'j, F 2 , F r 2 are thirty-six coefficients 

 which are determined by the six equations (16), with i = : and tho 

 six equations (8), modified by (9); with i successively put = 1, 2, 

 3, 4, 5 ; with the given values substituted for b % t , $ 1? i, j/i, g\ in 

 them; and with for 02, %2 &c., their values by (16). 



9. Our result proves that every path infinitely near to the orbit is 

 unstable unless 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 con- 

 sideration of paths very nearly coincident with an orbit. 



