170 THEORY OF JUPITER'S SATELLITES. [7 



Now compute 6 2 , b 3 ...b s by successive steps from the sequence equation, 

 which when s = - assumes the form 



and we have the choice of proceeding backwards from 6 10 and b,, or for- 

 wards from b t and b^ If we try the latter method, 



given /; = 2'2588406 



6, =07543069, 



we derive 6, =0'3632098 



b 3 =0-1923505 

 b t =0-1065085 

 b, =0-0605299 

 b, =0-03499318 

 b, =0-02047747 

 b s =0-01209361 

 b, =0-00719524 

 6 10 = 0-00430918. 



The agreement with the values of b s and 6 10 already calculated is not 

 very good, and we can trace the reason for this ; for if in the sequence 

 equation 



we denote bjb^i by p { , and a small error in this ratio by Ap i( we have 

 the relation 



_ i + s 1 A^ 

 *?*+*- i- s+ 1 p *' 



so that 



_ s(s - 



but 



f s(s+l) ... (s + i-2) 



therefore ^-' A = ' 



