7] THEORY OF JUPITER'S SATELLITES. 171 



Hence if we take i large enough, an original error is increased if we 

 derive p { from p lt but it is diminished if we derive i\ from p t . Let us 

 then derive &, 6 7 , etc. from 6 10 and 6 9 ; 



given b ia = 0'004297246 



b, =0-007187308, 



we find 6 8 =0'01208930 



b 7 =0-02047389 

 1 6 =0-03499074 

 b. =0-06052825 

 b t =0-1065074 

 b. =0-1923497 

 b, =0-3632093 

 b t =0-7543069 

 6 =2-2588406. 



In this case the agreement with the values of b, and & already calculated 

 is perfect. 



Consider now the application of the foregoing results to the disturbing- 

 function 



_ m r __ m'r cos (Q ff) 



~ ~ 



for the disturbances of a body m at the position (r, 6) by a body m' at 

 (r f , ff] in the same plane. 



Let r=a(l+x), 9 =nt +e +y =1 +y, 



where x, y, x', y' are supposed small; also let 



_ _ 1 __ a cos (I I') 

 {<r + a' 2 - 2aa r cos (I - 1')}* ~ ~~^ 



Then if 



where a refers to the inferior satellite and a! to the superior we have 



222 



