7] THEORY OF JUPITER'S SATELLITES. 159 



multiplied by A 2 



s (s+l) s(s + l)(s + 2) 3 . 2 6| s ( s +l)(s + 2)(s + 3) 4 . 3 

 1.2 1 .~2 . 3 " F72 a 1.2.3.4 " 1~2 * 



_____ 

 (1 - a 2 ) 8 f".~2 ~ (1-a 2 ) 2 ' 



and so on, the transformed expression being 



_ ___ _ _ 



(l-a')' H-a- 1.2 (1-a 2 ) "T72T" 



But taking the first differences of the coefficients 





, * + i (s + i)(s + i+l) (s + i) (a + i+l)(s + i + 2) 



' i+l' (i+l)(i + 2) ' (t+l)7Tf2y(t + 8) 

 we have 



s-l s-l s + i s-l (s + i) (s + i+l) 



i + l' t'+l + 2 ' i~+l (i + 2)(i+3) 



Taking second differences 



(s^l ) (s - 2) (s- l)(s- 2) s + i (s-l) (,-2) (s + i)( s + 



(i+l) (i+2)' ([,+ }) (i + 2) i + 3' (i+l) (i + 2) (i + 

 The law of succession is evident ; we have 



and 



.s^+ n ... I'.s' + ^-n a 1 r 



+ 



(!-')' 



+ 1 .~2~(i+T)"(tT2JT (r-a 3 ) 2+ "'_r 

 This expression is very useful for computing b t for large values of i. 



We observe that the expression within square brackets is unchanged 

 if we write 1 s for s. Hence if 



cos <> + cos 



Lt 



we have 



