7] THEORY OF JUPITER'S SATELLITES. 163 



and so on. By these formulae we find the derived functions of any order 

 from those of orders next below. We require to calculate independently 

 two of these functions, say the derived functions of b and \. 



Thus in the case of first derived functions we find ~ and -j-^ from 



da. da. 



the formulae 



which may be written more simply 



db a db^ 



db, db, 



= Q, ~^j ^ A&Un t/T ) 



CvQt CbCL Ct 



and by differentiating these we can find the values of -j-| , -r-l , etc. 

 J da. da. 



Let us now return to the case from which we started, that is to 

 say, the case when s = -. The quantities b and 6j are then expressible 



a 



by means of elliptic functions. 



We have 



1 (^ dd> , 1 (* cos<bd(!> 



& = - ' * - 



1 T 2ir 

 ~TT)O l- 



o (1 - 2 



Assume sin(# 0) =a sin 0, 



so that cos (6 (f>) = ( 1 a 2 sin 2 0)*, = A, suppose. 



Then cos(0-<j>)(d0-d<{>) = acos0d0, 



or Ae& = (A acos0)d0. 



Also cos < = cos (0 <f>) cos + sin (0 <j>) sin 



= A COS 0+0.8111*0, 



212 



