7] THEORY OF JUPITER'S SATELLITES. 151 



we see that the result of substituting the term e cospt for rSr/a 2 is to 

 produce the following terms on the left-hand side of the final equation 



e cos pt \p- - <f ( I - 4/+ 6/c 2 ) + 3/c 2 f- 



+ ..<*,+,,)[ 



It is to be especially remarked that if p is nearly equal to q the co- 

 efficient of the term ecos(2qp)t arising from the term e cospt in rSr/<r 

 will be very small. 



Now suppose another term in rSr/cr to be e 1 cos(2qp)t, then the 

 result of substituting this term will be found at once by putting e l in 

 place of e, and 2q p in place of p ; hence will arise the following addi- 

 tional terms on the left-hand side of the final equation, viz. : 



e, cos (2,/ -p) t [(% -p)' - ,f(l - 4/+ 6/6) + W p (i q _ p )~\ ' 



Again suppose another term in r8r/a 2 to be e. 2 cos (2q + p) t, then the 

 terms on the left-hand side of the final equation arising from this will be 



which may also be derived from the last by changing p into p. 



