32 The Rev. Brice Bronwin on the Theory of Planetary Disturbance. 



d^ _p d^ ^_^ ^ ^ 

 dt ~ dT dt dr dt dr' 



d-<l> _ ^ d (dip dip d<j) d(f)' 

 d^t ~ TT\dt 'd'r~Yt dr" 



Let the integrals of these relative to t be 



(1) 



/(t) and /,(t) being the arbitraries of the integration. If we make m' = ; 

 then £ = 0, and \ = \^ . But in this case X is a function of t without t. Con- 

 sequently, \,, f, and (p, are functions of t without t ; and, therefore, — - = 0. 

 Hence, all the terms in the second members of (1) vanish, and we have 



^0 and -j~ being the values of £. and ^ when m' = 0. 



dip 



Now if we make the disturbing force to vanish in (2) and (4) of the first 

 section, they become 



d^ _ ho 

 dr 

 d% 



hU' 



1 4- '^ - _^ 



dr -h,^y' 



(2) 



the last of these being derived from the first by putting for ^o its value. These 



equations, being integrated, give us f„ and ^o, or !„ and 0„. 



h ^ 

 To mtegrate the second of these, make ?o = -^ (1 +7), when it becomes 



To abridge, make ^ = 1 + e^", + iE. + &c., where £',,£,, &c., are known 



