Tlie Rev. Brice Bronwin on the Theory of Planetary Disturbance. 31 

 The above, subtracted from this, gives 



'd7Jt^^<'iTt=dt~^''^' 



or, 



d ( , d\\ dh „ ^ 



drY'Tt)=Jt-^P^- (10) 



This is singular, and very worthy of notice. This theory, as we have 

 treated it, gives rise to many very interesting formula. 



Rut „i^^ - 2 (^\ , p o y, -idX^ dt • fir 



Therefore (10) becomes 



Or, since A|-=4-, 



^.(*-*-|)--«-'4*-^'-- 



hdt h^P^-Ji^T- (11) 



2. In the preceding section we have found all the fundamental formula 

 and indeed more than are absolutely required. We now proceed to discuss' 

 some points preparatory to integration. 



We might immediately integrate (11) relative to^; but the integral of the 

 second member would contain a very great number of terms with r in their 

 coefficients, or of the form rf{t). We must, therefore, proceed otherwise. 



Make^ = < + a,, f = T + rf,. Then ^-^ ^^-1 ^'^'^ c; k .> .- 



.J, T y. xjieu j^-^, ^-^+"^- Substituting 



these values in (9) and (11), neglecting terms involving the fourth power of 

 the disturbing force, and putting 



1 J, &n^(m_ Idh 1 Qn d(p^) „ 



hP" +2A dr -^'kdi-k^P^-J -d7-^^' 

 they become 



