The Rev. Bkice Bronwin on the Theory of Planetary Di.stiirhance. 39 



0' rf^(5°) e/i0 t/-(p^=) d4> 



2/«, ^/t Jt I * rfT/ "^ h, c/t= V '/t / ^ h, dr 7h' ^ 



2 dr' 2h, dr" dr ' 



^ _ /^ _ , #^ .U^ d^_]_M d4>^ 



dr ^'^ ■' [dr ^ dr'r^ dr dr 2h dt dr' 



and 



€n c/(p°^) d£ tn^ d(j) cP<f, 

 2/l, dr dr' h^ dr dr ' 



We must now put these values of iS and S -j- in (11) of tlie last section, 



UT 



and we shall find as the result, 



dt drA dr) 2h, dr dr\ ^^drj^ h, dr'\ dr T 2 dr' 



en d'ipT) ^d^ Qn^ dj^ dr^ 



h, dr' 2 dr "^ h, dr dr'' ^ > 



We can do nothing with the last in its present form, on account of the 



terms which contain and — . But happily these may be driven out by a little 



transformation. Neglecting terms containing the third power of the disturbing 

 force, (4) gives 



/CO. ^ _ 1 . CON '^ _ , d(S^ t/0 d'cp \ dhd<j> Snpf d'<p 

 ^' ^ drdt -^^ ^ dr '^ dr dr dr' ^ 2h dt dt h^ dr' 



en djp"^) rf0_ 

 2h^ dr dr 



^0 _ , (/0 ^ _ d'4> iJ^ d'(t> ]^dh d^ _ Z_npJ_ rf> en djp";) d<j) 



drdt - dr drdt ^ drdt ^ Tt ~dr' ^ 2ii It llr h~ 7f? 2h^ dr d^ ' 



With this value we eliminate (5") from (4), in those terms where the dis- 

 turbing force rises above the first power, and we thus find 



