26 The Eev. Brice Bronwin on the Theory of Planetary Disturbance. 

 When m' = 0, let the integrals of (1) be 



'"I =/ (*> ^0, eo, TTo, e„), Vo= (p (t, floi eo, TTo, e„). 



And when the disturbing force is restored, let 



r =/ (t, a, e, tt, e), v = (p {t, a, e, it, e). 



Thus r and v are the same functions of t as when m' — 0, the elements a, e, &c., 



being variable, and determined in the usual manner. And, t being a new time, 



we make 



P =f (t, a, e, TT, e), \ = <f) (t, a, e, tt, e). 



We further suppose, 



'■/=/(^- «/' <^,' "^z- ^/)> ^/= (^' '^n «/' '^z' '/) ; 



the elements a^, e^, &c., and A^, to be presently introduced, being constants, z a 

 function of t, and f a function of t and t. To these we must add the assumed 

 relations, 



r = r,^, P-P,^, u = w^ + ^nt, X = \, + Qnt. 



By changing t into t, we change p into r, X into v, />, into r^, X^ into v^, 

 I into /^, and f into z. 



From what has been given above, we have necessarily ?■'; -j-^ =h^, 



p] -J- = hi. And from the way in which a, e, &c., are found, i/r = 0, Iv = 0, 



the characteristic 8 denoting the variation relative to a, e, &c. ; therefore, 

 d\ 



But p'~^ p]e —' = p'e ?^ $ = Ke ^. consequently h,^ ^ = A; or, 



^^^° ^' jt = ^' I' + ^-^ - ^'^^ £ S + ^-'^^ = ^''^-' jt + ^'^^'^^ = ^^ ■' ''' 



dz _ h Znr] ,„. 



di ~hp~l^' ^ ^ 



p- -r- = A- 



