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LXIII. On the Variable Elements of a Disturbed Planet^ and 

 the Equations of its Motiofi on the Plane of the Orbit. By 

 the Rev. Buice Bronwin*. 



THE following method of determining the elements of a 

 disturbed planet, based upon a theorem of Lagrange, 

 leads to a useful result. R being the perturbating function, 

 the well-known equations of the planet's motion are 



d'^x lux dK _ d^y u.y ^R _ ^ "^ 



d^z fj.z ^^ _ rs 



7^'^ ~^'^~d^ ~ 



(A.) 



Let X, y, z be determined from 



\dt^) + r^ - ^' \dtV + r^ -"' \dtV ^ 1^ -"• • ^""'^ 

 These values of x, y, z will satisfy («.), however the ele- 



ments may vary, since t alone varies in \-t-^ ), &c. They 



may therefore be made to satisfy (A.). But we have six quan- 

 tities to determine and only three equations : we must then 

 assume three conditions; let them be 



8.r = 0, 8^ = 0, 8sf = 0, .... {J).) 

 % denoting the variation of the elements only. Then 

 dx (dx\ - d^x (d^x\ Idx _ 



Tt = Kdth ^'^•' dW = \w) + -dF^ ^^' 



Put these values in (A.) and subtract («.), there result 

 ^dx d^ ,ldy dV. ^ U% ^R 



-d¥ + ;^=^' li -^ 7^=^' -dt^ + ^=^' • • (^-^ 



where d refers to the variation of ^ only, 8 that of the elements. 

 If A X, Ay, A s; denote variations indeterminate, and inde- 

 pendent one of another, we easily form from (b.) and (c.) the 

 following, which is equivalent to (b.) and (c), and contains 

 the whole solution of the problem : 



. idx ^ Adx , . ddy . Ady 

 dt dt ^ dt ^ dt 



, (B.) 

 Leaving out ARdtf this is independent of t. For since 



. ^dz ^ Adz ^ . r^ .^ ^ ( 

 + A2-r- — 82;— 5— - + ARrfj^ = 0. 



dt dt J 



dA X _ Adx dZx _^dx » 



~~dr"~dr' iu~~dr'^'^"' 



* Communicated by the Author. 



