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U.—On the Theory of Planetary Disturbance. By the Rev. Brice Bronwin. 



Read November 30, 1848. 



1. In this paper I shall consider, with M. Hansen, the disturbance as af- 

 fecting the radius vector (or rather the mean distance) and the mean longitude ; 

 and I shall first, after his manner, employ two times. Various formula) not 

 noticed by him (one of them fundamental) are given, in the hope that they 

 may some time be made useful. The principal equations are investigated in 

 a way that leads to some very elegant formula;, and the elimination of the 

 quantities containing both the times is effected in a very simple manner. In 

 finding the latitude, I propose to introduce the latitude itself and the reduction 

 into the distui-bance function ; by which means the part of that function de- 

 pending on the inclination of the orbit to the fixed plane is greatly simplified, 

 and the determination of the latitude and reduction rendered easier. But it is 

 not my intention in the present paper to develope the functions in series of sines 

 or cosines. 



The well-known difibrential equations of a planet's motion, referred to the 

 plane of the orbit, are 



^J^ t^ clR 

 df r'^i^^ dr 



,dv , , rdR 



> (1) 



,dv [dR , 



di=''^^'^-\^v'^'^ 



E^a^cosx-r'.-^ _;cosx = ""^^-j'^±^'- 



r -= '^ (r - - 2?-V cos x + r-)' ' a. r'r ' 



where x, y, and z are the rectangular co-ordinates of the disturbed {z = 0), and 

 .c', y', and s' those of the disturbing body. The meaning of the other symbols 

 is obvious. 



VOL. xxri. E 



