THE 0KB IT OF URANUS. 7 



h, the coefficient of any term of "'- A', so that we have 



m 



n * 1 h xr 



It = 2 - cos N 

 i 



m' Ix-ini; here the mass of the disturbing planet. 



X, -the mean distance of the planet from the node, or the mean value of V. 

 u, the distance of the perihelion from the node. 

 /, the mean anomaly. 



/, the mean longitude, or the mean value of v. 

 4, the aii^le of eccentricity so that e = sin >//. 

 r,, the radius of the planet in the undisturbed ellipse. 

 r,, the quotient of r divided hy the mean distance, which is a function of the 



eccentricity and mean anomaly only. 

 T, the time alter the epoch Ib50, Jan. 0, Greenwich mean noon, counted in Julian 



centuries. 



v, the integrating factors of the periodic terms, or the ratio , JVbeing the change 



of the angle in unit of time. 



/, the eccentric anomaly, and, in the tables, the argument of latitude. 

 \\ - have for the value of R 



R= -- .- (cosvcosv'-l-sinvsinv'cosy) 



V r 1 I?/-/ (cos v cos v'-l-sin v sin v'cosy)-}-/* f* 



or. if we suppose r replaced by its value in p, namely 



r=c' 



ire sliall have 



R n,' f (v, V, p, p', y). 



AVith tliis value of R it is well known that the differential equations for the longi- 

 tude and radius vector of a planet are 



j " ' i r V T__i 



(//" </<* r 



. rfH? </r <7w _ 5/2 



</<* dt dt 6v ' 



If we multiply the first of these equations by 2 - ^ and the second by 2 ^ and 



ctt ctt 



add them together, putting, for brevity, 



and then integrate, we shall have 



