THE ORBIT OF URANUS. f 



h, the coefficient of any term of ^ R, so that Ave have 



m' 



liz=Z COS iV 



m' being here the mass of the disturbing planet. 

 X, the mean distance of the planet from the node, or the mean vahie of v. 



0, the distance of the perihelion from the node. 

 g, the mean anomaly. 



1, the mean longitude, or the mean value of w. 

 4-, the angle of eccentricity so that e = sin i^. 



rj, the radius of tlie planet in the undisturbed ellipse. 



j-j, the quotient of r^, divided by tlie mean distance, wliich is a function of the 



eccentricity and mean anomaly only. 

 T, the time after tlie epoch 1850, Jan. 0, Greenwich mean noon, counted in Julian 



centuries. 



V, the integrating factors of the periodic terms, or the ratio , iV" being the change 



of the angle in unit of time. 

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

 We have for the value of Ji 



7?= ^(cosvcosv-|-smvsmv cos^) 



V t"^ — 2r?"'(cosvcosv'-|-sin v sin v' cos y)-}-/^ *' 



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

 wc shall have 



n = "''/ (^% "^''■. P' p'' y)- 



AVith this value of 7? it is well known that the differential equations for the longi- 

 tude and radius vector of a planet are 



dh- _„ dv" ^ fi (1 + w) ■^ . 



(1) 



"" de~'"'de^' r ~^<9p' 



, d'^v . , dr dv dR 



de ^ dt dt ' dw 



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



(Xt etc 



add them together, putting, for brevity, 



dR dp SR dv ,n\ 



op dt cv dt 



and then integrate, Ave shall haA-e 



