THE ORBIT OF URANUS. 



Let us, for convenience, replace x and y by two other variables and n connected 

 with them by the equations 



x a^ 



y = an cos 4". 



and v; are then functions of the eccentricity and mean anomaly only, and may be 

 developed according to the multiples of the latter. Substituting the last three 

 expressions in the preceding value of r 2 <5p it becomes 



If we put r* for the value of r when the mean distance of the planet is put equal 

 to unity, so that r l5 like and Y/ contains only the eccentricity and mean anomaly, 

 we shall have 



} (13) 



We must now express and YI in terms of the time, or of the mean anomaly. 

 Putting for the present u for the eccentric and v for the true anomaly, we have, 

 by the theory of the elliptic motion, 



x = r cos v = a (cos u e), 

 y = r sin v = a cos ^ sin M, 



from which follow 



= cos u e, 

 vi = sin u. 



As and YI are to be expressed in the form 



= | Ip t cos ig, 



= 'Z sin 



the finite integrals extending to all values of i from oc to -}-cc, we shall deduce 

 general expressions from p t and q t arranged according to the power of the eccer- 

 tricity. Since 



u = g -f- e sin u, 



we have by Lagrange's theorem 



cos u = cos g e sin 2 g 

 or 



using the notation 

 We then have 



r 



n\ = 1.2.3 n= F(n + 1). 



0! = 1! = 1. 



