THE ORBIT OF URANUS. 9 



F(j being the equation of the centre, and </>// the part of p depending on the eccen- 

 tric ity in the elliptic motion. It follows that if we express the developed expression 

 for R as a function of X, X', g, g 1 , , rf, which we may do by putting 



<j = X g, G' = X' g 1 ; 



9 I rf '. 



we shall have by successive differentiation 



//,' dR dv dR 



<9Jt dv d*. dv 



6V <3v' ^ ,'v 2 



f-lt dR dp < R (6) 



dr> dp dr> ~ /y 

 d /-"It 



' dp* dr>~ (' ,;- 

 etc. etc. etc. 



and in general 



d\ m dp* dv"* 'dp* 



Thus, by expressing the developed R in the above form, we may find the derivative 

 of any order with respect to v, v 1 , p and p, by taking the corresponding derivative 

 with respect to X, X', and '. 



The developed R is usually expressed in the form 



R = I cos (W + tt +/'J + j") 



a, being the mean distance of the outer planet, whether disturbing or disturbed, 

 and h a function of e, <!, a, and y. Substituting for w its value in g, this equation 

 will become 



Putting for brevity 

 the formulae (6) give 



a, 



=-2 **(.'+ co. Jf (7) 



and in general 



5--"'A 

 < a, cos 



March. 1873. 



