THEORBITOFURANUS. 9 



Fg being the equation of the centre, and <pg the part of p depending on the eccen- 

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

 for li as a function of A, X', g, g\ », »', which we may do by putting 



6) = X — <7, (J = ;i' — ^; 

 = 0", d =^ c°'; 



we shall have by successive differentiation 



dR _ dR dv_ _ 6R 

 dX ~ ^v ^X~~6y' 

 6'R _ ^R^ Sv _ d-R 



dR _ SR 6? _ SR (6) 



<9x) dp dr) (9p 



6-R _ 6^R dp _ d-R 

 dr^ dp^ dry ^p^ 

 etc. etc. etc. 



and in general 



eX"^ (5»" dW" do'" SV" 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', p and p', by taking the corresponding derivative 

 with respect to X, X, » and »'. 



The developed R is usually expressed in the form 



^ =z 2 — cos ii'X + i-k -\-j'J +yu) 



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

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

 will become 



= 2 ^* cos ((^'+y') ;i' + (^+i) X -j'g' -jg). 



R 



Putting for brevity 



the formulae (6) give 



dR _ m'h .•,.-,• -KT 

 — 2 (t+j)smiV 



dv cti 



-—-=—2 (t+;)-cosiV (7) 



av" «! 



2 »i -K— cos N 



dp dn 



and in general 



^v" <5v'"' (9p" ^p'^ = S ± »«' (* +i)"(*'+i')" -Q^r^h- s^n ^ 



rin + n'" 



March, 1873. 



