30 THEORBITOFURANUS. 



quantities of the first order, in which case d and r may be assumed equal. These 

 values are 



^cp = sin r ^,p -\- cos r ^q 

 sin ^ ^T = cos <p (cos r /^ — sin t ^.q) 



the terras dependent on ^j)' and tq being omitted because, being purely secular, 

 they may be included in the mean values of <p and r. Substituting in the expres- 

 sion for hjS 



cos 138(3 = cos (^ \ sin v ^q — cos V(^p\. (37) 



In the case of all tlie larger planets both cos [3 and cos (p may here be put equal 

 to unity, when the expression for 8(3 will become 



S(3 = s'mv<'q — cosv^p. (38) 



To develop tliis expression in purely periodic terras we must substitute for v its 

 value in terras of the mean longitude or mean anomaly, namely, 



5 



■y = Z -)- 2e sin </ -j- e^ sin 2<j -\- etc. ; 



suppose the terms of 8p and ^.q depending on any argument, N to be 



^p = — a, sin N — «,. cos JV , oq. 



8q = a', sin N -\- a\ cos N 



and put 7t for the longitude of the perihelion, so that 



then, to terms of the first order with respect to the eccentricities, we have 

 ^/3 = — e ((/^. cos 7t -(- a\ sin n) sin N ^ e («^. cos n -\- a\ sin 7t) cos N 



+ 3 1 C«* + «'c) cos 7t + {a, — a,) sin n \ sin (A^+ <;) 

 + I ! («c — O cos 7t + (rt'e + «J sin 71 1 cos {N-\- g) 

 + 3 1 («* — <•) cos 7t 4- («', + a,) sin tt | sin (.V — (j) 



+ 3 \ («e + «'») COS 7t + («'e «J slu 7t J COS (iV </) 



+ 2 ^ ! (*^» + **'<;) COS 7t -j- («', — rt„) sin 7t j sin {N-\- 2g) ^ ^ 



+ 3 e j (a, — a'J cos 7t + « + aj sin tt | cos (^ + •2g) 



+ I^K"" — "'^-^ coSTt + (a'., -|- a,.) sin 71 ( sin (^ — 2g) 

 + I e j (a, + a'J cos 7t + [a, — a,) sin 7t | cos (iV— 2//) 



The point of the orbit from which 7t and ?' are counted is entirely arbitrary, 

 and, in considering the action of but a single planet, it will be most convenient to 

 count them from the common node, in which case n must be replaced by u, and 

 ("p and 8q by hk and ^y;.' Thus, deducing tlie perturbations of the latitude imme- 

 diately from the formulae (27), we shall have 



h^ = sin V hr; — cos v hk. 



