100 SECULAR VARIATIONS OF THE ELEMENTS OF 



CHAPTER II. 



ON THE SECULAR VARIATIONS OE THE NODES AND INCLINATIONS 

 OF THE ORBITS. 



1. The secular variations of the nodes, and the inclinations of the orbits, are 

 determined by the integration of a system of differential equations which are 

 entirely similar in form to those from which the eccentricities and perihelia were 

 obtained. 



If we denote by <p, q>, <£>", &c, the inclinations, and by 6, 6', 6", &c, the longi- 

 tudes of the nodes of the planets, Mercury, Venus, the Earth, $cc, and put 

 tan <|> sin 6=p, tan <p' sin 6'=p' tan q>" sin 0"=p" &c, 1 (oqa\ 

 tan <|> cos Q=q, tan <?>' cos &=(/ tan q>" cos.0"=g" &c. ; j 

 we shall have the following system of differential equations for the determination 

 of p, p'i p"-, &c., q, q', q", &c. 



— " \ (0,l)-)-(0,2)-|-(0,3)-|- &C. \p (°^)p' (°. 2 )p" (0.3)2)'" & C -' 



(It K ) 



■&■=- { (o,i)+(o.2)+(o,3)+&c. J q +(o,i) 8 '+(o,iy+(o..)^ + & C ., 

 d<L = I (1 , o)-L.(i , s)+(i , s)-j- &C. j p' — (l , o)p — (1 , 2)p"— (1 , s)p'"— &C, 

 ^=-{(.,»)+(M)+(M)+&C.J ? '+(i,») 2 +(i, ! )g"+(i 1 %"+&C, > {E) 



-2-= < (2,0)-|-(2,l)-f-(2,3)-f-&C. Ijj"— (2,0)2>— (2,l)2>'— (2,3)2)'"— &C, 



^=-|(»<o)+(., l )+(..8)+&c.J g rJL.(»,i) 2 +(. 1 i>/+(.,,)5»_&a 



&c. 

 To integrate these equations, we shall suppose 



p=N sin (gt+p), p'=N' sin (gt+@), p"=N" sin (^+/8), &c, "I , 36 ~ 

 q=N cos (gt+(3), q'=N' cos (^+/3), g"=IV" cos (gt-\-@), &c. j V ' 



If we substitute these values of 2>, 5, p', q\ &c., in equations (E), they will become 

 — 5 rJV=J(o,i)+(o,j)-fo,s+&c.}iV— (o,i)JV— (0,2)2^— (o,s)J\r— &c. 

 _ fl riV = {(i,o)+(i,2)-|-i,8+&c.|iV — (i,o)2V — (i,a)JV— (i,s)2\r— &c; . .-^ 

 — flriV w ={(s,o)+(«,i)+«,s-|-&c.jiV w — (s,o)iV— (s,i)iV— (2,3)iV'"— &c; ' v 

 &c. 



These equations are similar in form to equations (B), except that g is negative. 

 They will produce, by eliminating N', 2V", &c, an equation in g, of the eighth degree. 



