32 



SECULAR VARIATIONS OP THE ELEMENTS OF 



h =N- sin (gt+^+N, sin (gj+pj+2^ sin (^+/? 2 )-f & c . 

 h'=N' sin (^+/3)+iVi'sin fat+pJ+Wsm (<M+&)+&c. 

 ^"=i^"sin (gt+(3)+Wsm (gJ+pJ+NJsm (^+/3 2 )-j-& c . 

 &c; 



^ (C) 



Z ==# cos (^-|-/3)-J-JVi cos (g 1 t-\-(3 1 )-\-N 2 cos (s^+/? a )+&c., 



Z'=iV'C0S (^-f^+iV/cOS Q^+^+i^'coS (&<+&)+&<!., 



l"=N"cos (gt-\-(3)+N;'cos (flr 1 «+/3 1 )+JV r a 'cos (&<+&)+&<;., 

 &c. 



& /?n &, &c., being arbitrary constant quantities. These equations contain twice 

 as many constant quantities as there are roots g, g x , g 2 , &c, of the differential 

 equations (A). These constant quantities are indeterminate by analysis ; but they 

 may be deduced from the values of h, h', h", Sec, I, I', I", &c, at any given epoch. 

 If we suppose t=0, in equations (C), and also suppose the first members to be 

 known quantities, the preceding equations will give the values of these constant 

 quantities by direct elimination. But in order to determine them more conveniently, 

 we shall resume the differential equations (A), and we shall multiply the first, third, 

 fifth, &c, by Nm-^-na,. N'm'-^-n'd, N"m"-^-n"a", &c, and they will become 



^§=^{ ( °' ,)+<, ' ,)+(0 ' !)+&c - 



—N~ | CT'+FT~ 2 lZ"+F^r+&c„ 

 , T , m' dh' ,„«*'(, x i / xi, x , p 



-N'~ { (TT 3 z +[I3 r +EI ]r+&c., 



, T „ m" dh" - T „ m" \ \ . N ' ... N , 



-N"^, [ E3Z+[EI]Z'+[E3F+&c., 



■ (130) 



&c. 



If we add these equations, and in their sum substitute the values of iV^(o,i)-[- 

 (o,a)-f(o,8)-}-&c.}, i^{(i,o)-f (i,2)+(i,s)+&c.|, &c, deduced from equations (B), 

 we shall obtain 



m dh m' dh' . A7 „ m" dh , . 



n'a' dt 



n"a" dt 



= 9 \m^ + N'l'^ J + N"l^ + &c] 



+n\ z'(- m '[I3-^[^)+?'(^[Eo]-^[o 3 ) + &c. 



i im f 7 / m i i in' , , \ , , / m" , , in' , , \ 



(131) 



