XXXIV 



THE THEORY OF THE LONG INEQUALITY 



we shall find — - — = 0; and the same is true if for R 2 and S 2 we substitute R 3 and $ 3 and so 

 de 



on. If therefore we reject all the above terms, and also all terms of the third order except 



such as will be divided by at*, 



Si? ^W? s x ( dS » L dR A I 



dR 



J, /dff 2 d/? 3 \ » , 



T& e+ ^ + {-di + *r)^ e 



d'zzr 



to which we may add the following terms of the third and fourth orders, which will be divided 

 by co 3 and oj 4 respectively, 



* The part of SR which is due to the variation of the incli- 

 nation and node is correctly given by the expression 



™(6y + 6y') + i?L{&n + &n>), 



when we consider only the principal terms of R containing any 

 given argument (kn-k'n')l; and such terms only are con- 

 sidered in this section. But the expression is not true of all 

 terms. Referring to the value of / given in Art. (2), we may 

 put it in the form 



7 = {cos(0 + 0'-2Sl )-cos(0-0')Jsin 2 ii, 



+ {cos(0 + 0' -2 Sl')-cos (6 - 6')} sin* i »', 



-}{cos(e+e'-ol- il' )- cos (0 - 0'- & + &')} sin i sins', 



omitting the term of the fourth order. Now the principal terms 

 in R, involving the inclinations, of the form (kn-k'n')t can 

 arise only from the terms in I of the form cos (6 + 0' + Q) com- 

 bined with the terms of the factor J B + B x cos (0 — 0') + &c, 

 (see Art. 7); for the order of these will be k~k'; but the 

 order of the terms of that form arising from the other parts of / 

 must be at least *; ~ k'+ 2. Confining our attention, therefore, 

 to the former, 



/ = cos (0 + 0'- 2 S\, )sin a i» + cos(6 + 0'-2 o l') sin 2 ji', 



- J cos (0 + 0' - Si, - Si') sin i sin «'. 



Now let the planet be referred to its own plane at the epoch ; 

 then at any time t, i is entirely due to perturbation in the in- 

 terval. Let Rp (fig. 9) be the plane of reference; rp the posi- 

 tion of the orbit at the time t ; then i = L p ; Q_ = II'+ Rp ; 

 «'= y'+ iy'= Z j ; Y=IT+ Rq = H'+ <5IT ; L r = y + $y ; Rr 

 = - ill. It must be observed that Rp is not necessarily small, 



although it is due entirely to perturbation. Substituting these 

 quantities in the above equation, we have 



SI = 2 sin (0 + 0'- 211') sin 2 $ y'. in', 

 + cos(0 + 0'-2II') sin iy'.iy', 

 - J {cos (0+0-211') cos Rp+ sin ( 0+0'- 211') sin Rp] sin i sin y', 

 from which we must eliminate cos Rp, sin Rp, and sin t. 



By the triangle Rpr, 



cos Rp = cos Rr cos rp + sin Rr sin rp cos r, 



whence eliminating rp, 



„ sinr „ „ sin rRp 



cos Rp = - — r cos rRp cos Rr H ; — — cos r ; 



r sin • sin t 



.•. sin » cos Rp = -sin(y + iy) cos y + sin y cos(y + <5y), 

 = — sin Sy = — Sy, 

 also sin i sin Rp = sin r . sin Rr = — sin y . SU. ; 

 .'. substituting these in SI, 



SI = 2 sin (0 + 0'-2n') sin 2 jy'. SW, 

 + cos(0 + 0'-2n')sinJy'.ay', 

 + 1 cos (0 + 0'- 211') siny'. Sy + i sin (0 + 0'-2n') sin 2 •/. MI, 

 = 2sin(0 + 0'-2n') sin 2 Jy'(«n + «n'), 

 + cos(0 + 0'-2n') sin Jy' (Sy + Sy). 

 But this is the expression which would be obtained from the 

 value of I given in Art. (3), {omitting cos (0 - 0')} if y and n' 

 were supposed to receive the increments Sy + Sy' and in + in ; 



.•.i/ = ^(iy + iy') + ^,(in + an'); 

 .-. SR = ™, (iy + iy') + ^(SU + SW). 



dW 



The expression is also true for the constant term of R of the 

 second order. 



