the Motion of a Disturbed Planet. 21 



i^_ = 1 +£ cos (u— *r ) / h dh 



fi r ° v w firj 



+ cos y / (cosTrt?^— esin7r£?7r) + siny / (sinTrJtf-f ecos7r£/7r). 



Let the constant quantities X ana g be the same functions 

 of a constant time t which o and r are of / ; then putting the 

 former in place of the latter, we may put them under the sign 

 of integration, changing t into t after the integrations are per- 

 formed. This will change the last equation into 



/V , 2 Phdh p . 



JL.— \+e n cos(o— 7r n ) / h/ cos(X— ii)de 



fi r v fij g J 



-t- J e sin (A— it) dir. 



But fhdh = — C-j-hdU 



hdt . , x dR (2hdt . x dr . . . WR 



de— sm(y — 7r)-v 1 cos(u — w) H — sinfy — it) )-=-, 



fi K dr \ pr fi /do 



. hdt , AR /2hdt . . . dr , .\dR 



dii— — cosfy — it)-^ I sin a- w] cosfy — it) I — . 



fie dr \ fire v fie / dv 



d R 

 The coefficients of —z— are put under the above form for 

 ay * 



convenience. Substituting these values, we find 



T=^ + ^° cos ("-*») + I^f^r Si " < X -"^ '' dt 



1 /"ilR! . , . , 2kdt , . 2kdt"\ 



+ ^^i sin(x " u) — r"Ht7 }• 



To abridge, let this be written 



r =f- 2 + TTCOs(y- 9 r ) + P. 



hr? h 



But if £ t be the progression of the apse, 



cos(y — ir ) =cos(y — St — ir + §t) = cos(y — Gt — 7r Q ) 

 — £/ sin (y — 6/ — 7r ), 

 neglecting higher powers of 8 1. Therefore 



— = JTZ i 1 + ^0 COS ( U — *'— *o) — e ^ Sm (° — S/ - 7T )} + P. 



Terms similar to the above, containing £ in their coefficients, 

 will arise from the development of P ; and § must be so de- 

 termined as to drive them out, which will be easily done. 

 We may always neglect terms involving the higher powers 

 of/. 



