is-) 



tan 



Seft Elements of a Planet on the Plane of its Orbit. 



^(ArSu- AoSr) + '^^^ (Aw 8r - ArS^r)! 



+ Ay6'^ — AASy + A R^if = 0. J 



If u be the eccentric anomaly, 



7- = a ( 1 — ^ cos ?/), S r = ( 1 — e) 8 « — « 8 ^, 

 making m = after taking the variations. Also 



U — TT /l + 



3 w^ 



M— esinM = 7^/ + s— TT, (1—^)8?^= 8a + 8s— 87r = 8s — Stt 



^ 2a 



by making / = 0, as it has been proved we may do. There- 

 fore by elimination, 



With these values of 8 r, 8 o, {g.) becomes, making 



^(A«87r-A7r8fl) + ^(As8«-Aa8s) 

 + ^^^ (A e 8 TT — A TT 8 ^) + A R iZ^ = 0. 



(I.) 



By making the coefficients of A a, A e^ &c. separately equal 

 to nothing, the equations (D.), (E.) and (I.) will give all the 

 elements. 



If e' be the eccentricity, tt' the longitude of the apse, and s' 

 the epocha on the fixed plane, such as would be introduced 

 by the integration of the differential equations relative to that 

 plane, I find by a comparison of the values of the co-ordi- 

 nates on the plane of the orbit and the fixed plane, 

 e cos (3— tt) = e' cos (9 — w'), e sin (d — tt) = ^ cos i sin (5 — w') ; 

 and also g — £' = d— fl true to quantities of the order e^ inclusive. 

 Perhaps this last, like the two former, is strictly true. It is 

 evident therefore that e' and w' at least differ from e and tt, and 

 that the same values cannot be used lor either set indiscrimi- 

 nately without the hazard of error. I have not room to dis- 

 cuss this subject at length, but I wish to invite attention to it. 

 The main object of this paper is the formulae (G.), which I 

 have found by a different method elsewhere. 



Gunthwaite Hal), Penistone, Yorkshire, 

 September 30, 1844. 



