THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 85 
Hence since k? (1 + m) is included in X, Y, R, S, we have 
5 
is 20. hh 5 ds 
— = fh, [ze cos (f —o)—1+ tee (cos (f—o) — 1) | af 
2 
(18) 
+ 2hop.sin (f—o da 
dr 
If we write h,. a cos *@ in place of & in equation (18), we have the same ex- 
pression for = as that given by HANSEN. 
Equations (17) and (18) are fundamental in HansEn’s method of computing the 
perturbations. We will now give Hansen’s method of deriving them. 
Using the same notation as before, we have, since 
a _ 1+ecosf 
; 
cos *¢ 
also 
r cos? g, 
Geeele=e\costac 
hence 
ra _ 1+ecosf COs *¢, 
dae cos ch lemencosias 
Using f + ™— x in place of f, and developing, we get 
r.a __ r+reos f.e cos (y—m)+r sin f.e sin (y—7) 
Pity A, COS “Gy 
Let us put 
esin (y— 7%) = 1 C08 "hp, 
(19) 
€ cos (y—™) = £ cos*q + 43 
since €é = sing, we have 
C08 "p = Cos) (1 — 2ey E — cos *y &* — cos “hy 7”). 
