OF UBANUS AND NEPTUNE. XXXU1 



dR . dR r dR\ 





.dR <PR . of-i? , d 2 i? » , A . m 

 Now o — - = — -— da + -— — de + — — d (wtf + e) + &c. 

 cte aaae aeae ae 



if, therefore, we put 



dlff cM? <fi? 



+ "TT7 oo + 3-7-r- be + 3-7-7- d (w * + e') + &c, 

 da de de de de de 



. dR . d/? . rfii., 

 d\B ■= - - da + -- de + — - d (ra£ + e) + &c. 

 aa ae ae 



dR . , dR . , dR ., , 

 + —. ba + —r be + —, b(nt + e) + &a, 

 da de de 



then § — ■ = — - — -, the brackets written round bR, signifying that ba, ba, be, de', &c. are 

 ae de 



to be considered constant in differentiating bR. Hence 



d (SR) et .rfr**\ 



^HC-T^niifS- 



40. The only parts of R which it will be necessary to retain are those of the first order 

 involving X, and those of the second and third orders involving !2\ and 3\ respectively, which 

 we have represented by R 1} R 2 , and R 3 ; and we shall retain only those portions of the 

 variations of the elements depending on these terms of R, and which we shall represent by d\a, 

 b 3 a, b 3 a, &c. ; so that we may put R = R + R 2 + R 3 and ba = d\« + S t a +$ 3 a, and similarly, 

 for the other elements. It will be convenient to put b (nt + e) = £ + be where £ is that part 

 which is twice integrated, and therefore divided by w 2 , and be represents the other terms which 

 are only once integrated. 



If we consider those terms of bR only, which involve R u and the variations of the ele- 

 ments depending on R L , except £, and JY, 



bu = —r d,a + — — V + - — b/sr + — — d,e 

 da de dor de 



rfi?! , o\R, , dR, dR x , 



+ 77 «i° + -yr oi« + -7—, oi-sr + —y d,e ; 

 da de dw de 



and if the values of d\a, &c. be substituted, and if we then differentiate considering ^a, b L a, 

 constant, we shall fi: 

 Similarly, if we put 



. d(bR) 



&c. constant, we shall find = 0. 



de 



dR, dR 2 dR 2 d&« 



bR = -3— b 2 a + — — b 2 e + — — do-jsr + -j- b^e 

 da de dw de 



dR 2 . , dR 2 , dR 2 . , dR 2 . , 



[E] 



