OF URANUS AND NEPTUNE. XI 



aM 7 f 1 \ 



a a a 



,,, , aitf M 



a Jl/ W = . 



a 



Also, generally, a'N' = , and a'K' = — . 



a a 



11. The values of the fundamental quantities bfi\ b^\ &c. were calculated in the fol- 

 lowing manner. First 6„(i> and 6,(4) were determined by the formulae, 



6 (D = - F(a), 6,<*> = — {F(a) - E(a)}, 

 ir air 



(see Pontecoulant, Theorie Analyt. Liv. VI. chap, ii.), and the others by the formula 



k+ 2& + 1 a fc 2* + l k x 



_ . . db d> d6,(J> 



The quantities a — — , a -— — , were found from the formulae, 

 du da 



db W a»6 (J) _ aft/J) d^i) d6 (i> 



aa 1 - or da da 



and the higher differential coefficients of b %, b^i were found from these by differentiating them. 



The others were calculated by the formula 



a ^dir ~ a db *i!r = (k ~ 1} KJi) + (k + 1) b ^ m ~ * kab * (i) > 



and the higher differential coefficients by successively differentiating this formula. 

 The values of bj§\ &c. were calculated by the formulae 



6 fc (l) + 6 fe+1 (».i*il(6 fc (i),6, +l (4)), 

 (1 - a)" 



b k (i)-b k+l m^^±\ A b k (i) + b k Ji)y, 

 (1 + ay 



and a — , &c. by the formulas, 

 da 



d&fctt) db k Ji) _ 2&+J_ / db k d) db k+l U\ 2a 



da da (1 - a y \ da da ) 1 - a K + " 



db k d) db k Ji) 2ft + 1 I db k U) db k Jih 2a rum E ,, K 



da da (l + a) 2 \ da da J 1 + a V +1 ' 



b2 



