The Astronomer Royal's Reply to Professor Challis. 147 

 dr \_ r r* J 



q — •LI L cos $ + ^r, 



r 

 where y}r is a function of the angles which determine the posi- 



tion of r, and of t. If we make — = L ~ = m. — = n. 



r r r 



we may consider f asa function of /, m, n, t. 



Using therefore <p = z ■■ * ~ — - + iM since cos # = — V 



£tu. ,/ g /(>— «o , /'fr-api 



rf* 2 *\ ? + r* J 



\ i s r 6 i A J 



+ ^^ 2 * 



Similarly, 

 ^<E _^ f 2f(r~at) f'(r-at) ~\ 

 dy ' \ r 4 I r 3 J 



Then 

 rf»_ /(r-flQ , o f 2f(r-at) f {r-aj)\ d + 

 dz~ P 2- +2 1 F~ + r 3 J + ~dl 



£♦ _ o 2 f _ g/(r-g<) v /Ifcrfg 1 



d«« \ r 4 "* ? J 



+ ~3 f , 8/(r-a/) 6f'(r-at) ] f"(r-at) -} 



dz c < 



The sum, or^+^-f-^s 

 L2 



