Prof. Challis on a Mathematical Theory of Tides. 29 



earth westward at the rate of the moon's apparent diurnal 

 motion. 



It remains now to indicate in what manner the small quanti- 

 ties neglected in the first approximation may be taken into ac- 

 count. If we omit, for the present, the terms of the second order 

 with respect to the velocity, the terms neglected in the value of 

 p were 



■ Gk 2 a) 2 ?- 2 2 



ir {a — r) 2 + — - cos 2 X. 

 2a 2 



The complete differentiation of these with respect to time would 



add in the value of f -f- ) the terms 



(i) 



— (a — r)u! + ft) 2 r cos 2 Xu ! — —- sin 2Xw' 

 a v 2 



dr 



— ( u' — (a — r) -r- ) + ft) 2 cos 2 Xu 1 -f ©V cos 2 X—r-> 

 a V dr J dr 



and in that of - the terms 



dr 



it being supposed that — '-r — =0, because with sufficient ap- 

 proximation 



, Smb 2 

 rw' 



-^3— sin 2X sin 2(6— fit). 



-m 



The new terms in the value of — •* — are 



dU 



( (a— r) + (D z r cos 1 h)—Q — sin 2X 



du f aPr . ^ dw 1 

 Id 



(!) 



d.CZ 



and those in the value of -—7- are 



dX 



Gk , . a a ^ \du ! 



( — — (a—r)+ a> 2 r cos 2 X )-=- ft> 2 r sin 2 Xu ! 



&) 2 ?- . dw' 2 . , 



=- sm 2X -jr &rr cos 2Xw. 



2 dX 



All the additional terms are expressible as definite functions of 

 r, 6, X, and t by means of the first approximations to the values 



