30 Prof. Challis on a Mathematical Theory of Tides. 



of u\ v'j and w ! ; and, after being so expressed, more correct 



values of these velocities can be obtained by new integrations. 



Gk 

 It may here be stated that the term — (a—r) 2 might have 



&a 



been included in the first integrations. I obtained, in fact, by 



so doing the following results : 



,, N (1-f k)a+[q + k)r „, ,, ..-A+?) 



«'= jjrr-<£(r)cos 2 A,sin2(0 — fit), 



"' =- a^( r ~ 2 ( a+ ^ a -'"))^?) cosXcos2 ^-^' 



w 1 — — V sin X tan 2(0 — fit). 



When all the terms in u', v' t w' which contain the first power 

 of m have thus been obtained with sufficient exactness, we can 

 proceed to the investigation of the terras containing m 2 , by first 

 substituting, in those terms of the second order with respect to 

 velocity which occur in the complete values of 



wi\ w*\ d .(f) 



fdp\ \dt) \dt) \dt) 



P > \dt)' dr ' d6 ' dX ' 



the values of u', v', w' given by the antecedent approximation, 

 and then effecting new integrations. 



I have thus shown how the problem of ocean-tides, under the 

 simple form in which it has been proposed in this communica- 

 tion, may be solved with as much accuracy as we please. As I 

 have neither the leisure nor the inclination to go through the 

 details of the second approximation, I have only indicated the 

 proper process, in the hope that younger mathematicians may 

 be induced, by the great importance of an accurate solution of 

 this problem, to undertake the requisite analytical calculation. 

 The problem has acquired special interest since M. Delaunay 

 suggested, in the course of his researches on the acceleration of 

 the moon's mean motion, that the length of the sidereal day may 

 be affected by tidal action. But the verification of this view evi- 

 dently demands that the mathematical investigation employed 

 for the purpose should be conducted in strict accordance with 

 the principles of hydrodynamics, and be carried to terms of the 

 second order with respect to the disturbing force. As far as I 

 can judge without the aid of such investigation, it seems to me 

 possible that the increment of pressure at high water on eastern 



