Prof. Challis on a Mathematical Theory of Oceanic Tides. 435 



into heat by the operation of electrical resistance ; but the absorp- 

 tion in this way cannot take place instantaneously, requiring as 

 it does a time comparable with the time-constants of the circuits 

 concerned. So far, indeed, is a condenser from itself absorbing 

 electrical energy, that in many cases it actually prolongs the du- 

 ration of motion ; for an oscillatory current, in consequence of 

 its smaller mean square, sustains itself twice as long against the 

 damping action of resistance as a comparatively steady current of 

 the same maximum value. 



Terling Place, Witham, 

 May 5, 1870. 



LXIII. Supplement to a Mathematical Theory of Oceanic Tides. 

 By the Eev. Professor Challis, M.A., F.R.S., F.R.A.S* 



XN two preceding communications, contained in the Num- 

 bers of the Philosophical Magazine for January and April 

 of the present year, I have attempted to solve the problem of 

 lunar tides on the suppositions that the earth is wholly covered 

 by water of uniform depth, and that the moon has no declination. 

 My object in attacking the problem under this form was to dis- 

 cover the correct mode of employing for the solution the general 

 equations of hydrodynamics, the effects of friction and of imper- 

 fect fluidity being left out of account. Notwithstanding the 

 simplicity of the conditions, the difficulty of ascertaining the 

 proper process of reasoning is so great that it is not surprising 

 success should be attainable only by slow degrees and after re- 

 peated failures. Having good reason to be dissatisfied with my 

 first attempt in the Number for January, I proposed two other 

 solutions in that for April. The first of these fulfilled the given 

 conditions of the problem, and at the same time satisfied the 

 general hydrodynamical equations ; but as it led to the result 

 that there would be an infinite tide if the depth of the supposed 

 uniform ocean were 8^ miles, I thought that it must on that 

 account be rejected. In the other solution this result was evaded 

 by the introduction of certain special considerations, which, how- 

 ever, conducted to values of the resolved velocities which did not 

 satisfy the equation 



d^ d_^> *£ 



dx* + dy* + dz* K ] 



The solution being liable to objection on account of this defect, 

 I endeavoured to show that, as it satisfied a variation of that 

 equation, it was not necessary that it should satisfy the equation 



* Communicated by the Author. 



2F2 



