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V. A Mathematical Theory of Tides. By the Rev. Professor 

 Challis, M.A.. F.R.S., F.R.A.S* 



THE problem of the tides of oceans, which was pronounced 

 by Laplace to be "le plus epineux de toute la Mecanique 

 Celeste/' is also one which, when put in its simplest form, espe- 

 cially requires to be treated in strict accordance with the neces- 

 sary principles of hydrodynamics. In the following attempt to 

 give it such a solution, on the supposition that the whole of the 

 solid part of the earth is covered by water of a certain limited 

 depth, I commence with employing the general equations appli- 

 cable to the motion of an incompressible fluid. The mode of ap- 

 plying these equations in this problem, and the subsequent course 

 of the reasoning, are, as far as I am aware, such as have not been 

 adopted before. 



The resolved parts of the velocity V at the point xyz at the 

 time / being u } v, w in the directions of any axes of rectangular 

 coordinates, we have for an incompressible fluid the known 

 general equation 



du dv dw _ . > 



dx dy dz 



This equation expresses that the quantity of fluid in a given 

 small rectangular space remains the same, in whatever way the 

 fluid passes through it. I assume as an axiom the existence of 

 surfaces of displacement (that is, of surfaces which cut at right 

 angles the directions of the motion), which may also be disconti- 

 nuous in any manner so long as they are not broken or their 

 tangent planes do not at any point make a finite angle with each 

 other. Consequently the equation udx + vdy + wdz = 0, which is 

 capable of satisfying these conditions, is the general differential 

 equation of the surfaces of displacement, provided the left-hand 

 side be integrable of itself. If it is not, it must, according to 

 the above axiom, admit of exact integration by means of a factor. 



In that case, the factor being -, the general differential equa- 

 tion of the surfaces of displacement is 



— dx + - dy + — dz = 0. 



\ A A, 



In both these cases, as I have elsewhere shown, 



du dv dw _ dV v / 1 1 \ 

 dx + a)/ + dz~~~di + \r T + 7 , r 



dy being the increment of the velocity at a given instant corre- 

 * Communicated by the Author 



