2o6 Proceedwgs of the Roynl Irish Academy. 



and in the same time there passes through the wall (2, 4) the quantity 

 (. 4- I de) X rW dt X (. sin 6' 4- ^-^^^^dO' 

 The difference of these quantities is 



rHO' dcj,' dt U sin 6' ^, + zv cos 6' + v sin 6' ^\ 



which is equivalent to 



r- sin 6' d6' d4' dt I ^^ + vz cot 6'\ (i) 



The sum of («) and (b) is the excess of inflow or outflow, in the 

 time dt, through the four walls of the trapezoidal prism. Now, as the 

 bottom of the sea is fixed and allows no inflow or outflow, the sum of 

 (a) and (h) must be equal to the area of the trapezoid, multiplied by 

 the rise or fall of the surface (taken with its proper sign). 



This volume will be 



r^ sin e'dd'dcji' dt x u. (c) 



Hence, adding (a), (b), and {c) together (with a proper sign for u), 

 we obtain 



d(wz) d(vz) , ^, ^ ,-r^s 



This is Laplace's famous Equation of Continuity, and is identical 

 with that given by him {Mec. Cel., vol. i. p. 104), when the notation 

 is changed into his notation. 



Equation (D) may be thus written : — 



I div dv , ^\ ( dz dz\ ^ 



The second part of this equation vanishes when the sea has a con- 

 stant depth ; in which case the Equation of Continuity reduces to the 

 form 



where 8 is the conetant depth of the sea. 



Every conceivable problem, in tidal motion and oceanic current 

 circulation, is theoretically solved by equations A, B, C, and D ; 

 and the only further difiiculties are practical, arising from the imper- 

 fection of our mathematical knowledge. 



