MR. G. I. TAYLOR ON TIDAL FRICTION IN THE IRISH SEA. 7 



The amount of energy in the fluid which crosses the element of surface dS during 

 the time dt is the sum of these two. The amount in the fluid which crosses the 

 whole surface, S, is therefore 



s, ...... (12) 



where the integral is taken round the curve, s* The amount of energy which 

 crosses the surface, S, in time dt is the sum of (ll) and (12), that is, 



= pgdt I ~Df>ra,meds + [ %pv sin 6 dt (2gh* + ~Dv 2 + hv 2 ) ds. . . (13) 



We shall now assume that h is small compared with D. This is true for the Irish 

 Sea where the average maximum rise of tide above the mean sea-level is about 6 feet 

 (one fathom) at spring tides, while the average depth is over 40 fathoms. 



It is evident also that since the order of magnitude of a must be that of ch/D, 

 where c is the velocity of a tidal wave in water of depth D (i.e., c = vgD), 



* It has been suggested to me that a term should be added to allow for the potential energy of the 

 entering water due to the moon's attraction. This appears to- be a misapprehension. Potential energy is 

 only a mathematical expression used in finding the work done on matter by certain systems of forces. 



The work done in time St by the moon's attraction on the liquid contained in any surface which is fixed 

 relatively to the earth is 



where ft is the potential due to the moon's attraction, <li- \* an element, of volume, and the integration 

 extends throughout the volume. If the linear dimensions of the surface are small, so that ft does not van- 



appreciably throughout its volume (this may be taken as true for the Irish Sea), then \\\P <h' = M, the 

 mass of the liquid contained in the surface at any time. 



The total work done by the moon in a complete period is - I M dt = I M dil. 



Integrating by parts - Mrfft = - [Mft] + ft rfM, where [Mft] represents the change in the product 

 Mft during a complete period. This is evidently equal to 0. Hence 



L i s the rate at which water enters the volume and ft '-=- is the potential energy of the entering water. 

 dt dt 



In calculating the work- done by the moon on the waters of the Irish Sea we could therefore use either 

 expression A or expression B, but we must not use both. 



At a later stage in this paper (see p. 18) the work done by the moon's attraction has been calculated 

 from expression A. The potential energy, due to the moon's attraction, of the entering water has 

 therefore been left out at the present stage. 



