MR. C4. I. TAYLOR ON TIDAL RPJCTION IN THE IRISH SEA. 
7 
The amount of energy in the fluid which crosses the element of surface during 
the time dt is the sum of these two. The amount in the fluid which crosses the 
whole surface, S, is therefore 
sin ^D^+e^(D + ^)} ds,.(12) 
where the integral is taken round the curve, 5.* The amount of energy which 
crosses the surface, S, in time dt is the sum of (ll) and (12), that is, 
|* ^pv sin 0 dt [g (D + /a)^ -\-g]d—gT)'^ + v^ (D + A)} ds 
^ l"* 
= pg dt I DA V sin Bds+ J ^pv sin 6 dt {2gh^ + + hv^) ds. . . (13) 
We shall now assume that A 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 y must be that of c/^/D, 
where c is the velocity of a tidal wave in water of depth D {i.e., c_= \/gT>), 
* 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 
- Sf 
(IQ. 1 
ST 
(A) 
where 0 is the potential due to the moon’s attraction, dc is an element of volume, and the integration 
extends throughout the volume. If the linear dimensions of the surface are small, so that fi does not vary 
appreciably throughout its volume (this may be taken as true for the Irish Sea), then 
mass of the liquid contained in the surface at any time. 
The total work done by the moon in a complete period is - j" M f/i = - j M dil. 
d.'V = M, the 
Integrating by parts - j M = - [MR] + j R c/M, where [MR] represents the change in the product 
MR during a complete period. This is evidently equal to 0. Hence 
- [ .M dil = I" R dM = j" R ,]f .(B) 
is the rate at which water enters the volume and R is the potential energy of the entering water. 
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. 
