﻿The Rev. T. K. Abbott on the Theory of the Tides. 121 



rotation of the earth in one second, the force acting on the water 

 may be supposed unchanged while it passes from a to a' ; and 

 its effect during that interval (i. e. in this quadrant the retar- 

 dation) will also be proportional to cp or its double cf, and to 

 the time, that is, to a a!, or the angle at 0, a a!. Now the 

 angle at = the angle at /, being in the same circle ; and this 

 angle multiplied by cf = the small perpendicular cd, or pp\ 

 which is parallel and equal to it. Therefore the whole retarda- 

 tion since leaving B is proportional to the sum of all the abscissas 

 pp' — that is, to B//. This represents the defect from the 

 greatest eastward velocity ; and after passing its mean value at 

 the middle point s it represents a velocity which, relatively to 

 the earth, is westerly. The velocity of the current relatively to 

 the earth is represented by p s. 



Now if at any point in the supposed canal a thin section be 

 taken, the quantity of water entering this section in a given time 

 is proportional to the depth and the velocity. If the water flows 

 in a little more rapidly than it flows out, it is clear that the in- 

 crease in the quantity contained in the section, and therefore the 

 increase in depth, will be proportional to the difference between 

 these two velocities and to the whole depth. This holds as long 

 as the change is small compared with the whole depth. If this 

 be supposed uniform throughout the canal, the increase in it 

 (that is, in the height of the tide) at a' is therefore proportional 

 to the retardation ; and since the tide began to rise at B, where 

 the velocity began to diminish, it follows that Bp' is also pro- 

 portional to the height of the tide at a' above its lowest point. 

 It is easy to deduce from this construction the corresponding 

 formulas. For calling B, r, we have Bj» = r(l — cos 2 o>). But 

 ps=%r— Bp = ^r(2cos 2 co—1) = ^rcos2a>. And since sB 

 is proportional to the mean height, the defect from this height 

 is proportional to p s, and therefore to cos 2co. 



I have one remark to add with respect to the former paper. 

 It was observed there that, in proving that without friction there 

 would be low water under the moon, it was assumed " that the 

 ocean is carried round by the earth in its rotation.'" It appears 

 •to be a priori an admissible supposition that this is not the case, 

 but that the ocean is in a state of equilibrium under the moon's 

 action while the earth rotates. But this would obviously imply 

 an apparent movement of the whole body of water, relatively to 

 the earth, with a velocity equal and opposite to that of the earth's 

 rotation, i. e. at the equator there would be an apparent current 

 of about 1000 miles per hour. As this does not correspond to 

 the fact, the supposition is practically inadmissible. But when 

 friction is considered, it appears theoretically inadmissible also. 

 For in this case friction would be continually acting in the same 



