﻿of Tidal Action, 223 



sphere equal to the earth in size and having the density of water ; 



k would then be equal to r ; and according to the most recent 



estimates of the earth's density g would indicate a velocity of five 



r 4 

 feet a second generated in a second of time. Now-^ is nearly 



equal to Tg^gVoW* anc * ^ e P ro ^ uct 7I " 2 ^ sm 2A will represent 

 a quantity of matter which, with the unit of measure I have as- 

 sumed, will be 3ooooo(y r ^ ne velocity due to the force F in a 

 second of time will be expressed by the following fraction of a 

 foot per second : 



2T64 80 000 • 



This insignificant force acting on the moon for three millions of 

 years would change her velocity a little more than 1 per cent. ; 

 and through the indirect influence on her orbit an increase of 

 about 3 per cent, would be then occasioned in her period of re- 

 volution around our planet. If in this estimate I have assigned 

 too low a value for the height of the equatorial tides, there is 

 an ample compensation for the error by giving to the angle A 

 the value necessary for producing a maximum effect. 



The relation already exhibited between the change in lunar 

 and terrestrial motion may be also deduced by investigating the 

 loss in the earth's rotation from the reciprocal attraction of the 

 moon on the protuberant tidal waters in a channel either coinci- 

 dent with the equator or parallel to it. To arrive at an approxi- 

 mate estimate of this loss, in the first case the tangential force 

 proceeding from lunar attraction must, with the notation already 



used, be expressed by — — n3 — ; and the momentary 



decrease in the momentum of terrestrial matter from the action 

 of this force on the small portion of the fluid represented by 

 bryd(j> will be 



Sk~gbyr*m sin 2(A — c£)c?(£ dt 



2D 3 -. • . . 



(13) 



On making the substitutions already employed for y and k*gh, 

 the formula becomes 



3Chm? s cos 2</> sin 2(A - $)d(f> dt 

 2D 3 

 3C# mr 2 sin 2<j> sin 2(A — <ft)*ty dt 

 + _ 2D 3 



(14) 



Reducing and integrating with reference to d(f>, taking the limits 

 of (j>=0 and <£ = 27r, there results 



37rC/?mr 2 sin2A^ 3ttCAW 2 cos 2Adt 



2D 3 _+ ~ "2D 3 ' ' ' \-J 



