176-178] CANAL THEORY OF THE TIDES. 287 



towards the point of the earth's surface which is vertically beneath 

 the moon, provided 



f=frMa/D* .............................. (3). 



If E be the earth's mass, we may write g 7^/a 2 , whence 

 /_3 M (a\* 



j-2'~E'\D) 



Putting M/E = sV, a/D = ^, this gives f/g = 8'57 x 10-. When 

 the sun is the disturbing body, the corresponding ratio is 



It is convenient, for some purposes, to introduce a linear 

 magnitude H, defined by 



El*=fl9 .............................. (4). 



If we put a = 21 x 10 6 feet, this gives, for the lunar tide, H = 1-80 ft., 

 and for the solar tide H = *79 ft. It is shewn in the Appendix 

 that H measures the maximum range of the tide, from high water 

 to low water, on the ' equilibrium theory/ 



178. Take now the case of a uniform canal coincident with 

 the earth's equator, and let us suppose for simplicity that the 

 moon describes a circular orbit in the same plane. Let f be the 

 displacement, relative to the earth's surface, of a particle of water 

 whose mean position is at a distance x t measured eastwards, from 

 some fixed meridian. If n be the angular velocity of the earth's 

 rotation, the actual displacement of the particle at time t will 

 be f + nt, so that the tangential acceleration will be c 2 f/cfa 2 . If we 

 suppose the 'centrifugal force' to be as usual allowed for in the 

 value of g, the processes of Arts. 166, 174 will apply without 

 further alteration. 



If n' denote the angular velocity of the moon relative to the 

 fixed meridian*, we may write 



*b = n't + as/ a + e, 

 so that the equation of motion is 



The free oscillations are determined by the consideration that ( 

 is necessarily a periodic function of x, its value recurring whenever 



* That is, n' = n-n lt if w x be the angular velocity of the moon in her orbit. 



