176-178] CANAL THEORY OF THE TIDES. 287 



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



the moon, provided 



(3). 



If E be the earth s mass, we may write g = yE/a 2 , whence 

 f 3 M 



Putting M/E = , a/D = ^, this gives f\g = 8 57 x 10~ 8 . 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 



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 a?, 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 d^g/dt 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 f 

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



That is, n = n-n lt if Wj be the angular velocity of the moon in her orbit. 



