406 NEWTON S PRINCIPIA. 



or be less than 14 miles, there will be low tide immediately 

 under the luminary. 



In this instance there was only a semi-diurnal tide, but 

 when the solar path is no longer restricted to the equator, 

 and the channel is any position on the earth, we have both 

 diurnal tides and tides of longer period. The diurnal tide 

 does not exist when either the canal is equatorial, or the 

 luminary is on the equator. When it exists, the phase at 

 the instant of transit depends on whether the depth of the 

 sea be less than 3^ miles, and the luminary on the oppo 

 site side of the meridian or not. If both these hold or 

 both fail, there will be low water at zenith transit ; if one 

 fails and the other holds, high water. 



There are also tides of long period ; that is to say, the 

 relative mean level in different parts of some canals is not 

 the same as if the disturbing body did not act. The mean 

 level thus depends on the position of the luminary. 



The state of the tide will manifestly be affected by fric 

 tion ; it appears that if we wish to calculate the height of 

 the tide, at any time t, we are to proceed as if calculating 

 the height (on the supposition of no friction) at a later 

 time t-\-a, with the luminaries in the position they were 

 in at a preceding time tfi 9 where and /3 are two quan 

 tities given by the theory, and depending on the amount 

 of the friction. 



In considering the effects of two bodies, we see that, 

 according to this theory, they do not produce effects which 

 have the same ratio as in the equilibrium theory. If in 

 that theory, p be the ratio of the lunar to the solar high 

 tide, and ? the ratio given by the wave theory, then 



where n, n are the angular velocities of the sun and moon 

 relative to a fixed line on the earth, b the radius of the 

 earth, and k the depth of the sea. This ratio will be 

 different according to the different values of k. This 

 explains the fact, that in different seas the ratio of the 



