200] .DIRECT AND INVERTED TIDES. 541 



186) % = Bs 



where by insertion in 185) we find 



From 180) we obtain 



188) 77 = ^irri cos (m t - Jcx). 



The coefficient of the cosine is positive or negative according as ok 

 is greater or less than m, so that we have, according to circumstances, 

 high or low water under the moon. In the former case, the tides 

 are said to be direct, as in the equilibrium theory, in the latter they 



are inverted. But '- is the ratio of the time period of the force, 

 m 



or half a lunar day, to the time required for a free wave to travel 

 half around the earth, and the tide is direct or inverted according as 

 this is greater or less than unity. Equation 188) is the analogue of 

 equation 50), 44. Inserting the values of the constants in 188) 

 we find that the canal theory gives the height of the tide as given 

 by the equilibrium theory in 172) (which we also obtain by putting 

 m = 0), multiplied by the factor 



-f-Y 



(ak) 



exactly as described for the system with one degree of freedom on 

 page 155. If we introduced into our equations a term giving the 

 effect of friction we should obtain a change of phase, as in 44, of 

 amount other than a half -period, or inversion. 



In order to determine the directness or inversion of the tides, 

 let us insert the values of m, k from 184) in 188), by which we 

 find that the tides are direct or inverted according as we have the 

 upper or lower sign in the inequality 



189) 0&^r 8 a> 2 cos 2 V>. 



Supposing the lunar day to be 24 hours, 50 minutes, the earth's 

 circumference forty million meters, we find at the equator the critical 

 depth, determining the inversion, to be 20.46 kilometers, or 12.7 miles. 

 As the depth is less than this, the tides are inverted. For any depth 

 less than the critical depth, there will be a latitude beyond which 

 the tides will be direct. Accordingly we see that even if we consider 

 the ocean to be composed of parallel canals separated by partitions, 

 the tides will be very different in different latitudes, so that if the 

 partitions be removed, water will flow north and south. We thus 

 obtain an idea of the complication of the actual motion of the tides. 



