The Tides in Relation to Geophysical and Cosmic Problems 513 



If ft = 0, we have again equation (IX. 12) for the case without friction. 



As nt + l + e measures the hour-angle of the moon past the meridian of 

 any arbitrary point A on the canal it appears that high water will follow the 

 moon's transit at an interval t x given by nt l = %■ 



If/? 2 > 1 we should in the case of very small friction, according to (XV. 8), 

 have x = 90°, i.e. the tides would be inverted. If the frictional influence 

 becomes noticeable the values of % w iU De between 90° and 45°, and the time 

 of high water is accelerated by the time equivalent of the angle 90° — %. On 

 the other hand, when p 2 < 1 the tides would be direct, and the value of % lies 

 between 0° and 45°, the time of high water is retarded by the time equivalent 

 of this angle. These two cases are illustrated in Fig. 212. In the direc- 

 tion M, the moon is supposed to be in the plane of the equator. It is 



*~M 



*~M 



Fig. 212. Tidal friction acting on a tide wave in an equatorial canal (above p > 1; below 



when p < 1 :). The curved arrows indicate the sens of rotation of the earth; in the direction 



M is the moon generating the tide. 



evident that in each case the attraction of the disturbing system on the 

 elevated water is equivalent to a couple tending to decrease the angular 

 momentum of the rotating system composed of the earth and sea. 



The retarding effect of the moon upon the rotation of the earth exists, of 

 course, still today, although it will have been much stronger in ancient times. 

 This retarding effect makes the day grow longer; the amount is still an open 

 question, 1 sec in 100,000 years is regarded as a maximum value (Delaunay). 



In the same manner the frictional effect could be used to explain the secular 



33 



