SPECIAL CASE. 93 



Equations (32) are very instructive, for they prove rigorously, under 

 the hypotheses we have adopted in this section, that the rates of change 

 of both the day and the month are proportional to the rate of the loss of 

 energy, however it may be lost. That is, if tidal friction is now almost 

 exclusively in the ocean tides, as Darwin supposed,^ in so far as the earth- 

 moon system satisfies the hypotheses of this section it is quite immaterial 

 whether the energy is lost in the manner described by the spherical har- 

 monic analysis when applied to the viscous theory, or whether it degen- 

 erates after the waves have been time after time reflected from the con- 

 tinents and have run into narrow bays or into the high latitudes. The 

 relation of the tidal wave to the moon is not directly involved as it is in the 

 elementary geometrical discussions of tidal friction, though of course the 

 rate, and therefore the phase, of the friction depends upon the viscosity 

 of the water. This would increase one's faith in the spherical harmonic 

 analysis for such an earth and ocean as we have if it were not for the fact 

 that the irregularities in the depth of the ocean and in the outlines of the 

 continents undoubtedly greatly change the whole amount of friction. 



The moon sets up motions in the waters of the ocean, but not all of the 

 energy possessed by this water is lost. At the succeeding disturbance of 

 the same region by the moon the phase of the tidal deformation still per- 

 sisting may be such that the moon's attraction will tend to destroy it rather 

 than generate a new wave; or, the phase may be such that the moon v/ill 

 augment the tide. On a world covered with oceans of many dimensions and 

 depths we should expect to find places where the natural periods of oscilla,- 

 tions in water basins are such that the moon's disturbance builds up consid- 

 erable tides, and others where they are kept low. In the former case the 

 friction of the water prevents their becoming excessively large; if the water 

 were entirely frictionless they would increase until the resulting alteration in 

 their period would lead to their destruction by the moon's disturbing forces.^ 



One method of finding the present rate of tidal friction, at least so 

 far as it is due to ocean tides, is to compute from tidal observations in all 

 parts of the earth, and from the frictional properties of water, the actual 

 waste of energy.^ If one were to observe the energy manifested when the 

 tide runs through a strait on our coasts, he would be apt to overestimate 

 the work the moon is doing upon the earth. In the first place such condi- 

 tions are quite exceptional, and in the second place only a very small part 

 of that energy degenerates into heat. When the run of the tide ceases the 

 kinetic energy has very largely become potential, and it becomes kinetic 

 again when the tide runs out. If the outgoing tide has the same energy 

 as the inflowing tide there has, of course, been no loss, and, according to 

 equations (32) and (33), there is no tidal evolution in such a system as 

 we are considering in this section. 



In the units employed the kinetic energy of rotation of m^, the kinetic 

 energy of revolution of mj and m^ about their common center of mass, 



» 2, pp. 483-484. 



' For a discussion of the observational e^^dence see Harris, U. S. Coast and Geod. 

 Survey (1900), app. 7, pp. 535-699. Also Chamberlin's paper, ante, pp. 5-59. 

 ' See paper by MacMillan, ante, pp. 71-75. 



