30 THE TIDAL PROBLEM. 



similar way on the antipodal wave, tending to increase its amplitude and 

 to reduce its surface gradient. On its return it will fall so far behind the 

 original wave as to have appreciable incommensurate effects. The waves 

 of intermediate period will have corresponding intermediate effects. In 

 so far, therefore, as water-waves react upon the lithosphere and develop 

 waves in it, these, while cooperating with the original waves commensur- 

 ably for a time, will tend to distribute the oscillatory action into broader 

 amplitudes. This may be looked upon as a tendency to develop a distribu- 

 tive series of small pulsations in lieu of the original more concentrated one. 

 It is observed that small pulsations attend the incoming and outgoing of 

 the tides, but they have not, so far as I know, been made the subject of 

 sufficient study to determine whether they are systematic or irregular, 

 and whether their periods are at all in accord with the natural periods of 

 the lithosphere or not. 



It is assumed in the foregoing that tidal pulsations will move through 

 and over the lithosphere at the same rate as seismic pulsations, which 

 probably does not involve any essential error, though pulsations vary some- 

 what in their speed, even when of analogous classes. But only the general 

 order of velocity is of special moment here. 



According to the mathematical investigations of Lamb, there should 

 be a double series of modes of oscillation in the spheroid derivable from 

 the initial impulse, of which one set should spring from the compressional 

 waves and another set from the distortional waves, and these should differ 

 in period. Only the gravest periods have been cited above. Without 

 attempting to determine what these shorter periods are in the case of the 

 lithosphere, it is probably safe to say that such as are commensurate with 

 the tides of any body of water would cooperate to build these tides up and 

 such as are incommensurate would have the opposite influence. Now if 

 the tides of each body of water are essentially individual and are radically 

 influenced by the breadth, depth, and configuration of the water-body, 

 it is not improbable that different species of both series of natural pulsa- 

 tions may cooperate with the tides of different oceans and assist in their 

 perpetuation and development. 



If the distribution of the strains developed by increasing or diminishing 

 attraction takes place at a velocity similar to seismic vibrations, even the 

 larger tidal movements of the lithosphere will act almost simultaneously 

 with the tide-generating forces, for no strains of the same phase will extend 

 more than 4,000 miles from the center of development of that phase. The 

 extreme movement from the center to the circumference of the strained 

 area would occupy, at the observed rate of compressional waves, less than 

 12 minutes. In so far, therefore, as retardation of the earth's rotation is 

 dependent on lag of the tide, it will be inconsequential for this class. 



Relative to the tides of an elastic earth, Darwin says: 



The other hypothesis considered is that the- earth is very nearly^perfectly elastic. 

 In this case the semi-diiirnal and diurnal tides do not lag perceptibly, and the whole of 

 the reaction is thrown on the fortnightly tide, and moreover there is no perceptible tidal 

 frictional couple about the earth's axis of rotation. From this follows;,the remarkable 

 conclusion that the moon may be undergoing a true secular acceleration of motion of 



