DEDUCTIONS FROM THE TIDES. 41 



Now, it being thus rigorously shown that the energy dissipated by the 

 reaction between the tide-producing bodies must be taken from the mechan- 

 ical energies possessed by the bodies, respectively, in the proportions 

 given, and it being further shown that the evolution must take this direc- 

 tion and no other, it is clear that it will be possible to determine the effect 

 of the water-tides on the earth's rotation if we can estimate the total 

 energy dissipated by the tides, for the proper proportion of this can then 

 be subtracted from the kinetic energy of the earth's rotation. We can of 

 course use only imperfect data at present, but the uncertainties of these 

 can be covered by allowances so as to approximate the true order of mag- 

 nitude, and if the value does not prove to be a critical one, the order of 

 magnitude may be as decisive for all geological purposes as a precise deter- 

 mination. If the value proves critical, the serial method may be applied 

 to make the results cover the whole range of uncertainty. By thus dealing 

 with the whole of the friction, and by applying it by means of the rigorous 

 laws of energy, we not only avail ourselves of a radical mode of treatment 

 but avoid the tangle of special interpretations and the discrepancies of tidal 

 theories. 



Preparatory to an attempt to compute the total friction, it is worth 

 while to note that an altogether exaggerated impression of the friction 

 of the tides is inevitably conveyed by their association with wind-waves, 

 river-currents, sea-currents, and other water-movements. As all of these, 

 or nearly all of these, are the products of energy communicated to the earth 

 by the sun's radiation, they may be assumed to be neutral in their rota- 

 tional effects. If all these adventitious elements be removed in imagination, 

 and the hydrosphere be made to take on a perfect calm, save as affected 

 by the tidal forces, the picture of the frictional effects will be radically 

 transformed, as may be seen by simple inspection. In the mid-ocean a 

 water-particle will merely describe a circuit of a few feet in twelve hours, 

 its movement on its fellow particles which are pursuing a somewhat similar 

 circuit being almost imperceptibly slow; the movement on the bottom will 

 be very sHght. On a shore shelving at the rate of 1 foot in 50 feet, and 

 with a tide of 5 feet, the edge of the tide would advance 250 feet in about 

 6 hours, or a little over 40 feet per hour. On the exceedingly low slope 

 of 1 foot in 800 feet, the advance of a 5-foot tide would be only 666 feet 

 per hour. There are of course concentrations of motion in bays, straits, 



cations of which were worked out after this was written. In the accompanying graphic 

 illustration by Dr. A. C. Lunn, the relations of the factors are somewhat differently 

 arranged. In the upper diagram, fig. 5, the line " i2 = M^" represents the angular velocity 

 of the rotation of the earth; the Hne, w, the angular velocity of the moon in its orbit; r, the 

 distance between the centers of the earth and moon; N, the relative days in the month; 

 X = Mj, the orbital moment of momentum of the system. In the lower diagram, E repre- 

 sents the total energy of the earth-moon system; Ky, the kinetic energy of the earth's 

 rotation. 



The crossing of A and u, where the angular velocity of the earth's rotation and the 

 angiilar velocity of the moon's revolution are equal and they move as though a rigid body, 

 is somewhat over 9,000 miles from the line of reference at the left. From this point of 

 crossing to the left, where the distance of the centers is declining as shown by r, the total 

 energy is declining as shown by E in the lower diagram and loss of energy promotes move- 

 ment to the left. To the right of the crossing of ^ and w the centers move apart, the 

 total energy declines, and loss of energy promotes movement to the right. 



