DEDUCTIONS FROM THE TIDES. 39 



periphery ^5CZ). These would be precisely compensatory — outside action 

 neglected — if there were no loss of energy by dissipation. This loss reduces 

 the mechanical action of the system by precisely the mechanical equivalent 

 of the heat lost. 



In the actual system the distribution of the loss of energy is necessary 

 to a complete solution. When the earth-moon tides alone are considered, 

 it is clear that the lost energy must come either from the rotation of the 

 earth or the revolution of the moon, or from both partitively. Now the 

 moment of momentum of the system must remain constant and the distri- 

 bution of the loss of energy must be such as to meet this requirement. 

 Just what this distribution must be in a given case depends on the con- 

 figuration of the system which then obtains, and is a question of celestial 

 mechanics rather than of tidal theory. For the purposes of the present 

 discussion it is of decisive moment to know whether the configuration of 

 the earth-moon system at present is such that the earth and moon may 

 recede with reduced rotation of the earth if the dominant phase of the tides 

 is what has been called retardative, or may approach with accelerated 

 rotation of the earth if the phase is what has been regarded as accelerative, 

 or whether the configuration of the earth-moon system is such that it can 

 move in one direction only when loss of energy by tidal friction takes place, 

 bearing in mind that all tides give rise to friction and dissipation of energy. 

 For a solution of this I appealed to Dr. Moulton, who found that, under 

 the present astronomic relations of the earth and moon, any loss of energy 

 by their interaction requires that the bodies recede from each other and 

 that the rotation of the earth be diminished. This he finds to be rigorous 

 under the laws of energy, and it seems to follow as a necessary inference 

 that the special phase of the tidal action which caused the loss of energy 

 is imnaaterial. While this determination at the time was wholly indepen- 

 dent, it was soon recalled that Sir George Darwin had made a similar deter- 

 mination so far as the dynamic relations of the earth and moon under loss 

 of energy are concerned. It does not appear, however, that he drew the 

 inference we have just drawn, for this seems to exclude any differential 

 effect dependent upon the positions or phases of the tides, other than such 

 as is expressed in the amount of friction involved, which is dependent 

 solely on the amount of the water-movement and not on its phase. 



There are assignable configurations of the system in which a movement 

 in the opposite direction would take place if energy were lost by interaction. 

 Such a configuration would obtain if the centers of the earth and moon 

 were within 9,000 miles of each other or their surfaces about 4,000 miles 

 apart. The precise figure reached by Dr. Moulton, all computable influ- 

 ences being taken into account, is 9,241 miles. Dr. Lunn computed this 

 independently, using slightly different values for some of the factors— essen- 

 tially those of Darwin— and neglecting some inconsequential ones, and 

 found the distance 9,113 miles. From this the conclusion seems inevitable 

 that if, for any reason, the moon were separated from the earth within 

 this distance, the tidal interaction of the earth and moon would tend to 

 bring them together, an adverse tendency which the fission theory must 

 faCe in addition to those cited before. 



