INTRODUCTION. 83 



In questions of cosmogony, where immense intervals of time are in- 

 volved, the problem of tidal evolution is obviously one of great importance, 

 unless it shall some time be shown that it is not a sensibly efficient factor. 

 The two most obvious methods of determining its efficiency are by direct 

 attacks from the mathematical standpoint, or by comparing its certain im- 

 plications with as many facts given by observation as possible. The first 

 is mainly the method of Darwin, and he has written what will certainly 

 always be an extremely important chapter in the question when considered 

 in the broadest possible way. His results can be improved, apparently, 

 only by a determination of the physical properties of the earth as a whole, 

 and by an estimate of the loss of energy in the ocean tides. While there 

 is hope for the former from seismic vibrations, certain astronomical phe- 

 nomena, and the character of the crustal deformations as revealed by 

 geological studies, the results are not now so well established that they do 

 not need support from other sources. The second method, that of com- 

 paring the positive implications of the tidal theory with observed facts in 

 as extended a way as possible, is broadly speaking that adopted in this 

 paper. Since there can be no test of time-results except on the basis of 

 other doubtful hypotheses, and since it is impossible to draw any certain 

 conclusions in the questions involving the time, this variable has been 

 entirely eliminated from the discussion except in section 14. In a general 

 way it may be said that the energy of the system has been taken as the 

 independent variable, for it is known that under any sort of friction it 

 must degenerate into heat. The results are characterized by certainty 

 so far as they go, but as compared with Darwin's they are in most cases 

 much less explicit as to particulars. The discussion is mostly attached 

 to the fundamental equations of moment of momentum and energy. After 

 the work was well advanced it was found that Darwin had applied fun- 

 damentally the same methods to illustrate his results in a paper supple- 

 mentary to his main series and published in a different serial (No. 5 in the 

 hst previously given). The variables he used were different from those 

 employed here, but, though for certain purposes they may be more conve- 

 nient, nevertheless, for the sake of complete independence, those originally 

 selected have been retained. 



When the discussion is based simply upon the moment of momentum 

 and energy equations, the number of quantities to be determined is greater 

 than the number of the determining equations. To attain the greatest 

 simplicity the general problem has been divided up into a number of 

 special cases covering altogether the entire field. In this way each special 

 problem is very easily understood and the question as a whole is much 

 illuminated. 



