SUMMARY. 127 



XVI. SUMMARY. 



The object of this investigation has been to examine the theory of tidal 

 evolution in order to find out, if possible, not what might take place under 

 certain assumed conditions, but how important this process has been in the 

 actual development of our system. The aim has been to avoid, as far as 

 possible, assumptions regarding the uncertain factors depending upon the 

 physical conditions of the bodies involved. In order to compare the theory 

 with the actual facts the various methods of testing it have been carried to 

 quantitative results. 



A large part of the discussion has been made to depend upon the com- 

 ponents of the moment of momentum and upon the energy of the system. 

 In section II the moment of momentum equations and the energy equation 

 are developed, and they are perfectly rigorous so long as the two bodies 

 are subject to no forces except their mutual attraction. Under this con- 

 dition the three components of moment of momentum are rigorously 

 constant, and the three equations which express these conditions are fixed 

 relations among the various quantities which define the dynamical state 

 of the system. The energy equation is a relation among the same quan- 

 tities, but unlike the components of the moment of momentum the energy 

 diminishes by friction. These relations are too few to determine the changes 

 which will actually take place, but they give important information about 

 them. They are particularly valuable, for they are true whatever the 

 physical conditions of the bodies involved. 



One of the conclusions reached by Darwin was that it is probable that 

 the earth and moon have developed from an original mass by fission. One 

 critical test of this hypothesis is the determination of the smallest distance 

 at which the bodies could have revolved around each other consistently 

 with the present moment of momentum and energy. This test has been 

 worked out quantitatively, first with the problem simplified so that the 

 conclusions are absolutely certain under the hypotheses; then the effects 

 of various modifying conditions, which seem more or less probable, have 

 been examined, one after another, and their influence upon the final result 

 determined. The results reached are so near the border line separating what 

 is favorable to the theory from that which is unfavorable, that it is impor- 

 tant in applying this test to determine accurately the constants upon which 

 the system depends. One of these is the rotational moment of momentum 

 of the earth, which depends only upon the law of density of the earth as an 

 uncertain factor. In section III the constants of the density according to 

 the Laplacian law are worked out. It is found that according to this law 

 the density varies from 2.75 at the surface to 10.84 at the center. While this 

 is probably not an exact expression for the earth's density, the inherent 

 probabilities as well as the actually observed precessional phenomena lead 

 us to conclude that it is not sensibly in error for the purposes of this dis- 

 cussion. By the same law the surface and central densities of the moon are 

 respectively 1.65 and 6.51, 



In section IV the moment of inertia for the Laplacian law of density 

 is found, and it comes out 0.336 times the mass instead of 0.4 times the 

 mass, as in the case of a homogeneous body. 



