128 THE TIDAL PROBLEM. 



Section V is devoted to a consideration of the problem in which the 

 two bodies revolve in circular orbits undisturbed by exterior forces, and 

 in which the rotational moment of momentum and energy of one of the 

 bodies are so small that they may be neglected, while the axis of rotation of 

 the other is perpendicular to the plane of the orbit. Although the condi- 

 tions assumed in this section are not exactly fulfilled in any physical prob- 

 lem, still they are near enough those prevailing in the earth-moon system to 

 throw much light on what may possibly have taken place. But the chief 

 value of this investigation is that the variables are so few that the results 

 are precise, except as to the time rate at which the possible changes will 

 take place. The thing of greatest interest is that the rates of change of 

 revolution and rotation are proportional to the rate of the loss of energy 

 through friction, and are not directly dependent upon the phases and 

 lags of the tides or the surface peculiarities of the tidally distorted body. 

 These results show that, so far as the hypotheses upon which they are 

 founded apply to the earth-moon system, if we could from the direct tidal 

 observations calculate the rate of loss of energy in tides raised by the moon 

 upon the earth, then we could compute the rate of tidal evolution at the 

 present time. While this problem undoubtedly presents serious difficulties, 

 they are not more formidable than those of assigning to the earth a physical 

 constitution which shall agree reasonably with the truth. 



Assuming that friction is proportional to the height of the tide and its 

 velocity relative to the surface of the tidally distorted body, and that the 

 loss of energy is proportional to the square of the friction, equations are 

 developed, (33), giving the rates of change of the periods of revolution and 

 rotation. They involve only one unknown constant depending upon the 

 physical constitution of the distorted body. 



In section VI the equations of section V are applied to the earth- 

 moon system. The influence of the sun is neglected, later computation 

 showing that its effects upon the rotation of the earth are now about one- 

 fifth as great as the moon's. The rotational moment of momentum and 

 energy of the moon are small because of the small mass of the moon, its 

 small dimensions, and its slow rotation. Using the numerical data, it is 

 found that the moon's rotational moment of momentum is less than one 

 thirty-thousandth that of the earth. Since the eccentricity of the moon's 

 orbit is small and the cosine of obliquity of the ecliptic not much less than 

 unity, it is seen that the conditions of this investigation really approxi- 

 mate rather closely to the actual earth-moon system. It is found that the 

 month has always been increasing and that it can not pass beyond 47.7 

 of our present days, at which period the month and day will be equal and 

 the system move as a rigid body. There is no way of telling by this investi- 

 gation how long a time will be required for the system to reach that state. 

 But it is a more interesting fact that the month can never have been less 

 than 4.93 of our present hours, this being the period of revolution when 

 the distance from the center of the earth to the center of the moon was 

 9,194 miles. Consequently we must suppose that when the moon broke 

 off from the earth it was at this distance from it, or 5,236 miles from its 

 present surface. Or, including the radius of the moon and supposing that 



