92 THE TIDAL PROBLEM. 



Since this value of P must also satisfy (28) we have 



whence 



M = ~m,^ (31) 



as the value of M for which there are two equal roots of (28). For greater 

 values of M (28) has two distinct real roots. The roots are the abscissas 

 of the points b and c, curve I, fig. 10. The smaller root corresponds to a 

 maximum of E, and the larger to a minimum. When the roots are equal 

 the curve has a point of inflection with tangent parallel to the P-axis. 



We may express the rates of change of P and D in terms of the rate of 

 change of E by differentiating (23) and (24) and solving. We find 



dP SPW dE dP _ PD^ dE 



'di"~~27:iP-D) dt dt 2m,7i(P-D) dt 



(32) 



Consider the case first where the direction of rotation and revolution 



dE 

 of mj are the same, i.e., when D>0. Since -^ can be different from zero 



only when P%D, then when P>D both P and D must increase whatever 

 may be the character of the tides as determined by the physical condition 

 of Wi; and when P<D both P and D must decrease. When nii rotates 

 in the negative direction, i.e., when D<0, P must always decrease and 

 D always numerically increase. When P = D equation (23) becomes 



-^(Pi— ilfP + mJ=0 



for which, by (28), E is a. maximum or minimum. 



We are supposing the orbit a circle and the axis of mj perpendicular 

 to this orbit. Consequently when P = D there can be no change in the 

 motion of the system due to the tides. Therefore the right members of 

 (32) must carry (P—Dy as a factor, where / > 1. The exponent / can not 

 be fractional for then the rate of change of P would be imaginary for P<D. 



Consequently j is 2 or some greater positive integer. The velocity of the 



p J) 



tide with respect to the surface of w^ is , and the tidal force is pro- 



portional to ^. If we assume that the friction is proportional to the height 



of the tide and the first power of its velocity over the surface of Wj, and 

 that the loss of energy, i.e., the work done against friction, is proportional 

 to the square of the friction, equations (32) become 



(33) 

 where Cj is a constant depending upon the physical condition of m^. 



