276 Mr. G. H. Darwin. [Mar. 18, 



the latter the parts between the maximum and minimum of energy be 

 annihilated. 



These three figures may be interpreted as giving the various stabilities 

 and instabilities of the system, just as was done in the first case. 



The solution of the problem, which has been given and discussed 

 above, gives merely the sequence of events, and does not show the rate 

 at which the changes in the system take place. It will now be shown 

 how the time may be found as a function of x. 



Consider the equation 



*L=j± sin °\ 

 dt g \ n) 



/is here the angle of lag of the sidereal semi-diurnal tide of speed 2n ; 

 then by the theory of the tides of a viscous spheroid, tan 2/= 2?? /p, 

 where p is a certain function of the radius of the planet and its density, 

 and which varies inversely as the coefficient of viscosity of the 

 spheroid.* 



Since by hypothesis the viscosity is small, / is a small angle, so that 

 sin 4/ may be taken as equal to 2 tan 2/. Thus, sin 4//?j is a constant, 

 depending on the dimensions, density, and viscosity of the planet. 



It has already been shown that t 2 varies as x~ 12 , and (J is a constant, 

 which depends only on the density of the planet. Hence, the above 

 equation may be written 



dt y h 



where K is a certain constant, which it is immaterial at present to 

 evaluate precisely. 



Since n — h—x and Q==aT 3 , we have 



Kdt= - xl ° dx _ y 



or Kf = — I — - — — — -j- a const. 



The determination of this integral presents no difficulty, but the 

 analytical expression for the result is very long, and it does not at 

 present seem worth while to give the result. The actual scale of time 

 in years will depend on the value of K, and this is a subject of no 

 interest at present. 



It will, however, be possible to give an idea of the rate of change 

 of the system without actually performing the integration. This may 



* " On the Bodily Tides of Viscous and semi-elastic Spheroids," &c. '■''Phil. 

 Trans.," Part I, 1879. p. 13, § 5. 



