2 GO 



Mr. G. H. Darwin. 



[Mar. 18, 



Therefore fL log e=--l . f^^h n£y 



dx ° x 2 x±-hx*+l y J 



These two equations are integrable as they stand, except as regards 

 the last term in each of them. 



It was shown in a previous paper that the whole energy of the 

 system, both kinetic and potential, was equal to \ [n* — a; -2 ].* 



Then integrating (12) and (13), and writing down (7) and (8) 

 again, and the expression for the energy, we have the following equa- 

 tions, which give the variations of the elements of the system in terms 

 of the square root of the satellite's distance, and independently of the 

 time. 



log ? =log — — + p f 'V^ — : + const, 

 n—x \x— fix 6 -\-i 



j -•• - 1 1 x ( x ~~ty _^ _|_ const. 



^-/^ 3 +l 



. h—x. 

 n=h—x. 



(14). 



2E=(h-xy—±- 



X i 



When the integration of these equations is completed, we shall 

 have the means of tracing the history of a fluid planet of small 

 viscosity, attended by a single satellite, when the system is started 

 with any given moment of momentum h, and with any mean distance 

 and (small) inclination and (small) eccentricity of the satellite's 

 orbit, and (small) obliquity of the planet's equator. It may be re- 

 marked that h is to be taken as essentially positive, because the sign 

 of h merely depends on the convention which we choose to adopt as 

 to positive and negative rotations. 



These equations do not involve the time, but it will be shown later 

 how the time may be also found as a function of x. It is not, how- 

 ever, necessary to find the expression for the time in order to know 

 the sequence of events in the history of the system. 



Since the fluid which forms the planet is subject to friction, there- 

 fore the system is non- conservative of energy, and therefore x must 

 change in such a way that E may diminish. 



If the expression for E be illustrated by a curve in which _E7 is the 

 vertical ordinate and x the horizontal abscissa, then any point on this 

 " curve of energy " may be taken to represent one configuration of 

 the system, as far as regards the mean distance of the satellite. 



* " The Secular Effects of Tidal Friction," &c, " Proc. Hoy Soc," Xo. 197, 

 1879, eq. (4). 



