1879 J On the Secular Effects of Tidal Friction. 171 



Q = n the planet's angular velocity round its axis ; or since n—y and 



Q~i= x, therefore y—~- 



x 6 



By substituting this value of y in the equation of momentum (3), 

 we get as before 



x *-hx*+l=0 (5). 



In my paper on the " Precession of a Viscous Spheroid,"* I 

 obtained the biquadratic equation from this last point of view only, 

 and considered analytically and numerically its bearings on the history 

 of the earth. 



Sir William Thomson, having read the paper, told me that he 

 thought that much light might be thrown on the general physical 

 meaning of the equation, by a comparison of the equation of con- 

 servation of moment of momentum with the energy of the system for 

 various configurations, and he suggested the appropriateness of 

 geometrical illustration for the purpose of this comparison. The 

 method which is worked out below is the result of the suggestions 

 given me by him in conversation. 



The simplicity with which complicated mechanical interactions may 

 be thus traced out geometrically to their results appears truly remark- 

 able. 



At present we have only obtained one result, viz. : that if with 

 given moment of momentum it is possible to set the satellite and 

 planet moving as a rigid body, then it is possible to do so in two 

 ways, and one of these ways requires a maximum amount of energy 

 and the other a minimum ; from which it is clear that one must be a 

 rapid rotation with the satellite near the planet, and the other a slow 

 one with the satellite remote from the planet. 



Now, consider the three equations, 



f>=y+x (6), 



Y=(h-xy—^ ...... (7), 



x*y=l . . . . . . . . (8). 



(6) is the equation of momentum; (7), that of energy; and (8) 

 we may call the equation of rigidity, since it indicates that the two 

 bodies move as though parts of one rigid body. 



Now, if we wish to illustrate these equations geometrically, we may 

 take as abscissa x, which is the m. of m. of orbital motion; so that the 

 axis of x may be called the axis of orbital momentum. Also, for 

 equations (6) and (8) we may take as ordinate y, which is the m. of 

 m. of the planet's rotation ; so that the axis of y may be called the 

 axis of rotational momentum. For (7) we may take as ordinate Y, 



* Of which an abstract appears in " Proc. Roy. Soc," No. 191, 1878. 



