172 



Mr. G. H. Darwin. 



[June 19, 



which is twice the energy of the system ; so that the axis of Y may 

 be called the axis of energy. Then, as it will be convenient to exhibit 

 all three curves in the same figure, with a parallel axis of x, we must 

 have the axis of energy identical with that of rotational momentum. 



It will not be necessary to consider the case where the resultant m. 

 of m. h is negative, because this would only be equivalent to reversing 

 all the rotations ; thus h is to be taken as essentially positive. 



Then the line of momentum, whose equation is (6), "is a straight 

 line at 45° to either axis, having positive intercepts on both axes. 



The curve of rigidity, whose equation is (8), is clearly of the same 

 nature as a rectangular hyperbola, but having a much more rapid rate 

 •of approach to the axis of orbital momentum than to that of rotational 

 momentum. 



The intersections (if any) of the curve of rigidity with the line of 

 momentum have abscissas which are the two roots of the biquadratic 

 aj*— fcu s +l=0. The biquadratic has, therefore, two real roots or all 

 imaginary roots. Then, since x=Q~^, it varies as r, and, therefore, 

 the intersection which is more remote from the origin, indicates a con- 

 figuration where the satellite is remote from the planet; the other 



