174 



Mr. G. H. Darwin. 



[June 19 



gives the configuration where the satellite is closer to the planet. We 

 have already learnt that these two correspond respectively to minimum 

 and maximum energy. 



When x is very large, the equation to the curve of energy is 

 Y=(h— x) 2 , which is the equation to a parabola, with a vertical axis 

 parallel to Y and distant h from the origin, so that the axis of the 

 parabola passes through the intersection of the line of momentum 

 with the axis of orbital momentum. 



When x is very small the equation becomes Y=— — , 



x 2 



Hence, the axis of Y is asymptotic on both sides to the curve of 

 energy. 



Then, if the line of momentum intersects the curve of rigidity, the 

 curve of energy has a maximum vertically underneath the point of 

 intersection nearer the origin, and a minimum underneath the point 

 more remote. But if there are no intersections, it has no maximum 

 or minimum. 



It is not easy to exhibit these curves well if they are drawn to scale, 

 without making a figure larger than it would be convenient to print, 

 and accordingly fig, 1 gives them as drawn with the free hand. As 

 the zero of energy is quite arbitrary, the origin for the energy curve 

 is displaced downwards, and this prevents the two curves from cross- 

 ing one another in a confusing manner. The same remark applies 

 also to figs. 2 and 3. 



Fig. 1 is erroneous principally in that the curve of rigidity ought 

 to approach its horizontal asymptote much more rapidly, so that it 

 would be difficult in a drawing to scale to distinguish the points of 

 intersection B and D. 



Fig. 2 exhibits the same curves, but drawn to scale, and designed 

 to be applicable to the case of the earth and moon, that is to say, 

 when h=4i nearly. 



Pig. 3 shows the curves when h—1, and when the line of momentum 

 does not intersect the curve of rigidity ; and here there is no maxi- 

 mum or minimum in the curve of energy. 



These figures exhibit all the possible methods in which the bodies 

 may move with given moment of momentum, and they differ in the 

 fact that in figs. 1 and 2 the biquadratic (5) has real roots, but in 

 the case of fig. 3 this is not so. Every point of the line of momentum 

 gives by its abcissa and ordinate the square root of the satellite's 

 distance and the rotation of the planet, and the ordinate of the 

 energy curve gives the energy corresponding to each distance of the 

 satellite. 



Parts of these figures have no physical meaning, for it is impossible 

 for the satellite to move round the planet at a distance which is less 

 than the sum of the radii of the planet and satellite. Accordingly in 



