1880.] History of Planet and Single Satellite. 277 



be done by drawing a curve in which the ordinates are proportional 

 to dt/dx, and the abscissas are x. The equation to this curve is then 



j^dt_ — x v » 



dx x 4, — 7wr 3 + l 



The maximum and minimum values (if any) of dt/dx are given by the 

 real roots of the equation 



lis**— 1W + 15=^0. 



One of such roots will be found to be intermediate between a and b. 

 and the other greater than a. 



Fig, 



Diagram illustrating the Eate of Change of the System. 



Fig. 8 shows the nature of the curve when drawn wdth the free 

 hand. It was not found possible to draw this figure to scale, because 

 when h=2'6 it was found that the minimum M was equal to "85, arid 

 could not be made distinguishable from a point on the asymptote A, 

 whilst the minimum m was equal to about 900,000, and could not be 

 made distinguishable from a point on the asymptote C. 



The area intercepted between this curve, the axis of x, and any pair 

 of ordinates corresponding to two values of x, will be proportional to 

 the time required to pass from the one configuration to the other. 



Where dt/dx is negative, that is to say, when the satellite is falling 

 into the planet, the areas fall below the axis of x. This is clearly 

 necessary in order to have geometrical continuity in the curve. 



The figure shows that the rate of alteration in the system becomes 



VOL. XXX. U 



