1880.] 



History of Planet and Single Satellite. 



261 



Then such a point must always slide down a slope of energy, and we 

 shall see which way x must vary for any given configuration. This 

 consideration will enable us to determine the sequence of events, 

 when we come to consider the expressions for i, e, j, n in terms of x. 



We have now to consider the further steps towards the complete 

 solution of the problem. 



The only difficulty remaining is the integration of the two expressions 

 in the first and second of (14). From the forms of the expressions 

 to be integrated, it is clear that they must be split up into partial 

 fractions. The forms which these fractions will assume will of course 

 depend on the nature of the roots of the equation re 4 — 7k£ 3 -f-l = 0. 



Some of the properties of this biquadratic were discussed in a 

 previous paper, but it will now be necessary to consider the subject 

 in more detail. 



It will be found by Ferrari's method that 



I 4 2\* / I 4 2A* / 



where X 3 — 4\— h 2 =0. 



By using the property ft) (Ai = 4\, this expression may be 

 written in the form 



l{x+\{U-h)Y + {\Q£-h) v / l + 2/a-f}2]x 



Which is of course equivalent to finding all the roots of the bi- 

 quadratic in terms of h and \. 



Now let a curve be drawn of which 1$ is the ordinate (negative 



Fig. 1. 



The ordinat.es are drawn to one-third of the scale to which the abscissae are drawn. 

 VOL. XXX. T 



