ON THE CAPTURE OF COMETS BY PLANETS. 



517 



Table I. 



15. Using these values of 0, s, and A we may now represent graphic- 

 ally the dependence of @ upon the other two variables d and h tor each 

 specified value of <j. Let d and h be Cartesian coordinates, then for each 

 point of the coordinate plane there is a value of @. The ambiguous sign 

 will be fully satisfied by giving positive and negative values to h. For 

 an assumed value of @ we shall have a curve whose equation is (13), 

 and each point of this curve represents values of d and h for which the 

 total action of the planet upon the comet will be to reduce the energy of 

 the comet a constant amount. This locus will be called an isergonal 

 curve. 



16. Faisceau of isergonal ellipses. — The equation (13) of the isergonal 

 curve may be written 



4m@ (A cos e + 11 sin^ 6) = s {A? ■}- d"^ + li^ sin^ 6), 



and this is the equation of an ellipse. As @ changes its value we may 

 treat it as a parameter, and we have a faisceau of similar isergonal 

 ellipses, each ellipse symmetrical with the axis of h. The radical axis of 

 the faisceau A cos 6 + h sin^ =-0, and the imaginary ellipse A'^ + d'^ 

 + h"^ sin^ ^ = 0, are theoretically two members of the faisceau. For 

 points on the radical axis @ = oc, and therefore for this locus there is no 

 change in the energy of the comet. 



17. Centre and area of tlie isergonal ellipse. — The centre of the isergonal 

 ellipse is upon the axis of h ; making d ^^ and solving for h, we have 



k = '^J^±^^Ji-(oose-A 



s siu 6 



(!-(< 



2m@ 



)y 



(14) 



The first term of the second member of (14) is the ordinate of the centre, 

 and the second term is the semi-axis major of the ellipse. The ratio of 

 the axes being 1 : sin $, and As^ being = mr, the area of the ellipse will 

 be equal to 



18. Maximum action of the planet. — For two particular values of @ 

 the isergonal ellipses become points. These values of @ result if the 

 maximum effect of the planet in increasing and in decreasing the 

 energy of the comet takes place, and they are obtained by making 



