ON THE CAPTURE OF COMETS BY PLANETS. 513 



5. If we assume two epochs, one before and one after the perturbation, 

 at which the comet is equally distant from the planet, the term 2mfa- jr^ 

 is the same at both instants, and it disappears from the value of A. 

 Therefore 



P Pi 



But by the well-known formulas from the law of gravitation, 



and 



hence 



' \@, ®J 



that is, from (1) 



1 1 4m cos <f> sin a 



'@/ @ Fs 



This equation is valid whatever be @, the major axis of the orbit C^, and 

 may be used to determine the major axis of either orbit from the elements 

 of the other. My present purpose is, however, to study the action of 

 Jupiter in changing orbits that are originally parabolas, and hence in 

 general @^ will be taken infinite. In that case 



® = I ^V— .... (-2) 



4m. cos 9 sm a 



It will be found that the second member of (2) depends on to, d, and 

 h, and these are known quantities when the elements of C/ and g) are 

 given. The use of the equation is, moreover, greatly simplified and 

 enhanced by the fact that the plane of the planet's orbit is involved only 

 in so far as that it must contain the tangent to B at E. 



6. In the second member of (2) all the factors are positive except 

 cos <^ ; hence if <j><ij7r, @ is positive and the orbit (£ is an ellipse ; but if 

 ^>^7r, @ is negative and C is ^^ hyperbola. This result may be thus 

 expressed : If the comet passes in front of Jupiter the kinetic energy of the 

 comet is diminished ; if it passes liehind the planet the hinetic energy of the 

 comet is increased. The reason for this may also be given iu general 

 language. If the comet passes in front of the planet the comet's 

 attraction increases the velocity, and hence increases the kinetic enei'gy 

 of the planet, and vice versa. But the total energy of the two bodies is 

 constant, so that when that of the planet is increased that of the comet 

 is diminished, and vice versa. 



7. It is desirable now to transform the value of @ given in equation 

 (2) so as to be able to determine the major axis of the new orbit of the 

 comet directly from the circumstances of its initial approach to the 

 planet before perturbation ; in other words, to find @ in terms of m, d, 

 and h. For this we must find in terms of w, d, and h values for .<r, p, a, 

 and <^. 



1891. L p 



