518 



REPORT — 1891. 



the two values of h equal to each other in (14), that is, by making 

 cos 6 — o — 7=. = ± 1. 



h = 



Since at the same time li = 2m@/s, we obtain 

 A , _ As 



±1 



, and @ = 



2to(cos ^ ± 1 



(15) 



Let h' and h", and @' and @", be the positive and negative values of h 

 and @ in (15), and we may construct the following table of their values. 

 As in Table I , Jupiter is assumed to be the perturbing planet. 



19. Explanatio7i of Table 11. — The meaning of the numbers in this 

 table may be explained by an example. If a comet moving in a parabola 

 passes near to Jupiter, and the directions of the two original motions at 

 nearest points of the orbits make an angle of 10°, then the greatest 

 action of Jupiter (during the whole period of transit) in diminishing the 

 velocity of the comet in its orbit about the sun will take place if the two 

 orbits actually intersect (d = 0), and if the comet in its unperturbed 

 orbit arrives first at the point of intersection at the instant when Jupiter 

 is distant therefrom •01250 (the earth's mean distance from the sun 

 being unity), the resulting semi-axis major of the comet's orbit about 

 the sun will be 3-04. 



On the other hand, the greatest effect in increasing the velocity of 

 the comet will take place when the two orbits actually intersect, and the 

 comet in its unperturbed orbit reaches the point of intersection later than 

 the planet and when the planet is distant therefrom 0-15174. The semi- 

 transverse axis of the resulting hyperbolic orbit about the sun will be 



20. Resulting orbits of maximum perturbation. — The position of the 

 relative orbit about Jupiter in these cases of maximum perturbation for 

 given values of w is easily determined. From the equations (7), (6), 

 and (15) 



tan a = B/A = h sin 6/ A = sin 6l/(cos 6 ± 1). 



The positive sign gives 2a = 0, and the negative sign gives 2a = tt + 0. 

 But the angle 2a in the first case is the angle of the asymptotes enclosing 

 the branch of the hyperbola described about Jupiter by the comet. 

 Since the two original orbits intersect, the plane of the relative orbit 

 contains the planet's path, so that the comet passes directly in front of 

 the planet, and being turned backward leaves Jupiter exactly in the 



