1920.] 
Gifford.—The Origin of New Stars. 
49 
Example 3 — If t he initial kinetol is of the parabolic kinetol, 
k -M. 
k ~io’ 
.'. e — \/ { 1 — '36 sin 2 a}. 
The values of e for different angles of projection, and the distance at 
the point of nearest approach in terms of the semi-major axis, are given 
in figs. 7 and 8. 
Example 4 .—A body is projected from a point as far from the Sun as 
the orbit of Neptune, and its initial kinetol is ^ of the critical : at what 
angle must it be projected in order that it may just graze the surface of 
the Sun ? 
If it were projected at an angle of 90° with the line from it to the Sun, 
f 
its nearest approach would be (see p. 43)—-.that is, about 719 times the 
9 
Sun’s radius. If the angle is a, this distance is reduced to 
719 (1 - Vl — -36’ 
sin 2 aV 
If this equals unity, V 1 — *36 
718. 
Sin a 719’ 
•36 sin 2 a 
: 1 - 
1437 
(7i9) 2 ; 
/718> 2 
4719. 
Sin a - 
10^1437 
6 X 719 
•0879. 
a = 5° 3' approx. 
If gravity did not act, the necessary angle would be - radius. 
The angle is increased approximately '0879 X 6452 times—approximately 
569 times. 
That is, the chance of collision is increased 323,600 times by the action 
of gravitation. 
(ii.) K = k. 
If the initial values of the actual and parabolic kinetol are identical 
the eccentricity will be unity and the curve a parabola whatever the angle 
of projection. The distance of nearest approach, however, will vary when 
the angle changes. 
Let the particle be projected from the point P with parabolic velocity, 
and let S be the centre of force. (See fig. 9.) 
Then A is the point of closest approach, and 
SA = iSX = i(SP-SN) 
- |SP (1 - cos 2 a) 
= SP sin 2 a. 
That is, the distance at the point of nearest approach varies as the square 
of the sine of the angle of projection, and takes all values from 0, when 
the impact is direct, to r Q , the initial distance, when a = 90°. 
4—Science. 
