1920.] Gifford.—The Origin of New Stars. 51 
To find the eccentricity, draw the hodograph, and, as before, 
9 4 k{k — K) sin 2 a 
e 1 = - w - ; 
e = Vl + c sin 2 a, 
where c is a constant. The distance of nearest approach = a(e — 1). 
Since a is constant, this varies as 
Vl + c sin 2 a — 1. 
If the starting-point and the velocity are known c can be easily found. 
We can then plot the values of e — 1 for all values of a, and then read off 
the distance of nearest approach for any particular angle of projection in 
terms of the major axis or of the initial distance. 
We notice that if a = 0° the nearest distance vanishes, that it 
increases as the angle of projection increases, and reaches the value r Q when 
a = 90°. 
Since the stars appear to have proper motions averaging perhaps 20 miles 
a second, and the parabolic velocity at a distance from our Sun equal to 
that of a Centauri is barely one-twentieth of a mile per second, it follows 
that in the great majority of cases of actual collision the paths of approach 
will be hyperbolic orbits of very great eccentricity. Such orbits differ 
little from straight lines. The difference of velocity just mentioned gives 
k = 160000 K. If the body was projected at right angles to the line of 
force this would give e the value 319999. 
For any other angle of projection 
e = y {1 + 4 X 160000 X 159999 sin 2 a} 
= K / {1 + (320000 sin a) 2 } approx. 
= 320000 sin a unless a is extremely small. 
If a has a value as great as 30", e is > 3200, and e — 1 is practically the 
same value. 
The distance at the point of nearest approach is given approximately 
in such cases by the simple formula 
r Q sin a, 
which means that the path is not appreciably altered by the attractive 
force, and remains practically a straight line. 
In all questions relating to the collisions of stars it is important to 
remember that any kinetol or thermatol due to proper motion persists 
throughout, and in determining the final values must be added to those 
produced subsequently by gravitation. The consequent changes in the 
velocity are, however, comparatively slight. For instance, if a body passes 
the distance of a Centauri with a velocity of 25 miles per second, instead 
of one-twentieth of a mile per second, which is the parabolic velocity at that 
distance, and moves so as just to graze the Sun’s surface, its original 
thermatol of over 193,000 calories is added to the final thermatol, but its 
final velocity is changed by less than a mile per second—viz., from 381-284 
to 382-106 miles per second. 
The consequences which follow the collision of two stars will form the 
subject of another article. 
