1920.] 
G-ifford.—The Origin of New Stars. 
43 
If we denote tlie initial values of the parabolic and actual kinetol 
by K and k, we have 
- = 2 (K — k) ; 
a 
. . [i K 
therefore the major axis = 2 a = g-^ = g-^ • r Q , 
since 
Now, r 0 = a (1 + e) 
- = K. 
(1 + e) 
K 
K- k 0 
2K-2 k 
)• = 2r„; 
e — 
K 
1 = 1 
2Jc 
K 
The minor axis 26 is given by 
b 2 = a 2 ( l-e 2 ) 
2/1 I \2 
= a 2 (1 + e) 2 -—— 
1 + e 
= r 2 • 
° K — & ’ 
26 = 2r. 
\/_ = 2 / ; ; \/ 
& 
K — k K K-r 
The £ distance Jbetween the bodies at the instant of nearest approach is 
(1— e) a, and the initial distance is (1-f-e) a ; therefore 
nearest approach 1 — e 
k 
initial distance 
nearest approach = 
1 -f- e K — k’ 
k K . 
~- 7 • r n = 7 maior axis. 
K — k° k J 
Example 1 .—If k is given in succession the values 
111 1 
10 ’ 100 ’ 1000 ’ 10000 
of K respectively, 
. 1W , ,, , 4 49 499 4999 
e will take the values 5Q! — , 
, . . xl | 10 100 1000 10000 . 
the major axis the values —, times r 
9 ’ 99 ’ 999 ’ 9990 
3 3'/IT 3\/m 3VITTT 
o ? 
the minor axis the values g, ^ , 500 . 5000 
2 2 VlT 2Vffl 2vT[Tl 
or —--- 
3’ 33 ’ 333 ’ 3333 
and the distance of nearest approach the values 
Io_ and Jo_ 
9 ’ 99 ’ 999 5 ' 9999 * 
of the major axis, 
of r Q , 
