44 
The N Z. Journal of Science and Technolocy. 
[Feb. 
Conversely, in order that the distance of nearest approach may be 
t'oj tuo> tuooj or Toboo initial distance, the initial values of k must 
Lp _i_ _A__3_or>_A_<-)( |\ 
11’ 101’ lOOl’ 10001 
Example 2 .—To find the velocity with which a particle must be pro¬ 
jected, at right angles to the attractive force, when at a distance from the 
Sun equal to that of Neptune, in order that it may just graze the surface 
of the Sun. 
The distance of the Sun is 6452 times the Sun’s radius ; therefore 
k 
1 
K — k 6452 ’ 
.-. K = 64537c; 
therefore the particle must have a kinetol 
i i 
of the parabolic, and 
therefore a velocity of 
V6453 
6453 
of the parabolic velocity, which 
80-33 
we see from the table is 4-7578 miles per second. The particle, there¬ 
fore, will just graze the surface of the Sun if the initial velocity is 
0-05923 mile per second. 
Corollary : If the velocity at the distance of Neptune is less than 
0-05923 mile per second a collision is inevitable whatever the angle of 
projection. 
(ii.) If the actual initial kinetol = J the initial parabolic kinetol, then 
e = 0. The curve is a circle, the two foci coinciding with S. The distance, 
therefore, never varies from its initial value. 
(iii.) If the actual initial kinetol > \ the initial parabolic kinetol, but 
< the initial parabolic kinetol, then S is the nearer focus and AS = a (1— e ); 
K 
r o = 
1 — e 
2(K- k) 
2 (K — k) 
r Q (1 — e) 
K 
2 — 
A K 
1. 
The starting-point is the position of nearest approach. 
(iv.) If k = K, the eccentricity = 1, the curve is a parabola with A as 
vertex and S as focus, the axes are infinite, and A is the point of nearest 
approach. 
(v.) If k > K, the curve is an hyperbola. At every point of the path 
® 2 = A- + - 
V r a 
or 
t = l„ o 2 — — = — K ; 
2 a 2 ° r 0 
2 a — 
fX 
k — K 
K 
therefore the major axis 
