46 
The N.Z. Journal of Science and Technology. 
[Feb. 
IV. Let a particle be projected from a point P in a direction which makes 
an angle a with PS, where S is the centre of attraction. (See fig. 5.) 
(i.) K > k. 
Let the initial velocity be v Q and the initial distance r Q . Since in all 
positions in the curve 
/X 2 (X 
a r 
fX 
2 . 
2(K — k); 
fx K 
a 
. 2 a 
K — k K -k °* 
That is, the major axis is constant whatever the angle of projection. 
To find the eccentricity and the minor axis of the path, draw the hodo- 
graph BHD, of which 0 is the pole and C the centre. (See fig. 6.) 
CO 
Then 
e — 
CH 
OH = v Q , HC = 
7r 
V o r o Sill a 
, and Z OHC = - — «, 
Now CO 2 = CH 2 + OH 2 - 2CH . HO sin 
2 fx 
a 
= CH 2 + ^ 0 2 - : 
2 [X 
v 0 r 0 sin a 
v Q • sm a 
CO^ _ 
*’* CH 2 “ 1 
— 
[X‘ 
v q 2 t q 2 sin 2 a 
1 — 
4 k (K — k) 
K 2- 
sm“a; 
• 1 
» • l 
Now 
e* = — 
4A;(K — k) . 
sm 2 a. 
KV 
b 2 — a 2 ( 1 — e 2 ); 
2v / A;(K — k) . 
. b = a • - = -sm a 
l\. 
= t \I ^ 
K* K- k 
= V 
sm a 
k 
K — k r ° Sm a * 
That is, the minor axis is directly proportional to the sine of the angle of 
projection. 
xt / fi — fy . 2 
Now e = v 1-- sm < 
I K 2 
a 
— y {1 — c sin 2 a}, 
where c is a constant. Therefore e diminishes as a increases. 
