POLES AND POLARS. 
233 
and therefore (1) becomes 
c cos a 
^ = (e + cosa). (r sin# 
c sin a 
sin a. ( r cos# — 
1 + e cos a. 
1 + e cos a/ 1 
or 
r {(e + cosa) sin# — sina cos#} = 
1 + e cos a ’ 
or 
c 1 + e cosa , . a , . fQ ^ 
- = ; . e sm 0 + sin 0 — a) , 
rv* /3 Ö1T1 « 1 N 
e sm a 
which is the required equation to the normal. 
2. If the normal through any point P in an ellipse cuts the 
major axis in 6r, to prove that, S being the focus, 
SG = e. SP. 
Section XXIII. 
Poles and Polars. 
1. There are two ellipses the axes of which are in the same 
straight lines. To find the locus of a pole of the exterior ellipse 
the corresponding polar of which is always a tangent to the 
interior. 
Let a, be the semi-axes of the interior, and a, /3, of the 
exterior ellipse. Let the coordinates of the pole be cc ; , ?y ( , and 
let x, y, be those of the point in which its polar touches the 
interior ellipse. 
Then, the axes of the ellipses being taken as axes of coor 
dinates, the equation to the polar will be 
but, since it is a tangent to the interior ellipse, its equation will 
also be 
(2). 
Since the equations (1) and (2) must be identical, we have 
x 
hence, by the equation