MATHEMATICAL SECTION. 139 
then every point on the equal probability ellipse (3) corresponds 
to a point (#,, y,) on the circle: #7 + y7 = 7", (6) 
which is the reduced equal probability circle. 
Counting directions from the right of the x - axis, let 
a = direction of (x,y) (7) 
i= ia “ (@,,Y,), or reduced direction of (z,y) (8) 
h wots y fy O &x oo fy 9 
peewee ee sri sais (9) 
also = = 7 COS a, (10,) 
Bee ca 
y= 7 sin a, (10,) 
€ € s 
whence dz = ~ cosa, dr — ~r sin a,da, 
S é 
een a ae 
dy = — sin a,dr + — r cos a,da, 
Transforming, then, (2) to the new variables, 7 and a,, we must 
replace: 
dedvan e,.¢,rdrda, 
e? 
and thus obtain 
rdrda, — kad 
Ont & 2c — probability to hit a point of which (r, a,) 
is the reduced point. (11) 
Vg. 
A» 
FAN 
an 
Fig. 1 exhibits 24 shots of equal probability, on an equal proba- 
bility ellipse, and their reduced positions evenly distributed over 
the reduced circle. 
