140 PHILOSOPHICAL SOCIETY OF WASHINGTON. 
The probability to hit anywhere on the perimeter of an equal 
probability ellipse of mean semi-diameter, 7, is found by integrating 
(11), with respect to a,, through a circumference. It is 
re 
r € . 
Let n, = number of shots on area of equal probability ellipse of 
semi-diameter 7, and n = total number; then 
n ar Afi co =f ue MB 
T= f- we 22@=1—e 26” op Bo 2 —tt = 2e7 (18) 
0 o 
Let r =p; if n,=3n, thon} =e 22.5. p= 7/212 (14) 
The ellipse : 
2 2 
+ 4=212 (15) 
x y 
is then an even chance ellipse, which is hit or missed with equal 
probability. Eliminating <« between (13) and 14), we obtain: 
(55)"= (@)" as 
These formule agree with Herschel’s in form, and have, also, the 
same signification, in case the precisions of sighting and leveling are 
equal, for in that case the ellipses (3) and (15) become circles and 
r, p their radii, respectively. Herschel employs these formule for 
determining the skill of a marksman, which he defines to be =-, 
from the number of shots that have fallen on a circle of radius r. 
Correspondingly, we should have to count the shots that have 
fallen on an equal probability ellipse, the axes of which have the 
unknown ratio —., which, as yet, we have no method of finding; 
€ = 
therefore formule (14) and (17) cannot be employed in their gen- 
eral signification. If, nevertheless, we count the shots on a circle 
of radius r and compute a value for p and ¢, we shall come as near 
to their true values as the problem requires, especially if the precis- 
ions of sighting and leveling are not very different. This can be 
