138 PHILOSOPHICAL SOCIETY OF WASHINGTON. 
Mr. C. H. KuMMELL read a communication ertitled 
THE THEORY OF ERRORS PRACTICALLY TESTED BY TARGET- 
SHOOTING. 
[ Abstract. ] 
Sir John Herschel treats a special case in which shots of equal 
probability are in circles. According to Liagre’s theory target 
shooting is compounded of two distinct operations, viz., sighting 
and leveling, each of which is liable to errors, independently fol- 
lowing the ordinary linear law of error. Some reasons for the in- 
dependence of these operations are that for sighting the direction 
of the wind, which does not affect the leveling, must be regarded; 
and that, on the other hand, leveling only is affected by the range. 
The consequences of Liagre’s theory will now be developed. 
Let x = error of sighting and ¢, its mean error; 
y = error of leveling and «, its mean error ; 
then it follows that 
x 
d — 52 
wi e 26x" — probability to hit anywhere at distance z from 
sighting axis. (1) 
d a 
we gree probability to hit anywhere at distance y from 
y g | 
leveling axis. (1,) 
ded Se 
— - — 2 
’ —— ates 3 “Go probability to hit the point (2, y). (2) 
FE y7 
This probability is the same for any point on the ellipse: 
Pr 2 9? 1 
3 a a= where © = oy Ce 0 e”) (3) 
x Ag 
This I shall call, then, an equal probability ellipse; its semi-axes 
are: 
ae fy 
A oe : (4) 
and r = mean semi-diameter (which is equal to its conjugate). 
r 
€ 
Assume ar and : pee ae (5) 
