E.. L. De Forest—Law of Error in Target-Shooting. 3 
ing it to be accidental. Then if the actual difference falls within 
this amount, or does not much exceed it, we may presume that the 
probability of error is really the same in all directions, and that 
formula (5) may be properly used, the axes being taken horizontal 
and vertical, simply because those directions are most convenient. 
On the other hand, if the actual difference between p,* and ,” is 
much in excess of its probable value, we must. presume that the ob- 
served elongation of the group of shot-marks is due to constant 
causes, likely to have a similar effect on future shots, so that (1) is 
the most suitable formula to express the law of error, and the axes 
assumed should be the free axes of the group. It has seemed to me 
that the question whether formula (1) should be used, and if so, 
whether the codrdinate axes should be made coincident with the free 
axes, is not a mere matter of opinion, but should be decided by some 
definite test like the above, applied to an extended set of observa- 
tions. The most suitable observations for this purpose within my 
reach are those given by Didion at the close of his Calewl des Proba- 
bilités appliqué au Tir des Projectiles. Paris, 1858. 
His first table gives the positions of.125 shot-marks made by 
spherical bullets, fired from a rifled pistol at 50 metres, under a 
charge of one gramme of powder. The weapon was placed on a 
rest, and aimed ata point 0:430 metres above the centre of the target. 
The positions of the shot-marks were referred to axes taken hori- 
zontally and vertically through that centre. The arith. mean of 
their ordinates is the ordinate of their centre of gravity, and the 
arith. mean of their abscissas is its abscissa. We easily find the 
abscissa uw and ordinate v of each shot-mark, referred to axes taken 
horizontally and vertically through the centre of gravity, and ex- 
pressed in centimeters. The sums of their squares and of their 
products are 
[w*] = 44211, [v"] = 55766, [wv | = — 6655. 
The angle m which a free axis makes with the U axis is given by 
2 
fee esac Za (6) 
[w*]—[e*] 
Hence log. tan 2~=:06140, and 
Gg VAT ai" or = 1149 si"; (7) 
These two values, differing by 90°, represent the inclinations of the 
two free axes of X and Y to the U axis. Denoting them by gy’ and 
y' +90°, the codrdinates of a shot-mark referred to the free axes 
will be 
x= ucos p+ vsin @’, yY =v COs g' — usin GY’, 
(8) 
