146 PHILOSOPHICAL SOCIETY OF WASHINGTON. 
Much less proper, it would seem to me, to regard the position of 
the axes unknown, and to compute their most probable position. If 
center and axes are to be determined, 2’ y’ denote the co-ordinates 
of ashot from a random origin and position of axes, and w the 
angle of turning the latter into their most probable direction ; then 
the most probable co-ordinates of a shot are: 
e=2,+2 cosw+ysinw; y=y,+y cosw — x sinw. 
Imposing the conditions of a minimum for [27] and [y’], we find 
v, = — =; (LeJeosw+ fy] sin w) 
(33) 
y= — = (iy cos w — [2’] sin w) 
[?yJ—— (ef 
tan 2u = ——_,; —*_______ (34) 
[2] ——{e'P — [y+ — FP 
These formule have, however, their proper place in the theory of 
Andre’s “ Fehler-ellipse.” 
Fig. 2 exhibits an ideal distribution of 45 shots. Each ring con- 
tains 6 shots, leaving 3 shots between the outer ring and infinity. 
The dotted circle is that of the most probable shot, and the dashed 
one that of the even chance shot. 
The following table refers to the combined target record of the 
Irish team at 800 yards range, in the international shooting match 
of 1874, at Creedmoor : 
