1 FE. L. De Forest—Law of Error in Target-Shooting. 
and the sums of their squares are 
[a] = [2"] cos* p+ [v*] sin’ ¢' + [wv] sin 29’, t (9) 
[y*] = [v*] cos* pg’ + [u?] sin’? p’— [wv] sin 29’, 
from which we readily find 
[a*] = 41172 [y*] = 58806. 
The squared q. m. errors are therefore 
p'= ea — 332°03, p2= BEaE = 47424, (10) 
where 7 is the number of shots. 
Since the codrdinates of the centre of gravity are the arith. means 
of those of the shot-marks, the semi-axes of the ellipse of probable 
error in the position of the centre of gravity are 
p p 
a=11774-—+= 1°92, 6= 171714 —== 229) 11 
VA = (11) 
Mr 
(Analyst, viii, p. 77). In the case we are considering, the horizontal 
and vertical coérdinates of the centre of the target referred to the 
centre of gravity are 
u = °02, Ui 825); 
and when referred to the X and Y axes they are by (8) 
a 3°44, ==] "ON. (12) 
These values compared with a and 0 in (11) show that the actual dis- 
tance of the centre of gravity from the centre of the target is much 
greater than it probably would be if it were purely accidental. 
Hence, to represent the probabilities of error in future shots, the sur- 
face (1) should in this instance remain with its vertex at the centre 
of gravity, and not be shifted to the centre of the target, unless the 
aim is corrected at the same time. 
Having thus determined the most probable position of the origin, 
we wish next to know whether the actual difference between p,* and 
p, is much in excess of what might be expected if it were acci- 
dental. Its probable amount may be found approximately as fol- 
lows. When the q. m. error ¢ is computed from m observations, the 
probable error of this determination is known to be 
é 
6745 ye (13) 
and consequently the probable error of & will be 
67458, / 2. (14) 
Mv 
