MATHEMATICAL SECTION. 141 
shown analytically by proving that the probability of hitting the 
area of the circle 
x _- y? — ia 
differs from that of hitting the equal probability ellipse 
ger Ss ea 
es a mee 
by terms of the fourth order, with respect to the difference between 
the mean errors of sighting and leveling. 
In computing p by (17) the radius (or mean semi-diameter) 7 is 
left arbitrary ; it is, however, not at all indifferent; for if we take 
it very small or very large it will give very unreliable values of p. 
There must then be a certain magnitude of r giving the most re- 
- 
re 
: ue : rdr — ss 
liable value of p, anditis that which makes P, = —-é 22 4 
‘ ie aay r 
maximum. This gives the condition: 0 = zrworse 
Thus the most favorable value of r for determining p is the 
2 
2 
mean error « and the ellipse on =1 (18) 
= My 
is the ellipse of the most probable shot. 
Placing 7 = ¢ in (13), we have 
I 
pe 2 OR06E8.... 
n 
I 
.n, = ( —e z) n = 0.39847 ...n=0.4n nearly (19) 
The most probable shot is, therefore, the distance of the (0.4n)th 
shot from the center nearly; also the mean of the (0.4n + m)th, 
and the (0.4n — m)th shot should, if m is not too large, give a fair 
value of the most probable shot. 
Solving (13) for «, we have also 
Cee 
2! n (20) 
n—N, 
From the definition of e, and es it is obvious that 
psa gf a 
