54 
MINUTES OF PEOCEEDINGS OF 
Then the probability of striking at a distance = + y above the centre, 
will be 
= -4= e-iVdy .(11) 
V 7T 
The probability therefore of hitting a very small area of the target, 
situated as at a on the figure, will be the multiple of the two probabilities 
on which it depends, i.e. 
-El e-W* * '+Mdxdy .( 12 ) 
22. Now suppose this very small area to be situated as at b , i.e. close 
to the central point, making x and y each = 0. The probability of hitting 
it, which may be taken in practical language to be the same as that of hitting 
the centre of the target, will be 
= dxdy .....(IB) 
7r 
and substituting for p and q their values in Equations (8) and (9), the 
probability becomes 
= 2a?***.< 14 > 
Erom which we see that the accuracy of a gun is inversely proportional 
to the product of the mean vertical and horizontal errors , i.e. 
Accuracy of gun oc — 
.( 16 ) 
23. We can now shew that this determination exactly corresponds with 
the measure of accuracy adopted by Captain Noble, namely, the area of the 
probable rectangle formed on the horizontal target . Eor if we project the 
marks of the shots from the normal to the horizontal position, by means of 
Equation (2), and then deduce from them the area of the probable rectangle, 
we shall find this area always proportional to hh, so long as the descending 
angle of the shot remains the same. And, moreover, if the product hlc ) and 
consequently the area of the probable rectangle, be constant, the accuracy 
of the gun will be equal, whatever proportions the sides may bear to each 
other. And we thus see that the definition of accuracy we have proceeded 
on, coincides with Captain Noble’s. 
* Or making dy = dx ; we may say that the probability of hitting a very minute square covering 
the centre of impact, whose side = dx, will be = —i— dx 2 . For a circle whose diameter = dx, the 
2irhk 
probability will be 
iihlc 
dx 2 
(15) 
