58 
ARTILLERY FIRE. 
+ • • • • + {l +^2 } p "-* + ^-i] 
=,7~r{( w ~ i).«+(«■-!)/), + -'^pb 3 
, n- 1 , 7 
+ + 
= * +Pi + 2P2 + ip 3 + ■ ■ ■ + -^—-.Pn-z + —U: p n -i yards. 
n — Z n —1 
Accordingly the n th shot should be fired for a range of 
r-Pi-lPt- iPa - ■ ■ ■ - -^ 2 ^- 3 yards. 
But, from the assumption made, the (n — ]) th shot was fired fora 
range of 
r-pi—hp 2 —$ 4 > 3 —• • • ^^-8 yards; 
hence the correction for the n ilx shot is 
1 
—-j Vn-l) 
n—'' 1 
or, assuming the law to hold in regard to the corrections to be applied 
to the first n — 1 shots, the law also holds in regard to the correction to 
be applied to the w th shot. Since the law is certainly true for small 
values of n it is thus shewn to be true universally. 
When n is large, the distance 
X+Pl + hPz + • • • + — Pn 
n 
approximates closely to the range appertaining to the gun when it is 
laid for a range r. Suppose this range to be r x so that r x is the distance 
from the gun of the centre of impact when a large number of shots 
are fired for the range r, then 
x = r 1 -p 1 —ip i —. . . -~^Pn 
and we see that if x = r x or the gun was correctly laid for the first shot, 
Pi + iPn + . ■ 
n 
or the sum of the successive corrections, is zero. 
It is important to remark that, with this systematic method of correc¬ 
tion of fire, the corrections become of less and less importance as the 
firing proceeds ; this is, indeed, the natural result of any sound method 
of correcting fire. The gun may be regarded as oscillating about its 
trunnions, the amplitude of the oscillations becoming smaller and 
smaller, and gradually settling down at the elevation which will cause 
the object to be at the centre of its fire. 
If the target be moving so that the range is practically constant the 
above principles apply without alteration. 
If the target be moving so as to diminish the range by k yards 
