CONCENTRATING THE EIRE OE A GROUP OF GUNS, ETC. 
309 
A 1 and A s (but opposite in direction) and, since the magnitude of 
small angles, such, as are here in question, is approximately proportional 
to their sines, the correction will vary inversely as the range R , and 
directly as the distance cl between the pivots and as the sine of the 
angle made by the line of fire of the centre gun with the line joining 
the pivots. For a given group d is of course a constant, so that the 
correction will vary inversely as the range and directly as the sine of 
the angle A x A 2 B which depends on the training. 
Let w represent the angle of correction, then 
sin w = ^ sin A a A e B, 
or R — sin Aj Ao B. 
sin w 
Now if we assign a definite value to w. —-becomes a constant (= K 
sin w 
suppose). Then for any given value of A l A 2 B we get the correspond¬ 
ing value of R at which w has the value assigned; that is we can find 
for every successive angle of training the corresponding range for 
which the correction will be equal to w. 
The above equation when w has a fixed value becomes of the form 
R = K sin 6 
and this is a polar equation to a circle, the origin being on the circum¬ 
ference. and the initial line a tangent: — — = K being the diameter 
sin mi 
of the circle, R the length of a line joining a point on the circumfer¬ 
ence to the origin, and the angle 6 , the angle between that line and 
the initial line. This relation points out an easy way of showing the 
corrections. 
Fig. 2. 
0 
This figure is not drawn to scale. 
In Fig. 2, BD is at right angles to ABO, ADO is a semicircle with 
