Aucust 17, 1899] 
NATURE 367 
Since any practical method of scoring must be rapid and 
easily understood by people who are not mathematical, it would 
probably be asking too much if it were proposed to treat each 
result in an accurate manner, simple though the required arith- 
metic is; but some modification of the accurate method could 
probably be made sufficiently simple for general use, which 
would give a much truer estimate of the goodness of the shoot- 
ing than that now in use. 
The accurate plan of estimating the value of any series of 
shots consists, in mathematical language, of finding the dis- 
tance of the centre of gravity of the group from the centre of 
the target, and taking the radius of gyration of the group about 
its centre of gravity. The goodness of the shooting will then 
be measured by the reciprocal of the sum of the squares of 
these quantities, each multiplied by a constant, and it will 
presently be shown that if, as in an ordinary match, the object 
is to hit the centre of the target, these constants are equal. 
A convenient way of finding the centre of gravity and radius 
of gyration is to have the target divided into 100 squares by 
‘eleven vertical and eleven horizontal lines (see Fig. 4), the 
position of the shot being recorded by naming the square through 
which it passes ; (for instance, a shot in the fourth vertical row 
and fifth horizontal row would be recorded as 4'5). 
If we call the number of the vertical row x, and the number of 
the horizontal row y, very simple algebra will prove—in fact, it is 
obvious—that for a series of 7 shots the coordinates (% . 2) of 
square containing the centre of gravity of the shots will be 
h=3x/n k=Zy/n, 
where =x denotes the sum of all the «’s, and Sy denotes the sum 
of all the y’s. 
Since the score does not record the position of an individual 
shot with greater accuracy than the width of a single square 
this is equivalent to the assumption that each shot passes through 
the centre of the square it hits, and that the origin of the co- 
ordinates is in the centre of the square at o (off the target) (see 
Bigs 4). 
Thus the coordinates of the centre of the target will be 
sa Sev 
and the distance of the centre of gravity from the centre of the 
target is 
R= V(Bx/2—5°5)° + (By/n — 5°5)*. 
The radius of gyration of the group about an axis normal to 
the plane of the target, and passing through the centre of 
gravity is 
p= NV 3x7/n =? + 3y"/n — ke. 
.To examine the relative importance of the closeness of the 
shots to one another and the distance of their mean from the 
centre of gravity, consider the effect of slightly varying each of 
the quantities R and p. 
The question to be answered is: ‘‘If of two groups one is 
represented by R and p, and the other by R+@R and p-—dp, 
which gives evidence of the best shooting ?” 
In Fig. 3 let C be the centre of the target, and G the 
centre of gravity of the group, and 
Pp PQR acircle described with radius 
p about G, so that CG=R, G P=p; 
then if CG P=8, the distance (7) of 
P from C is 
v= \R2—2Rp cos 0+’, 
differentiating with respect to R and 
p, we have 
dr jdr__p—Reos@, 
Q aR/ dp R=p cos 0’ 
and integrating this with respect to 
6 from 7 to O we have for the relative 
mean values of dy, caused respectively by alterations of @R and 
— dp in the values of R and p, 
Ea Ae o 
o@R/ Jy aps R 
If the two groups are equally good, the mean value of dR 
must be equal to minus the mean value of dp. 
This leads toa simple relation between R and p, viz. R® +p? 
= constant. Thus any group of shots for which the sum of the 
NO. 1555, VOL. 60] 
Fic. 3. 
squares of the mean distance of the group from the centre and 
its radius of gyration is constant is equally good. 
This may be stated more concisely by saying that when the 
object is to hit the centre of the target, the merit of any series 
of shots is inversely proportional to its moment of inertia of the 
group about the centre of the target. 
If for convenience it is decided to make the score 100 when 
the moment is unity, the worth of any given series will be 
represented by 
100 
R*+p" 
I give below an actual target with the results analysed in the 
way described. 
If a slide rule is used, the arithmetic takes about five minutes. 
Ido not for a moment suppose that such an analysis would be 
practicable at ordinary rifle matches, but it does seem possible 
that coordinate targets might become popular, and some simple 
way devised of using the more precise information they would 
afford. ; 
As far as finding the moment of inertia of the group about 
the centre of the target is concerned, this might be done more 
simply by the use of polar coordinates, only the mean square of 
the distance from the centre being wanted for that purpose ; but 
the Cartesian coordinates are more convenient when the close- 
ness of the grouping has to be considered. 
EXAMPLE. 
Coordinate Target. Ten shots at 100 yards. Target 10 inches 
square. 
Wee ce SG 7 1G le) 
10 10 &h Score 
9 [ 9 50 errs? 
8 ai —— 
6 
f “i He f 2 6 
sm) 7, 5 3/5] 8 
% al < OTe Nee 
+ 
| 3 7 5 6 
3 + — 8 6 6 
2 ih 2 9 5 6 
| | TO 5 5 
OTP Sa GO! 7 Lo) 
Fic. 4. 
To analyse the score it is convenient to arrange the results in 
the following form ; # and g being the number of shots on each 
vertical and horizontal row. 
x|p px y\¢@ w EEE ee 
r= |e = Loa ir a a 
2\|— = 24 = ie a. a 
alee Sala em We i 
aM |) a Mle Wie = a a 
5|6| 30 5| 2 10 150 ae 
6/2 12 6| 7 42 i ee 
7 | 1 | 7 Gv ai 7 49 49 
3 || = = 8] — ce iis a 
ON lar oars z eS 
10 | — = CO ae inn SA pe 
Sx 5 ZY gq) 22" 9g -7| 39" ‘I 
* 5°3 SES bee 28°7) 5 35 
Sor Mees 2g 2 (=) 281 (2) 3 ° 
= h =5'5 - 5'5 Ppp | Ne ed 
| s 6 C 
2 : ‘aiaeee 
R2=(:2)?+(-4)'="2 ae tae 
A 100 
This Counts fear or 84. A, MALLOcK. 
