AN ELONGATED PROJECTILE 
593 
1 ^2 (i — 1 \ ^ 1 density of iron # 
5 \ m 6 / x 2 + 1 a — y density of air 
where a — y is given by the formula above, giving n the number of 
calibres to one turn for the rifjjng at the muzzle of a gun firing shell 
of which the length is x calibres, and the cavity 4 th of the linear dimen¬ 
sions of the external surface. 
The formula is obtained on the hypotheses that gravity is neglected, 
that the surrounding medium is frictionless and incompressible, and 
that the external surface of the shell is a prolate spheroid, and the 
results obtained agree fairly with what is observed in practice. 
For instance, the Krupp 40 cni gun fires a steel shell 2’8 diameters 
long, and the rifling at the muzzle makes one turn in 45 calibres. 
As the steel shell is to be employed against armour, we may assume 
that the linear dimensions of the cavity are one-half those of the 
external surface, as in our Palliser projectiles, and therefore m — 2 
and x — 2'8. 
We have 
v^ 2 — 1 = 2-61534, 
x + vA 2 — 1 = 5'415 34, 
log e (x + \/x 2 —■ 1) = 1'68924, 
1 0 2 — l)f = 6-38890, 
x 
X = 7-32294, 
- = 1-86810, 
(-■) 
(-» 
s/x 2 — 1 = 9*19104, 
•A 2 — 1 = 5'45484; 
and therefore by the formula 
6-38890 ^ 9-19104 - 5-06772 
a 7_ ~ 7-32294 — 1-68924 X 5-4.548 + 1/68924 
6-38890 
5-63370 
4'12382 
744408 
= •654535 ; 
and therefore 
o 1 9 31 
32 X 
1 w 1 
8'84 X -654535 
X 7-8 X 816; 
and working this out by logarithms, we shall find n — 45*753. 
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