AN ELONGATED PROJECTILE. 
123 
supposed frictionless and incompressible; and for the particular case 
of the prolate spheroid of semi-axes a and c, 
A+ O’ 7 
and a' = 
G_ 
IA 
A—C 
j 3 — a 3 
c 3 + a 2 
/ c 2-a 2 \* # 
V 3 + w ; 
+ c 3 —a 2 
( 2 A+ C) - A + C 
C" f 
where 
/* aD _ c 1 
Jo (a 3 +W + *)* ~ tf 2 ( C 3 -« 3 j 2 (c 3 -« 3 )§ 1U * € c __ ’ 
/ %G0 dA. _ 1 , c+Jc 2 —a 2 2 * 
~J 0 (« 3 + A) (c 3 + A)t ~ ( C 2 -a 2 )f ge ” c(c 3 -« 3 ) * 
■(4) 
Where, as in practice, the fraction ^ is so small that its square may 
be neglected, we have 
tan 2 p — ^ = 4 ^ ( Cl _ c 3 ) 
tt 2 6* 4 
_ . y. fa vl , i ^+n- 1 v 
^ W Wa X ^ ^ ^ 3® r 2z4i 
= 4 
l j. ^ 
+ /r 7 
1+ a 
(a — y) « 3 
WVc* 
W‘ 
V+ZrW 
~ 
, r' . ffi V 
= V (a ~ 7) ^ 
w 
+ higher powers of , which are neglected. 
.(B) 
From equations (4) and (5) Captain J. P. Cundill, R.A., has cal¬ 
culated a table of values of a — y and the corresponding value of n 
for service projectiles, and the results obtained appear to agree very 
fairly with what is observed in practice. 
It may be noticed from the formula that, on the hypothesis of the 
incompressibility of the medium, the value of n is independent of 
(1) the velocity, (2) the calibre, or length of bore; so that, for similar 
projectiles, one value of n would do for all guns in the service. 
When, however, the velocity is high and the projectile is large, the 
compression of the air cannot be neglected, and the air behaves as if 
its density were increased; so that less rotation is required than that 
given by the formula. 
For instance, the 80 and 100-ton guns are rifled at the muzzle with 
a twist of one turn in 50 calibres, while the formula would give one 
turn in 40 calibres as requisite for common shell three calibres long. 
* ‘‘ Quarterly Journal of Mathematics,” Vol. XVI. “Mathematical Papers of the late George 
Green,” edited by the Rev. N. B. Ferrers, p. 322. 
