122 
ROTATION FOR STABILITY OF 
and dropping the fraction sin a, which equalled to zero would imply 
perfect centering, 
Ctf'fJL — C^fJ? cos a = -3 (Cj — c 3 ) 
or c 4 cos cl/jl" — c^rjji + — (c 4 —> c 3 ) 
cos a 
w* 
cos a 
“ 0 ; 
a quadratic equation in ^. 
Solving this quadratic. 
/* = 
)• ± / y/je 8 V> - 4 ^ («! — c 8 ) c 4 w 3 j 
2c 4 cos a 
.( 2 ) 
and therefore the least admissible value of r, in order that the roots of 
this quadratic should not be imaginary, is given by 
c 6 W = 4 ^ (Cj — c 3 ) c 4 w%, 
or 
^=4?( Ci - C3 )^. 
If the shot had been fired from a gun of calibre 2 a, the rifling at 
the muzzle making one turn in n calibres, and ft being the angle of 
rifling at the muzzle, then 
tan P=l = f= 3a \/% (c > - Cs) $ •. (3) 
If W— weight of shot, 
W = weight of air displaced, 
then c 4 — W + W a, 
c s = W+ W'y, 
c 4 = Wk - 1 3 + JTV-fa 1 , 
c 6 = W&i 
where k 1} k are the radii of gyration of the shot about OA and OC, and 
k\ of the air displaced (supposed rigid) about OA; a, y, a being 
certain quantities depending only upon the external shape of the body. 
The only body for which a, y, and a' have been, as yet, determined 
by mathematicians is the ellipsoid, the surrounding medium being 
