ROTATION OP PROJECTILES. 
107 
of C and A, and thence of a and y, we require the semi-axes of the 
confocal spheroids of the medium envelo ping t he bod y. Th ese semi¬ 
axes are denoted by the expressions yV -j- A and \/$ + A; c and a 
being the semi-axes of the spheroidal body. 
The method of obtaining the value of c u (or, as it may be written, page 578. 
c 4 ) is also omitted, on account of its intricacy.* It will, however, be ^vaiufof 
seen subsequently that c 4 may be put as equal to Mh i 2 — i.e., the moment c 4* 
of inertia of the body about an equatoreal axis. This means that the 
motion of the medium due to the angular velocities p and q of the 
body need not be taken into account, any more than that due to r, 
which, as previously noted, is nil. 
The expression “ lines fixed in space ;; may be put more fully as Page 578 . 
“ lines fixed in direction in space/'’ Last lme ' 
Passing now over the somewhat intricate calculations contained in 
pages 579-583, and only noting that the symbols x, y,; x, y, &c., are 
x- i -i dx dy. d 2 x d 2 y , ,, „ , 
respectively the same as ^ ^ > we come to the funda¬ 
mental equation 
If sin a ~ 0, the conditions of perfect centring and of perfect pro¬ 
jectiles are attained. (For remarks on this state of things, see above.) 
We thus come to the equation 
which represents the state of things given by the helical path which 
the projectile actually describes. 
This equation, then, gives the solution of the original problem—viz., 
“ How much spin is necessary to ensure stability in flight V } 
Solving the equation for ft, we find that 
2c 4 cos a 
Hence, to give real values of /*, 
r\ 2 must not be less than 4 F 2 cos 3 ac 4 
^ For details of this pethocl gee Ferrers, in “ Quarterly Journal of Mathematics,” 1874. 
