THE AERODYNAMICS OF A SPINNING SHELL. 
24 1 
of Qt corresponding to the ends of half-periods can be determined from the symmetry 
of the curve, with the exception of the start of the first half-period. This must, of 
course, be near the muzzle, but need not be actually at the muzzle, and its precise 
0 1 2 3 4 5 6 7 
Scale of cms. 
Fig. 4. Curves showing the observed angular motion of the axis with (sinJS, as polar 
co-ordinates (\p = —<£o— ffit). The circles show the observed points and the curves are 
drawn freehand through them. 
For I. (5) 1 cm. represents 0-01 in sin|-8. 
For III. (16) and IY. (8) 1 cm. represents 0-02 in sin-IS. 
position must be guessed. The values of sin-|a can be obtained from a rough curve 
drawn freehand through the observations. Each half-period is then analysed separately 
and the only additional constant (when (3 = 0) required for computing a curve, by 
the approximate formula 
sin = sin |-a cn (K— \Qt, k), 
is the value of k or sin k. This may be approximated to with the help of the following 
artifice. Draw by eye a cosine curve (fig. 3, III. 16, first half-period) to have third 
order contact with the observed sin |^-curve at the maximum and to cut the axis 
at T. Then the ratio of NT to the half-period NP must be nj 2 K. For all curves of 
the form y = A cn (K— fix), for different values of k, have' third-order contact at the 
maximum ordinate, and a cosine curve is the limiting case, in which Iv = \n. A first 
estimate of k can usually be made by this method to the nearest 5 degrees, when 
