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MESSRS. R. H. FOWLER AND C. N. H. LOCK ON 
in (35) are effectively unity, and the equation reduces to that of an ellipse in polar 
co-ordinates with axes sin |a, sin|/ 3 . This tends to a straight line as the limiting 
form when /3 -> 0 . 
The same (simplified) relation between 8 and f was obtained in (A)* for the case 
of small yaw only. The elliptic motion may be calculated most easily by recognising 
that \!/ is the auxiliary angle 0 of equation (33) so that 
<P — <po + + 0. 
The conditions under which this approximation is valid may be seen by reference to 
equations ( 11 ) and ( 12 ) which are satisfied by ff, /q 2 and cq 2 . We notice that/y is 
never small compared to unity and tends to infinity as q, and therefore g tends to zero ; 
while /q 2 may be comparable with rq 2 unless k is small. Thus the approximation is 
really only valid for s > 1 , in which case it applies even if (3 is not small compared 
to a. Finally, if 1 — k 2 is small, equation ( 12 ) shows that /q 2 will be small compared 
to cq 2 , and the third bracket in (35) will be more important than the second near a 
minimum of y ; this indicates that the shape of the (y, \L) curve there approximates 
to an hyperbola instead of to an elongated ellipse ; the curve may also no longer be 
re-entrant, the total change of f in one period differing from +tt. 
Examples of all these results may be seen in fig. 4, in which the observations for 
three different rounds are plotted with (y, g,) as polar co-ordinates. For round I. (5), 
for which (3 is small and the shell just stable, we find the expected elongated ellipse-like 
curve, with a slowly-developing minor axis caused by the damping factors. In III. (16) 
k — 80 degrees for the first half-period, diminishing to 40 degrees for the third ; a 
considerable minimum yaw developes and the curve is less like an ellipse, though the 
maxima are still nearly 180 degrees apart. Finally, in IV. ( 8 ) k = 85 degrees, falling 
to 75 degrees, and the shape of the curve near the minimum clearly resembles an 
hyperbola ; we may guess then the angle between the two maxima is somewhere about 
100 degrees instead of 180 degrees. 
Lastly, a word must be said about the observational determination of Q, which is, 
of course, theoretically determined by the muzzle velocity, twist of rifling and moments 
of inertia of the shell. In all cases the slope of the ^-curve over the first half-period, 
or rather more, is uniform and well determined. Since (3 appears to be really very 
small initially one may expect from theory the slope of this part of the curve to be 
|Q whatever the value of k. The agreement between this observed slope and the 
calculated value of i } is satisfactory. 
§ 8. The Method of Analysis. 
The method of analysis of the sin Id-curves will now be explained with reference 
to fig. 3. After the observed values of sin |d have been plotted against {It, the values 
* Loc. cit., p. 346, equation 4.06. 
