350 MESSRS. R. H. FOWLER, E. G. GALLOP, C. N. H. LOCK AND H. W. RICHMOND: 
In practice the values of Q, obtained by the two methods, were in satisfactory 
agreement, except for the shells with centre of gravity forward, whose dynamical 
constants were considerably altered by the set back of the lead block on firing. For 
these shells the slope of the observed 0 -curve was taken as defining Q. The value of 
B after firing was deduced from this value, and the position of the centre of gravity 
was determined by equations ( 2 . 21 ) and ( 2 . 22 ). 
The curve showing the true relation of <p to Qt must pass through the true values 
of 0 , which differ from the observed values only by integral multiples of 27r. It 
remains doubtful whether the value of 0 increases or decreases by the amount i r 
radians in passing through a minimum of <5, in addition to its steady increase at rate 
|-Q. This question is settled by the divergence of points near the minimum of (5 
from the straight line of fig. 11 ; thus, if the points lie above the straight line in 
approaching a minimum, there will be an increase of amount 7 r, and vice versa. In 
this way a continuous curve may be drawn which is consistent with the equation 
(4.06). Specimens of the curves obtained in the analysis of the actual observations 
are : shown in fig. 12 . The portions of the curves in the neighbourhood of the 
maxima will then coincide approximately with a series of parallel straight lines at 
distances apart of ir radians. The method can only fail in one case when none of the 
points diverge appreciably from the straight line of fig. 11 . This indicates that the 
value of the minimum /3 is indistinguishable from zero, while the value of 0 changes 
almost discontinuously by ± 7 r at the time of the minimum. It is then immaterial 
whether the change is taken to be positive or negative.* 
The observed values of 0 in the neighbourhood of the minimum also yield 
information as to the value of /3i/a 1 and the instant at which the minimum occurs. 
Let P be any observed point on a 0 -curve which diverges measurably from the 
nearest straight portion of the 0 -curve ; lying above it by A degrees. Let t 0 be the 
time of occurrence of the nearest minimum, and Sp 2 the change in p 2 between the 
minimum and P. Then, by (4.06), 
(4.101) cot A = 77 -wv cot <k> 2 . 
v ' AW 
If, in this equation, A, t 0 , Sp 2 , and (£ 0 ) are regarded as known, we can at once 
obtain a value of /3. By adjusting the value of t 0 we attempt to reconcile the one or 
more values of f3 obtained in this manner and also the value demanded by the 
^-curve. By combining all the available evidence in this manner, remembering that 
the d-curve is nearly symmetrical about a minimum, and the 0 -curve at the same 
time halfway between two straight portions, we can draw fairly precise final curves, 
* The rapid changes or discontinuities in the values of 0 and S', which occur when 8 is very small or 
zero, are due to the singularity which occurs at the origin of polar co-ordinates. The motion of the shell 
is, of course, in all cases continuous. 
