228 
MESSRS. R. H. FOWLER AND C. N. H. LOCK ON 
and have been used in both papers for extending the results down to low velocities, 
as in figs. 1 and 2, here. 
Shells of four different types, I.-IV. were used. Types I.—III. were of the same 
external shape (form A), with three different positions of the centre of gravity. 
Type IV. was of a different external shape (form B). The details are given in (A).* 
The experimental data, which have already been discussed in (A), consist of the 
mass, principal moments of inertia, and position of the centre of gravity for each type 
of shell ; rough values of the forward velocity over the whole range of 600 feet from 
the muzzle of the gun ; the spin of the shell, deduced from the rifling of the bore; the 
yaw and orientation of the shell’s axis at a number of points along the range, deduced 
from the shape and orientation of the holes punched by the shell in cardboard targets. 
These cards were set up at intervals of about 60 feet for all the unstable rounds, and 
it appears from figs. 12 of (A) and figs. 3 and 4 of this paper that they were close enough 
together for satisfactory curves to be drawn through the observed points representing 
the variation of the yaw S and its azimuth <p over the whole range. 
§ 2. The Equations of Motion. 
It is convenient to recapitulate the notation of (A). Suppose that OA denotes the 
direction of the axis, OP the direction of motion of the centre of gravity of the shell ; 
then AOP = S, and <j> is the angle that the plane AOP makes with a fixed plane 
through OP. M (= u sin S) is the couple in the plane AOP which tends to increase S, 
A, B and N are the principal moments of inertia and the axial spin of the shell, and we 
write Q = AN/B. Then the equations of motion will be taken in the formf 
S' 2 + cf) rj sin 3 j ^ d cos S — E,.(l) 
sin 2 d + Q cos S = F,.(2) 
where E and F are constants. The various assumptions underlying equations (1) 
and (2) are discussed in detail in (A). If ,u is constant the equations are, of course, 
of the same form as the ordinary integrals of energy and angular momentum for a 
spinning top, and the complete solution hi elliptic functions is standard. 
When M is an arbitrary (odd) function of $ the top solution no longer applies, but 
a solution in elliptic functions is still possible if M has the form X sin $ 1 — Y (1—cos cf)}, 
where X and Y are constants. This more general form allows the first two terms in 
the expansion of an arbitrary M to be catered for and can represent M adequately 
* Loc. cit., p. 316 and fig. 6. See also fig. 1, liere. 
t (A), loc. cit., p. 334, equations 3.404, 3.405. For the underlying assumptions see (A) Part I., pp. 301 sqq., 
311 sqq. These equations are, strictly speaking, not referred to fixed axes, hut are approximate equations 
referred to axes changing direction with OP. Dashes denote differentiations with respect to the time t. 
