326 MESSRS. R. H. FOWLER, E. G. GALLOP, C. N. H. LOCK AND H. W. RICHMOND: 
the results are suitable for use only under certain conditions. Equations of type a 
are valid when the shell is sufficiently stable and the yaw is small; type /3 when the 
shell has settled down to a non-periodic motion in which the yaw may be large, the 
initial oscillations being damped out; and type y when the motion of the centre of 
gravity is nearly rectilinear. 
Equations of these types cannot be solved exactly, and the method of approximation 
used to obtain a solution is different in each case. The equations of type a are used 
for the analysis of the jump card experiments, for all sufficiently stable rounds, and 
could be used to compute the entire motion in any trajectory whose initial elevation 
is less than 45 degrees. Equations of type /3 have been used to compute the latter 
part of a twisted trajectory at an elevation of 70 degrees. Equations of type y have 
a limited application in analysing the jump card records for rounds which are nearly 
or quite unstable. 
3.01. Note on the Vector Notation. —All letters which represent vector quantities 
will be in clarendon type, to distinguish them from scalar quantities in the ordinary 
type. The three components of any vector A, referred to right-handed rectangular 
axes 1, 2, 3, are written A x , A 2 , A 3 . 
If A and B are two vectors, their vector product is denoted by [A . B]. This 
represents the vector whose components are 
(A 3 B 3 —A 3 B 2 ), (A 3 B 1 -A 1 B 3 ), (A^-AjB,). 
It is perpendicular to the plane containing the two vectors in the direction of the 
axis of the right-handed screw, which turns from A to B, its modulus being equal to 
the product of the moduli of A and B into the sine of the angle between them. The 
scalar product of the two vectors is written (A . B), and is equal to the scalar 
quantity 
AjBi + A 2 B 2 +A 3 B 3 ; 
it is also equal to the product of the moduli of A and B into the cosine of the angle 
between them, being positive when this angle is acute. For simplicity, we denote 
(A. A) by (A) 3 , which is equal to the square of the modulus of A. 
Constant use is made of the following identities :— 
(3.011) [A. A] = 0, ([A. B] . A) = 0. 
(3.012) [[A . B]. C] = (A . C) B-(B . C) A. 
(3.013) ([A . B] . [B . C]) = (A . B) (B . C)-(B ) 2 (A . C). 
§ 3.1. I'he General Vector Equations of Motion. 
We take a system (l, 2, 3) of right-handed axes of reference, see fig. 9, whose 
origin is O, the centre of gravity of the shell, and whose angular velocity at any 
