300 MESSRS. R. H. FOWLER, E. G. GALLOP, C. N. H. LOCK AND H. W. RICHMOND: 
undisturbed medium. In sucli circumstances the path of the particle lies in a vertical 
plane and is called the plane trajectory* This theory would be exact for a shell if 
the axis of the shell always pointed along the tangent to the path of its centre of 
gravity. The total reaction between the air and the shell would then, as required, 
take the form of a single force, called the drag, acting by symmetry tangentially to the 
path of the centre of gravity, and depending only on the velocity and shape of the 
shell and the state of the medium. The equations of motion resulting in this simple 
case are insoluble in finite terms for the actual law of resistance of the air; in 
practice they are capable of rapid numerical solution to any desired degree of 
accuracy, by a variety of methods of step-by-step integration, when the drag has 
been specified with corresponding accuracy. 
In order to specify the drag completely it is necessary to consider with some care 
what are the variables on which the drag for a given shell can depend to an 
appreciable extent. This question is, as yet, by no means settled, and a few of the 
more important considerations are summarised in § 1.01. This fact does not concern 
us here to a very serious extent; an incomplete specification of the variables on 
which the drag (or, in the general case, the complete force system) depends will only 
invalidate the results of observation when an attempt is made to apply them to 
widely different conditions of the state of the resisting medium, or of the motion of 
the shell. The validity is unaffected when the experimental conditions are 
approximately repeated. It may be assumed that, in this case of symmetry, a fairly 
adequate expression for the drag is given by the equation 
(1.001) R = pv 2 r 2 f R (v/a), 
where R is the total drag, p the density of the air (or other medium), r the radius of 
the shell, v the velocity of the shell, and a the velocity of sound in the undisturbed 
medium; all these quantities, of course, are to be measured in a consistent set of 
units. In the numerical work in this paper the foot, pound, second system will be 
used. 
Since pv 2 r 2 has the dimensions of a force, the function f R is a numerical coefficient, 
independent of the system of units chosen, called the drag coefficient. Existing 
determinations of this coefficient as a function of v/a are very inadequate from a 
scientific point of view ; satisfactory ones could now be made. We shall not be 
concerned here with the determination of this coefficient, whose value we shall oidy 
require roughly in the analysis of our experiments. We may therefore regard f R as 
known for all values of the argument from 0 to 3, for shells of the particular external 
shapes which we use, moving through dry (or not too nearly saturated) air, whose 
temperature is not too widely different from 0° C. 
* From the point of view of this paper, we regard the whole theory of the plane trajectory as 
“ classical,” though its adequate treatment was only evolved during the last years of the war. 
