5i8 SCIENCE PROGRESS 



and p is the density of air, and r the radius of the shell. The 

 function / is numerical only, and the so-called drag coefficient, 

 determined experimentally and with some precision when vja 

 ranges from o to 3. 



The shell is discussed as a rigid solid of revolution, its axis 

 of symmetry being coincident with a principal axis of inertia. 

 The classical theory is confirmed so far as the divergencies of 

 the axis of the shell from the tangent to its part are concerned, 

 but the paper also makes a determination of the magnitude and 

 effect of these divergencies. 



The angle between the axis and the path of the centre of 

 gravity is generally known as the yaw. The main force com- 

 ponents are (i) the drag, acting through the centre of gravity 

 directly against the motion ; (2) the cross-wind force, perpen- 

 dicular to the drag and in the plane of the yaw ; and (3) a 

 moment in the plane of the yaw. We have given the type of 

 formula used for (i). Similar forms are used for (2) and (3), 

 the functions / being elucidated by experiment, partly in wind 

 channels and partly otherwise. Complications ensue when the 

 shell has axial spin, and into these we do not enter. In practice, 

 the direction of the axis of the shell relative to the direction of 

 motion has rapid changes, and needs the introduction of a new 

 couple called the yawing moment, again given by a similar 

 formula. The authors make the assumption that (i), (2), and 

 (3) are not appreciably affected by the existence of this new 

 couple, and they justify the assumption d posteriori. 



It is clear that the dynamical equations of motion of a body 



under the influence of such force-components are of a novel 



type, and, in relation to their solution, interesting problems of 



pure mathematics arise, as we have stated already. Three types 



of equations are dealt with, expressed in terms of different sets 



of co-ordinates, and two are solved. The nature of the problem, 



in any set of co-ordinates, will be clear. By the use of certain 



complex variables, much simplicity can be introduced into the 



differential equations, which are of an interesting form and with 



coefficients dependent on quantities which in practice are very 



large or very small. They are of a new type, and the ultimate 



object is the determination of asymptotic expansions for their 



solutions. In this way they enter very definitely into the 



domain of the pure mathematician. The authors use Horn's 



method, which is fairly familiar now. If an equation is reduced 



to the form 7. 



d y 



where k is large, and zi; is a function of x, the asymptotic 



solutions are , ^ . „ ... 



y = {Ae'^"" ->t-Be-'^'')y\r 



