Motion of a Spinning Projectile. 349 



considering i£ the resistance always acted contrary to the 

 motion o£ 0, which is what we have assumed in rinding the 

 equation to the trajectory. That is, AOq is the same angle 

 a as in fig. 2. The angles x and y are small component 

 angular deflexions of the axis, x being the lag of the axis 

 behind the line of flight, and y the deflexion to the right as 

 seen from the gun, the spin of the shot being assumed to be 

 right handed. Suppose a wind is blowing with velocity w 

 perpendicular to the plane of the trajectory, and, assuming 

 that Oq is the direction of the velocity of relative to the 

 earth, let OQ be the direction of its velocity relative to 



the air, so that tan/3=— . Although Oq is not the actual 



u 



direction of motion of 0, yet it will avoid complications 

 to assume that it is at present, and we can correct our 

 results afterwards. Moreover, we shall assume that the 

 resistance acts parallel to the motion of relative to the 

 air, that is, parallel to QO. If the shot were spherical, 

 this last assumption would be quite true, but for a long shot, 

 such as a bullet, it is certainly not true, but we shall con- 

 sider later what modification is necessary to correct this 

 assumption. 



34. Let us denote by B the moment of inertia of the shot 

 about its axis of symmetry OP, and by A its moment of 

 inertia about OY, which is perpendicular to OP. The axes 

 OP, OY, and one perpendicular to both of these, are a set 

 of principal axes of the body. 



35. We shall denote by v the number of radians through 

 which the shot has turned, in the right-hand direction, 

 about its own axis OP, relative to the moving plane PON. 

 The shot has angular velocities (a — x) about NO ; y about 

 OY ; and v about OP, this last being in the right-hand 

 direction. When these angular velocities are resolved along 

 the three principal axes OP, OY, and the third axis not 

 shown in the figure, the components are 



v— (ol—x) sin#; y; (d — x)cosy; 



the third axis being taken on the side of the plane AOY 

 opposite to N. Therefore the kinetic energy of the 

 motion is 



JJ = iA{f + (^-^) 2 cos 2 ?/} + |-B{z)-(«-^)sin2/} 2 . 



The force P can be resolved into components P sin /3 and 

 P cos /3 parallel to ON and qO respectively. The com- 

 ponent P cos /3 can be resolved again into P cos /3 cos x and 

 P cos /3 sin x parallel to PO and YO respectively. Denoting 



Phil. Mag. S. 6. Vol. 34. No. 202. Oct. 1917. 2 B 



