Motion of a Spinning Projectile, 355 



Therefore the particular integrals corresponding to the wind 

 velocity w are 



# = 0, 6 = 0, -v 



iv _ w_ _ to > , . . . (47) 



^~~ u it u } ) 



and these results are independent of the particular law of 

 resistance. But it is worth while to note that, with our 

 particular law of resistance, 



dri to du w w^ ,.„ N 



di = -u*dt = l T or v • • • < 47a > 



so that the side wind makes the plane of the tangent to the 

 trajectory rotate with one or other of two constant angular 

 velocities. 



46. The result expressed by 



r)-y = (3 



means that the shot points its nose directly against the 

 relative velocity, for /§ is the angle between the true and 

 relative velocities, and (77 —y) is the angle between the true 

 velocity and the axis of the shot. And, furthermore, the 

 result expressed by 



w 



shows the nose of the shot points in a fixed direction. Thus 

 the shot, by keeping its axis in a fixed direction, always 

 faces the relative wind. 



47. The solution we have just obtained represents only a 

 part of the motion of the shot. The remaining part, which 

 has to be superposed on this motion, will now be considered. 

 The nose of the shot cannot, of course, suddenly face the 

 wind as soon as the shot leaves the gun, but it will be seen 

 that the axis begins by gyrating about the line of relative 

 velocity as a mean position. 



48. Having accounted for the effect of /3, we may now 

 drop it from our equations, since they are all linear. Then 

 let us put 



z = x + iy, 

 where t = \/ — 1. Now multiplying both sides of (43) by 



