Also 

 that is, 



or 



Again, 



or 



Motion of a Spinning Projectile. 351 



— Hex, 



cit ox d% 



d 



dt 



I — A(a — x) + Bcoy} = Rex, 



Aai + B (oy — Ret- = Aa. 

 ^ BU dU n ( ., ~ 



Ay — BwX — Hey = — Bwa -I- R.c/3. 



37. We will now return to the modifications of these 

 equations that are necessary in consequence of our wrong 

 assumptions. Since the resistance does not always act in 

 the plane AOY, it is clear that the centre of mass of the 

 shot will deviate from this plane to some extent, so that the 

 velocity of 0, instead of being along Oq as we have assumed, 

 will make a small angle, which we shall denote by 77, with 

 the plane AOY, and we shall suppose rj to be positive in the 

 same direction as y. Moreover, the velocity of will make 

 another small angle with the plane NO*/, this small de- 

 flexion being likewise the result of the fact that the resistance 

 does not act along the line of the velocity of ; this latter 

 angle will be denoted by e, and will be reckoned positive 

 towards Op from Oq. The new assumptions amount to the 

 same thing as saying that the line of flight is not in the 

 position Oq, but makes small component angles (x — e) and 

 (y—rj) with OP instead of x and y as we assumed earlier. 

 The angle /3 must still be regarded as the angle between the 

 true and the relative velocity, and is not now measured 

 from Oq, but from the new line of true velocity, so that 



tan 8 is still — . 

 u 



38. Furthermore, the line of resistance to an elongated 

 shot does not act along the line of the velocity of the shot 

 relative to the air. It acts in the plane containing the axis 

 and the relative velocity, but for an ordinary shot it will 

 make a greater angle with the axis than the relative velocity 

 makes. Suppose 6 is the angle between the relative 

 velocity and the axis OP, that is, 6 is the angle which 

 has components (y — rj -+ j3) and (x— e), and suppose 0' is the 

 angle between the resistance and the line of relative velocity. 

 The relation between 6' and depends on the shape of the 



2B2 



