370 CARNEGIE INSTITUTION OF WASHINGTON. 



(3) The greater the density of the air the less the stability of the projectile. Hence, 

 if a projectile is stable near the gun it will be stable throughout its flight. Only 

 if it is near instability and fired at a high angle can there be an exception. 



(4) The longer the projectile of a given diameter the less its stability. 



(5) For a given velocity of translation, the higher the rate of spin of a projectile the 

 greater its stability. 



(6) Two similar projectiles of different dimensions fired with the same velocity from 



guns rifled one turn in a given number of calibers have the same stability. 



(7) If the length of a projectile near instability is increased by the factor C the pitch of 

 rifling must be increased by the factor \/c to preserve the stability. 



The motion of the axis of the projectile can be analyzed into two periodic 

 motions, one between the two circles Ci and C2, and the other around the 

 point P. If Ci is the smaller of the two circles, the curve described by the axis 

 is always tangent to C2 and is described in the direction of spin of the pro- 

 jectile. It may be tangent to Ci in either direction, depending upon initial 

 conditions, and it may have a cusp on Ci. For given circles Ci and C2 the 

 axis may make many oscillations between Ci and C2 without completing a 

 circuit about P, or it may make a circuit about P with each oscillation, accord- 

 ing to the initial conditions. In firing from well-constructed and well- 

 mounted guns, the circle Ci is verj^ small, and hence the two cases are physically 

 not much different from each other. 



The periods of oscillation of the axis of a projectile between the circles 

 Ci and C2 and about the point P have the following properties: 



(1) The period of oscillation between two given circles, Ci and C2, varies nearly in- 

 versely as the velocity of the projectile. 



(2) The period of oscillation between two given circles, Ci and C2, is greater the greater 

 the density of the air. 



(3) The period of oscillation between two given circles, Ci and C2, is greater the longer 

 the projectile. 



(4) The period of oscillation between two given circles, Ci and C2, is shorter the more 

 rapid the spin of the projectile. 



(5) If the circle Ci is small, the period of oscillation between Ci and C2 is shorter the 

 larger C2. 



(6) The two periods of oscillation around P are both shorter the smaller C2. 



(7) The two periods of oscillation around P are both longer the longer the projectile. 



(8) The two periods of oscillation around P are both shorter the higher the rate of spin 

 of the projectile. 



