Motion of a Spinning Projectile. 363 



61. We have found that the shot is stable with its nose 

 foremost provided 



m 2 T 2 >rf, 



that is, provided 



4A* u 2 gA ' 

 which may be put in the form 



BV>4AK/b (75) 



If we write N for the number of calibres of forward 

 motion of the shot for one turn on its axis at the muzzle of 

 the gun, then, d being the diameter of the shot, 



(O 



(76) 



2tt Ncf ■ 



so that the condition for stability of the axis becomes 



-d^ttV 4A/cZ»W 

 N 2 d 2 ~ ^ 



* < / AW^ 2 clJ (7/} 



To ensure stability throughout the motion we must make 

 N 2 less than the least value of the right-hand side of (77), 

 and this occurs when u = u . According to our earlier 

 assumption that the shot may be treated as a cylinder for 

 the purpose of finding its moments of inertia, the above 

 condition for stability becomes, when u = u , 



We have already shown (section 8) that I is proportional 

 to h for similar shot, and c is clearly also proportional to A, 

 whence it follows that I is proportional to c. Moreover, / is 

 the same for all similarly shaped shot. Thus the condition 

 for stability of the axis gives the same minimum value of N 

 for all similar shot of different sizes. Even if we take the 

 accurate moments of inertia, instead of assuming the cylin- 

 drical form, it is still true that the condition for stability gives 

 the same minimum value of N for similar shot. Since f 

 depends on the shape of the nose the similarity referred to here 

 is absolute similarity of shape, and especially of the nose. 



