MATHEMATICAL PHYSICS. 369 



MATHEMATICAL PHYSICS. 



Moulton, F. R., University of Chicago, Chicago, Illinois. Research Associate 

 in Applied Mathematics. (For previous reports see Year Books 5, 6, 8-19.) 



The current year has been spent in completing the study of the question 

 of stability of artillery projectiles, and in putting all the investigations on 

 exterior ballistics in form for publication. The entire work is nearly ready 

 for publication, and it will be sufficient to give a brief outline of the results 

 obtained respecting stability. 



The dynamical elements upon which the oscillations of a projectile depend 

 are: Ai, the moment of inertia with respect to a transverse axis through the 

 center of gravity of the projectile; A 3 , the moment of inertia with respect to 

 the axis of the projectile; w, the angular rate of spin of the projectile; and Mi, 

 M 2 , M 3 , the moments of the exterior forces. 



Let the oscillations of the projectile be described by the curve traced out on 

 a unit sphere by a line coincident with the axis of the projectile. Let P be 

 the point on the sphere toward which the projectile moves. Then the axis 

 of the projectile describes curves which, neglecting the curvature of the 

 trajectory and the damping effects of the resisting forces, are contained 

 between two circles Ci and C2 about P as a center. If the curves are both 

 near P, the oscillations of the projectile are small. Since the projectile leaves 

 the gun almost exactly nose on, one of the circles must be near P.' The other 

 may be far removed. If it is, the projectile undergoes large oscillations, or 

 even tumbles and the motion is unsatisfactory. 



Let 6 represent the angle between the axis of the projectile and the direction 

 of its motion. Let the moment of the exterior forces about a transverse axis 

 of the projectile through its center of gravity be AiM sin 0. Then the quantity 





:) = 



plays an important role in all considerations of stability. 



Whatever the value of X 2 , the projectile will have large oscillations if it is 

 not started so as to have initially small transverse rotational velocity. A 

 more important fact is that if 4 X 2 much exceeds unity the oscillations will be 

 large, however small the initial transverse velocity may be. Hence the projec- 

 tile and gun must be so designed that X 2 shall be sufficiently small. 



The quantities Ai and A 3 depend upon the shape and distribution of the 

 mass in the projectile, co depends upon the rifling of the gun and the speed of 

 the projectile, and M depends upon the density of the air, the speed of the 

 projectile, and unknown properties of its shape. The theory which has been 

 developed leads to a method of determining M in case of any particular type 

 of projectile by firing through card-board screens. The other quantities upon 

 which X 2 depends are all known. 



It follows from the form of X 2 that projectiles in respect to the question of 

 their stability have the following properties: 



(1) For velocities of the projectile below the velocity of sound, M varies nearly as 

 v 2 . Since w varies as v 2 , the stability of a projectile is independent of its velocity 

 so long as the velocity remains below that of sound. 



(2) Since the resistance of the air to a projectile varies as a higher power of the velocity 

 than the second for velocities near that of sound, a projectile may be stable for 

 velocities below that of sound and unstable for velocities above that of sound. 



