March 5, 1920] 



SCIENCE 



225 



are checked. Hence a trajectory can be com- 

 puted taking account of variations of air 

 density with height, and satisfying at all 

 points the assumed law of retardation. 



Since the retardation depends on the rela- 

 tive velocity of air and projectile, winds can 

 be allowed for by considering the motion 

 relative to the air at every point. This in- 

 volves the principal of moving axes. It im- 

 plies however, that the projectile is a sphere 

 or that the retardation is independent of the 

 angle which the projectile presents to the air, 

 or else that the projectile always turns nose 

 on to meet the air. We know, however, 

 definitely that an air stream of a few miles 

 per hour at right angles to the axis of a 

 projectile may have several times as great 

 a force as the same stream would exert 

 along the axis, and that a spinning projectile 

 can not turn quickly to meet every wind that 

 blows, even though the wind may have but 

 small influence upon the angle at which the 

 air meets the projectile. 



It was this method of short arc computa- 

 tion which Professor Moulton applied to the 

 problem of exterior ballistics when he was 

 made head of that branch in the Ordnance 

 Department. For his courage in setting aside 

 the long-established, revered but rather em- 

 pirical method in use in the War Department, 

 and in introducing a logical, simple method 

 of computing trajectories, and for his energy 

 in initiating and pushing through certain ex- 

 perimental projects, he deserves great com- 

 mendation. Valuable contributions to the 

 method were made by his associates, notably 

 Bennett, Milne, Eitt. Professor Bennett de- 

 vised a method which has a number of points 

 of merit. It is the one now used at the 

 Aberdeen Proving Groimd. Professor Bliss 

 gave an inclusive method of computing varia- 

 tions in range, altitude and time due to 

 changes in air density, winds, muzzle velocity. 

 Dr. Gronwall greatly simplified and extended 

 the work by Bliss, and made other important 

 contributions. In short, leaving out of ac- 

 count the question as to the correctness of the 

 law of air resistance, the variation of that 

 resistance with the angle of attack of air and 



projectile, leaving out the motion of pre- 

 cession and nutation which are dependent 

 upon the transverse and longitudinal moments 

 of inertia of the projectile and its rate of 

 spin — ^leaving out these factors the mathe- 

 matical basis for finding the trajectory of a 

 projectile is secure. 



But the system of forces tmder which a 

 projectile moves is not the simple one implied 

 by the equations just given. Por a projectile 

 is a body spinning rapidly about an axis prob- 

 ably nearly identical with its geometrical 

 axis. It emerges from the gun either with a 

 small yaw, or with a rate of change of yaw, 

 or both. (By yaw is meant the angle between 

 the axis and the direction of motion of the 

 center of gravity.) As in the case of a top, 

 processional motion results. If the motion is 

 stable, precession accompanied by nutation 

 continues. If unstable, the axis is driven 

 farther from its original direction imtil the 

 projectile is " side on " to the air, or " base 

 on " to the air. In short, the projectile tum- 

 bles. Loss of range and great dispersion are 

 the results. 



The condition for stability may be taken 

 the same as that for a top spinning about an 

 axis nearly vertical, viz., 



where 



A =■ moment of inertia about the axis 



of spin 

 £ = moment of inertia about an axis at 



right angles 



iV^ = frequency of spin in radians per sec. 



M sin ^ = moment of force about an axis 



through the C.G. at right angles 



to the axis of spin, where Q is the 



yaw %. e., the angle between the 



axis of the shell and the direction 



of motion of the center of gravity. 



The rate of orientation of the yaw or the 



processional velocity is given by 



i> ^ AN -^ B{\ + cos6). 



The relation given for stability, viz., that 



4Bm 



>1, 



