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XLVIII. Resistance of the Air at Speeds below One Thousand 

 Feet a Second. By A. F. Zahm, Ph.D* 



MANY attempts have been made, since the time of Newton, 

 to determine the law of air-resistance at all speeds from 

 zero up to that of the swiftest projectiles, but w T ith varying 

 results. Newton himself taught that the resistance is a 

 quadratic function of the velocity, which he proved, for 

 moderate speeds, by observing the time of fall of spheres 

 dropped from the dome of St. Paul's Cathedral. The most 

 accurate ballistic researches made since his time seem to verify 

 this law for speeds below one to two hundred feet a second, 

 and above thirteen to fourteen hundred ; but for intermediate 

 velocities there is much divergence of opinion, some experi- 

 menters maintaining that the resistance varies as the square of 

 the velocity, others that it varies as the cube, or according to 

 some more complex relation. 



The present paper offers some facts in favour of a law not 

 generally admitted. In 1842 Col. Duchemin published a 

 remarkable bookf, in which he derived analytically the ex- 

 pression TL = av 2 4- ?>r 3 , for speeds below 1400 feet a second ; R 

 being the resistance, v the velocity of the projectile, a and b 

 constants. This equation expresses quite closely the experi- 

 mental results of the present research, in which an effort has 

 been made to develop a more accurate instrument for measuring 

 the velocity and, especially, the acceleration of projectiles. 

 The speeds employed have not exceeded one thousand feet a 

 second, though I hope to extend them to upwards of thirteen 

 hundred. The results are published now because the work 

 may not be resumed for a long time. The investigation was 

 begun at the Johns Hopkins University in January 1895 ; and 

 it gives me pleasure to express here my thanks to Prof. Rowland 

 and Prof. Ames for valuable suggestions during the pre- 

 liminary experiments. 



For speeds above one hundred feet a second, the funda- 

 mental principle of measurement has been the same in all 

 modern ballistic apparatus. The projectile's time of transit is 

 recorded for three or more equidistant points along the 

 horizontal part of its path, and thence its velocity and resistance 

 calculated. Usually electric-wire screens are fixed at the 

 several points of the trajectory, and the instant of their rupture 

 recorded by some form of electric chronograph. The best 

 of such instruments may record the time of rupture 

 truly to one ten-thousandth of a second, or less. But the 



* Communicated by Prof. J. S. Ames. 



t Les Lois de la Resistance de VAir sur les Projectiles. 



