64 PROCEEDINGS OF THE AMERICAN ACADEMY. 



a great number of experimenters under a great variety of physical 

 conditions, and a resume - of the results at which they have arrived can 

 be found in the articles of Finsterwalder on Aerodynamik and of 

 Cranz on Ballistik in the fourth volume of the Encyklopadie der 

 Mathematischen Wissenschaften. 2 



That under otherwise given conditions the air resistance, when v 

 is large, is a complicated function of v, is shown by the practical 

 formulas based on experiments made with rotating projectiles of the 

 standard Krupp form. For a projectile of this kind of given size, in 

 free air, the expressions are av 2 , bv 3 , cv 5 , dv 3 , ev 2 , fv 1 - 7 , gv 1 - 55 , accord- 

 ing as v, measured in meters per second, lies in one or other of the 

 intervals between the values 50, 240, 295, 375, 419, 550, 800, and 

 1000. The constants are different for projectiles of different diameters 

 and vary with the temperature of the air, the barometric pressure, and 

 other circumstances. 



In order to determine the resistance which the air offers to a given 

 body moving uniformly through it at a comparatively small velocity, 

 v, many different observers have made use of the whirling table in 

 some form. The phenomenon to be studied is in any case a very com- 

 plex one, since the moving body drags with it, as it moves, a certain 

 mass of air, and the viscosity of the air contributes an uncertain amount 

 to the quantity to be measured. It appears, however, from the ex- 

 periments of Schellbach, von Loessl, Langley, Recknagel, Hagen, 

 and others, 3 that when proper corrections have been made for the 

 effect of the wind which the table takes with it as it turns, the air re- 

 sistance varies as the square of the velocity 4 for all values of v between 

 50 and 0.2. For velocities much less than 20 centimeters per second 

 the viscosity of the air appears to determine the resistance which is 

 approximately proportional to the velocity. It is well to remember 

 that a solid sphere, to take a concrete example, moving in an infinite 

 homogeneous liquid at rest at infinity, in a straight line, with constant 



2 Leipzig, B. G. Teubner, 1903. 



3 Schellbach, Ann. d. Phys., 143, 1871. Recknagel, Zeitschrift d. Ver. 

 deutsch. Ing., 30, 1886. F. v. Loessl, Die Luftwiderstandsgesetze. Langley, 

 Experiments in Aerodynamics. Cranz, Aeussere Ballistik, Leipzig, 1895. 

 Thiesen, Ann. d. Phys., 26, 1885. Mach und Salcher, Wiener Berichte, 1887, 

 1889. 



4 Mohn, Grundzuge der Meteorologie, Zweite Auflage, p. 137: " Durch 

 vergleichende Versuche iiber Druck und Geschwindigkeit des Windes, hat man 

 gefunden dass der Winddruck dem Quadrate der Geschwindigkeit propor- 

 tional ist." On page 138, however, the pressure of the wind in kilograms per 

 square meter is given as 0.15, 1.87, 5.96, 15.27, 34.35, 95.4, according as the 

 velocity in meters per second is 0.5, 4, 7, 11, 17, or 28. 



