66 PROCEEDINGS OF THE AMERICAN ACADEMY. 



where a is the radius of the sphere, M' , the mass of the displaced 

 liquid, and / 2 = -rrp/fi T : Mq is the mass of the sphere. 



Such a sphere, oscillating under the action of this resistance and 

 a restoring force (b 2 6) proportional to the displacement, would have 

 an equation of motion of the form 



*•£+.»«•£ + »-* w 



where M = Mq + M" : all the coefficients are to be considered con- 

 tant, since b 2 is fixed, but they would be different for a different 

 period of oscillation. 



For an infinitely long cylinder of revolution also, oscillating in a 

 viscous liquid, in a direction perpendicular to the axis of the cylinder, 

 Stokes found an equation of motion of this same familiar form which 

 had long been used to explain the behavior of pendulums, though it 

 had been founded on a theory quite different from his. As early as 

 1828 Bessel 7 had pointed out the necessity of allowing for the inertia 

 of the air which accompanies a pendulum in its motion, and the work 

 of Sabine, Dubuat, Poisson, Baily, Plana, South, and others, had 

 made it clear that in practical cases the moment of inertia of the swing- 

 ing system might be twice that of the pendulum bob, and that the 

 "resistance " of the air might be accounted for in many practical cases 

 by assuming it to be proportional to the first power of the angular 

 velocity. This equation had been used by Gauss for determining the 

 motion of swinging bar magnets, as has been already mentioned, and 

 it still forms the foundation of much modern work, as, for instance, 

 that on the properties of damped d'Arsonval galvanometers. 8 



If, however, a swinging magnet presents to the air a relatively large 

 surface, or if the magnet is provided with a large mica damping vane, 

 it often happens that the resistance of the air cannot be satisfactorily 

 explained on the assumption that it is proportional to the angular 

 velocity at every instant, and that at the beginning of the motion it 



7 Bessel, Untersuchungen fiber die Lange des einfachen Secunden Pendels, 

 Berlin, 1828. Bottomley, Phil. Mag., 23, 1887. Graetz, Reibung, Winkel- 

 mann's Handbuch der Physik, I. O. E. Meyer, Pogg. Ann., 113, 1861; 125, 

 1863; 142, 143, 1871; 148, 1873. Wied. Ann., 23, 1887. Kundt und War- 

 burg, Pogg. Ann., 155, 1875. Crookes and Stokes, Proceedings Royal Society, 

 1888. 



8 Dorn, Ann. der Physik, 17, 1882; 35, 1888. F. Kohlrausch: Ueber die 

 Inconstanz der Dampfungsfunction eines Galvanometers und ihren Einfluss 

 auf die Absolute Widerstandsbestimmung mit dem Erdinductor, Ann. der 

 Phys., 26, 1885. Schering, Ann. derPhys., 9, 1880. Jaeger, ' Instrumenten- 

 kunde, 1903. Dorn, Ann. der Physik, 17, 1882. 



