614 PROCEEDINGS OF THE AMERICAN ACADEMY. 



perimectal conditions, any quantities neglected in the solution of the 

 ditferential equations involved are really negligible. 



Professor Stokes, in his discussion of the motion of an infinite viscous 

 fluid disturbed by a sphere oscillating about a vertical diameter, neglects, 

 in the general equations expressing the motion, terms which are of the 

 second degree; e. g., the square of the velocity of the fluid at any point 

 is neglected. The assumption tliat these terms are negligible leads to 

 a comparatively simple expression relating the logarithmic decrement, 

 the density of the fluid, the inertia of the whole pendulum, the time of 

 swing, and /j. the coefficient of viscosity. But it has been shown that in 

 the case of the sphere the terms which Stokes has neglected are not by 

 any means always negligible.* These terms, it appears, are of the order 

 of magnitude of V'^ap/fi, where Fis the velocity of the fluid at any 

 point, a the radius of the sjjhere, p the density of the fluid, and fi the 

 coefficient of viscosity. In order, then, that Stokes's solution may be 

 valid, the quantity Vap//x must be small. Fand a are the quantities 

 which can be varied in this expression, and if they can be made suffi- 

 ciently small, the determination of p. should be rendered merely a question 

 of manipulation. 



Two very good tests can be applied to determine whether the desired 

 conditions have been obtained. In the first place, if V'-ap/p. is too 

 great to be neglected, then as Fchanges so does the relation between the 

 velocity with which the solid body moves and the resistance with which 

 it meets, and therefore the logarithmic decrement must change as the 

 velocity changes. If, therefore, the period of vibration remains constant, 

 the decrement measured when the pendulum is moving through lai-ge 

 arcs should be different from that measured when the arcs are small. 

 Thus, if the pendulum be given a certain amplitude of oscillation at the 

 start, and the decrement be measured from arc to arc, there should be a 

 gradual change in its value as the arc of swing diminishes. Again, when 

 the period of oscillation, the radius of the sphere, and the angle through 

 which the sphere moves are known, the maximum value of Fcan be 

 calculated and compared with its square. Also, by using one of the 

 values of p, given above, the value of the quantity Vap/p. can be 

 calculated. 



Stokes's discussion of the sphere oscillating in an infinite viscous fluid 

 leads to the following expression for the logarithmic decrement: 



* Whitehead, Quar. Jour. Math., 23, 1889. Lord Rayleigh, Phil. Mag., 36 

 1893. 



