of a Solid Body in a Viscous Fluid. 629 



the ordinary solution of a damped harmonic vibration requires 

 modification when the initial conditions are taken into 

 account, but no explicit solution of this nature seems to have 

 been given ; in certain experimental refinements, the dis- 

 turbance may be of some importance. In the following 

 notes, I have worked out in detail first the simpler case of a 

 plane oscillating between two fixed planes. The problem 

 can be solved by various methods : by normal functions, or, 

 more readily, by operational methods. I have chosen to use 

 it as an example of a type of integral equation, for which 

 reference may be made to a previous paper*. In this case 

 the equation of motion is an integro-differential equation of 

 Volterra's type, and it can be solved by a repeated appli- 

 cation of Whittaker's method which was used in the simpler 

 cases ; the solution may be of interest apart from the parti- 

 cular problem. The results are then verified by using 

 Bromwich's method of complex integration. Finally, the 

 solution is indicated for a sphere oscillating within a fixed 

 outer sphere, and the results are discussed in connexion 

 with the experiments to which reference has already been 

 made. 



2. Suppose that a viscous liquid can move in laminar 

 motion between two fixed planes x= +/i. Let the plane of 

 yz be a thin rigid barrier of mass a per unit area, and let it 

 be acted on by an elastic force parallel to Oy such that, if the 

 liquid were absent, the plane would vibrate with a natural 

 period 2ir/p. Further, suppose the motion starts from rest 

 with the plane displaced a distance a from its equilibrium 

 position. The equation of motion of the plane is 



^- 2 KS)o + <^=°'- • • • w 



where v is the fluid velocity. 



Now if the plane of yz has a velocity V(t), the fluid 

 velocity may be written in the form 



v= | l^e-n^t,^ sin 5™ ry(. T )^ T/%) . (2 ) 



Taking the value of 'dv/'d-x' for .i- = 0, integrating by parts 

 and noting that in this problem V(0) = 0, equation (1) gives 



* Supra, p. 620. 



