﻿Motion of a Sphere hi a Viscous Fluid. ;*>;53 



by the motion of the frame when the sphere has reached the 

 •centre of the trough. The duration of the exposure can be 

 varied to suit the velocity of the sphere, and the length of 

 the trace gives the velocity as before. About half a dozen 

 photographs were required to cover the whole field, and these 

 were combined and measured by the following method. A 

 circle of 2*5 cm. radius was drawn in the middle of a sheet 

 of squared paper, which was then set up on a drawing-board 

 in front of a projecting lantern. One of the plates was 

 placed in the lantern and a magnified image of the plate 

 thrown on the paper, this was then adjusted so that the mean 

 position of the ball during the exposure coincided with the 

 •circle drawn on the paper, the magnification being adjusted 

 so that the diameter of the image was exactly 5 cm. The 

 length of each trace was then measured with a pair of 

 ■dividers and its direction marked on the paper. Each plate 

 of a series was put into the lantern in turn, and thus the 

 whole field of motion was mapped out. 



The motion is best represented by means of Stokes's 

 current function yfr. If the motion be referred to cylindrical 

 coordinates z, -sr, the line traced out by the centre of the 

 sphere being taken as axis of z and the corresponding 

 velocities being denoted by it, u\ then yjr is defined by 



w ~ 1 <W r „_ 1 ^± 



W =r~ U — ^T^5 



and hence the curves ty = const. give the direction, and their 

 distances apart divided by ijs give the magnitude of the 

 velocity at any point. The values of yjr for the median line 

 ^ = and for the surface of the sphere were first calculated 

 from the measurements of velocity, and starting from points 

 giving yfr = "2, *4, &c, the curves were drawn in the direction 

 of the velocity as marked on the paper, and thus the stream- 

 line diagrams of figs. 2-8 were obtained. In each diagram 

 the straight line at the bottom is fori|r=0and each suc- 

 ceeding curve is for values of yfr increasing by an interval 

 of *2, the velocity of the sphere being always taken as 

 unity. 



The results obtained are shown in the accompanying 

 •diagrams and photographs which we may now proceed to 

 describe in detail. 



The first two diagrams figs. 2 and 3 and the corresponding 

 photograph A (PI. IX.) refer to a sphere moving with a 



