ON SOME CASES OF FLUID MOTION. 



413 



For the sake of showing the manner in which f{r) alters with r, I have calculated the following 

 values of the function. The expression (6) shows that /'('') = "> when r = 0; and /'()•) is 



also =0 when r = 1, since / (-) =f(r). 



The experiments to which I have alluded were made with a wooden box measuring inside S 

 inches by 4 square. The box weighed not quite lib., and contained about i^lbs. of water, so 

 that the inertia of the water which had to be overcome was by no means small compared with that 

 of the box. The box was suspended by two parallel threads 3 inches apart and between 4 and 

 5 feet long : it was twisted a little, and then left to itself, so that it oscillated about a vertical 

 axis midway between the threads. The points of attachment of the threads were in a line drawn 

 through the centre of the upper face parallel to one of its sides, and were equidistant from the 

 centre. The weight of the box when empty, the length and distance of the threads, the time 

 of oscillation, and the known length of the seconds' pendulum are data sufficient for determining 

 the moment of inertia of the box about a vertical axis passing through its centre. When the 

 box is filled with water the same quantities determine the moment of inertia of the box and the 

 water it contains, whence the moment of inertia of the water alone is obtained by subtraction. It 

 is suposed here that the centre of gravity of the box coincides with the centre of gravity of its 

 interior volume. In the following experiments a different face of the box was uppermost each 

 time. In Nos. 1 and 2 the long edges of the box were vertical, in Nos. 3 and 4 they were hori- 

 zontal. In all cases the inertia determined by experiment was a little greater than that resulting 

 from theory : the difference will be given in fractional parts of the latter. The difference was 



— in No. 1, — in No. 2, — in No. 3, and — in No. 4. On referring to the table at the end 

 21 13 17 21 " 



of the last paragraph, it will be seen that the ratio of the moment of inertia of the fluid to what it 



would be if the fluid were solid is about three times as great in the last two experiments as in the 



first two. 



I had expected beforehand to find the inertia determined by experiment a little greater than 

 that given by theory, for this reason. In the theory, it is supposed that both the fluid itself and 

 the surface of the box are perfectly smooth. This however is not strictly true. The box by its 

 roughness exerts a tangential force on the fluid immediately in contact with it, and this force 

 produces an effect on the fluid at a small distance from the surface of the box, in conse(|uence of 

 the internal friction of the fluid itself. We may conceive the effect of this force on the time of 

 oscillation in a general way by supposing a thin film of fluid close to the surface of the box to be 

 dragged along with it. Conse<iuently, the moment of inertia determined by experiment will be a 

 little greater than it would have been had the fluid and the surface of the box been perfectly 

 smooth. 



