436 
PROFESSOR G. G. STOKES OK MR. CROOKES'S EXPERIMENTS OX 
difference of position of the plate relatively to the bounding wall of the bulb, a differ¬ 
ence however which is trifling on account of the smallness of the angle through which 
the plate oscillates. The forces therefore arising from the viscosity tend in the limit 
to vary entirely as the angular velocity, and not, as is the case when the index of 
friction is comparatively small, partly also on the angular acceleration. The result 
therefore will be that the oscillations are retarded by a force varying as the angular 
velocity, and producing therefore a subsidence of the motion such that the propor¬ 
tionate rate of change of the arc is proportional to the coefficient of viscosity. 
Mr. Ceookes’s experiments were carefully made from pressures as high as the 
atmospheric pressure downwards. At first there is a very evident decrease of sub¬ 
sidence as the pressure decreases, except in the case of hydrogen, in which it is ver} r 
small, we may say insensible. Then it remains very nearly constant for a considerable 
range of exhaustion, and at last, for extreme exhaustions, it rapidly fades away. 
In the second of these three stages the condition of ideal simplicity above mentioned 
is doubtless approximately attained. If however we confined our attention to this 
part only of the series, the lower part, although so carefully made, would remain 
unutilised; and further, we should remain uncertain whether in taking the logarithmic 
decrement as proportional to the viscosity our approximation was not too rough. 
The determination of the motion of the gas corresponding to a given motion of the 
vibrating solid, and thereby the determination of the forces which the gas exerts on 
the solid, forms a perfectly definite problem, the solution of which, if it could be 
effected, would lead to a determination of the coefficient of viscosity from the observed 
influence of the gas on the motion of the plate. But although in these slow motions 
the terms in the hydrodynamical equations which involve the squares of the velocities 
are insensible, so that the equations may be taken as linear, the problem is one of 
hopeless difficulty except in a few simple cases. In the paper referred to, I have given 
the solution in the case of a sphere vibrating in a mass of fluid either unlimited or 
confined by a concentric spherical envelope, and in that of a long cylindrical rod 
vibrating in an unconfined mass of fluid. In the latter especially of these cases the 
solution involves functions of a highly complicated form. For a lamina such as that 
employed by Air. Crookes, the problem could not even be solved if the fluid were 
regarded as perfect, much less when the viscosity is taken into account. 
But though we are baffled in the attempt to give an absolute solution of the problem, 
theory indicates the conditions of similarity of the motion of the gas in two different 
cases, and enables us thereby to compare the viscosities when those conditions are 
satisfied. 
The bulb, vibrating plate, and torsion thread being always the same, the two things 
that varied from one experiment to another were the nature and the pressure of the 
gas, not the temperature, which for the present was kept constant. The moment of 
inertia of the lamina was sufficiently large to allow the time of vibration to be nearly 
the same in the different experiments. For the present I will suppose it constant, 
