THE DECREMENT OF TITE ARC OF VIBRATION OF A MICA PLATE. 
437 
reserving to a later stage the consideration of the correction to be made for its 
variation. 
Let p be the density of the gas, p the observed pressure, D the density under a 
standard pressure, /x the coefficient of viscosity, and let accented letters refer to another 
gas. The dimensions of the terms in the equations of motion show that in comparing 
two cases in which the nature and pressure of the gases alone differ, the motions will 
be similar provided 
nr 
P ~ p° pB p'D' . 
( 1 ) 
This condition being satisfied, the resultant pressures of the gas on the solid will vary 
as /x or as p ; and as the logarithmic decrements (/) will vary in the same proportion, 
we shall have 
l V l V 
pD p’B’ 
( 2 ) 
The equations (1) and (2) are such that when one is satisfied so is the other. It 
will be convenient to regard (2) as giving the condition of similarity, and then (1) or 
£ P 
l V 
( 3 ) 
gives the ratio of the viscosities at the two corresponding pressures in the two gases. 
The times of vibration were practically constant when once the exhaustion was 
pretty high, at least until the very highest exhaustions were reached, when it fell off 
a very little ; but at atmospheric pressure and at low exhaustions it was somewhat 
greater, though not much. Its variability will not affect the results obtained by 
the above method, provided only the times are the same in the two experiments of 
each pair, which was very approximately the case. Nevertheless it may be well to 
consider the correction to be made in consequence of the inequality of the times. 
Let r be the time of vibration from rest to rest, then in comparing two similar 
systems the time-scale must be varied, so as always to be proportional to t, and the 
hydrodynamical equations show that for the condition of similarity we have, in place 
of (1), the equation 
ff T . 
P ' 
/ / 
/XT 
(4) 
As the two dynamical systems are not similar as a whole, but only the gaseous 
parts of them, we must have recourse to the equation of motion of the vibrating 
lamina. Let 6 be the angle of torsion, I the moment of inertia, n 2 10 the force of 
restitution, which will be proportional to the angle of torsion, provided at least the 
l 2 
