444 
PROFESSOR G. G. STOKES OK MR. CROOKES’S EXPERIMENTS OX 
the table of experiments for air ’1059. The logarithmic decrement for kerosoline at 
54 millims. calculated from that for air at 484‘3 millims., is T059 X‘0390='04024, 
which comes very near the observed number ‘0404. It is curious that we should 
apparently be able to calculate very approximately an unknown vapour density from 
observations on the decrement of the arc of vibration of a vibrating lamina. 
Before considering the falling off of viscosity at high exhaustions, I would point 
out a result of theory which is of some interest in connexion with the forms of 
Mr. Crookes’s curves. 
At p. [34] in the paper of mine already referred to I have given in equation (61) 
an expression (which has to be subsequently multiplied by p) for the resistance to a 
sphere vibrating in a viscous fluid within a concentric spherical envelope. When the 
index of friction is sufficiently great, as will be the case when the exhaustion is high 
enough, this expression may be developed according to ascending powers of m, which 
will be convergent even from the first. It will be found that the successive terms 
will be multiplied by 
m ", nr, m", m?. . . 
where m 2 is a pure imaginary multiplied by p-r-p ; and from the mode of treatment 
there adopted it will readily be seen that the terms fall alternately on the arc and on 
the time. 
The same thing may, however, be shown to be true generally, independently of the 
form of the vibrating body. It is here supposed, as has been all along, that the 
motion is sufficiently slow to allow us to neglect squares and products of the com¬ 
ponents of the velocity, or of their differential coefficients. For if p-Pp be very great, 
we may imagine the hydrodynamical equations solved by successive substitution. 
First we should neglect the terms in p, and solve the equations ; then substitute in 
the terms multiplied by p the result of the first approximation and solve again, and so 
on. And though we cannot actually solve the equations, still this consideration shows 
that the solution must be of the form 
ap-\-l)p-\-c— -f-d—. . . 
P A 
where a, b, c, d . . . involve neither p nor p. And by adopting the artifice for the 
introduction of the time employed in the paper it readily appears that the terms fall 
alternately on the arc and on the time. Hence taking the two most important terms 
only in the expression for the effect on the arc we shall have for l an expression of 
the form 
2 TQO 
ci p —b c or ct p —b c — p~. 
Hence in a curve plotted with / and p for coordinates, the tangent as p diminishes 
