THE DECREMENT OF THE ARC OF VIBRATION OF A MICA PLATE. 439 
Now n is 7r-F t, and qr is the Napierian logarithmic decrement, or l-~ M, M being 
the modulus of the qommon system. Hence (6) becomes 
AMIZt 3 
7r/j(M 2 7r 2 + Z 2 ) 
and as B is the same in the two systems compared we have 
l T“ V t'~ 
pMV + P" + 
( 7 ) 
an equation which takes the place of (2), and serves to define corresponding densities, 
and then (4) gives the ratio of the viscosities at those densities, or say, at the corre¬ 
sponding pressures. If we eliminate the ratio of p to p between (4) and (7) we get 
f (tt~w + 1*)=+n 
IT (T 
( 8 ) 
which takes the place of (3). 
The ratio of the factor 7 t 3 M 3 +F to 7 t 3 M 3 alone but little exceeds unity; thus even 
for oxygen at 760 millims. pressure it is barely l - 0085, and of course the ratio of that 
factor for one gas to the factor for another gas at the corresponding pressure will 
differ from unity still less. Hence it is almost a needless refinement to keep in this 
factor at all. However, even if we retain it in (8) it is quite superfluous in (7), which 
merely determines what densities are to be deemed to correspond in seeking the 
logarithmic decrements. For until extreme rarefactions are reached, to which the 
above investigation no longer applies, the logarithmic decrement changes so slowly 
that a small error in the density of one gas which is deemed to correspond to a given 
density in another will make no sensible error in the logarithmic decrement. And not 
only may the factors above mentioned be omitted, hut as the ratio of r to r will differ 
but little from a ratio of equality, the formula (7) may he dispensed with altogether, 
and the simpler formula (2) employed. But when the logarithmic decrements have 
been found, in determining the ratio of the viscosities from (8) it is better not to dis¬ 
regard the quantities by which the ratios of r to t and of 7r 3 M 3 -)-^ 2 to 7r 3 M 3 +Z' 3 differ 
from ratios of equality. And if we now wish to know more precisely what densities 
or pressures do correspond, we may obtain them from (4). 
In the numerical calculations which follow, the difference in the times of vibration 
(r) at corresponding pressures in the different gases is neglected, and likewise the 
difference between the ratios of the factors M 3 7 t 3 +Z 3 and a ratio of equality. The 
general effect of this omission, which is very minute, will be considered in the end. 
The values of D adopted were, hydrogen, 1 ; air, 14'42 ; oxygen, 16 ; nitrogen, 14 ; 
carbonic anhydride, 22 ; carbonic oxide, 14. 
