672 Prof. M. Smoluchowski-Smolan on the 
exact similarity in the sense above defined, which we are 
going to set out in detail in what follows. 
§ 5. In order to get the necessary conditions to be fulfilled 
by similar motions of gases, let us substitute the new variables 
770 . 
me, My, Mz, NU, NV, Nw, —t, h@, bp, and the coefficients 
a By, yx for the variables 2, y, 2, u, v, w, t, 6, p and 
coefficients R, w, « respectively. Hvidently the coefficient k 
cannot be changed ; motions of gases with different values of 
k cannot display exact similarity. All the following consi- 
derations are limited, therefore, to gases with the same value 
OL 
We may take into account the variability of viscosity and 
conductivity, too, by putting Pht, yh*« instead of p, «, in 
supposing these coefficients to be proportional to the eth 
power of temperature. According to Barus and Puluj, ¢ has 
the value 4 for the viscosity of air and hydrogen ; the kinetic 
theory requires an identical value for conductivity; experi- 
mental evidence, though, seems to point to a somewhat smaller 
number, 0°57 according to Hichhorn and Miller; but this 
difference, the reality of which is by no means beyond doubt, 
could have no appreciable influence, except in careful special 
experiments. 
By substitution of those variables in (2) and (3) there 
result the conditions of similarity (in the case of no external 
forces) : 
abm?  b  Smhe . mb Bmzh? inte 
=> = i 
Bae nde 
hn n n* n n? n 


which can be reduced to three relations :— 
i ets 
ao = PRIN iy Se se fede 
. fa 

As the first equation tells, similarity is possible only for 
gases which have a common value of a Th: restriction, 
K 
however, is not of great importance in practice, since it is 
fulfilled by itself with sufficient approximation for some 
gases, as the following table shows [wu and « for air being 
taken as unity, and the molecular weight M being substituted 
1 | 
for =| :— 
