670 ‘Prof. M. Smoluchowski-Smolan on the 
which is characterized by a considerable generality, and is 
the only hitherto known general method of dealing with such 
problems. Its practical use, unfortunately, is rather limited, 
and we see no way yet leading to explicit solutions of the 
most important problems of aerodynamics, such as resistance 
of bodies in rapid motion, or flow of gases through wide 
tubes, where turbulent motions play a prominent part. 
§ 3. There are to be noted, however, some results obtained 
by application of the fundamental equation (2) to such cases 
as the last-named one, which present some interest, as cor- 
recting in several respects the common opinion on the thermal 
effects in outflowing gases. 
Thus it cannot be proved, as I have shown (loc. cit.), that 
stationary flow of a perfect gas leaves its temperature un- 
changed. This is true only fon the mean temper ature of the 
outflowing @ gas—in parts where its motion is sufficiently slow 
and uniform, supposing the walls of the tube to be heat 
insulators,—but its different stream-lines may be heated or 
cooled. 
If the gas is flowing out from a closed reservoir, its tempe- 
rature, after having passed the “rapids,” will be identical— 
with the same restrictions—with that of the gas contained 
in the reservoir, cooling down according to the adiabatic 
formula. But, contrary to common opinion, the use of this 
formula for viscous gases in motion is erroneous. It has to 
be replaced by an approximate equation, valid under certain 
restrictions for every stream-line: 
ER u? + v? + w? 
pa Yen ee 
which is not identical with the adiabatic law, except for 
ideal gases. 
These results concerning the thermal effects do not give us 
much help, however, in unravelling the complicated laws 
of such motions themselves, as defined by our system of 
equations. It does not seem probable, indeed, that theory 
of aerodynamics will soon surpass the experimental methods 
in efficiency ; and we must still apply to these latter ones as 
the chief sources of knowledge for the present. 
§ 4. The more important seems to me a simple method of 
reasoning, founded on the above equations, by which exact 
conclusions can be derived in many cases, and which often 
proves useful by verifying experimental results or by ex- 
tending their range. ‘This is the method of “ mechanical ” 
or “ dynamical similarity,” which, being closely allied with 
