668 - Prof. M. Smoluchowski-Smolan on the 
velocity of sound, and St. Venant & Wantzel’s experiments* 
on effusion of gases, extended and confirmed by many subse- 
quent authors, by which these phenomena have been shown 
to differ widely from what the isothermic theory maintains. 
In fact, the isothermic theory has to be limited to the few 
exceptional cases of slow viscous motion, where conduction 
of heat is prevalent; transpiration through capillary tubes 
and slow oscillations of pendulums seem to be the only 
examples of practical importance. 
In the two cases alluded to, and in most others, heat of 
adiabatic compression + is a prominent factor ; but, in general, 
the heat produced by internal friction is no negligible quantity 
either. It was sufficient in Joule and Kelvin’s plug expe- 
riments to annul the cooling by expansion, and in other 
experiments of those authors (Kelvin, Math. Phys. Papers, i. 
pp- 851, 400, 445) to produce considerable heating effects. 
§ 2. In hitherto published papers and treatises on aerody- 
namics, isothermic formulas and adiabatic ones are to be 
found, but no proofs of the thermic supposition underlying 
them ; some authors, after explaining both theories, satisfy 
themselves with the statement that reality probably will be 
contained between those thermic extremities—a rather rough 
and unsatistactory way of speculating. 
It is impossible, indeed, to develop a reasonable theory 
of these phenomena, unless we unite the mechanics with the 
thermodynamics of the subject. Such is the starting-point 
for the following considerations, being contributions to what 
may be called “ exact ” aerodynamics. 
The thermodynamics of our case are contained in an 
equation which appears, in its complete form, for the first 
time in 1894, derived from somewhat specialized kinetic 
considerations by Kirchhoff and Natanson, in a more general 
way by Neumannt. It follows easily from the principle of 
conservation of energy: by equating the increase of internal 
energy (caloric, kinetic, potential) of an element of mass, on 
its path: 

Dire u? 4-9? + w? | 
Di | i Gita Iem +U], 
* J.d.l Ecole Polyt. xvi. p. 92 (1839) ; see ea. gr. Wilde, Phil. Mag. xx. 
p- 531 (1885), xxi. p. 494 (1886). 
+ In acoustics it is predominant, above all others, there taking it 
into account is sufficient for a first approximation, and this is the only 
part of aerodynamics, therefore, where a systematic theory has been 
built up. 
t Kirchhoff, Vorlesuny. ti. Warme, p. 194 (1894); Natanson, Bull. 
Acad. Cracovie, 1895; Neumann, Gott. Ber. 1894, p. 19. 
