















Principles of Aerodynamics. 673 
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The two remaining conditions define two of the four 
variables: velocity, dimension of length, pressure, tempe- 
ture, for any special gas, the two others remaining arbitrary, 
which implies a much greater variety of applications than in 
the case of hydrodynamics. 
§ 6. The simplest example demonstrating the principle of 
similarity (4, 2) is the common formula for the velocity of 
sound, V7kRO, this velocity being independent of pressure 
and dimensions. That condition is valid, too, for the higher 
order of exactitude, where the influence of those variables 
has to be taken into consideration, as in Kirchhoff’s formula 
for the propagation of sound through narrow tubes, or for 
Earnshaw’s and Riemann’s results concerning waves of sound 
of finite amplitude; but in these cases pressure is supposed 
to assume values corresponding to (4,3). 
Besides, the following applications may serve to show the 
use of similarity. 
§ 7. The only hitherto known theoretical result about 
resistance of moving bodies, agreeing approximately with 
experiments on pendulums, is its proportionality to the first 
power of velocity, in the case of extremely slow motion. For 
a greater speed, experiments have proved approximate propor- 
tionality to the second power ; but when the speed approaches 
the range of velocity of sound, the increase of resistance is 
much more rapid, and, after crossing this range, siower again*. 
Besides, resistance is commonly supposed to vary in pro- 
portion to superficial dimensions, although experiments have 
not been in very close accordance with this supposition, and 
in proportion to the density of the gas, although there is no 
experimental evidence yet whatever for this rule. Neither has 
the influence of temperature nor of pressure been investigated, 
nor have gases other than air been tried. 

* Useful information about these subjects is to be found in ZEncyclo- 
padie d. math. Wissensch. iv. 2, p. 160 (Finsterwalder), p. 190 (Cranz), 
Leipzig, 1903. 
te) 
Phil. Mag. 8. 6. Vol. 7. No. 42. June 1904. z2Z 

