
Principles of Aerodynamics. 671 
the well-known method of dimensions, seems to have been 
used for the first time in hydrodynamics by Helmholtz, but 
has not yet been made use of for “ exact ” aerodynamics. 
It consists in examining what solutions can be derived 
fo) 
from a known solution, by magnifying the variables in certain 
constant proportions and. bya suitable change in the constant 
coefiicients. 
As examples* of, partly exact, partly approximate, appli- 
eation in hydr -odynamics ee be quoted :—The criterion for 
the validity of Poiseuille’s law (Helmholtz, Wied. Ann. vii. 
p- 375 (1879); Reynolds, Phil. Trans. elxxiv. (1883), clxxxvi. 
(1895)); or the criterion for the formula of viscous resistance 
fora sphere moving through liquid (Lamb, Hydrod. p. 533), 
and Froude’s calculation of : ship resistance. There ought to be 
mentioned, too, Boussinesq’s investigation (Journ. de Physique, 
1. p. 69, 1902), on the cooling effect of currents raised in 
various liquids by a heated body—which I believe to be 
erroneous, however, as depending on the equations for ideal 
liquids, and wanting complete alteration in the manner of 
§ 14.—and two papers of Helmholtz, connected with “ rough” 
aerodynamics. 
One of them (Ges. Abh. iii. p. 8309) contains his most sug- 
gestive theory of cloud-waves; it belongs to the class of 
s approximate ” applications, however, as resting on the 
equations for irrotational motion of incompressible fluids, 
with neglect of viscosity and capillarity, which seem to play 
no insignificant part in the formation of waves on water by 
wind. 
The other paper (Ges. Abs. i. p. 158), dealing with dyna- 
oD 
mical similarity in aerodynamics in general, and with the 
analogy of ship-resistance to balloon-resistance in particular, 
gives cause to most serious objections. The general considera- 
tions are limited by the supposition of isothermal compressi- 
bility, excluding by itself nearly all practical applications, and 
by the tacit supposition of small changes of pressure. ‘The 
application to the comparison of ships and balloons is vitiated, 
besides, by neglect of viscosity (which would imply no 
resistance at all for constant velocity in liquids), by a fatal 
slip in the numerical suppositions, lastly by neglect of 
gravitational ship-waves, differing w idely from compressional 
balloon-waves. 
Generally speaking, there may exist some rough analogy 
between motions of liquids and gases in some cases, but no 
* Fuller details concerning this question are given in a paper contained 
in Prace mat. fiz. Varsow, xv. (1904). 
