66 Fluid Friction on Even Surfaces. 
case in which the plane is indefinitely wide, and in which the 
edge conditions are negligible. The integrals, if sufficiently 
general, will be of great importance to the science of surface- 
friction, and may at once be applied to the mass of accurate 
data that, for a generation, has been accumulating in the 
laboratories of the marine engineers. 
Department of Mechanics, 
The Catholic University of America. 
fluid Friction on Even Surfaces, 
Note by Lord RAYLEIGH. 
Iy connexion with such experiments as those of Froude and 
Zahm respectively on flat surfaces moving tangentially 
through water or air, it is of interest to inquire how much 
can be inferred from the principle of dynamical similarity. 
Dynamical similarity includes, of course, geometrical simi- 
larity, so that in any comparisons the surfaces must be 
similar, not only in respect of their boundaries, but also in 
respect of their -roughnesses, at any rate until it is proved 
that the restriction may be dispensed with. Full geome- 
trical similarity being presupposed, the tangential force per 
unit area, I’, may be regarded as a function of a the linear 
dimension of the solid, p the density of the fluid, v the 
velocity, and v the kinematic coefficient of viscosity. It is 
assumed that the compressibility of the fluid does not come 
into account. As in a similar problem relating to the flow 
of fluid along pipes*, the method of dimensions shows that, 
if it be a function of the above quantities only, 
P=pv jf(auy),s . . . >» =n 
where 7 denotes an arbitrary function, whose form must be 
obtained, if at all, from other considerations. If F be inde- 
pendent ‘of v, 7 is constant, and F is proportional to the 
density and the simple square of the velocity. Conversely, 
if F be not proportional to x’, 7 is not constant, and the 
viscosity and the linear dimension enter. 
If the general formula be admitted, several conclusions of 
importance can be drawn, even though the form of / be 
entirely unknown. For example, in the case of a a given fluid 
(p and v constant), F is strictly proportional to v?, provided 
a be taken inversely as v. Again, if the fluid be varied, we 
may make comparisons relating to the same surface (a con- 
stant). For if v be taken proportional to v, f remains un- 
affected ; so that F is proportional to pv* simply. For air 
the kinematic viscosity is about 10 times greater than for 
* Phil. Mag. xxxiv. p. 59 (1892) ; Scientific Papers, iii. p. 575, 
