Friction on Even Surfaces. 6) 
for a surface 8 to 20 feet long, and n=2°-00 for a plane 2 feet 
long; I find n=1°85 for all lengths from 2 to 16 feet. By 
Froude’s measurements the friction varies as the power 0°83 
of the length for varnished planes 2 to 20 feet long; I find 
it to vary as the power 0°93. With a varnished board 2 feet 
long, moving 10 feet a second, the ratio of our coefficients of 
friction for air and water is 1:08 times the ratio of the 
densities of those media under the conditions of experiment. 
But in some respects Froude’s results are quite unlike 
mine. For several surfaces he finds the skin-friction to vary 
as the square of the velocity, or nearly so, which is the re- 
lation I observed in a turbulent current, and when the friction- 
board was vibrating slightly. He finds the friction of calico 
about twice that of a varnished surface; I find that glazed 
eambric has about the same friction as a varnished surface. 
But if the cambric is roughened so as to expose a fine down, 
the friction is very much increased. | 
The fact that, for some surfaces, the coefficients of friction 
for air and water are roughly as their densities is of con- 
siderable interest. It is well known that the head resistances 
of the two fluids are directly as their densities; and if their 
friction coefficients also bear that ratio, the total resistances 
must be approximately as their densities. Hence the data 
obtained from water-resistance measurements on such sur- 
faces may be fairly well applied to estimate the air resistance 
on various-shaped models. 
It is not, however, self-evident that the surface-friction of 
any two fluids is proportional to their densities, and should 
not be taken for granted. It happens to be roughly true for 
varnished wooden surfaces in air and water; but is wholly 
untrue for calico surfaces. In default, therefore, of an 
adequate physical theory of surface-friction, the magnitude 
in any given case can be determined only by direct ex- 
periment. 
To complete the theory of the skin-friction board two steps 
further remain to be taken. First, the equations of motion 
for a viscous fluid must be integrated, to tind the velocity at 
all points in the disturbed region about a thin material plane. 
Then the speed of flow must be measured at all points next 
the plane and at some distance away. The writer expects 
soon to map the stream-lines and measure the velocity. If, 
then, the equations can be integrated, so as to give the speed 
as a function of the space coordinates, the computed and 
observed values can be directly compared. It is hoped that 
some one may obtain sufficiently general solutions of the 
equations to be of practical value; particularly for the simpler 
Phil. Mag. 8. 6. Vol. 8. No. 43. July 1904. jit 
