252 REPORT—1874. 
owing to the combined effect of the greater power of the speed to which the 
resistance is proportional, coupled with its less rapid declension in terms of 
length of surface, with a length of 50 feet the mean resistance of the tin- 
foiled surface is barely less than that of the varnished surface, and its 
resistance per square foot at the 50th foot is the greater of the two. 
It is true that this apparent anomaly probably in part depends on the fact 
that the coating of the longer surfaces with the foil was not so easily effected 
as that of the shorter, and therefore perhaps their smoothness was less perfect 
and their resistance somewhat increased ; yet, making every reasonable allow- 
ance for this, the anomaly is still remarkable. 
Again, no rational explanation presents itself of the differences in the law 
of variation of resistance in terms of length, exhibited by the rougher and 
more highly resisting surfaces. The resistance, for instance, of the medium 
sand alters disproportionately little towards the end of the plane, nor do any 
of these resistances exhibit as marked an excess of decrease in that direction 
as might have been expected. Partly, no doubt, this is owing to the diffi- 
culty in securing uniformity of coating; but also, it must be admitted, that 
the law which really governs the decrease has yet to be discovered, though it 
can hardly be doubted that it depends somehow on the current created by the 
passage of the surfaces. 
I shall conclude the Report with some remarks on what appears to me to 
be the rationale of the declension of resistance in terms of length of sur- 
face. 
It is certain that any surface which, in passing through a fluid, experiences 
resistance, must, in doing so, impress on the particles which resist it a force in 
the line of motion equal to the resistance. Now, we cannot regard a fluid as 
anchored to the shore or bottom by lines of tension or of thrust which are 
snapped or crushed by the force which causes motion ; but, on the contrary, 
we must assume the resistance offered by the particles of fluid to be purely 
dynamic, and to be dependent on and correlative to their weights and the 
velocities imparted to them. 
This being so, it is quite certain that the operating force, which (whatever 
be its amount) must be precisely equal to the resistance when the speed is 
steady, will in each unit of time, say in each second, generate a given definite 
amount of new momentum, estimated in the line of motion, in the system of 
particles on which it operates. The force must, in fact, generate some- 
where and somehow in the surrounding fluid the momentum which exactly 
corresponds dynamically to the universal law connecting force and mo- 
mentum. 
That law may be expressed as follows :— 
If F be the force in pounds which operates in a given direction, 
W the weight operated on in pounds, 
V the velocity in feet per second, 
t the time of action, 
32-2 ft. 
g the force of gravity = 
1” > 
Fgt 
ant ar 
then V W 
For the momentum, therefore, we have 
WV SEG ides os oe os 05 G) O (1) 
and this is equally true, whether it be the result of a small force acting 
