EXPERIMENTS ON SURFACE-FRICTION. 253 
on a large mass, or vice versd, or of a single force acting on a succession of 
masses. 
The expression, therefore, quantifies the momentum which must be gene- 
rated in each second in the surrounding fluid, by the transit of a surface the 
resisting force of whichis F. In some shape or other, there must be left behind 
it, in each second, new momentum to that extent, existing either in the shape 
of a narrow and rapid current, or a broad and slow one, or one of graduated 
speed and corresponding volume. 
This last supposition is clearly the most reasonable one, and it is approxi- 
mately in visible accordance with fact; and, without speculating on the modus 
operandi by which the motion is communicated, it becomes easy by help of 
this supposition to put an approximate value on the breadth of the current 
produced under any given circumstances. 
It will be seen presently that if the surface is long, the current thus esti- 
mated must be of considerable breadth ; and if this be so; instead of finding 
it difficult to explain why the resistance per square foot grows less as the 
length is increased, the perplexing question is, how the rate of declension is 
so slow. For a little reflection obliges us to see that it is the motion of the 
surface relative to contiguous particles, and not relative to distant ones, that 
governs the resistance ; and if these contiguous particles are already possessed 
of considerable velocity, concurrent with that of the surface, their resisting 
power must plainly be impaired. 
When we proceed to trace the genesis of the momentum in detail, as it 
must exist in the completely generated current left behind by the surface, if 
we select at any point an element or strip of current parallel to the line of 
motion, and possessing the velocity v in feet per second in that line, we see 
that in that element the quantity of matter newly put in motion per second 
will, at that point, be a portion of the strip, (V—v) feet in length (that being 
the length left behind by the surface), while the velocity impressed on it is v ; 
and if all the dimensions be in fect, taking the depth of the current parallel 
to the surface as unity, and the thickness or breadth of the element as dh 
(hk being the distance from the plane of the surface), we shall have for 
the weight of the element, dw =o (V—v) dh, & being the weight of a 
cubic foot. 
- Now if we assume that the current possesses a velocity =V at the plane 
of the surface (that is to say, that the particles in contact with the surface 
have the same speed as the surface), and that where h=H, then also v=0, 
the intermediate gradation of speed being uniform, we have 
vV@= h) 
Ns sep 2 
hence 
dw= ove dh ; 
H 
and if M be the momentum, 
aM=vdu= 9" “(H—A)nah : 
oV?(Hh? he 
we M=tr(--3); 
