MATHEMATICAL THEORY OE STREAM-LINES. 
301 
and the difference between those quantities — that is, 
RV-*|(V-W')A‘ = A(V-W')(vW'-A-'), .... (84b) 
is the energy added to that of the eddies in each second. If, as before, we have 
W'=±V, the preceding equation takes the following value, 
gCV» 3gCV 3 
16 ^~ 1 % ’ 
so that one fourth of the work of friction is expended in producing the current in the 
wake, and the other three fourths in producing eddies. 
If the velocities V and W' of the ship and her wake, and the amount of eddy-resist- 
ance E, are given, the sectional area C of the wake may be calculated from equation (7b). 
The elementary streams of which the wake is composed move astern relatively to the 
ship with a velocity less than that of an undisturbed current in the ratio expressed by 
V-W' 
— y — ; and hence they occupy a transverse area greater than they would do in the un- 
disturbed state in the ratio expressed by 
I_ = l+ W' 
n ' 
V-W 
V-W ' 5 
(85) 
which, when W'=^V, becomes = 2 . This causes a certain modification in the forms of 
the stream-lines outside the wake, which might be represented by taking for the surface 
of an imaginary disturbing solid a surface midway between the skin of the vessel and 
the outer surface of the frictional layer, followed by an indefinitely long cylindrical tail 
of one half of the sectional area of the wake ; but the detailed investigation of this will 
not now be entered on. 
Mr. Froude a few years ago pointed out that the most perfect propeller for driving a 
ship against skin-resistance, would be one which should act solely on the particles of the 
wake, driving them astern so as just to take away their forward velocity and no more. 
The velocity of such a propeller relatively to the ship would be equal and opposite to 
her speed V ; and the energy expended in working it would be simply EV, equal to the 
work done by the ship, through friction, on the water. It would thus be a propeller 
free from “ slip” and free from waste of power. It would stop the following current in 
the wake, and would at the same time impress on the water an additional quantity of 
energy in the form of eddy-motion, equal to the energy taken away in stopping the 
current ; so that the total energy impressed on the water in each second would be the 
same as before. 
It would preserve to the stream-lines the shape which they would have in the absence 
of friction. 
A propeller of the most efficient kind possible, producing the same forward thrust E, 
by acting on previously undisturbed water so as to impress a backward velocity W" on a ’ 
current of the sectional area B, would move it astern relatively to the ship with the 
2 t 2 
