478 APPLIED MECHANICS 
The weight of water eeipate the vanes is now W=wAv, and the 
total pressure on the vanes is P=Te —v,)= “co 0). 
Useful work done per second =Puv,= WENA, — 04). " 
2 
Kinetic energy of jet per second =wAv- on 
o 20, ae 
red ear 
For a given value of v the efficiency will be a maximum when 
Efficiency =o — 0) + wAv - 
v,(v—v,)isamaximum. Let y=v,(v—,), then be =v—2v,, therefore — 
s 
the efficiency is a maximum when v = 2v,, and the maximum efficiency is 
S(0- 4v) =4, or 50 per cent. rs 
The action of a series of vanes will perhaps be better undetstgil 
by reference to Fig. 775. . This does 
not represent a practical contrivance, 
and it is designed to illustrate the 
principle only. A frame, carrying 
a series of vanes at intervals q¢ apart, 
travels parallel to the jet, and each 
vane in turn is swung into the jet 
at the same point. The vanes are 
perpendicular to the axis of the jet. 
At (a) the first vane has just come 
in front of the jet. At (d) the 
second vane has just come into 
action, cutting the jet in two. The 
forward part of the jet will continue 
moving until its rear end B overtakes 
the vane in front of it. At (c) B, 
which is moving faster than the 
vanes, is overtaking the vane in 
front of it, and at (d) B has over- 
taken the vane in front of it, and 
that vane therefore ceases to act. 
Referring to (0), let a be the dis- 
tance which the front vane will have 
to travel before it ceases to act : 
after the second vane has come into 
action. Then g +z is the distance travelled by a point in the jet while 
: +2 v 
a vane travels the distance x Hence a at in ae and the number 
1 ' 
ev 
| oat 
2 5 iy 
of vanes in action at one time = a =]+ 
The total pressure on one vane is, by Art. 414, “ae — 0). Therefore 
