GENERAL PRINCIPLES OF HYDRAULICS 479 
the total pressure on all the vanes is “0v- 0)? x — - = (0 - 1), as 
already shown in another way. Fig. "715 is drawn for the case where 
v= 2v,. 
“Tt is easy to show that in the general case, Case IV. of the preceding 
Article, but with a succession of vanes instead of one, the total normal 
— on all the vanes in oe at one time is 
P= (osind v, sin $), 
and that the efficiency is a8 Bo sin @—v, sin ¢), also that the maxi- 
mum efficiency is } sin? 0 Eas vsin 0 = 2v, sin ¢. 
416. Impact of a Jet on a Cup.—The axis of the jet is supposed 
to coincide with the axis of the cup, 
and the effect of friction will be 
neglected. 
Case I. Cup at rest.—The water will 
leave the cup in a direction tangential 
to the surface at the lip of the cup, as 
_shewn in Fig. 776, and the velocity of the 
water as it leaves the cup will have the Fi. 776. 
Same magnitude v as the velocity of the 
jet, but its direction will have been turned through an angle 180° — 6°. 
Loss of momentum of water te second in the direction in which it 
_is moving before striking the cup= -~ + cos 6). 
Therefore P =~ + cos 0) = “nl + cos 6). 
It 0-0, pa vA 
Case II. Cup moving in same direction as jet with velocity v,.— 
_ Relative velocity of jet and cup=v—v,, and this will be the relative 
_ velocity of water and cup as the water leaves the cup. Hence the loss 
of momentum of water per second in the direction in which it is moving 
before striking the cup is aC —v,) (1+ cos 4), and this is equal to P. 
But W =wA(v —»v,), therefore P= we —v,)? (1 + cos 6). 
Useful work per second = Pv, = wg yet (1 + cos 8). 
Eificieney = "(9 — —,)?(1 + cos 0) + wAv: 5 sy ae —v,)?(1 + os 6). 
The efficiency str be a maximum when v= ed 
8 0 
Maximum efficiency = 971 +cos 0) = = cos? = - 
417. Reaction of a Jet.—When a jet of cross sectional area A issues 
from a vessel with a velocity v, the momentum given to it per second is 
