= Nesin o- —v, sin $) =P, 
second. OD=v,, and OE=velocity of 
GENERAL PRINCIPLES OF HYDRAULICS 477 
Casz IV. Same as Case III., except that the vane is moving parallel 
to itself with a velocity v, in a direction making an angle } with the 
vane {Fig, 774).—Loss of ‘momentum per 
second in direction of normal to vane 
‘Let the vane BC move to B/C’ in 1 
vane in direction of jet. 
OE sin @=OD sind. Therefore 
OE-= OD sin } _ », sin p Fig. 774. 
sin 6 sin 0° 
Relative velocity of jet and vane in direction of jet =v — fens. 
sin 
x _ysingd\_ wA, . 7 | 
a W wA(» Ts ) on AG sin 0 —v, sin ¢), 
wa . F 
ae 6— 2 
and aaa ae’ sin 6 —v, sin ) 
Useful work done per second 
ia -  wAysingd, -» 9 «ino 
= Pv, sin = meagre v, sin $)*. 
Kinetic energy of jet per second = casi 
: _ wAr, sin p 9— 2, wArs 
Efficiency rune 1 (v sin 0 —v, sin ¢)? + > 
_ 2% 8in $y, sg _y. si $y? 
= San OO” sin 6 —v, sin $)?. 
In the same way as in Case II. this efficiency can be shown to be 
, sin 0 
a maximum when died vr & 
The maximum efficiency = fd sin? 0, 
27 
Case IV. is the general case from which the others may easily be 
deduced. For example, Case II. may be deduced from Case IV. by 
putting 6 and ¢ each equal to 0. 
In the foregoing demonstrations the losses due to friction and the 
production of eddies have been neglected. 
415. Impact of a Jet on a Succession of Vanes.—In the preceding 
Article the jet was supposed to impinge on a single vane, and it was seen 
that the amount of water arriving at the moving vane was less than the 
amount delivered by the nozzle. If, however, a series of vanes come in 
turn in front of the jet, each vane entering the jet at the same point, 
the vanes will receive the whole of the water discharged by the nozzle, 
and the useful work done will be increased, and the efficiency therefore 
raised. For example, consider Case IT. of the preceding Article with a 
succession of vanes instead of one, 
