GENERAL PRINCIPLES OF HYDRAULICS 481 
The line of action of R passes through O, the intersection of the 
of the jets at B and C. | 
The foregoing result may be obtained in another way. At B the 
reaction of the jet on the vane is F = We in the direction CO. The resultant 
of these two forces, obtained by the parallelogram of forces OHKL, is 
Case II. Vane moving parallel to itself in a given direction with a 
welocity v, (Fig. 779).—The jet moving in the direction BD with velocity 
wv meets the vane BC at B. The vane is 
moving in the direction BE with velocity 
4) Make BD=v, and BE=v,. Com- 
q ete the parallelogram BEDH. Then 
_BH=v, is the direction and magnitude 
of the relative velocity of the water and 
vane ; therefore in order that there may be 
no shock at entrance, BH must be the 
- direction of the tangent to the vane at B. 
The water moves over the vane with 
| the relative velocity v,, leaving the vane at 
C, where it has a velocity v, in the direction 
CK tangential to the vane at C, and a 
velocity v, in the direction CL parallel to FiG. 779. 
BE. Make CK=v,, and CL=v,. Com- 
_ plete the parallelogram CKNL. The diagonal CN =», is the direction 
_ and magnitude of the absolute velocity of the water leaving the vane at C. 
3 Draw BS parallel and equal to CN. Join DS. Then DS is the 
_ change, in magnitude and direction, of the velocity of the water while 
_ passing over the vane. If R is the resultant force on the vane due to 
the impact of the jet, then R=" - DS, where W is the weight of water 
impinging upon the vane per second. Draw ST perpendicular to and 
_ meeting DH produced at T. Then if P is the component of R in the 
direction of the motion of the vane, p=" -DT. | 
; 
| CN, the absolute direction in which the water leaves the vane, should 
_ be perpendicular to CL, the direction of motion of the vane. CN has 
_ then no component in the direction CL. The component of CN in the 
_ direction CL in the case of a revolving vane is called the velocity of whirl 
_ at exit, and for maximum efficiency this should be zero. 
; If CN is perpendicular to CL, then BS and ST are in the same 
straight line, and DT =v cos 8, where 8 =angle BDT =angle DBE. Then 
) 
) 
W 
Ba-7 vcos 8. But W=wA(v—v,cos f), where A is the area of the 
section of the jet, and v, cos 8 is the velocity of the vane in the direction 
of the motion of the jet. Hence P= “mae — v, cos Byv cos P. 
2H 
