Mr. J. L. Woodbridge on Turbines. 315 



For the element of time, dt, we will take the time occupied 

 by one of the liquid prisms in passing along the vane through 

 a distance equal to its own length, or outwardly a distance dp, 

 thus making t a function of p, the latter being considered the 

 independent variable. We have 



dt 7 dp dp 



dp r dp vsinY 

 dt 

 In the figure let aa! be the distance passed through by the 

 turbine in the time dt, then 



aa!=Q)pdt=<ap -j- dp. 



The mass m will have two motions ; one along the vane, 

 the other with the wheel perpendicular to the radius. By 

 changing its position successively in each of these directions, 

 both its velocity with the wheel and its velocity along the 

 vane may suffer changes both in amount and direction, and 

 will give rise to the following eight possible reactions : — 



I. B}' moving from a to a' in the arc of a circle. 



1. wp may be increased or diminished ; 



2. wp may be changed in direction ; 



3. v may be increased or diminished ; 



4. v may be changed in direction. 



II. By moving from a! to <% along the vane. 



5. oap may be increased or diminished ; 



6. (op may be changed in direction ; 



7. v may be increased or diminished ; 



8. v may be changed in direction. 



By the conditions imposed, 1 and 3 are zero. For the 

 others we have : — 



No. 2. By moving from a to a!, the velocity (op is changed 

 in direction from ak to a'k 1 in the time dt. The momentum 

 is moop, and the rate of angular change is 

 kaM _ oadt _ 



~dF~~dT~ co > 



and hence the reaction will be maPp in a direction radially 

 outwards. This is the centrifugal force as designated by most 

 writers. Resolving into two components, we have 

 ma? p sin y along the vane, 

 maPp cos 7 normal to the vane. 



No. 4. In moving from a to a', the velocity along the vane, 

 f, is changed in direction from at to a't' at the rate (o as 



Y2 



