8 



water a previously determined velocity and direction, and in 

 having no surrounding casing. 



As previously stated the thrust is lbs. per lb. of water 



per second flowing through the screw at any point by the 



ordinary theory, and the total work done is this multiplied by 



V + K , • j. 11 u- i. v 2 -K 2 , ,, . 

 , longitudinally, which amounts to , or halt the 



change of vis visa, in accordance with the usual accepted laws 



of mechanics ; likewise the work done circumferentially is , 



and radially, if there be any radial flow, , where \\> 



stands for the angle made with a tangent to the circle of radius 

 r by the actual path of the water, projected on the screw disc. 

 This can be seen by inspection of Fig. 2, which represents 

 isometrically the motion of a particle of water at A in the 

 direction, and with the velocity, AD, the angle ACD being \p. 

 AC is v, the velocity resolved parallel to the axis, CB is rw, the 

 circumferential velocity ; and BD isrcotam/-, the radial velocity. 

 The whole work done is done by the engine, and consequently 

 we can equate the sum of the longitudinal, circumferential, and 

 radial, work to that done by the engine at radius r, which has 



.ft 



been previously shown to be 



. . r 2 o> 2 rVtanV v 2 -K 2 r^ft 

 giving — - + T -+- 



2g zg zg 



i 



COS 2 )/' 



Multiplying this across by 2g, and writing - — j-j for i + tan 2 \p 



gives — z-7 + v 2 —K 2 =2r 2 (titt. 



b COS 2 !/' 



It may be well to note that it is unnecessary to take account of 

 the circumstance that the point D maybe higher or lower than 

 A } so that there may be work done against gravity when, as in 

 a screw propeller, the axis is horizontal, because if D be (as in 

 the figure) higher than A, there is, at the opposite side of the 



