558 PROFESSOR RANKINE ON THE THERMAL ENERGY 



contrary ; and consider what resultant forces are exerted by the stream on the 

 two parts into which those two cross sections divide the tube. The mass of 

 matter which flows through each cross section of the tube in an unit of time is 



pwda ; 



and in each unit of time a mass of matter of that amount has its velocity reversed. 

 The force required in order to produce that reversal of velocity is of the following 

 amount in absolute units, 



2pw 2 da ; 



and such is the amount of each of the pair of inward pressures which the tube 

 exerts on the stream, and of each of the pair of equal and opposite outward 

 pressures exerted by the stream on the tube, tending to pull it to pieces. It may 

 be called the centrifugal tension of an elementary stream. 



The velocity of the particles flowing in the stream may undergo periodical 

 fluctuations, positive and negative alternately ; these will cause periodical varia- 

 tions in the centrifugal tension ; but the mean value of that tension will continue 

 to be that given by the formula. 



The mean intensity of the centrifugal tension, in a direction tangential to the 

 stream, is found by dividing the amount given in the preceding expression 

 by the collective area, 2d<j, of the two cross sections, giving the following result, 



pw 2 . 



Suppose now that the stream is cut by an oblique sectional plane, making the 

 angle 6 with a transverse section. Then the area of that oblique section is 

 greater than that of a transverse section in the ratio of 1 : cos 6 ; and the amount 

 of the component tension in a direction normal to the oblique section is less than 

 that of the total centrifugal tension in the ration of cos 0:1; whence it follows, 

 that the mean intensity of the component centrifugal tension in a direction 

 making an angle 6 with a tangent to the stream is 



pro 2 cos 2 . 



Next, suppose a vessel of any invariable volume and figure to be filled with 

 vortices or circulating streams, the velocity of steady circulation being w, and the 

 mean density p. The centrifugal force will cause a pressure to be exerted in all 

 directions against the inside of the vessel. To determine the mean intensity of 

 that pressure, irrespectively of periodical variations, conceive the contents of the 

 vessel to be divided into two parts by an imaginary plane, and consider what 

 will be the mean intensity of the force with which the circulating streams tend 

 to drive asunder the portions of matter at the two sides of that plane. The 



