226 REPORT— 1875. 



portion in whicli the sectional area of the pipe at B is less than that at A ; and 

 therefore while passing from A to B the forward velocity of the fluid is being 

 increased. This increase of velocity implies the existence of a force acting in the 

 direction of the motion ; that is to say, each particle which is receiving an increase 

 of forward velocity must have a greater fluid pressure behind it than in front of it ; 

 for no other condition will cause that increase of forward velocity. Hence a par- 

 ticle of fluid, at each stage of its progress along the tapering contraction, is passing 

 from a region of higher pressure to a region of lower pressure, so that there must be 

 a greater pressure in the larger part of the pipe than in the smaller, and a diminution 

 of pressure at each point corresponding •with the diminution of sectional area ; and 

 this difference of pressure must be such as to supply the force necessary to establish 

 the additional forward velocity required at each point of the passage of the fluid 

 through the contraction. Consequently, difi'erences of pressure at different points 

 in the pipe depend simply upon the velocities at those points, or, in other words, 

 on the relative sectional areas of the pipe at those points*. 



It is simple to apply the same line of reasoning to the converse case of an enlarge- 

 ment. Here the velocity of the particles is being reduced through precisely the 

 same series of changes, but in an opposite order. The fluid in the larger part of 

 the pipe moves more slowly than that in the smaller, so that, as it advances into the 

 enlargement, its forward velocity is being checked ; and this check implies the ex- 

 istence of a force acting in a direction opposite to the motion of the fluid, and each 

 particle being thus retarded must therefore have a greater fluid pressure in front of it 

 than behind it ; thus a particle of fluid at eacli stage of its progress along a tapering 

 enlargement of a pipe is passing from a region of lower pressure to a region of higher 

 pressure. As is well known, the force requfred to produce a given change of velocity 

 is the same, whether the change be an increase or a decrease. Hence, in the case of 

 an enlargement of a pipe, as in the case of a contraction, the changes of velocity 

 can be satisfied only by changes of pressm-e, and the law for such change of pres- 

 sure will be the same, mutatis mutandis. 



In a pipe in which there is a contraction and a subsequent enlargement to the 

 same diameter as before (see fig. 11), since the diflerences of pressure at difierent 

 points depend on the diflerences of sectional area at those points, by a law 

 which is exactly the same in an enlarging as in a contracting pipe, any points 

 which have the same sectional area will have the same pressures, the pressm-es 

 at the larger areas being larger, and those at the smaller areas smaller. 



Precisely the same result will follow in the case of an enlargement followed by 

 a contraction (see Plate IX. fig. 13)t. 

 This proposition can be illustrated by experiments performed with water. 

 Figs. 14, 15, show certain pipes, the one a contraction followed by an enlarge- 

 ment, and the other an enlargement followed by a contraction. At certain points 

 in each pipe, vertical gauge-glasses are connected, the water-levels in which 

 eeverally indicate the pressiu-es in the pipe at the points of attachment. 



In fig. 14 the sectional areas at E and P are equal to one another. Those at 

 and K are likewise equal to one another, but are smaller than those at E and P. 

 The area at I is the smallest of all. Now, if the water were a perfect fluid, the 

 pressures P Q and E D would be equal, and would be greater than H and K N. 

 C H and K N would also be equal to one another, and woidd be themselves greater 

 than I J. 



The results shown in fig. 15 are similar in land, equal pressures corresponding to 

 equal areas. 



As water is not a perfect fluid, some of the pressure at each successive point is 

 lost in friction; and this growing defect in pressure is indicated in the successive 

 gauge-glasses in the manner shown in figs. 10, 17. 



As the pressure of the perfect fluid in the pipe at any point depends upon the 



* See Appendix, Note B. 



t In a perfect fluid, we may say, in a sense, tliat the vis viva of each particle remains 

 constant. If the iiarticlc is stationary, the vis viva is entirely represented by the pressure ; 

 if it be imder no pressure, the vis viva is entirely represented by the velocity ; if it be 

 moving at some intermediate velocity, the vis viva is partly represented by the pressure 

 and partly by the velocity. 



