TRANSACTIONS OF THE SECTIONS. 337 



sectional area at that point, it follows that the amounts of the pressures are inde- 

 pendent of tlie distance, as measured along the pipe, in which the area of the pipe 

 alters ; so that if in the pipe shown in Plate IX. fig. 18 the areas at all the points 

 marked A. are equal, if also the areas at all the points marked B ai-e equal, and so 

 also with those at and D, then the pressures at all the points A will be the same, 

 the pressures at all the points B will be the same, and so with those at and D. 



Since, then, the pressure at each point depends on the sectional area at that 

 point and on that only, it is easy to see that the variations in pressure due to the 

 now, are not such as can cause any total endways force on the pipe, provided its 

 sectional area at each end is the same. 



Take the pipe shown in fig. 19. The conical portion of pipe AB presents 

 the same area of surface effective for endways pressure as does the conical 

 portion HI, only in opposite directions. They are both subject to the same pres- 

 sure, being that appropriate to their effective mean diameter J. Consequently 

 the endwaj's pressures on these portions are equal and opposite and neutralize one 

 another. Precisely in the same way it may be seen that the endways pressures 

 on B C, D, D E exactly counteract those on G H, F G, E F ; and in precisely the 

 same way it may be shown that in any combination whatever of enlargements and 

 contractions, provided the sectional area and direction of the pipe at the two ends 

 are the same, the total endways effect impressed on the pipe by the fluid flowing 

 through it must be nil. 



In the experiment I am about to show you, the several propositions which I 

 have been elucidating will be seen to be verified step by step, if due allowance be 

 made for the effect of friction. 



A cistern (see Plate X. fig. 20), in which a definite head of water is maintained, 

 discharges itself through a continuous series of pipes, which, in their local changes 

 of diameter, exhibit the several characteristic features which have been under con- 

 sideration. 



From « to e at the outlet end, we have a contraction followed by an enlargement ; 

 from e to g the diameter is imiform ; from A to / we have an enlargement followed 

 by a contraction. At the various critical featm-es are fitted gauge-glasses such as 

 have been described, so that the level at which the water stands in each indicates 

 the pressure in the pipe at the point of attachment. 



The series of pipes is laid out on an inclination which represents the mean resist- 

 ance due to friction, or the " head " lost by friction, between the cistern and the 

 outlet — in other words, the hydraulic mean gradient. 



The mean diameter of the contracted part between a and c has been so deter- 

 mined by well-known hydraulic rules, that when it is compared with the adjoining 

 parallel pipe, the hydraulic gi-adient shall be the same in each. 



You observe that while the levels at which the water stands in the several gauge- 

 glasses, correspond from end to end with the gradient from the head in the cistern 

 to the head at the outlet, when examined in detail they verify thi'oughout the 

 propositions I have been establishing. 



For if for the moment we regard the gradient as vu'tually level, the depressions 

 of the several columns below it due to varying velocity of flow should be inversely 

 as the fom'th powers of the several diameters ; but the local frictional gradient 

 should be is inversely as the fifth power of the diameter, and thus steepest where 

 the diameter is smallest. And, broadly speaking, the results plainly conform to 

 these rules. As a quantitative verification I point out that by careful calculation 

 the mean diameter, and therefore the gradient from a' to c', is the same as that for 

 the parallel pipe from e' to y'; and the result agrees exactly with the calculation. 



In dealing with pipes of varying sectional area I have hitherto treated only of 

 the modifications caused in the forward motion of the particles of fluid ; for I have 

 limited the argument to cases where the alteration in sectional area of the pipe is 

 so gradual that, practically, the only alteration in the motion of the particles is 

 that in their forward velocity ; but I have previously shown that tortuosity in a pipe 

 of uniform diameter does not introduce endways pressure, provided the initial and 

 terminal directions are the same ; and it is easy to see that an elongated system of 

 such gradually tapered pipes as we have been considering may bo also tortuous 



