THE LAWS OF FLUID RESISTANCE. 103 



The frictional gradient, according to well-known 

 hydraulic rules, has a definite law of variation in terms 

 of diameter and velocity, consequently it has been possible 

 by calculation to so arrange the diameters of the pipes 

 that the parallel pipe e g should, according to the rule, 

 have the same frictional gradient as the pipe a c, and as 

 we see that the gradients are in fact the same, the result 

 not merely illustrates but verifies the propositions. 



In the pipe h k I we have the smallest diameter at the 

 two ends h and I, and the largest diameter at the middle 

 point k, and consequently we have the smallest pressures 

 denoted by the water levels at h' and I', at the two ends, 

 and the greatest pressure in the middle denoted by the 

 water level at k', and we again have the fall or gradient from 

 end to end due to friction. 



These experiments afford a good verification of the pro- 

 position which I have just now explained, namely, that in 

 a frictionless fluid flowing through a pipe of varying 

 diameter, the pressure at each point depends on the sec- 

 tional area at that point, there being equal pressures at 

 the points of equal sectional area. Hence if in the pipe 

 shown in Fig. 13 the areas at all the points marked A are 

 equal, if also the areas at all the points marked B are 

 equal, and so also with those of C and D, then the pres- 

 sures 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 C 

 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 

 show that the variations in pressure due to the flow are 

 not such as can cause any total endways force on the pipe, 

 provided its sectional area at each end is the same. 



Take for instance the pipe shown in Fig. 14. The 

 conical portion of pipe A B presents the same area of 

 surface effective for endways pressure as does the conical 

 portion H I, only in opposite directions. They are both 

 subject to the same pressure, being that appropriate to 

 their effective mean diameter J. Consequently the end- 

 ways pressures on these portions are equal and opposite, 

 and neutralise one another. Precisely in the same way it 

 may be seen that the endways pressures on B C, C D, D E, 

 exactly counteract those on G H, F G, E F ; and it may be 



