172 



NATURE 



[Dec. 30, 1875 



the bend divided at that point into two bends, and there joined 

 together by an infinitely short piece of straight pipe. 



If, then, the tortuous pipe I have above referred to h; s its 

 ends at A and B parallel to one another, as shown in Fig. 4, it is 

 clear that the tensional forces at its ends balance one another, 

 and the pipe, as a whole, does not tend to move endways. 



Note B. 



The law regulating these changes of pressure due to changes 

 of velocity can be best understood by considering the case of a 

 stream of perfect fluid flowing from a very gradually tapered pipe 

 or nozzle placed horizontally and connected with the bottom of a 

 cistern, as shown in Fig. 34. Let us suppose that nt the points 

 B and C the sectional areas of the pipe are severally twice and 

 four times that at the point of exit A. 



At the point of exit A the fluid is under no pressure whatever, 

 since there is no reacting force to maintain any pressure ; each 

 particle of fluid in the issuing jet is rushing forward on its own 

 account, neither giving nor receiving pressure from its neighbours. 

 We know, however, what force it has taken to give the velocity 

 which the fluid has at the point of issue A, and we measure this 

 force by the pressure or head of fluid, lost. In the case we are 

 considering, this head is represented by the height of the fluid in 

 the cistern, or by the height AD. 



Within the cistern, at the point E, on the same level as A, 

 the point of issue — at this point E within the cistern, we have 

 in effect the whole pressure due to the head of fluid equal to AD, 

 but we have no velocity, at any rate the velocity is so small as 

 to be inappreciable ; and at the point of issue A we have no pres- 

 sure at all, but we have what is termed the "velocity due to the 

 head." 



Let us suppose that at the points A, B, C, and E, gauge- 

 glasses or stand pipes are attached so that the fluid in each may 

 rise to a height corresponding with the pressure within the pipe 

 or nozzle at the point of attachment. 



The gauge-glass attached at A will show no pressure, thus 

 indicating that the entire head AD has been expended in pro- 

 ducing the velocity at the point A. 



At the point B, as the sectional area is twice, the velocity is 

 one-half that at A. Now the head required to produce velocity 

 varies as the square of the velocity to be produced ; in other 

 words, to produce half the velocity requires one quarter of the 

 head ; thus of the whole head AD available, one quarter only, 

 or GD, has been absorbed in developing the velocity at B, and 

 the remainder of the pressure, which will be represented by the 

 head BG, will be sensible at the point B, and will be exhibited 

 in the gauge-glass attached at that point. 



Again, as the pipe at C is four times the area that it is at A, 

 it follows that, of the whole head AD, one-sixteenth part only, 

 or HD, has been absorbed in developing the velocity at C, and 

 the remainder of the pressure, which will be represented by the 

 head CH, will be sensible at the point C, and will be exhibited 

 in the gauge-glass attached at that point. 



In the case I have chosen for illustration, the small end. A, of 

 the nozzle, is open and discharging freely, and the pressure at 

 that point is therefore m7. But the absolute diff"erences of pres- 

 sure at each point of the pipe or nozzle will be precisely the 

 same (as long as the same quantity of fluid is flowing through it 

 per second), however great be the absolute pressures throughout. 



Thus, suppose that from the end of the nozzle at A a pipe of 

 the same diameter, and of uniform diameter throughout its 

 length, is curved upwards, so that the end of it, I, is two feet 

 higher than A, as shown in Fig. 35, if the level of the cistern is 

 also raised two feet, namely to the level marked J, instead of D, 

 we shall have the same delivery of fluid as before ; and the 

 differences between the pressures at each point will be the same 

 as before. 



If we add 50 feet instead of 2 feet to the head in the cistern, 

 and raise I to 50 feet, instead of 2 feet above the nozzle, the 

 differences of head or pressure will still be the same, the head at 

 A being 50 feet, that at B being BG + 50 feet, that at C, 

 CH -I- 50 feet, and that at E (the cistern-level) ED + 50 feet. 



To put the case into actual figures, suppose the sectional area 

 at A to be I square inch, that at B 2 square inches, and that at 

 C 4 square inches, and suppose that the fluid is passing through 

 the nozzle at the rate of one-ninth of a cubic foot per second, 

 we shall have a velocity at A of 16 feet per second, to generate 

 which would require a difference of pressure between E and A, 

 equivalent to 4 feet of vertical head. The velocity at B will 

 be 8 feet per second, which would require a difference 

 between E and B equivalent to i foot of head. That at C 



will be 4 feet per second, and will require a difference of 

 pressure equivalent to 3 inches of head. If the pressure at A be 

 zero, the pressures at B, C, and E will be 3 feet, 3 feet 9 inches, 

 and 4 feet respectively. If the pressure at A be i foot, the 

 pressures at B, C, and E will be 4 feet, 4 feet 9 inches, and 5 feet 

 respectively ; and if the pressure at A be 1,000 feet, the pres- 

 sures at B, C, and E will be 1,003 feet, 1,003 feet 9 inches, and 

 1,004 feet respectively, always supposing the quantity of fluid 

 passing per second to be the same. If the quantity be different, 

 the absolute differences of pressure will be different, but will be 

 relatively the same. If, for instance, the quantity flowing per 

 second be doubled, the velocity at each point will be doubled, 

 and the differences of pressure quadrupled ; so that if the pres- 

 sure at A were again 1,000 feet, those at B, C, and E would be 

 1,012, 1,015, ^nd 1,016 feet respectively. 



To sum up— the differences of hydrostatic pressure at different 

 points vary as iAe differences of the squares of the velocities at those 

 points. 



Note C. 



Here again the argument given in the text suggests certain 

 other lines of argument which some persons may feel interested 

 in following out 



Suppose each and every one of the streams into which we 

 have subdivided the ocean, to be inclosed in an imaginary rigid 

 pipe made exactly to fit it, throughout, the skin of each pipe 

 having no thickness whatever. The innermost skin of the inner- 

 most layer of pipes (I mean that layer which is in contact with 

 the side of the body), the innermost skin, I say, of this layer is 



Eractically neither more nor less than the skin or surface of the 

 ody. The other parts of the skins of this layer, and all the 

 skins of all the other pipes, simply separate fluid from fluid, 

 which fluid, ex hypothesi, would be flowing exactly as it does 

 flow if the skins of the pipes were not there ; so that, in fact, if 

 the skins were perforated, the fluid would nowhere tend to flow 

 through the holes. Under these circumstances there clearly 

 cannot be any force brought to bear in any direction by the flow 

 of the fluid, on any of the skins of any of the pipes except the 

 innermost skin of the innermost layer. Now, remembering that 

 we are dealing with a perfect fluid which causes no surface- 

 friction, we know that the fluid flowing through this system of 

 pipes administers no total end\\ays force to it. But it produces, 

 as we have just seen, no force whatever upon any of the skins 

 which separate fluid from fluid ; consequently, if these are re- 

 moved altogether, the force administered to the remainder of the 

 system will be the same as is administered to the whole system, 

 namely, no total endways force whatever. But what is the 

 remainder of the system ? Simply the surface of the body, which 

 is formed, as I have already said, by the innermost skins of the 

 innermost layer of pipes. Therefore no total endways force is 

 administered to the surface of the body by the flow of the fluid. 



Lastly, let us recur for an instant to the case of fluid flowing 

 through the single flexible pipe. Here it was proved that the 

 flow of the fluid through it, if it was anchored at the two ends, 

 did not tend to displace any part of it, because the centrifugal 

 forces produced by the flow of the fluid, and which must act 

 exactly at right angles, or normally, as it is called, to the line of 

 pipe at each point, are exactly counterbalanced by a uniform 

 tension throughout the length of the pipe. If the flexible pipe 

 has variations in its diameter, the differences of quasi-hydrostatic 

 head appropriate to those variations are also normal to the sur- 

 faces of the pipe, being simply bursting-pressures. If, however, 

 these normal forces were directly covmterbalanced by equal and 

 opposite and normal external forces or supports, it is obvious 

 that this tension would be entirely relieved. Now, if we suppose 

 the system of pipes which we have several times already ima- 

 gined to surround the submerged body, to be flexible pipes 

 (instead of rigid pipes, as we have before imagined them), the 

 counterbalancing, or normal, external forces which exactly relieve 

 the tension are supplied to each pipe by its neighbour, except in 

 the case of the innermost skin of the innermost layer of pipes, 

 since this innermost skin has no neighbour. In this instance the 

 counterbalancing, normal, external forces are supplied by the 

 rigidity of the surface ot the body. Now we know that, since 

 the tensional forces produced by the flow of fluid through a 

 flexible pipe, whether of uniform or varying sectional area, have 

 no sum total of endways force, the counterbalancing forces which 

 exactly relieve this tension must also have no total endways 

 force ; and since the counterbalancing forces acting throughout 

 the whole system have thus no sum total of endways force, U 

 can be proved, as before in the case of the similar system of rigid 



