TRANSACTIONS OP TUB SECTIONS. 5i39 



of pressure quadrupled ; so that if the pressure at A were ag-ain 1000 feet, those at 

 B, C, aud E would be 1012, 1015, and 1016 feet respectively. 



To sum up — the differences of hydrostatic pressure at diflcrent points vary as the 

 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 

 ocefin to be enclosed 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 

 innermost 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 practically neither more nor less 

 than the skin or surface of the body. 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 dealii:g vnih. a perfect fluid which causes no siu-face- 

 frictiou, we know that the fluid flowing through this system of pipes administers 

 no total endways force to it. But it produces, <as we have just seen, no force what- 

 ever upon any of the skins which separate fluid from fluid ; consequently, if these 

 are removed altogether, the force administered to the remainder of the system, 

 will be the same as is administered to the whole system — namely, no total endwaj's 

 force whatever. But what is the remainder of the system ? Simply the sm-face 

 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 waa 

 anchored at the two ends, did not tend to displace any part of it, because the centri- 

 fugal forces, produced by the flow of the fluid, and which must act exactly at right 

 angles, or normallj^, as it is called, to the line of pipe at each point, are exactly counter- 

 balanced 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 surfaces of the pipe, being simply bursting-pres- 

 sm-es. If, however, these normal forces were directly coimterbalanced 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 imagined to surround the submerged body, to be flexible pipes, 

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

 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 

 of the body. Now we Iniow that, since the teusional forces produced bv 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, 

 it can be proved, as before in the case of the similar system of rigid pipes, that if we 

 remove the whole of the skins or sides of pipes, which separate fluid from iluid and 

 which are all therefore necessarily in perfect equilibrium, the forces acting on the 

 remainder, namely on those skins which are in contact with the surface of the body, 

 forces which therefore mav be considered as acting simply upon the body, must also 

 have no endways sum total. 



18* 



