TRANSACTIONS OF THE SECTIONS. 223 



flow thus obstructed would drive the pipe forward ; but if we endeavour to build 

 ■ up these supposed causes in detail we iind the reasoning to be illusory. 



I will now trace the results which can be established by correct reasoning. 



The surface being assumed to be smooth, the fluid, being a perfect fluid, can 

 exercise no di"ag by friction or otherwise on the side of the pipe in the direction 

 of its length, and in fact can exercise no force on the side of the pipe, except 

 at right angles to it. Now the fluid flowing round the curve from A to B will, no 

 doubt, have to be deflected from its course, and, by what is commonly known as 

 centrifugal action, will press against the outer side of the curve, and this with a 

 determinable force. The magnitude and direction of this force at each portion of 

 the curve of the pipe between A and B are represented by the small arrows marked 

 /; and the aggregate of these forces between A and B is represented by the larger 

 arrow marked G. In the same way the forces acting on the pai-ts B 0, C D, and 1) E 

 are indicated by the arrows H, I, and J ; and as the conditions imder which the fluid 

 passes along each of the successive parts of the pipe <ire precisely alike, it follows 

 that the four forces are exactly- equal, and, as shown by the arrows in the diagram, 

 they exactl}"^ neutralize one another in virtue of their respective directions ; and 

 therefore the whole pipe from A to E, considered as a rigid single structure, is 

 subject to no disturbing force by reason of the fluid running through it. 



Though this conclusion that the pipe is not pushed endways may appear on 

 reflection so obvious as to have scarcely needed elaborate proof, I hope that it has 

 not seemed needless, even though tedious, to follow somewhat in detail the forces 

 that act, and which are, under the assumed conditions, the only forces that act, on 

 a symmetrical pipe such as I have supposed. 



Having shown that in the case of this special symmetrically curved pipe the 

 flow of a perfect fluid through it does not tend to push it endways, I will now pro- 

 ceed to show that this is also the case whatever may be the outline of the pipe, 

 provided that its beginning and end are in the same straight line. 



Assume a pipe bent, and its ends joined so as to foriii a complete circular ring, and 

 the fluid within it running with velocity roimd the circle. This fluid, by centri- 

 fugal force, exercises a uniform outward pressure on every part of the uniform 

 cui've ; and this is the only force the fluid can exert. This pressure tends to tear 

 the ring asunder, and causes a uniform longitudinal tension on. each part of the 

 ring, in the same manner as the pressure within a cylindrical boiler makes a luii- 

 forni tension on the shell of the boiler. 



Now, in the case of fluid running round within lings of various diameter, just 

 as in the case of railway trains running round curves of various diameter, if the 

 velocity along the curve remain the same, the outward pressure on each part of the 

 circumference is less, in proportion as the diameter becomes greater ; but the cir- 

 cumferential tension of the pipe is in direct proportion to the pressure and to the 

 diameter ; and since the pressure has been shown to be inversely as the diameter, 

 the tension for a given velocity will be the same, whatever be the diameter. 



Thus, if we take a ring of doubled diameter, if the velocity is luichanged, the 

 outward pressure per lineal inch will be halved ; but this halved pressure, acting 

 with the doubled diameter, will give the same circumferential tension. 



Now this longitudinal tension is the same at every part of the ring ; and if we cut 

 out a piece of the ring, and supply the longitudinal tension at the ends of the piece, 

 by attaching two straight pipes to it tangentially (see Plate IX. fig. 2), and if we 

 maintain the flow of the fluid through it, the curved portion of the pipe will be under 

 just the same strains as when it formed part of the complete ring. It will be subject 

 merely to a longitudinal tension ; and if the pipe thus fonned be flexible, and fas- 

 tened at the ends, the flow of fluid through it will not tend to disturb it in any 

 waj'. Whatever be the diameter of the ring out of which the piece is assumed to be 

 cut, and whatever be the length of the segment cut out of it, we have seen that the 

 longitudinal tension will be the same if the fluid be moving at the same velocity j 

 so that, if we piece together any number of such bends of any lengtlis and any 

 curvatures to form a pipe of any shape, such pipe, if flexible and listened at the 

 ends (see fig. 3), will not be disturbed b}' the flow of fluid through it ; and the 

 equilibrium of each portion and of the whole of the combined pipe will be satisfied 

 by a uniform tension along it, 



17* 



