TRANSACTIONS OF THE SECTIONS. 225 



Now in travelling- through the lower bend of tlie catenary, the chain passes from 

 being nearly straight, to being sharplj' curved and immediately straightened again ; 

 and this change of form involves a continued pivoting of link within link, the fric- 

 tion being called into action by the tension which presses the surfaces together. 

 Each link thus in succession resists this pivoting with a definite force, and the 

 resistance, in effect, converts what appears to be a perfectly flexible combination 

 into one possessing a tangible degree of stiffness ; and the oblique attitude assumed 

 by the chain as it approaches the bend, and the slight back turn whicli it assumes 

 as it emerges from the bend, are alike consequences of this factitious stiffiiess. 



For, in virtue of gi-avity, the rimning chain, like the chain at rest, tends always 

 to maintain the original catenaiy ; and in virtue of its speed of rotation, it seeks to 

 maintain (not preferentially the catenary, but) whatever form it for the moment 

 possesses. Hence its departure from the true catenary was, as you saw, gi-adual. 

 But when the figure of equilibrium is once attained, the persistency of form im- 

 parted by velocity, serves to maintain this figme as indifferently as any other. 

 Hence the figure is that Iti which equilibrium subsists between tlie force of gravity 

 seeking to restore the catenary, and the factitious stifihess resisting the necessity 

 of bending and unbending. 



The slowness with which the form is assumed, and its steady persistency when 

 once assumed, alike boar witness to the truth of the proposition which it is the 

 object of the experiment to verify. 



The stream of fluid in the tortuous flexible pipe would behave in a strictly 

 analogous manner. 



It might at first sight appear that I have now the materials for the proof of my 

 chief proposition, the assertion of the unresisted progress of a submerged body ; for 

 such a body might be assumed to be surrounded-by a system of imaginary pipes, as 

 shown in Plate IX. fig. 8; and each of these pipes being in equilibrium endways, that 

 is to say, the flow of fluid through it not tending in the aggregate to move it endways, 

 neither, it might be said, would the flow of fluid tend to move the submerged body 

 endways. But this reasoning would not be sound. The pipes we have hitherto been 

 considering have been of imiform sectional area throughout their length, an assump- 

 tion which has been necessary to the treatment pursued, as the velocity has in each 

 case been assumed to be uniform throughout the pipe. The section of the pipe may 

 have been square, circular, trapezoidal, or any other form ; but the area of the sectic n 

 has been assumed to be the same throughout the length of the pipe. 



But pipes of uniform sectional area do not truly represent the flow of a fluid past 

 a submerged body. I shall presently ask you to consider the fluid as floVing past 

 the body through a system of imaginary pipes; but to render the assumption ad- 

 missible, the sides of the imaginary pipes must not be so placed as to interfere with 

 the established course of the fluid, whatever that may be; in other words, if, for 

 the sake of illustrating the behaviour of the fluid, we assume that it is divided into 

 streams or filaments flowing through imaginary pipes, we must accept such a form 

 for those imaginary pipes that their sides exactly follow the path of the adjacent 

 particles of fluid. 



Now such a rule may, and probably will, requhe the imaginary pipes to be of 

 varying- sectional area throughout their length. Therefore, before we can apply 

 the analogy of the flow of fluid through pipes to the flow of a fluid past a sub- 

 merged body, it is necessary to consider the behaviour of fluid in pipes of varying 

 sectional area. 



It is, I think-, a very common but erroneous impression, that a fluid in a pipe 

 exercises, in the case of its meeting a contraction (see fig. 9), an excess of pressure 

 against the entire converging siu-face which it meets, and that, conversely, as it 

 enters an enlargement (see fig. 10) a relief of pressure is experienced by the entire 

 diverging siu-face of the pipe. Further it is commonly assumed that, when passing 

 through a contraction (see fig. 11), there is in the narrow neck an excess of pres- 

 sure due to the squeezing together of the fluid at that point. 



These impressions are in no respect con-ect ; the pressure at the smallest part of 

 the pipe is, in fact, less than that at any other point, and vice versd. 



If a fluid be flowing along a pipe which has a contraction in it (see fig. 12), 

 the forward velocity of the fluid at B must be greater than that at A, in the pro- 



