112 Major Allan Cunningham on 



water would be of such great scientific interest, and its ulti- 

 mate result — a good formula for Discharge — would be of such 

 great practical use in engineering, that it is well worth while 

 to thoroughly test this new theory : it will be shown that the 

 approximate agreement above noticed is quite illusory. 



The primary result of the investigation is, as stated, a for- 

 mula for the velocity v at any point at a distance r from the 

 centre of a pipe of radius R flowing full, the central velocity 

 (in the same cross section) being v w viz. 



Upon different hypotheses, and by a quite different investi- 

 gation, M. Darcy proposed* the formula 



>=, .{i-m(~y}, 



and applied the same (his own) experimental results in verifi- 

 cation. 



It is quite clear that these two expressions, differing so 

 greatly inform, cannot both be correct rational formula?; and 

 yet they both give numerical results agreeing sufficiently with 

 Darcy's experimental results to have satisfied their proposers. 

 But the fact is, that the numerical test relied on from Darcy's 

 experiments is (though this seems to have escaped attention) 

 a very poor one. Darcy's velocity-measurements were made 

 at only five points in a vertical line through the centre of each 

 pipe, viz. at the centre and at points symmetrically above and 

 below the centre at ^ and § of the radius from the centre ; 

 thus embracing only the middle § of the diameter in question, 

 within which the change of velocity is very small, and the 

 velocity-curve (or locus of the equation) is therefore very flat. 

 Thus almost any very flat curve would agree tolerably well 

 with the observations in the middle § of the diameter, especi- 

 ally when the comparison is made (as in the present instance) 

 between ordinates measured in so large a unit as a metre, as 

 the difference (in metres) would then only be small decimals. 

 The dissimilarity of the two curves is in this case very striking, 

 their convexities being actually turned opposite ways. Thus 

 Moseley's curve is concave downstream with a cusp at the 

 middle, whilst Darcy's is convex downstream with an apse at 

 the middle. On plotting Darcj^'s observations they will be 

 found to give curves generally very flat, convex downstream 

 with an apse at the middle; so that Moseley's velocity-formula 

 does not agree with natm*e. 



All the rest of the investigation depends on this primary 



* Reeherches exp6rimentales &c, by H. Darcy, p. 128. 



