Hydraulic investigations. fjg 



means sufficient t» authorise the conclusions, which he has 

 drawn from them. lie directs the squares of the sines of His rule 

 the angles of flexure to* be collected into one sum, which, 

 being multiplied by a certain constant coefficient, and by 

 the square of the velocity $ is to show the height required for 

 overcoming the resistance. It is, however, easy to see, that fundamentally - 

 such a rule must be fundamentally erroneous, and its coin- 

 cidence with some experiments merely accidental, since the 

 results afforded by it must vary according to the method of 

 stating the problem, which is entirely arbitrary. Thus it 

 depended only on Mr. Dubuat to consider a pipe bent to an 

 angle of 144° as consisting of a single flexure, as composed 

 of two flexures of 72° each, or of a much greater number 

 of smaller flexures; although the result of the experiment 

 would only agree with the arbitrary division into two parts, 

 which he has adopted. This difficulty is attached to every 

 mode of computing the effect either from the squares of the 

 sines or from the sines themselves; and the only way of 

 avoiding it is to attend rilerely to the angle of flexure as 

 expressed in degrees. It is natural to suppose, that the ef- A different 

 feet of the curvature must increase, as the curvature itself tne0r7, 

 increases, and that the retardation must be inversely pro- 

 portional to the radius of curvature, or very nearly so ; and Sufficiently 

 this supposition is sufficiently confirmed by the experiments experiment/ 

 which Mi. Dubuat has employed in support of a theory so 

 different. It might be expected, that an equal curvature 

 would create a greater resistance in a larger pipe than in a 

 smaller, since the inequality in the motions of the different 

 parts of the fluid is greater ; but this circumstance does not 

 seem to have influenced the results of the experiments made ' 

 with pipes of an inch and of two inches diameter : there 

 must also be some deviation from the general law in cases of 

 very small pipes having a great curvature, but this devia- 

 tion cannot be determined without farther experiments. Of 

 the 2o which Dubuat has made, he has rejected 10 as irre- 

 gular, because they do not agree with his theory : indeed 4 

 of them, which were made with a much shorter pipe than 

 the rest, differ so manifestly from them, that they cannot be 

 reconciled : but 5 others agree sufficiently, as well as all the 

 rest, with the theory which I have here proposed, supposing 

 Vol. XXII.— Feb. 1809. I th« 



