132 Hydraulic Investigations. 



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 merely to the angle of flexure as ex- 

 pressed in degrees. It is natural to suppose, that the effect 

 of the curvature must increase, as the curvature itself in- 

 creases, and that the retardation must be inversely propor- 

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

 this supposition is sufficiently confirmed by the experiments 

 which Mr. 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 

 rery small pipes having a great curvature, but this deviation 

 cannot be determined without further experiments. Of the 

 25 which Dubuat has made, he has rejected ten as irregular, 

 because they do not agree with his theory ; indeed four of 

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

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

 reconciled : but five others agree sufficiently, as well as all 

 the rest, with the theory which I have here proposed, sup- 

 posing the resistance to be as the angular flexure, and to in- 

 crease besides almost in the same proportion as the radius of 

 curvature diminishes, but more nearly as that power of the 

 radius of which the index is f, Thus if p be the number of 

 degrees subtended at the centre of flexure, and q the radius 

 of curvature of the axis of the pipe in French inches, we 



shall 



