174 Dr. Young's Hydraulic Investigations. 
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 expressed in degrees. It is 
natural to suppose that the effect of the curvature must increase, 
as the curvature itself increases, and that the retardation 
must be inversely proportional to the radius of curvature, 
or very nearly so ; and this supposition is sufficiently con- 
firmed, by the experiments, which Mr. Dubuat has em- 
ployed 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 in- 
equality in the motions of the different parts of the fluid is 
greater; but this circumstance does not seem to have in- 
fluenced 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 deviation cannot be 
determined without further experiments. Of the 25 which 
Dubuat has made, he has rejected 10 as irregular, 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 the resistance to be as 
the angular flexure, and to increase 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 
Thus if p be the number of degrees subtended at the centre of 
