Page:Proceedings of the Royal Society of London Vol 1.djvu/319

Rh taken from Dubuat and Gerstner, With some of his own, the results of which are compared with the velocities calculated according to the formulae of Dubnat and of the author.

The next section treats of the resistance occasioned by ﬂexure of the channel. In this case Dubuat directs the squares of the sines of the angles of ﬂexure to be added together and multiplied by the square of the velocity, and considers the quantity thus obtained pro- portional to the height necessary for overcoming the resistance. But since the magnitude of this quantity is evidently dependent on the number of parts into which the angle is arbitrarily divided, the au- thor prefers attending merely to the aggregate angle of ﬂexure as expressed in degrees to which the resistance is proportional, but vaa ries also inversely as the radii of curvature, or more nearly as that power of the radius which is expressed by iv. A table which follows shows the comparative correctness of the author’s formula with that of Dubuat.

Dr. Young next considers the propagation of impulses through tubes, the elasticity of which supplies the want of elasticity of the ﬂuid contained, and admits the same mode of reasoning that is em- ployed in the case of elastic ﬂuids or solids: for if the elastic force of the tube be as the increase of its circumference, a certain ﬁnite height may be assigned, which would cause inﬁnite extension, and which may be called the modular column. The velocity of an im- pulse at any point will be equal to half that which is due to the height of this point above the base of such a column, and hence the time of ascent of an impulse will be twice as great as that of a fall- ing body ; and if the pipe be inclined, the ascent of an impulse will bear the same relation to that of a body moving along an inclined plane.

The magnitude of diverging pulsations is next examined, and the conclusions of Euler, Lagrange, and Bernouilli, who have demon- strated that the velocity of each particle of an elastic ﬂuid is as its distance from the centre of impulse, are supported by a new method of considering the subject.

When a wave is reﬂected from two surfaces distinctly opposed to each other, they evidently sustain equal pressures ; and if to one of these surfaces two others be opposed converging at the acute angle, the wave will be elevated higher as it approaches the angle; and if the height be supposed in the inverse subduplicate ratio of the cor- responding subtense of the angle, the pressure will then be equal to that upon the single surface opposed: and hence is an additional reason for inferring, that in all transmissions of impulses the intensity is in the inverse subduplicate ratio of the :extent of parts collaterally affected, and this in conformity to the law of the ascending force; but in the case of intersecting waves, there is observed to be a paradoxical deviation, which is deserving of further consideration.

From considering the effect of bodies moving along an open canal, the author infers, that by means of a contraction moving progressively along an elastic pipe, the quantity of ﬂuid impelled Will be very