HYDROMECHANICS 437 horizontal one, and hence he deduced the isochronism of these oscillations. From this Newton concluded that the velocity of waves formed on the surface of water, either by the wind or by a body thrown into it, was in the sub- duplicate ratio of their size. When their velocity, there fore, is measured, which can be easily done, the size of the waves will be determined by means of a pendulum which oscillates in the time that a wave takes to rise and fall, rnoulli. Such was the state of hydrodynamics in 1738, when Daniel Bernoulli published his Hydrodynamica, sive de Viribus et Motibus Fluidorum Commentarii. The germ of Daniel Bernoulli s theory was first published in his memoir entitled Theoria Nova de Mot-u Aquannn per Canales quocunque fluentes, which he had communicated to the Academy of St Petersburg as early as 1726. His theory of the motion of fluids was founded on two supposi tions, which appeared to him conformable to experience. He supposed that the surface of a fluid, contained in a ves sel which is emptying itself by an orifice, remains always horizontal ; and, if the fluid mass is conceived to be divided into an infinite number of horizontal strata of the same bulk, that these strata remain contiguous to each other, and that all their points descend vertically, with velocities inversely proportional to their breadth, or to the horizontal sections of the reservoir. In order to determine the motion of each stratum, he employed the principle of the conservatio virium vivarum, and obtained very elegant solutions. In the opinion of the Abbe" Bossut, his work was one of the finest productions of mathematical genius. The uncer tainty of the principle employed by Daniel Bernoulli, which has never been demonstrated in a general manner, deprived his results of that confidence which they would otherwise h&ve deserved, and rendered it desirable to have a theory more certain, and depending solely on the funda mental laws of mechanics. Maclaurin and John Bernoulli, who were of this opinion, resolved the problem by more direct methods, the one in his Fluxions, published in 1742, and the other in his Ilydraulica mine primum detecta, et demonstrata dirccte ex fundamentis pure mechanicis, which forms the fourth volume of his works. The method employed by Maclaurin haa been thought not sufficiently rigorous ; and that of John Bernoulli is, in the opinion of Lagrange, defective in perspicuity and precision. Alem- The theory of Daniel Bernoulli was opposed also by vt - the celebrated D Alembert. "Yhen generalizing James Bernoulli s theory of pendulums he discovered a prin ciple of dynamics so simple and general that it reduced the laws of the motions of bodies to that of their equili brium. He applied this principle to the motion of fluids, and gave a specimen of its application at the end of his Dynamics in 1743. It was more fully developed in his Traile des Fluides, which was published in 1744, where he has resolved, in the most simple and elegant manner, all the problems which relate to the equilibrium and motion of fluids. He makes use of the very same suppositions as Daniel Bernoulli, though his calculus is established in a very different manner. He considers, at every instant, the actual motion of a stratum as composed of a motion which it had in the preceding instant and of a motion which it has lost. The laws of equilibrium between the motions lost furnish him with equations which represent the motion of the fluid, Although the science of hydrodynamics had then made considerable progress, yet it was chiefly founded on hypothesis. It remained a desideratum to express by equations the motion of a particle of the fluid in any assigned direction. These equations were found by D Alembert from two principles, that a rectangular canal, taken in a mass of fluid in equilibrium, is itself in equilibrium ; and that a portion of the fluid, in passing from one place to another, preserves the same volume when the fluid is incompressible, or dilates itself according to a given law when the fluid is elastic. His very ingenious method was published in 1752, in his Essai sur la Resistance des Fluides. It was brought to perfection in his Opuscules Mathematiques, and was adopted by Euler. Before the time of D Alembert, it was the great object of philosophers to submit the motion of fluids to general formulae, independent of all hypothesis. Their attempts, however, were altogether fruitless ; for the method of fluxions, which produced such important changes in the physical sciences, was but a feeble auxiliary in the science of hydraulics. For the resolution of the questions concern ing the motion of fluids, we are indebted to the method of partial differences, a new calculus, with which Euler enriched the sciences. This great discovery was first applied to the motion of water by D Alembert, and enabled both him and Euler to represent the theory of fluids in formulae restricted by no particular hypothesis. The most successful labourer in the science of hydro- Dubuat dynamics was the Chevalier Dubuat, engineer in ordinary to the king of France. Following in the steps of the Abb6 Bossut (Nouvelles experiences sur la resistance des fluides, 1777), he prosecuted the inquiries of that philosopher with uncommon ingenuity ; and in the year 1786 he published, in two volumes, his Principes d Hydraidique, which contains a satisfactory theory of the motion of fluids, founded solely upon experiments. Dubuat considered that if water were a perfect fluid, and the channels in which it flowed infinitely smooth, its motion would be continually accelerated, like that of bodies descending in an inclined plane. But as the motion of rivers is not continually accelerated, and soon arrives at a state of uniformity, it is evident that the viscosity of the water, and the friction of the channel in which it descends, must equal the accelerating force. Dubuat therefore assumes it as a proposition of fundamental importance that, when water flows in any channel or bed, the accelerating force which obliges it to move is equal to the sum of all the resistances which it meets with, whether they arise from its own viscosity or from the friction of its bed. This principle was employed by Dubuat in the first edition of his work, which appeared in 1779, but the theory contained in that edition was founded on the experiments of others. He soon saw, however, that a theory so new, and leading to results so different from the ordinary theory, should be founded on new experiments more direct than the former, and he was employed in the performance of these from 1780 to 1783. The experiments of Bossut having been made only on pipes of a moderate declivity, Dubuat found it necessary to supply this defect. He used declivities of every kind, and made his experiments upon channels from a line and a half in diameter to seven or eight square toises. The theory of running water was greatly advanced by Prouy. the researches of Prony. From a collection of the best experiments by Couplet, Bossut, and Dubuat he selected eighty-two (fifty-one on the velocity of water in conduit pipes, and thirty-one on its velocity in open canals) ; and, discussing these on physical and mechanical principles, he succeeded in drawing up general formula?, which afford a simple expression for the velocity of running water. Eytelwein of Berlin published, in 1801, a valuable com- Eytel- pendium of Hydraulics, entitled Handbuch der ][echanik we i n - und der Ilydraidik, which contains an account of many new and valuable experiments made by himself. He in vestigates the subject of the discharge of water by com pound pipes, the motions of jets, and their impulses against, plane and oblique surfaces; and he shows theo retically that a water wheel will have its effect a maxi mum when its circumference moves with half the velocity
of the stream.
