136 THIRD REPORT— 1883. 



hypothesis of parallel slices, which assumes the velocity to be 

 the same, and in the same direction, and the density to be the 

 same at all points of any section transverse to the axis of the 

 vessel or pipe. This hypothesis is one of those that the theory 

 of hydrodynamics has borrowed from experience to supply its 

 defects. Lagrange has, however, shown theoretically* that it 

 always furnishes a first approximation, the breadth of the ves- 

 sel being considered a quantity of the first order, and the effect 

 of the adhesion of the fluid to the sides of the vessel being neg- 

 lected. It is right to observe, that in the problems M. Navier 

 has considered, this hypothesis might have been in a great 

 measure dispensed with : the expression he has given, — more 

 correct than that commonly adopted for the velocity of issuing 

 through a small aperture by which airs of different densities 

 communicate, — ^might have been obtained by employing the equa- 

 tion above mentioned of steady motion, as, in fact, Mr. Moseley 

 has done f. This would be a preferable mode of treating such 

 questions, because in every instance in which these auxiliary 

 hypotheses are got rid of, something is gained on the side of 

 theory. This memoir contains another hypothesis, which can- 

 not be so readily dispensed with. Theory is at present quite 

 inadequate to determine the retardation in the flow of fluids 

 occasioned by sudden contractions or widenings in the bore of 

 the pipe. It is found by experiments with water, that the re- 

 tardation is sufficiently represented by taking account of the 

 loss of vis viva which, on the hypothesis of parallel slices, will 

 result from the sudden changes of velocity which must be sup- 

 posed to take place at the abrupt changes in the bore of the 

 pipe. M. Navier extends these considerations to elastic fluids. 

 The theory manifests a sufficient agreement with the experi- 

 ments it is compared with, and is valuable on account of the 

 applications it may receive. 



II. The most interesting class of problems in hydrodynamics 

 are perhaps those which relate to small oscillations. Newton 

 was the first to submit the vibrations of the air to mathema- 

 tical calculation. The propositions in the second book of the 

 Principia, devoted to this subject, and to the determination of 

 the velocity of sound, may be ranked among the highest pro- 

 ductions of his genius. He has assumed that the vibratory 

 motion of the particles follows the law of the motion of an oscil- 

 lating pendulum. It was soon discovered that many other 

 assumed laws of vibration would, by the same mode of reasoning, 



* Mecanique Analytique, P&tt II. § xi. art. 34. 

 t Elementary Treatise, p. 204. 



