142 THIRD REPORT — 1833. 



by combining analysis with a delicate set of experiments, re- 

 sults are obtained which are a valuable addition to this part of 

 the theory of fluid motion. His experiments were made on a 

 tube open at both ends, and the column of air within it was put 

 in motion by the vibrations of a plate of glass applied close to 

 one end. The following are the principal results. Tlie nodes 

 are not points of quiescence, but of minimum vibration ; — the 

 extremity of the tube most remote from the disturbance is not 

 a place of maximum vibration, but the whole system of places 

 of maximum and minimum vibration is shifted in a very sensible 

 degree towards it ; — the distances of the places of maximum 

 and minimum vibration from each other, and from that extre- 

 mity, remain the same for the same disturbance, whatever be the 

 length of the tube. This last fact Mr. Hopkins proves by his 

 analysis must obtain. The shifting of the places of maximum 

 and minimum vibration is not accounted for by the theory : nor 

 is it probable that it can be, unless the consideration of the 

 mode of action of the vibrations on the external air be entered 

 upon, — an important inquiry, but, as I said before, one of great 

 difficulty. I think also that the effect of the vibrations of the 

 tube itself on the contained air ought to be taken into account. 



IV. The problem of waves at the surface of water is princi- 

 pally interesting as furnishing an exercise of analysis. The 

 general differential equations of fluid motion assume a very sim- 

 ple form for the case of oscillations of small velocity and extent, 

 and seem to offer a favourable opportunity for the application 

 of analytical reasoning. Yet mathematicians have not succeed- 

 ed in giving a solution of the problem in any degree satisfactory, 

 which does not involve calculations of a complex nature. We 

 need not stay to inquire in what way Newton found the velo- 

 city of the propagation of waves to vary as the square root of 

 their breadths : he was himself aware of the imperfection of his 

 theory. The question cannot be well entered upon without 

 partial differential equations. Laplace was the first to apply to 

 it a regular analysis. His essay is inserted at the end of a 

 memoir on the oscillations of the sea and the atmosphere, in 

 the volume of the Paris Academy of Sciences for the year 1776. 

 The differential equations of the motion are there formed on 

 the supposition that the velocities and oscillations are always so 

 small that their products, and the powers superior to the first, 

 may be neglected. The problem without this limitation be- 

 comes so complicated that no one has dared to attempt it. La- 

 place's reasoning conducts to a linear partial differential equa- 

 tion of the second order, consisting of two terms, which is 

 readily integrated; but on account of the difficulty of obtaining a 



