ON RECENT RESEARCHES IN HYDRODYNAMICS. 9 
_ The.theory of the two great solitary waves of Mr. Russell forms the sub- 
_ ject of a paper read by Mr. Earnshaw before the Cambridge Philosophical 
Society in December last*. Mr. Russell found by experiment that the hori- 
zontal motion of the fluid particles was sensibly the same throughout the 
whole of a vertical plane perpendicular to the length of the canal. He attri- 
buted .the observed degradation of the wave, and consequent diminution of 
_ the velocity of propagation, entirely to the imperfect fluidity of the fluid, and 
its adhesion to the sides and bottom of the canal. Mr. Earnshaw accordingly 
investigates the motion of the fluid on the hypotheses,—first, that the particles 
once in a vertical plane, perpendicular to the length of the canal, remain in 
a vertical plane; secondly, that the wave is propagated with a constant velo- 
city and without change of form. It is important to observe that these 
hypotheses are used not as a foundation for calculation, but as a means of 
ing a particular kind of motion for consideration. The equations of 
fluid motion admit of integration in this case in finite terms, without any 
approximation, and it turns out that the motion is possible, so far as the wave 
_ itself is concerned, and everything is determined in the result except two 
constants, which remain arbitrary. However, in order that the motion in 
question should actually take place, it is necessary that there should be an 
instantaneous generation or destruction of a finite velocity, and likewise an 
abrupt change of pressure, at the junction of the portion of fluid which con- 
stitutes the wave with the portions before and behind which are at rest, both 
which are evidently impossible. It follows of course that one at least of the 
two hypotheses must be in fault. Experiment showing that the first hypo- 
thesis is very nearly true, while the second (from whatever cause) is sensibly 
erroneous, the conclusion is that in all probability the degradation of the 
wave is not to be attributed wholly to friction, but that it is an essential cha- 
racteristic of the motion. Nevertheless the formula for the velocity of pro- 
pagation of the positive wave, at which Mr. Earnshaw has arrived, agrees very 
well with the experiments of Mr. Russell; the formula for the negative wave 
also agrees, but not closely. These two formule can be derived from each 
other only by introducing imaginary quantities. 
It is the opinion of Mr. Russell that the solitary wave is a phenomenon 
suit generis, in nowise deriving its character from the circumstances of the 
generation of the wave. His experiments seem to render this conclusion 
probable. Should it be correct, the analytical character of the solitary wave 
remains to be discovered. A-complete theory of this wave should give, not 
_ only its velocity of propagation, but also the law of its degradation, at least 
of that part of the degradation which is independent of friction, which is 
_ probably by far the greater part. With respect*to the importance of this 
_ peculiar wave however, it must be remarked that the term solitary wave, as 
| so defined, must not be extended to the tide wave, which is nothing more (as 
| far as regards the laws of its propagation) than a very long wave, of which 
the form may be arbitrary. It is hardly necessary to remark that the me- 
' chanical theories of the solitary wave and of the aérial sound wave are 
| altogether different. 
‘Ba Theory of River and Ocean Tides.—The treatise of Mr. Airy already referred 
| to is so extensive, and so full of original matter, that it will be impossible ~ 
ithin the limits of a report like the present to do more than endeavour to 
Lk ae 
_ pes re , — 
Bt pA ORE TAA PISS OS 
given as expressing the velocity of propagation of the phase of high water, which it is true is 
| not quite the same as the velocity of propagation of the crest of the wave; but the two velo- 
| cities are the same to the second order of approximation. 
-* Transactions of the Cambridge Philosophical Society, vol. viii. p. 326. 
i i arrived at, by the same reasoning, had the law not been restricted. This formula is 
