4 REPORT—1846. 
The researches of MM. Poisson and Cauchy were directed to the inves- 
tigation of the waves produced by disturbing causes acting arbitrarily on a 
small portion of the fluid, which is then left to itself. The mathematical 
treatment of such cases is extremely difficult; and after all, motions of this 
kind are not those which it is most interesting to investigate. Consequently 
it is the simpler cases of wave motion, and those which are more nearly con- 
nected with the phenomena which it is most desirable to explain, that have 
formed the principal subject of more recent investigations. It is true that 
there is one memoir by M. Ostrogradsky, read before the French Academy 
in 1826*, to which this character does not apply. In this memoir the author 
has determined the motion of the fluid contained in a cylindrical basin, sup- 
posing the fluid at first at rest, but its surface not horizontal. The interest 
of the memoir however depends almost exclusively on the mathematical 
processes employed; for the result is very complicated, und has not been 
discussed by the author. There is one circumstance mentioned by M. Planat 
which increases the importance of the memoir in a mathematical point of 
view, which is that Poisson met with an apparent impossibility in endea- 
vouring to solve the same problem. I do not know whether Poisson’s attempt 
was ever published. 
Theory of Long Waves.—When the length of the waves whose motion is 
considered is very great compared with the depth of the fluid, we may without 
sensible error neglect the difference between the horizontal motions of dif- 
ferent particles in the same vertical line, or in other words suppose the par- 
ticles once in a vertical line to remain in a vertical line: we may also neglect 
the vertical, compared with the horizontal effective force. These considera- 
tions extremely simplify the problem; and the theory of long waves is very 
important from its bearing on the theory of the tides. Lagrange’s solution 
of the problem in the case of a fluid of uniform depth is well known. It is 
true that Lagrange fell into error in extending his solution to cases to which 
it does not apply; but there is no question as to the correctness of his result 
when properly restricted, that is when applied to the case of long waves only. 
There are however many questions of interest connected with this theory 
which have not been considered by Lagrange. For instance, what will be 
the velocity of propagation in a uniform canal whose section is not rectan- 
gular? How will the form of the wave be altered if the depth of the fluid, 
or the dimensions of the canal, gradually alter? 
In a paper read before the Cambridge Philosophical Society in May 1837 t, 
the late Mr. Green has considered the motion of long waves in a rectangular 
canal whose depth and breadth alter very slowly, but in other respects quite 
arbitrarily. Mr. Green arrived at the following results :—If 8 be the breadth, 
and y the depth of the canal, then the height of the wave OC po y 4, the 
horizontal velocity of the particles ina given phase of their motion OC Bt y—* 
the length of the wave OC y®, and the velocity of propagation = gy. With 
respect to the height of the wave, Mr. Russell was led by his experiments to 
the same law of its variation as regards the breadth of the canal, and with 
respect. to the effect of the depth he observes that the height of the wave 
increases as the depth of the fluid decreases, but that the variation of the 
height of the wave is very slow compared with the variation of the depth of 
the canal. 
In another paper read before the Cambridge Philosophical Society in 
* Mémoires des Savans Etrangers, tom. iii. p. 23. 
+ Turin Memoirs for 1835, p. 253. 
} Transactions of the Cambridge Philosophical Society, vol. vi. p. 457. 
