TRANSACTIONS OF THE SECTIONS. 51 
agreed remarkably well with the experiments of Mr. Russell. His 
present more complete formule are such, that, for all cases in which 
the form of the canal is expressed by an equation between a power of 
y and a power of 2, they lead precisely to the results which he pre- 
viously obtained. One remarkable circumstance he pointed out, 
which is this; that if the form of the wave can be expressed by a 
single function, the velocity is the same in the middle of the canal as 
at the edges. Perhaps this apparently anomalous conclusion may arise 
from the circumstance that we assume the same form of wave to per- 
tain to all parts of the canal. Another result of his analysis is, that 
the height of the wave increases very rapidly as we proceed towards 
the edge of the vessel, to the detriment, as it would appear, of the 
height of the centre. i 
He next endeavours to obtain the motion of a wave in a canal, the 
depth of which is continually but slowly varying in the direction of 
the length. He here employs the method of the variation of parame- 
ters, and obtains the following results :— 
1. That the length of the wave is in direct proportion to the depth. 
2. That the velocity of transmission at any point varies as the square 
root of the depth. 
3. The elevation of the crest of the wave varies reciprocally as the 
total depth of the fluid. 
Lastly, he discusses the problem which Messrs. Poisson and Cauchy 
had solyed for the particular case, where the depth is very small or 
very great. It is well known that these philosophers undertook the 
solution of this problem in competition for the prize offered by the 
French Institute. As neither of these memoirs is printed exactly as it 
was originally delivered in, it would be hard for us to draw any con- 
clusions from the judgement pronounced by the judges. We know 
that M. Fourier found fault with Poisson’s solution on the ground that 
the function was limited in its value. The prize was accordingly 
adjudged to Cauchy. The plan which Poisson adopts, it is very easy . 
to understand ; viz. he finds a solution of the general equation of wave 
motion, and arranges it so as to make it coincide with a formula given 
by Fourier, which expresses the relation between a function and a par- 
ticular value of the function. 
M. Cauchy, on the other hand, demonstrates a formula slightly dif- 
fering from Fourier’s, and by means of it, he tgo expresses the general 
function in terms of the particular. Now it happens that M. Cauchy’s 
formula does not render necessary any limitation as to the depth of 
the fluid. Indeed, M. Cauchy himself discovered this, and published it 
in a note, but he never, to the author’s knowledge, made any further 
use of it. Some of the results obtained by both philosophers, are found 
by the author to be true without all the limitations which they have 
imposed. But there is one point of the utmost importance, viz. the 
determination of the length of the wave, for which their results are 
not satisfactory. Those who are acquainted with the analysis employed 
by the philosophers to whom he has referred, will remember that the 
integrals to be obtained may, by arranging in different forms, be made 
E2 
