50 REPORT—1840. 
He has also measured the velocity of the ebbing and flowing of the 
tidal current in various parts of the River Clyde, from Glasgow Har- 
bour to Port Glasgow. 
It appears from these observations that the tidal wave runs from Port 
Glasgow to Bowling at a rate or velocity of 14°56 miles per hour; from 
Bowling Bay to Clyde Bank at a rate of only 6°82 miles per hour; but 
from Clyde Bank to Glasgow Harbour at a rate of 10°85 miles per hour. 
The diminished velocity between Bowling Bay and Clyde Bank arises 
from the channel of the river being more crooked in that part than in 
any other portion of the River Clyde, thereby showing the great neces- 
sity of straightening and improving it. 
On the Theory of Waves. By the Rev. Professor KELLAND. 
The objects of the present communication are twofold : first, to pre- 
sent the subject of the Theory of Waves in its present form, and then 
to point out the difficulty which appears likely to impede its progress. 
1. The object of the theory is to account, on mechanical considera- 
tions, for such phenomena as are presented by the destruction of 
equilibrium of a fluid, and to obtain and to interpret mathema- 
tical expressions which give the velocity, form, impulses, and other 
circumstances of the motion. Our problem, then, divides itself into 
two parts; 1, the determination of the conditions which are pre- 
sented by the nature of the fluid on the hypothesis that it is in motion ; 
2, the investigation of the effects which will immediately follow the 
disturbance of equilibrium. The first of these problems alone pre- 
sents no considerable difficulty ; at least the difficulties are not such 
as to have deterred many writers from engaging to solve it. Laplace 
led the way, and was followed by Lagrange, Poisson, and others. As 
far as the author knows, however, all the writers on the subject con- 
fine themselves to two cases, viz. when the depth of the fluid is either 
very great or very small, in comparison with the length of a wave. 
Aided by their discoveries, he attempted, in a memoir read before the 
Royal Society of Edinburgh last year, and published in vol. xiv. of 
their Transactions, to supply the deficiency, and to obtain the equa- 
tions of motion of a regular set of waves, without imposing any restric- 
tion on the conditions. This was effected by assuming that a recipro- 
cating function can always be expanded in a series of sines and cosines 
of multiples of the space through which the whole series of values 
extends. This assumption, which is frequently made by Fourier and 
others, leads directly to the complete solution of the problem. The 
author has since applied himself to the completion of this part of the 
subject by solving the problem in cases in which the depth is variable, 
which had only been partially executed before. In the first place, he 
considers that case in which the depth is variable in the direction of the 
breadth, but uniform in that of the length. In his former memoir he 
contented himself with an approximate solution, founded on a particu- 
lar hypothesis. The results, approximate as they confessedly were, 
