300 Astronomical and Nautical Collections. 
has solved by a very intricate analysis, is capable of being exhi- 
bited in a much less embarrassed, and, we apprehend, even in a 
more accurate manner, by a mode of investigation, which is equally 
applicable to the tides of narrow seas and of lakes, and which may 
easily be made to afford a correct determination of the effects of 
“resistance, as well as a ready mode of discovering the laws of mo- 
tions governed by periodical forces of any kind; at least so far as 
these forces are capable of being represented by any combina- 
tions of the sines of arcs, which increase uniformly with the time. 
“ The essential character of this method consists in comparing 
the body actuated by the,given force to a pendulum, of which the 
point of suspension is caused to vibrate regularly to a certain small 
extent: the length of the pendulum being supposed to be such as 
to afford vibrations of equal frequency with the spontaneous vibra- 
tions of the moveable body, and the point of suspension to be car- 
ried by a rod of such a length, as to. afford vibrations of equal fre- 
quency with the periodical alternations of the force. It is then 
shown, that such a pendulum may perform regular vibrations, con- 
temporary with the alternations of the periodical force, and in- 
versely proportional in their extent to the difference between the 
length of the two rods: and that, whatever may have been the 
initial state of the pendulum, the motion thus determined may be 
considered as affording a mean place, about which it will at first 
perform simple and regular oscillations; but that a very small re- 
sistance will ultimately cause these to disappear:” so that the par- 
ticular solution of the problem, which indicates a series of vibra 
tions as they may be performed, is thus rendered general ; since 
every other initial state of the vibrations must ultimately terminate 
in this series. 
“« Now the sea, or any of its portions, may be considered as bodies 
susceptible of spontaneous vibrations, precisely similar to the small 
vibrations of a pendulum ; and the semidiurnal variation of the form 
which would afford an equilibrium, in consequence of the solar 
and lunar attractions, is perfectly analogous to the regular vibration 
attributed to the point of suspension of the pendulum. ‘The fre- 
quency of the simple oscillations of the sea, or of any of its parts, 
