THEORY OF TIDES. 
of the surface, and the velocity with which that figure succes- 
sively changes or advances, are the only causes which fur- 
nish the immediate forces concerned in producing the elemen- 
tary motions constituting the oscillations. Now, while the 
imaginary wave C advances a little into the situation N, it is 
obvious that, in consequence of the action of the forces which 
it represents, the surface must be elevated at N, where the sur- 
face of AM is depressed, and depressed in the half of the un- 
dulation next to C, where AM is elevated, so as always to di- 
minish the magnitude of the original undulation, without affect- 
ing its velocity, as it would do if the curve C crossed its absciss 
in any other points than those which correspond to the greatest 
ordinates of AM. And the rate of diminution will be such, 
that if it continued uniform, the wave AM would be lowered 
at M during the time that it passes over one-fourth of its whole 
breadth AL, by a quantity which is to the greatest ordinate cf 
C, the representative of the resistance, as the semi-circumfe- 
rence of a circle to its diameter : but since the resistance would 
vary with the height of the wave, the actual diminution would 
be expressed by a logarithmic quantity. Thus, if the greatest 
resistance be to the greatest propelling force as rtol,the 
fluxion, or rather variation, of the height q will be to that of the 
absciss x as 9O 0 X qr to 9O 0 , and — q = rax, — . rx, and 
9 
H. L. q = C — rx, or calling the primitive value of q unity, 
q = e~ rx , and-L = e r \ 
9 
London , June 1811 . 
E. F. G. H. 
