6 
MR. AIRY ON THE LAWS OF THE RISE AND 
elevation V of the surface, corresponding to the particle whose original horizontal 
k 
ordinate was x, is found by the equation V = I n order to find the eleva- 
* dx 
tion not of a given particle but at a given place, we must approximately express x, 
the original ordinate of the particle, in terms of x 1 , the ordinate of the place. After 
all these operations, putting m for the constant which makes m v t to change through 
360 ° in one complete tide, and putting It b for the coefficient of the first variable term, 
we find for the elevation of the water, 
f 33 
V = k < 1 — b sin ( mv t — m x') + b 2, . m x' . cos (mvt — m x') 
9 3 
-f- ^ b 3 . m 2 x ' 2 . sin (mvt — m x') + -j- b 2 . m x f . sin (2 m v t — 2 m x') 
32 
21 M 
— ^ b 6 . m x 
27 . 
•' . cos (3 mvt — 3 mx 1 ) — gg & 3 . m 2 x ' 2 • s i n (3 m v t — 3 m x') > 
where the approximation is carried to the third power of b. This expression supposes 
the canal unlimited at the end furthest from the sea. If the canal is stopped by a 
barrier, the expression changes its form. Putting a for the distance of the barrier 
from the sea, the elevation at the point x' is represented by 
= A- j 1 - 
c . cos (m a — m od) 
cos m a 
c 2 N 
sin in v t + 5 5 ( cos(2 ma — 2 mx') — cos 2 ma ) 
1 8 cos" 1 max'- ' / 
, 3 . c 2 .mod . sin (2 m a — 2 m x') ^ 
H 5 \ 1 cos 2 m 
1 8 cos^ m a 
vt | , 
or, putting k b for the coefficient of the first variable term, and omitting that term 
which does not vary with the time, 
V = k ^ 1 — b sin m v t + ~ b 2 . m x' . tan (m a — m x') . cos 2 mvt^ , 
where the approximation is carried to the second power of b. Neither of the sup- 
posed circumstances corresponds exactly to the case of a tidal river ; but it may with 
some reason be supposed to be represented by something intermediate to them ; the 
bridges and other impediments in the upper part producing in some degree the same 
effect as a barrier. The slope of the sides of the channel alters the magnitude of the 
coefficients, but does not appear to alter the general form of the expressions : the 
investigation, however, though not difficult, is so troublesome that I have not com- 
pleted it. Thus from theory we should expect the variable part of the converted de- 
pression to be expressed by a formula intermediate to the two following, the multi- 
plier k b being omitted : 
3 
+ sin (mvt — mx') — —b.mx'. sin (2 mvt 2 m x'), 
+ sin m v t — -rb .mx ' . tan (ma — mx ') . cos 2 m v t. 
