FALL OF THE TIDE IN THE RIVER THAMES. 
7 
To see clearly the import of this, suppose that our intermediate formula is to be half 
the sum of these two expressions. Its form then becomes 
nu? . / mx'\ 
+ cos — • sin l m vt — — J 
— ~ b.mx ' ^cos2mx' .§m27iivt — (sin 2 mx' — tan(ma — mx')')cos2mvt j- 
sin 2 mx' — tan [ma — m x 1 ) 
Let 
= tan (2 mx' — D), 
cos 2 m x 1 
then the expression may be put in the form 
( 77ZcX/\ 
mvt ) — C sin (2 m v t — 2 m x' + D), 
or, if m v t 
mx J 
= p, the theoretical converted depression is 
-j- B sin p — C sin {2 p — mx' + D). 
When m a — m x' is not large (as it certainly cannot be at Deptford), m x' is greater 
than D, and the theoretical converted depression, putting A for m x 1 — D, is 
+ B . sin^p — C . sin (2 p — A). 
This is precisely the same form as that given by the discussion of the observations. 
4th. According to the theory, the coefficient of the second variable term in the 
expression for the converted depression ought to be proportional to the range of the 
tide (for it contains b as a factor). In the discussion of the observations it appears 
that this coefficient increases more rapidly than b. This seems to render it probable 
that terms of sensible magnitude would be introduced by pushing the approximation 
to the fourth and higher orders. 
5th. Suppose then that the variable part of the converted depression is represented 
by sin 6 — c sin (2 6 — A), where 6, as appears from the theory, is m v t — m x', or is the 
value of the phase depending simply on the depth of the canal and the distance of 
the point of observation from the sea, or is that value of phase which would corre- 
spond to a shallow wave passing along the canal. The values of 6 for high and low 
water will be those which satisfy the equation cos 6 — 2 c . cos (2 6 — A) = 0. Solving 
this equation for high water, by successive substitution, we have as a first approxima- 
tion cos 6 = 0, or 6 = 270° ; cos (2 6 — • A) = cos (540° — A) = — cos A ; using this 
substitution in the second term, the equation becomes cos 6 -j- 2 c cos A = 0 ; or if 
6 — 270° + x, sin x + 2 c cos A = 0, or x = — 2 c . cos A nearly, and therefore the 
value of m v t — m x' for high water is 270° — 2 c cos A nearly. In the same way it 
is found that the value of mv t — mx' for low water is 90° 2 c cos A nearly. So far, 
therefore, as depends on this term, the high water is accelerated and the low water 
is retarded by nearly equal terms : and this acceleration and retardation are propor- 
tional to c, or to b, or to the whole range of the tide : and are therefore greater for 
spring tides than for neap tides. 
