THE REV. W. WHEWELL’S RESEARCHES ON THE TIDES. 
165 
may thus be five hours, or seven hours, or three hours. And this uncertainty deprives 
the establishment of all utility in such cases as the one before us, except we also take 
into account the diurnal inequality. 
If, after laying down the lunitidal intervals as ordinates, we draw a line cutting off 
the diurnal inequality, according to the method which we have previously employed, 
we find that the semimenstrual line gives intervals oscillating between the (mean) 
values of 3 h 39 m and 5 h 25 m . Hence the mean lunitidal interval or corrected esta- 
blishment (using terms explained in former memoirs) is 4 h 32 m : and the maximum 
amount of the semimenstrual inequality (positive and negative) is 53 m , and its total 
variation during a semilunation is 106 m ; which is larger than the variation at most 
places. Hence we see of what very wide changes the lunitidal interval at Petropau- 
lofsk is susceptible. If we take the maximum of the diurnal inequality (positive or 
negative) at two hours, the lunitidal interval may vary from l h 39 m to 7 h 25 m ; and 
thus the constancy of the establishment is quite obliterated. And even in this state- 
ment, we have not taken account of the effects of the parallax of the inoon, and the 
effect of her declination upon the diurnal mean. 
Diurnal Tide Wave . — As I have shown in former memoirs, we may represent the 
usual diurnal inequality at any place as the effect of a tide wave arriving at the shore 
once a day and superimposed upon the semidiurnal tide wave. We are naturally led 
to ask whether such a mode of representation is applicable to the tides now under 
consideration. The features which the diurnal inequality exhibits at Petropaulofsk are 
not, for the most part, inconsistent with such a representation. Thus, that the inequa- 
lity should affect high water and low water very differently, is easily explained. Nor 
is there any difficulty in accommodating the hypothesis of the diurnal wave to one of 
the most curious of the laws which we have discovered ; namely to this, that the 
inequality affects in the largest degree the time of high water and the height of low 
water. It is perhaps worth while to show this. In order to simplify the case as 
much as possible, I shall state and prove the following proposition. 
If the maximum of the diurnal wave coincides with the minimum of the semidiurnal 
wave, there will be a diurnal inequality of the time of high water, and of the height 
of low water, depending upon the proportion of the maxima of the two waves ; but 
there will be no diurnal inequality in the height of high water or in the time of low 
water. 
Let t be any time measured in (lunar) half-days from a given high water ; then the 
height (above mean water) of a semidiurnal tide may be represented (omitting inequal- 
ities) by a cos 2 cr t. This gives high water when t is 0, 1, 2, or any whole number, and 
l 3 
low water when t is — , or the half of any odd number. On the same supposition a 
diurnal tide might be represented by h cos * t, which is a maximum when t is 0, 2, 4, 
and a minimum when t is 1, 3, 5, &c. But since the maximum of the semidiurnal 
and of the diurnal tide do not necessarily coincide, the latter may be represented 
