166 
THE REV. W. WHEWELL’S RESEARCHES ON THE TIDES. 
more generally by b cos (vt — (3), which, when (3 is a quadrant, becomes b sin r t. In 
this case the maximum of the diurnal wave coincides with the minimum of the semi- 
diurnal : and the compound wave is a cos 2 t t + b sin x t. 
In this case it is proposed to find the diurnal inequality. For high and low water 
we have, by the rule for maxima, 
— 2 a sin 2 t + b cos n t — 0, or b cos t t — 4 a sin or t . cos v t. 
Hence 
b 
cos or t = 0 , or sin or t = - — 
* 4 a 
13 5 
The former value gives for t the values -775 -775 — , &c., which belong to the time of 
low water, and show that the time of low water is not altered by the diurnal wave. 
The other value gives for t a certain fraction x, and also the numbers 1 — x, 2 + x, 
3 — x, &c. ; these belong to the time of high water, and show that the time of high 
water is affected by a diurnal inequality, of which the maximum, positive and nega- 
tive, is x. 
13 5 
The height of low water is obtained by substituting, for t, the values -775 -$-> Sec. 
which give the depressions , a — b, a -f- b, a — b, See., showing that the height of low 
water is affected by a semidiurnal inequality, of which the maximum is b positive and 
negative. The height of high water is obtained by substituting, for t, x, 1 — x, 2 -f- x , 
3 — x, all which give the values a cos 2 n x + b sin it x, or a + ; showing that 
O CL 
there is no diurnal inequality of height in the high water. 
If the maximum of the diurnal wave do not coincide either with the maximum or 
the minimum of the semidiurnal, the compound wave will be a cos 2 r t + b cos 
(or t — (3) ; and there will be inequalities both of time and of height, both for high 
and low water. And the inequalities at high and low water will bear certain propor- 
tions depending upon (3 and — ; and conversely, if the proportions of the high-water 
and low- water diurnal inequalities of time and height be known, the quantities (3 and 
may be found. 
But all this is applicable so long only as the inequalities of high and of low water 
coincide in their cycles, reaching their maxima and vanishing together ; and as we 
have seen that, instead of doing this, they reach their maxima and vanish alternately, 
we are compelled to give up (in this instance) this mode of representing the diurnal 
inequality by means of a diurnal wave. 
Under the present aspect of the subject, I do not conceive that we can pursue this 
examination further with advantage, till we possess a longer series of careful observa- 
tions of the Petropaulofsk tides. The singular laws which the tides there follow 
makes it a matter of interest to science to have these laws still further established 
