58 REPORT—1885. 
The synthesis of the two parts of the K, tide has been performed in 
the harmonic method (Report, 1883), and we have 
(K,)=f,K, cos (t+ h—1’—}2r—«,). 
Then, writing f,k,=—K,, we have 
(K,)=K, cos (T+a—v'—47—c,}) «. . . © (52) 
We have next to consider what corrections to the time and height of 
high and low water are necessary on account of these diurnal tides. 
If we have a function 
h=B+Heos2(T—¢) +H, cos(nT—/), 
where 7 is nearly equal to unity, and H, is small compared with H; its 
maxima and minima are determined by 
sin 2(T—y)= a" sin (nT = 8): 
If T=T, be the approximate time of maximum, and T,+0T, the true 
time, then, since the mean lunar day is 24°84: hours, and the quotient when 
this is divided by 87 is 0-988, we have in mean solar hours, 
éT = o-ogsam sin (nT, —/3) 
And the correction to the maximum is 
,¢H=H, cos (nT, —/) 
Again if T=T, be the approximate time of minimum, and T,+cT, 
the true time, then 
yn 
oT, =0°9 : 
oT, 88 Ti 
sin (nT, —/) 
og Bar 154) 
And the correction to the minimum is 
oH=Hy, cos (nT, —/) 
In the case of the correction due to (O)+(P), 7 is approximately 
1——*_, and for the correction due to K,, ” is approximately a 
y74. 
VO: 
$8. Direct Synthesis of the Harmonic Expression for the Tide. 
The scope of the preceding investigation is the establishment of the 
nature of the connection between the older treatment of tidal observa- 
tion and the harmonic method. It appears, however, that if the results 
of harmonic analysis are to be applied to the numerical computation of a 
tide-table, then a direct synthesis of the harmonic form may be preferable 
to a transformation to moon’s transit, declinations, and parallaxes. 
Senvidiurnal Tides. 
We shall now suppose that M, is the height of the M, tide, augmented 
or diminished by the factor for the particular year of observation, accord- 
ing to the longitude of the moon’s node, and similarly K, generically for 
the augmented or diminished height of any of the smaller tides. As 
