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HARMONIC ANALYSIS OF TIDAL OBSERVATIONS. 99 
In the reduction of the S-series of tides, the numbers treated are the 
actual heights of the water exactly at the S-hours, and therefore no aug- 
menting factor is requisite. 
We must now explain how the harmonic analysis, which the use of 
these factors presupposes, is carried ont. 
If ¢ denotes T-time expressed in hours, and ” is 15°, we express the 
height h, as given by the averaging process above explained, by the 
formula 
h=A,+A, cos nt+B, sin n¢+Ay, cos 2nt+B, sin 2ut+ . 
where#is0,1,2.... 23. 
Then if = denotes summation of the series of 24 terms found by attri- 
buting to ¢ its 24 values, it is obvions that 
| Ajo 2h; A, =p Th cos vt; By=ys2h sin nt ; 
A g=5rh cos 2nt; Bz =7,5h sin 2ut; &e., &e. 
Since n is 15° and ¢ is an integer, it follows that all the cosines and 
sines involved in these series are equal to one of the following: viz. 
0, + sin 15°, + sin 30°, + sin 45°, = sin 60°, + sin 75°, +1. It is 
found convenient to denote these sines, as 0, +8), +S, +83, =S,, +S;, 
+1. The multiplication of the 24 h’s by the various $’s, and the sub- 
sequent additions may be arranged in a very neat tabular form. 
We append the form for the reduction of the M-tides, filled in for 
Karachi 1880-81, but abridged by the omission of some of the decimals. 
The columns marked M are the multipliers appropriate for each series. 
The columns I. and II. contain the 24 hourly values to be submitted 
to analysis. The subsequent operations are sufficiently indicated by the head- 
ings to the columns, and it will be found on examination that the results 
are in reality the sums of the several series indicated above. We believe 
that this mode of arranging the harmonic analysis is due to Archibald 
Smith, who gives it in the Admiralty manual on the Compass. The 
arrangement seems to be very nearly the same as that adopted by Everett 
(Trans. Roy. Soc. Edinb. 1860) in his reductions of observations on under- 
ground temperature. 
In most cases it is not necessary to deduce more than the tide of the 
speed indicated by astronomical theory, but we give the full form by 
which the over-tides are deducible. If we want only a diurnal tide, ther 
the only columns necessary are I. to VII. and IX. and X.; if only a semi- 
diurnal tide, the columns to be retained are I., IT., [II., XII., XIII., XV., 
XVI., XVII. 
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