228 REPORTS ON THE STATE OF SCIENCE, ETC. 
§ 12. Calculation of semi-diurnal tide.—The calculation of the semi-diurnal 
tide hour by hour is most expeditiously carried out in accordance with the scheme 
now to be explained.? 
Consider ¢,= SR, cos (o,t + ar), where ¢ is given in units of one mean solar hour; 
Y 
o, is the speed in degrees per mean solar hour, and is such that o,—30 is small. 
Then we may write 
¢,=C, cos 30°¢—S, sin 30°, 
where C,==R,. cos (o,—30¢+ a,j? 
Yr 
S,=3R, sin (o,—30t + a,)°= SR, cos (o,,—30¢ +a—90)°. 
r Yr 
Both C, and §, are slowly varying quantities because o,— 30 is small, and therefore 
interpolation can be used if C, and §, are calculated direct at convenient intervals of 
time. Supposing that we know C, and §, at intervals of twenty-four hours, then 
interpolation formule can be applied to give the values at intervals of six hours, 
and simple linear interpolation is usually sufficient to give the intermediate values 
at intervals of one hour; but even if this were not sufficient the principle of inter- 
polation can be used. By this method only two series of harmonic constituents 
have to be summed for intervals of twenty-four hours. 
We shall have occasion to use the speed denoted by 
Pr = 24 (oy = 30)° 
This is equal to the speed in degrees per mean solar day less 720°, and it is con- 
venient to speak of it as the ‘reduced speed’; we shall use T with p, to signify 
time measured in units of one mean solar day. 
The detailed procedure can now be considered. The methods of § 11 are applied 
with abacs constructed to give readings at intervals of p,; the abac for M, must be 
on a much more open scale than the others in order to read to three decimal places 
of a foot: a convenient scale is one-quarter inch to 0:1° and the abacs used are in 
four overlapping sections. Also the speed of K, is such that it is preferable to con- 
struct the abac for a speed 6p,—i.e. to read off at intervals of six days; intermediate 
values are obtained by increasing the appropriate value of & by p,, 2p,,.... 
When each cycle is completed it is desirable to verify that no omission has taken 
place, and this can readily be done by calculating independently the date (or day 
number) corresponding to the last reading of the cycle. Each cycle except the first 
adds either (»+1) or v readings according to whether 5 be less or greater than 7; 
these should be separately summed before any readings are taken from the abac, 
This check is very important indeed, for systematic error is fatal to success, and 
must be avoided. While we have now an assurance that each cycle ends on the 
correct date, the above check is not sufficient to ensure that a particular line has 
been drawn correctly. There is a check which can be applied on any day, but it is 
best to use it on the 15th, 30th and 31st days of each month, as these days are not 
covered by a check on tke sums, mentioned later. Each constituent is expressed as 
the sum of two others with amplitudes D, say: adding the two readings and 
subtracting 2D gives, say, R, cos (p,T + a,), a term of C; obtaining the corresponding 
term of S in the same way, the sum of the squares should be constant, and equal to 
R,”. This test should be made before the summations are commenced. 
After carrying out the summations for C and S a check is desirable to indicate 
casual errors of abac readings or of summations. The method of checking by 
successive differences is not very efficient in this instance, and use is made of the 
relation that the sum of (m+1) consecutive values of R, cos (p,T'+a,) is equal to 
the mid-value multiplied by a factor 
_ sin 3(m +1)pr 
sin dpr 
The test is best taken with m+1=7, whence we can test the values of C and Sina 
given month for the days 1-14, 16-29, inclusive; the remaining days are tested 
* The method as given here has been applied to the reduction of the second six 
months’ observations for Newlyn ; a slightly different method, not using ©, and §,, 
was used previously, ; aN 
