930 REPORTS ON THE STATE OF SCIENCE, ETC. 
specially as previously mentioned. The values of f, for the appropriate values of 
py are given in the following table :— 
S,: 7-000 L,: 6466 vp: 2674 
R,: 6-997 A, : 6294 N,: 2°349 
T,: 6997 M,: 4:720 bp: 0°393 
K,: 6983 28M: 4-720 2N: 0-131 
Multiplying by the above factors the corresponding terms in C on the 4th day of 
the month, and subtracting the constant due to the use of 1+ cos @ in the abac, 
the result should be equal to the sum of the values of C for the first seven days of 
the month; the differences between the two are usually less than 0-020 foot. 
The errors in the calculation of C and S as above rarely exceed 0:010 foot. 
The next step is to interpolate for values of C and 8 at intervals of six hours, 
and for this purpose interpolation formule are used; these are modifications of the 
ordinary central difference formula— 
Ug = Uy + HU, —Uy) + FX(H—1)(U,—U,—Uy+U-1)+ 1... 
which gives uw, in terms of the variates w_, Uy) %, %. It is supposed that wz lies 
between 0 and 1. If we apply this formula to the case e=% and uwz= cos (px +a) 
we get M cos (4p+a), where M=(18 cos 3p—2 cos 3p)/16. It is easy to show that M 
is always less than unity ; an improvement is to write 
Uy = Uy + Z(H — Ug) — B(Ug— Uy — Uy —%) 
whence M=(1+ 2k) cos 3p—2k cos 3p. We can now choose & to make M unity for 
a particular speed, or to make the possible error a minimum when several speeds 
are involved. In the present instance M,, the largest constituent, and N,, the con- 
stituent with the greatest value of p, need only be considered. Taking # succes- 
sively equal to ‘063, °064, :065, 066, we find the values of (M—1) multiplied by 
the appropriate amplitudes to be respectively —:003, —:001, ‘001, ‘003 for M,, and 
respectively —-005, —:005, —:004, —-003 for N,. Ignoring the signs, the value of 
k=-065 gives the least additive error, and therefore we adopt the interpolation 
formula— 
Cy= —'065C_,+°565 C,+°565C, —-065C,,. 
The errors when p is small are negligible whatever value we assign to % within the 
range ‘062 to ‘066. Except in rare instances the errors of interpolation will be less 
than ‘010 foot. 
A formula similarly derived could be used to get ©; from C_;, C,, Cy, and 
C,, but it is better to operate on the original series of values of C at intervals of 
twenty-four hours, as these have been carefully checked. We obtain 
C; = —-05C_, + °80C, + °30C, —-05C_, 
Cy = —-05C_1 + °30C, + 800, —-05C_, 
The maximum error is about ‘01 foot; this could be reduced a little by taking three 
decimals in the coefficients, but the simplicity of the formula outweighed any 
advantage to be otherwise attained. Obviously C; and C; can be calculated almost 
simultaneously by first calculating 
[—-05 C_1+ 300, +*300, —-05C,] and then adding 3C, or 3C,. 
It is helpful to write the primary values in coloured ink and to leave blak 
spaces for the values of C:, C,, and C;. As the interpolations are all separate 
calculations with no risk of systematic error, and as the smaller interval 
makes the successive differences to diminish more rapidly than for the primary 
series, all the interpolations can be checked by differencing the complete series 
- Cup Ox, Cary, 'C,,. . = 
” ‘The interpolations for C and $ for the intermediate hours are simply carried out 
by linear interpolation, and then we have the height of the semi-diurnal tide 
given by 
(= C, cos 30°¢—S§, sin 30°. 
There is no great necessity for checks to be applied at this stage. Systematic error 
is only possible by using the formula wrongly, and examination at intervals is 
sufficient to test this. 
43 
