TIDAL OBSERVATIONS. 4.99 
to the M hours. In thus averaging for the M tides every height which 
was recorded at a time within half an M hour before or after 0" M time 
was taken as if it had been observed at 0" M time, and so for 1", 2", 3, 
&c. of the M time. The correction on this was applied afterwards, as will 
be described later (§ 23). Four other averagings were performed according 
to the same rule for the K, L, N, O reckonings respectively ; each Averaging 
giving a group of twenty-four means. 
18. The next step was to find for each of these six sets of averages the 
coefficients A,, A,, B,, A,, B,, &c. of the harmonic formule, 
A,+A,cos nt+B,sin nt 
+ A, cos 2nt+ B, sin 2nt 
+A, cos 8nt+ B, sin 8nt, 
n denoting, as in § 1, the rate of increase of the hour-angle for each case, 
for instance y for the K tide, y—o for the M tide, and so on. The condition 
to be fulfilled is that the values of this formula calculated for t=0,t=1.., 
t=23 may agree as nearly as possible, on the whole, with the twenty-four 
numbers of the group (the sum of the squares of the differences to be a 
minimum*). The tabular forms and rules given by Mr. Archibald Smith, 
and published by the Admiralty, to be used for the harmonic reduction of 
the deviation of ships’ compasses, have been adopted mutatis mutandis, and 
have proved very convenient. 
19. If, instead of including only seventeen coefficients, A,, AL, Bones 
A,, B,, the calculation had been extended to A,,, B,,, A... so as to include 
in all twenty-four coefficients, the calculated values would necessarily have 
agreed with the twenty-four numbers given by observation. But there was 
no apparent probability that anything more than accidental irregularities and 
errors of observation could be represented by higher terms than A,, B,, and 
_ therefore these were the highest included. The following Table exhibits the 
results of this process. The columns headed “differences” preserve the 
residues, however, and may be referred to should further study of the subject 
indicate that useful results are to be derived from them. ‘The greatest of 
them is 037 of a foot, and the maxima in each column are only from ,1, to 
gy of a foot. 
Values of A,, A,, &c., to first Approximation. 
Ss) K L M N O 
(y—n) (y) (y-4¢-40) (y—-2) (y-$et4w) (y—2c) 
A, +0°0231 —0°2052 —0°0305 +0'0223 +oo181 —0'2963 
B, —0'0255 —0'0236 —o'0120 —0'0058 +0'0048 +0°0687 
A, +1°5598 —0'4540 —0'2276 —4°3176 +o'8191 —0*'0yo4. 
B, +0°9923 —o'oo061 +0°2669 +4°5037 —0°7342 —0°0007 
A, +0°0086 +0'0037 —o0096 —o'0138 —o'0008 —0'0073 
B, +0'0004 +0°0015 +070093 +0'0408 tool! +0'0078 
A +0'0295 —O'0127 —0°0457 —0°5443 —0'0094 +0'0030 
B, -+o'0009 —0o'0021 — 00927 —o'0878 —0'O122 -+0°0034. 
A. 00000 —o'oosI —0'0023 +0°0032 —0'00I13 +0'0022 
B; +0'0029 +0'0072 +0°'004.6 -+o"001g9 -+0°0052 —0'0074 
A, +0°'0017 —0'0008 — 00050 —071132 —0'0287 —o0'0062 
iB; +0°0068 +0°0027 —0'0079 —O'1rI4 —0'0024 00040 
* According to Laplace’s method of “least squares.” 
