TO ASSIST WORK ON THE TIDES. 931 
At each stage of the work the maximum error is less than ‘010 foot and the value 
of y, can be taken as correct to within ‘020 foot—the average error, regardless of 
sign, will be much less than this. 
§ 13. Calculation of diurnal tide.—The changes to be made in § 12 in order to 
adapt the work to the calculation of the diurnal tide are very few. We have 
¢,=C, cos 15°¢—§, sin 15°% 
where C,==Rr cos (or —15 t+4,)° 
rT 
S,=3R,. sin (or —15 t +.a,)° 
Tr 
The multipliers /* used in testing the summations are as follows :— 
8, : 7-000 M,: 6383 p,: 2°491 
Eeraoui J,: 6187 Q,: 2°167 
K,: 6997 O,: 4°553 2Q, :—°008 
00, : 4211 
The formulz of interpolation may be taken the same as for the semi-diurnal tide. 
(For Newlyn the diurnal tide is small, and it was unnecessary to use the 
multipliers f,: differences could be used on C,, C,, C,, ... .) 
§ 14. Analyses for quarter-diurnal tide.—It has previously been mentioned that 
the quarter-diurnal tide (¢,) at Newlyn has a simple relation to the square of the 
semi-diurnal tide ((,). ‘There are two methods for determining this relationship 
numerically, the first depending upon deduction from harmonic analyses for M, and 
M, over fairly long periods, and the second depending upon the direct correlation 
between (, and (,’. The former method assumes the existence of the relationship for 
all constituents, and, if it be known that this is legitimate, it is probably the best 
method in practice. The latter method is interesting and is of value when obser- 
vations over only a short interval are available; the results show whether the method 
is valid. Both methods have been applied to Newlyn observations, and the results 
are in satisfactory accordance. 
The phase shift is approximately the difference between the lag of M, and twice 
the lag of M,, and the reduction factor is approximately equal to the amplitude of 
M, divided by half the square of the amplitude of M,. The effect of the constituents 
L, and N, in contributing to the M, term in ¢? should be allowed for, but the 
corrections are small. 
The second method is as follows. Let the quarter-diurnal tide in the residue 
after removing the semi-diurnal tide be represented by 
¢,=2q, cos (¢,¢—h,) 
z 
Applying the §, least-square rule ® to twenty-four hourly values would give 
28 
a= 12, 2¢ , cos 60°% = rr cos (#, + &) + af "Gr COS (ky + E'y) 
b= 2 sin haath Gr Sin (hy + &,) eeu sin (2; + ’,) 
where f,, f’,, . and 2’, are dependent only on a, We may therefore write 
a= 2940 cos (2, +7,) 
b= 2HrGr sin (hk, + 1,) 
where g, and 7, are dependent only on o;,. 
* For technical terms and processes see Report for 1920. 
