218 REPORTS ON THE STATE OF SCIENCE, ETC. 
that no observations were better than those taken by the Ordnance Survey, 
and the situation of one of their stations (Newlyn, on the coast of Cornwall) 
was extremely favourable for the investigations because of its situation with 
respect to the Atlantic Ocean. The Survey kindly placed at the disposal of 
the Committee the records they had taken, and the accuracy of these was greatly 
appreciated. 
The powers of the predicting machines in summing the harmonic terms 
required were duly considered, but the evidence given in the Report for 1920 
was sufficient for the machines to be distrusted for this work. Tests of two 
predicting machines have been carried out, and the results show that even 
with very careful reading the errors are too great for their use in calculating 
hourly heights; the labour of reading the curves is also very great. The results 
of the tests are discussed in Part III. 
§ 3. The investigations were made possible by the invention of a scheme for 
the numerical calculation and summation of the harmonic constituents; this 
scheme very greatly reduced the labour of calculation, and the results of summa- 
tion of one set of constituents could be relied on to within about 0°01 foot. An 
account of this scheme is given in Part II., §§ 11-13. 
The first procedure was to remove the chief semi-diurnal constituents, or a 
first approximation to them. Examination of the analyses at neighbouring 
ports indicated the constituents M,, 8,, N,, K,, and L, as most worthy ot 
consideration. These were evaluated by certain inference methods (Part I1., 
§ 10) to a fairly good degree of approximation except in the case of M,, which 
was modified after the residue for one month had been obtained. Application 
of the scheme for the calculation and summation of harmonic constituents to 
these five constituents, and subsequent subtraction of the partial tide, resulted 
in a residue which was mainly quarter diurnal. The residue for the month of 
January, 1918, is given in fig. 1. There is obviously some semi-diurnal residue, 
as was to be expected, since only a first approximation to the semi-diurnal tide 
was removed. ‘here is also some diurnal tide, but this is not so prominent 
as the quarter-diurnal tide, and attention was first paid to the latter. 
§ 4. Now the quarter-diurnal tide does not correspond directly to the 
generating potential, as the quarter-diurnal constituents of the potential are 
very small. It is well known, however, that as a wave progresses in shallow 
water it changes shape and the front slope becomes steeper than the rear 
slope. If the departure from sinuity be not. too great, this change in form 
can be expressed by the addition of waves whose speeds are multiples of the 
speed of the primary wave. Theoretical considerations of a wave in a shallow 
canal have suggested that the quarter-diurnal constituents must have speeds 
which are either twice those of the primary semi-diurnal constituents or are 
equal to the sums of pairs of those speeds. The constituents usually analysed 
for are M,, MS,, and 8,, with speeds respectively equal to twice the speed of 
M,, the sum of the speeds of §, and M,, and twice the speed of §,. Now, on 
carrying out analyses by the usual methods for these constituents, and on 
subtraction of the partial tide compounded of them, it was found that they 
were quite inadequate to account for the whole, or a reasonable part of the 
whole, of the quarter-diurnal tide. Considerable attention was therefore 
directed to the matter, as it was known that these shallow-water effects were 
undoubtedly responsible for a large part of the errors in predictions. More 
constituents, as suggested by the theory mentioned above, were analysed for 
and calculated, but the rate of elimination was rather slow. It was ultimately 
found that there existed a very simple relation between the existing quarter- 
diurnal tide and the square of the semi-diurnal tide; a reduction factor and 
change of phase, applied to the quarter-diurnal portion of the square of the 
semi-diurnal tide, were sufficient to account so well for the quarter-diurnal 
tide that only a trace of it was left; this was possibly due to the incomplete 
(or approximate) semi-diurnal tide taken. This method was developed, very 
simple numerical formule were applied, and the quarter-diurnal constituents 
removed en bloc. 
1 For a résumé and criticism of these methods reference should be made to 
the Report for 1920 by Professor Proudman. 
