220 REPORTS ON THE STATE OF SCIENCE, ETC. 
§ 5. Tests have been made for Liverpool and Hilbre Island, and similar 
methods can be applied successfully to the sixth-diurnal tide as well as to the 
quarter-diurnal tide. 
(The methods of analyses for, and removal of, the quarter-diurnal tide are 
discussed in Part II., §§ 14-15.) 
§ 6. After removing the quarter-diurnal tide the residue is mainly diurnal 
and semi-diurnal, as may be seen from fig. 2. 
Inference methods for the chief diurnal constituents not proving very 
successful, resort was made to direct analyses of the residues, both for diurnal 
and residual semi-diurnal constituents. A summary of the usual methods of 
analysis, as applied to observations, is given in Part II., §§ 16-17, and modifica- 
tions of these for use with residues are afterwards discussed in § 18. The 
Report by Professor Proudman (1920) shows that the chief errors in the analysis 
of observations are due to the presence of large constituents, so that one 
‘constituent is imperfectly isolated. ‘The chief constituents having been prac- 
tically removed, modifications in analysis were possible. The usual method 
is to analyse only for expected constituents, and the resulting numbers are 
taken on trust; the methods given in § 18 do not assume the absence of per- 
turbing or of unknown constituents, but were designed so as to give unmis- 
takable indications if such be present. Further, they give valuable internal 
evidence as to the amount of trust to be associated with any derived ‘ harmonic 
numbers.’ This cautious procedure had its reward, as the results indicated 
that the residue contained constituents not included in Darwin’s schedules. 
For a full discussion of these matters reference is necessary to §§ 17-18 and 
figs. 7-9. It was not found possible, using only six months’ residues, to deduce 
the unknown constituents from these analyses, nor would it have been advisable 
to draw definite conclusions at this stage. Our sole index of reality lay in 
the residues. If, after taking out all the ‘Darwinian constituents,’ there were 
unmistakable signs of semi-diurnal constituents still remaining, then this would 
be regarded as sufficient proof of reality. On these coasts the diurnal tide is 
small, and no new diurnal constituents were suspected or found. 
The removal of the diurnal and semi-diurnal Darwinian constituents 
indicated by the analyses was carried out with the results shown in figs. 3 
and 4. 
§ 7. The order of procedure so far has been as follows :— 
(1) Removal of first approximation to the semi-diurnal tide ; 
(2) removal of quarter-diurnal tide ; 
(3) analyses ; 
(4) removal of diurnal tide; 
(5) removal of second approximation to semi-diurnal tide. 
No attempt has been made to analyse for, and remove, the long-period 
constituents, and it has not been found possible as yet to remove the residual 
semi-diurnal tide. 
§ 8. Uhe residual semi-diurnal constituents.—The residue illustrated by 
fig. 4 is clearly the difference between observations and predictions from 
Darwinian short-period constituents. (The matter of shallow-water effects and 
their inadequate representation will be dealt with separately in § 9.) On certain 
days it is seen that the residual semi-diurnal tide can give an error of + 6 inches, 
and statement 2, § 1, is justified. It may be stated that at Newlyn there is 
not a very great range of tide. 
The origin of these constituents remains obscure. Apart from the reasons 
given in Part II., § 18, the presence of non-Darwinian constituents may be 
readily shown. If we have two constituents, M, and §,, say, then the range 
of tide varies from day to day and we get the phenomena of springs and neaps : 
the time between two successive springs is equal to 360° divided by the difference — 
in speed (in degrees per mean solar day) of the two constituents; in the case 
mentioned this difference in speed is about 24°, so that the spring tides occur at 
intervals of approximately fifteen days. Now the greatest difference in speed 
between two Darwinian semi-diurnal constituents is about 52° per m.s.d., 
corresponding to ‘ spring tides’ at intervals of about seven days. Fig. 5 shows 
the variation of daily range of the semi-diurnal oscillation for 180 days. The 
