———— LL 
TO ASSIST WORK ON THE TIDES. 937 
to M,-time in the usual B.A. manner,‘ and were then treated in sections of ten lunar 
days. For each hour series in the section the heights were averaged, giving twenty- 
four values for submission to harmonic analysis, by the least square rule, for a 
semi-diurnal constituent. The results are shown in Table I., cols. 2 and 6; aver- 
aging these results in threes gives the values of A and B in cols. 2 and 7; these 
correspond to N=30, T=0,10, .... The diminution of range is obviously due te 
the diminution of the effects of other groups. If these results are plotted, as is 
done in fig. 7, a very striking effect is shown: there is a well-marked variation !n 
the values of A and B with a period somewhat less than three months. There is 
nothing in Darwin’s schedules of constituents to explain this perturbation; it 
cannot possibly arise from another group because a difference in speed of 12° or 
13° per day would require a true amplitude (R) about 0-7 foot! 
Thus the method has already shown the existence of oue constituent previously 
unsuspected : it may be repeated that the ordinary methods of analysis would have 
given one A and one B for an arbitrarily fixed interval, and thus no new constituent 
would have been revealed. We can now consider the calculation of the most trust- 
worthy values of A,, and B,,, corresponding to the constituent M,; shall we simply 
take the average of all the values of A, or shall we take the average over a definite 
interval of time determined by considerations cf relative speeds of M, and §,, say, 
or by the relative speeds of the perturbing constituents indicated in the figure? 
The problem is rather complex if we are limited to rigid arithmetical methods, 
obviously, if we choose an interval to satisfy one condition it may happen that the 
other constituents have their greatest effect init. Further, in this case we do not 
know exactly the speed of the perturbing constituent. The only satisfactory solu- 
tion of the problem is to use the idea of an ‘asymptotic mean’ and to discard 
altogether the idea of attempting to fix a criterion for choosing the interval for 
analysis. Suppose that we sum the values of A consecutively, writing down the 
sum & in col. 4, Table I., and then divide each sum by M, the number of contributory 
values of A, as in col. 5, then the asymptotic mean is the limiting value of this 
average (=/M) as M increases indefinitely ; 7.e. the asymptotic mean is defined, as in 
10(M—1) 
§ 18 (7), by L it SA 
Mo T=0 
Here the suffix T is used to denote the value of A in the thirty-day interval com- 
mencing on day T. The values of 5/M are plotted in Fig. 7, and the dotted lines 
give approximately the tendency of the curves. The oscillations about the dotted 
line diminish in range as M increases, and there is a tendency for the line to reach 
a constant value—the asymptotic mean. Of course, in this case, with only six 
months’ residues submitted to analysis, itis impossible to give a really definitive 
value to the asymptotic mean, but it is possible to give a much better value to A,, 
than would result from numerical methods with an arbitrarily fixed interval. These 
curves provide an interesting commentary on the accuracy of the ‘ constants’ usually 
obtained. 
In the case of M, there are no obvious semi-annual oscillations in A and B, 
though there may be some annual oscillation ; analyses from six months’ residues do 
not suffice for dealing with these. 
A very interesting case is that of the L, group. Darwin’s schedules of constituents 
indicate only L, and A,, and the latter would give semi-annual variations in L, (A, B). 
Table (II.) gives the values of A and B by the L, process, in columns 2 and 6 for 
N=30. The values of 3/M are illustrated in fig. 8. 
It is rather difficult to determine A.) and B,o satisfactorily because of the large 
perturbing constituents. It is an advantage to be able to remove even crude 
approximations to these before proceeding further, but in this case they are small 
and are allowed to stay. The value of p for A, has been given as —1:778° and we 
shall write 
(Scion 
the suffix here denotes that the variation is semi-annual. 
4 For technical terms and explanations reference should be made to the Report 
by Professor Proudman, 1920. 
