236 REPORTS ON THE STATE OF SCIENCE, ETC. : 
The application of the least square rule to (4) gives, for example, 
1 m—1 
SAcr= Lt m= Acos p;T - - 4 pene er) 
M—>coo” T=0 
Theoretically, therefore, we can obtain A,;, As... - and thence F,, nmr by (6)} 
application of the table given above determines the corresponding values of 
R and e. 
In the simple case considered by Darwin the periods are exactly a year and half 
a year, and he was able to apply the least square rule to twelve values with N =30 and 
T such that p,T is very nearly a multiple of 30°. The perturbation of one constituent 
upon another was negligible so far as constituents of the solar group were concerned. 
Some perturbation, however, was caused by M, with p=—24:38. Taking T at 
intervals of about 30°5 days on the average gives pT about —744°, or ‘apparently’ 
—24°, The residual effect of M, is therefore to give a perturbation which is approxi- 
mately annual in period. Darwin makes corrections for this perturbation. In the 
present paper we shall not have occasion to consider such perturbations, as we are 
not dealing with observations but with residues. 
§18. Analysis of residues—When the chief constituents have been subtracted 
from the tidal observations there is greater freedom possible in the details of 
analysis. In the analysis of observations there are two kinds of errors, the first being 
due to the methods of assignment, which introduce errors in the constituent sought 
whether other constituents be present or not. The second kind of error is due 
entirely to the presence of other constituents. In both cases the error is proportional 
to the size of the constituent producing it, and if the chief constituents be removed 
the methods of analysis can be applied to the residues without serious error. If we 
consider the errors due to the assignment as negligible or adequately corrected by 
the multiplying factor previously mentioned, then we can apply more generally 
Darwin’s method as given for the solar constituents. 
The exposition of Darwin’s method in § 17 can be generalised simply by writing 
‘ special time.’ for ‘mean solar time.’ If the heights are given (or obtained roughly 
by assignment) at intervals of one special hour, then the speed per special hour is 
15n° and p (in special time) is zero for the constituent appropriate to the special 
time. As an example the L,-group of constituents includes L, with speed = 29°5285° 
per m.s.h. and A, with speed =29°4556° per m.s.h. In special (or L,) time the speed 
of L, becomes 30° per L,.h. and the speed of A, becomes 
‘ ° 
29°4556 x 30" _ 99.9959° per L,.h, 
29°5285 
corresponding to p=1:7784°. The constituent A,, taerefore, will give approximately 
semi-annual variations in the A and B obtained by the ‘L, process.’ By the‘ L, 
process’ is meant the analysis as for L,. 
Darwin’s method is applicable wherever there are groups of constituents whose 
speeds are nearly equal. Tidal constituents are supposed to be separable from one 
year’s observations only, so that two constituents are not separable in a year unless 
their speeds, true or reduced, differ by a multiple of approximately 360° per mean 
solar year, corresponding to a difference of approximately 1° in p. Now the con- 
stituents occur in groups such that there is a difference in speed of about 12° per 
mean solar day from group to group. It is necessary to choose N such that one 
group has very little effect upon the results of analyses relating to another group. 
If we consider the account of the method in §17 it will be found that the ratio F/R 
depends chiefly upon 7’, and this vanishes for 24 (¢—15n)°=12° when N isa 
multiple of fifteen days. It will be sufficient, therefore, to take N =30, say, in all 
cases whether we are dealing with solar or special time. 
Darwin’s method is to take his thirty-day intervals running almost consecutively, 
but by so doing the most is not made of the material. In the present investigation 
the interval is taken as thirty days, but the intervals overlap, so that analyses are 
carried out for N=30 and T=0, 10,... This is obviously effected by taking 
N=10, T=0, 10, . .. ., and then averaging the values of A (and B) in threes to 
correspond to N =30. 
The methods are now easily explained by references to actual analyses and we 
shall first study the case of M,. The hourly heights (mean solar time) were assigned 
