934 REPORTS ON THE STATE OF SCIENCE, ETC. 
place of mean solar hours. A ‘special day’ is the time taken for the argument of 
the constituent in question to increase by 360° if it be diurnal or 720° if it be semi- 
diurnal. As this knowledge is lacking, the heights at the nearest solar hours are 
assigned and attributed to the special hours. A certain correction is made 
depending upon the assumption of a random distribution of the difference in time 
between special and mean solar hours. In dealing with large constituents this 
correction may not be adequate. 
(3) The length of record to be included in the process is determined by choosing 
an interval of time such that the effect of some one large constituent is made as 
small as possible. The effects of all other constituents are ignored, though this may 
not be justifiable if these constituents are large or if their speeds are very nearly 
equal to that of the constituent sought. 
The whole purpose of the methods described is that of determining simply two 
numbers A and B, and there is no internal evidence to show that we are entitled to 
attach any significance tu these numbers, though they are supposed, and are used, to 
define a constituent. There is nothing to indicate the presence or magnitude of 
perturbing constituents, and nothing to show whether the whole of the constituents 
have been dealt with or not. The whole process is repeated for each of the 
constituents expected to be present. As instruments for research these methods are 
singularly inefficient, the paucity and uncertainty of the results being in remarkable 
contrast to the magnitude of the work required. 
§ 17. Darwin's method of analysis of observations.—A method which ranks as a 
great improvement on those discussed in § 16 was introduced by Darwin in his paper 
on the abacus. The method is only applied to the group of constituents whose 
speeds are nearly equal to those of the principal solar constituents; the constituents 
K,, R,, S, and T, are all nearly equal in speed, and are treated over a short interval 
of time as having the speed of S,. For successive intervals the values of A,and B, 
would be constant if only S, were present, but in the case considered they vary 
harmonically, and analyses of the variations give the harmonic numbers for the 
four constituents. 
This method will now be considered in more detail. The exposition here given 
is different from that by Darwin, and the method has been generalised to some 
extent. 
Suppose that we are dealing with a constituent R cos (ot—e) whose value is 
given at intervals of one mean solar hour. Then the contribution to A, and B, in 
the analysis for solar constituents will be 
1 1 sta oe 
"= jon =R cos (ot —€) cos 15°nt = gy =R { cos (o— lbnt - €)° + cos (« + 16nt —€)° \ 
1 1 SSSR ee, sy CHEE 8 
Bn= jon =R cos (ot —e) sin 15°nt = oy =R { —sin (o—15nt—)+sin (o+ Tent —)° } 
where the summations are taken from ¢=24T to 24T + 24N —1, so that it is supposed 
that the summations begin at zero hour on day T and include all the hourly values 
of the constituent over N complete days. 
These may be written as 
An=f'R cos (e'—240T) +7’'R cos (e”’—240T), 
B,=/'R sin (c'—240T)+/’R sin (€”—240T), 
where 
fre sin 12N («—15n)° 
ee Ne EE f'= sin 12N(o + 15n)° 
24N sin 3(o— 15n)°’ 
~ 24N sin 3(o + 15)” 
e” =e—12N(o—15n)° + 4(¢—1dn)°? 
e’ =e—12N(o—15n)° +3(o + 15n)° =e’ + 152°, 
certain multiples of 360° having been ignored. 
