330 BEPOBTS ON THE STATE OF SCIENCE. — 1920, 



and to 



N sin{wK/ff„-l)7r} 1 L 2 V<r„ J 2i ffj 



on summatiou. 



If <rr/(r„ is sufficiently near to unity, (6) may be replaced by 



(6) 



,in{Nn(aK - l)r\ ^ Lj^^ h - A + ^ 1^1. - .1 , . (7) 



N«K/cr„-l)7r ( L 2 \<r„ /+24o-J )' ^^ 



; we should have obtained if we had replai 



which is exactly what we should have obtained if we had replaced (5) by the integral 



I _ I //r„ - \ h. fT.. I \ 



ds . . • (8) 



-i 



It is the general practice to choose N so as to make as small as possible the 

 residue (6) from some one large term of the type (3). 



When a year's record is to be analysed the value of N chosen is 369, and it is 

 then generally assumed that all the residues (6) may be neglected ; in other words, 

 that the isolation of the S series is complete. 



With a month's record Darwin's plan is to use for N both 30 and 27, and with a 

 fortnight's record both 15 and 14. It is assumed that the amplitudes of K, and P, 

 bear to those of Sj and K, respectively the equilibrium ratios, and that the lags of 

 Ej and P, are equal respectively to those of 8j and K„ while T, is supposed simply 

 proportional to Sj. The residues from all other constituents are neglected. 



For N = 369 and the constituents 



M^, N^, K^, K„ 0„ P, 

 the coefficient 



1 I sin { N w(cr,./(ro - l)7r| | - (9) 



N I sin { wCtj/o-q — 1)t } 



in (6) takes the values 



•000, -008, -010, 010, -000, -010 



respectively, and the corresponding coefficient in (7) takes the same values. 



For N = 15 and the constituents 



BIj, Nj, K2 

 (9) takes the values 



•016, -204, -989, 



respectively, and the only effect of replacing (6) by (7) is to give "200 instead of 

 •204. 



For N = 14 and the constituents 



^i> Op Pp 

 (9) takes the values 



•998, -014, -998 



respectively, which are not affected by replacing (6) by (7). 



The Isolation of the other Series. 



7. We shall now consider the isolation of the series which has <r for the speed 

 of its diurnal constituent, and shall refer to ZTrJcr as a ' special day.' For the 

 isolation to be carried out exactly like that of the S series we should require the 

 heights at intervals of time equal to a ' special hour.' The method of summation 

 we should then have is made the basis of the actual methods now under considera- 

 tion. With each place in the summations is associated a definite time, and this in 

 general is necessarily different from that of the height which is assigned to the 

 place. 



There are two ways in general practice of making the assignment, and we shall 

 refer to these as the B.A. and the Abacus assignments. The first is used by Roberts 

 and the Survey of India, while the second is that of Darwin's tidal abacus. 



