332 



REPORTS ON THE STATE OF SCIENCE. — 1920. 



may be calculated accurately by direct addition, and although this would be very 

 laborious it could be done once for all. 



Again, it is the general practice to try to choose N so as to make as small as 

 possible the residue due to some one large constituent, and then to neglect all the 

 residues. For example, when a year's record is available the B.A. plan is to take 

 N = 357 for the M series and N = 343 for the O series. Darwin's plan for a month s 

 record is to replace these numbers by 29 and 25 respectively, and for a fortnight's 

 record by 14 and 13 respectively. These values are taken on the basis of the formula 

 (17) with Ts omitted. 



Analysis of the Separate' Series. 

 8. From the isolation process for each series we have 24 values 



Co. Cp • • • Cll • • • ^23 



associated with times which differ by intervals of one special hour. Certain fractions 

 of these values are residues from other series. 



The usual method of analysing the Fourier's series into its separate terms is by 

 what we may call the ' least square rule.' If the series is expressed by 



C = A„ + K, cos i<rt - «,) + R2 cos (2<r^ - ej 

 the rule is given by 



+ Em cos (^m<rt — «„) • (1^) 



Rm cos f. 



28 



ft=0 

 23 



— > Cfc cos , 



12-^ '' 12 



A=0 



Km sin em = — 2. f'» ^'° "T^' 



h=0 



(20) 



and its application to the cases in question would be quite accurate if the isolation 



were perfect. 



We must therefore consider the effect on the results of imperfect isolation. 

 Taking the S series, the effect of a term of the type 



/ h 2w(ryir \ 



R cos 



(21) 



on Ag is 



1 sin (wtr^^/O R (,og 



24 sin (w(rrT/24(ro) 

 so that the effect of the term (3) on A^ is 



(23 MO-rir _ \ 

 2-4 -.7 7' 



L sin (N rtcr.^/O jj cos i f N - 1 + ^-"i " - - 4 • 

 4N sin (»Kr,.ir/24(r„) U 2»/ (r„ ) 



(22) 



(23) 



The effect of (21) on Rm cos «m is 



Rf sin{(wa-J<ro + m)irJ 

 24| sin{(w(rr/(ro + m)7r/24} 



sinj(«(rr/fo — "O'"'} 

 "*" sin{(»KrJ<ro — m)ir/24} 



f23/w^ 

 {24 \^^ + 



m>-')'--w- ■ '^' 



When n = m and o-^/o-o is near unity the second part of this is much larger than 

 the first, and it reduces to 



f23 ,„. _. , ^25^ 



R cos 



24" 



(s-O'-l 



