DIURNAL INEQUALITIES. FOURIER COEFFICIENT 



117 



CHAPTER IV. 

 DIURNAL INEQUALITIES. FOUKIEK COEFFICIENTS. 



31. From the rliumal inequalities, calculations were made of the Fourier coefficients answering to the 

 " waves " whose periods are 24, 12, 8 and 6 hours. The analysis of the diurnal inequality may be supposed 

 to proceed according to either of the two equivalent series 



<ii cos I + bi sin t + a-; cos 2t + b sin It + . . . 

 c t sin (t + ai) + c., sin (2/ + a 2 ) + . . . 



Here / is time, counted from local midnight, one hour being taken as equivalent to 15. The constants 

 with suffix 1 refer to the 24-hour term, those with suffix 2 to the 12-hour term, and so on. The a and 6 

 constants are calculated directly from the inequality tables. The mean a\, for instance, for a particular 

 season of the year is the arithmetic mean of the values of ! for the months composing that season. The 

 e (amplitude) and a (phase angle) constants are calculated from the corresponding a and b constants by means 



of the formulae 



a = tan" 1 (a>/b), c = a/sin a = ft/cos a. 



The c (or a) derivable from a seasonal diurnal inequality is not, as a rule, the arithmetic mean of the 

 c's (or a's) of the individual months which form the season. 



32. Tables XXIV to XXXII are devoted to the a and b coefficients. These were in all cases really 

 calculated to at least one figure further than is retained. It is, however, hardly necessary to remark that 

 even as thus curtailed they cannot be regarded as physical facts freed from observational uncertainties. 

 This reservation ought especially to be borne in mind in the case of the coefficients with suffixes 3 and 4, 

 which relate to the 8-hour and 6-hour waves. The differences between the values obtained for successive 

 months probably owe at least as much to the existence of " accidental " disturbances as to any real 

 difference between the magnetic conditions characteristic of successive months of an average year. 



In the case of the Declination, Horizontal Force and Vertical Force the values of a and b are recorded 

 for the individual months of the two years, as well as for the months of a representative year in which 

 common months of 1902 and 1903 are combined. Also two sets of values are given for Declination. Of 

 these the first set, comprising Tables XXIV and XXV, relate to the all-days' inequality data of Tables XII 

 and XIII ; while the second set, comprising Tables XXVI and XXVII, relate to the quieter-days' inequality 

 data of Tables XIV and XV. For Inclination only one table is given. When the sign to be attributed 

 to the numerical value of a constant is the same for each month and season it is indicated only at the top 

 and bottom of the column. 



TABLE XXIV. Declination (All Days). Fourier Coefficients. (Unit 1'.) 



