[jackson] MAGNETIC DECLINATION AT AGINCOURT 107 



analysis, owing to the facility with which we could study the variations 

 of the forces throughout the year. 



The absence of any natural force, nevertheless, does not impair 

 the usefulness of Fourier series, for by it if we are comparing inequalities 

 from different stations or for different seasons of the year, or for 

 different years we are enabled to indicate the nature of the differences 

 which exist in a more precise manner than otherwise. 



The Fourier analysis may be expressed in either of the equivalent 

 forms : 



ai cos t+b 2 sin t+a 2 cos 2t+b 2 sin 2t+ .... 



Ci sin (ai+t)+Co sin (a 2 + 2t)+ 



where ai, bi, Ci and ai, etc., are known as Fourier coefficients, and 

 where t represents the time counted from midnight at the rate of 15° 

 to the hour. The a, b constants are calculated directly from the 24 

 hourly values in the diurnal inequality and the c (amplitude) and a 

 (phase angle) coefficients are then deduced by means of the formulae. 



a n = tan -1 (a n /b n ) 



c n = v a n 2 + b n 2 



where n =1,2,3, &c. 



An increased value of a phase angle means an earlier occurrence 

 of a maxima or minima of the term involved. The method of deter- 

 mining these times is best shewn by an example. First, suppose 

 the 24 hour term to be 1/.26 sin (226°+t). Since a sine has its maxi- 

 mum at 90°, 450°, &c, we put 

 226° + t = 450° 

 .'. t = 214° 

 allowing 15° to the" hour we have 



t = 214°=14.3 hours or 2h. 18m. p.m. 

 The minimum occurs of course 12 hours earlier or at 2 h. 18m. a.m. 

 The term fluctuates between + 1'.26 and — 1/.26, its so-called 

 amplitude being 1/.26, but its range is double this or 2'. 52. 



As a second example suppose the 12 hour term to be 1'.51 sin 

 (18° + 2t). The first maximum is given by putting 

 18° + 2t = 90° 

 whence c = 36 = 2hr. 24m. 

 Therefore the first maximum ( + 1'.59) occurs at 2h. 24m. a.m. 

 and a second equal maximum at 2h. 24m. p.m. The two minima 

 (each — 1/.59) occur of course midway between the maxima or at 

 8h. 24m. a.m. and 8h. 24m. p.m. 



In Table II are given the amplitudes and phase angles of the 

 twenty-four, twelve, eight and six hour terms of the diurnal inequalities 



