Hour. 1 2 3 4 5 6 7 8 9 10 11 12
N. a.m. + 6 + 5 + 5 + 5 + 6 + 6 + 5 + 1 − 6 −14 −20 −20
p.m. −17 −12 − 6 − 1 + 3 + 6 + 9 + 9 + 9 + 8 + 7 + 7
W. a.m. − 2 − 2 − 3 − 4 − 6 − 9 −13 −17 −19 −13 − 3 +11
p.m. +20 +22 +17 +11 + 6 + 4 + 2 + 1 0 − 1 − 2 − 2

§ 20. Any diurnal inequality can be analysed into a series of Fourier Series. harmonic terms whose periods are 24 hours and submultiples thereof. The series may be expressed in either of the equivalent forms:—

a1 cos t + b1 sin t + a2 cos 2t + b2 sin 2t + ...

(i)

c1 sin (t + α1) + c2 sin (2t + α2) + ....

(ii)

Table XVI.—Ranges in Diurnal Inequalities at Falmouth (unit 1γ).

Jan.Feb.March.April.May.June.July.Aug.Sept.Oct.Nov.Dec.
N.212330393937373936322415
W.202446545555545651392415

In both forms t denotes time, counted usually from midnight, one hour of time being interpreted as 15° of angle. Form (i) is that utilized in actually calculating the constants a, b, ... Once the a, b, ... constants are known, the c, α, ... constants are at once derivable from the formulae:—

tan αn = an / bn; cn = an / sin αn = bn / cos αn = √(an² + bn²).