In the potential curves of the diagram the ordinates represent the hourly values expressed—as in Tables II. and III.—as percentages of the mean value for the day. If this be overlooked, a wrong impression may be derived as to the absolute amplitudes of the changes. The Kew curves, for instance, might suggest that the range (maximum less minimum hourly value) was larger in June than in December. In reality the December range was 82, the June only 57 volts; but the mean value of the potential was 243 in December as against 111 in June. So again, in the case of the Paris curves, the absolute value of the diurnal range in summer was much greater for the Eiffel Tower than for the Bureau Central, but the mean voltage was 2150 at the former station and only 134 at the latter.

8. Fourier Coefficients.—Diurnal inequalities such as those of Tables II. and III. and intended to eliminate irregular changes, but they also to some extent eliminate regular changes if the hours of maxima and minima or the character of the diurnal variation alter throughout the year. The alteration that takes place in the regular diurnal inequality throughout the year is best seen by analysing it into a Fourier series of the type

c1 sin(t + a1) + c2 sin(2t + a2) + c3 sin(3t + a3) + c4 sin(4t + a4) + ...

where t denotes time counted from (local) midnight, c1, c2, c3, C4, ... are the amplitudes of the component harmonic waves of periods 24, 12, 8 and 6 hours; a1, a2, a3, a4, are the corresponding phase angles. One hour of time t is counted as 15°, and a delay of one hour in the time of maximum answers to a diminution of 15° in a1, of 30° in a2, and so on. If a1, say, varies much throughput the year, or if the ratios of c2, c3, c4, ... to c1, vary much, then a diurnal inequality derived from a whole year, or from a season composed of several months, represents a mean curve arising from the superposition of a number of curves, which differ in shape and in the positions of their maxima and minima. The result, if considered alone, inevitably leads to an underestimate of the average amplitude of the regular diurnal variation.

It is also desirable to have an idea of the size of the irregular changes which vary from one day to the next. On stormy days, as already mentioned, the irregular changes hardly admit of satisfactory treatment. Even on the quietest days irregular changes are always numerous and often large.

Table IV. aims at giving a summary of the several phenomena for a single station, Kew, on electrically quiet days. The first line gives the mean value of the potential gradient, the second the mean excess of the largest over the smallest hourly value on individual days. The hourly values are derived from smoothed curves, the object being to get the mean ordinate for a 60-minute period. If the actual crests of the excursions had been measured the figures in the second line would have been even larger. The third line gives the range of the regular diurnal inequality, the next four lines the amplitudes of the first four Fourier waves into which the regular diurnal inequality has been analysed. These mean values, ranges and amplitudes are all measured in volts per metre (in the open). The last four lines of Table IV. give the phase angles of the first four Fourier waves.

Table IV.—Absolute Potential Data at Kew (12).

Jan. Feb. March. April. May. June. July. Aug. Sept. Oct. Nov. Dec.
Mean Potential Gradient 201 224 180 138 123 111 98 114 121 153 200 243
Mean of individual daily ranges 203 218 210 164 143 143 117 129 141 196 186 213
Range in Diurnal inequality 73 94 83 74 71 57 55 60 54 63 52 82
Amplitudes of Fourier wavesc1 22 22 17 13 18 9 6 6 9 7 14 30
c2 21 33 34 31 22 23 24 26 23 30 17 21
c3 7 10 5 5 3 1 3 2 3 6 5 7
c4 2 3 5 6 4 1 4 3 4 3 2 3
° ° ° ° ° ° ° ° ° ° ° °
Phase angles of Fourier wavesa1 206 204 123 72 86 79 48 142 154 192 202 208
a1 170 171 186 193 188 183 185 182 199 206 212 175
a3 11 9 36 96 100 125 124 107 16 18 38 36
a1 235 225 307 314 314 277 293 313 330 288 238 249

It will be noticed that the difference between the greatest and least hourly values is, in all but three winter months, actually larger than the mean value of the potential gradient for the day; it bears to the range of the regular diurnal inequality a ratio varying from 2.0 in May to 3.6 in November.

At midwinter the 24-hour term is the largest, but near midsummer it is small compared to the 12-hour term. The 24-hour term is very variable both as regards its amplitude and its phase angle (and so its hour of maximum). The 12-hour term is much less variable, especially as regards its phase angle; its amplitude shows distinct maxima near the equinoxes. That the 8-hour and 6-hour waves, though small near midsummer, represent more than mere accidental irregularities, seems a safe inference from the regularity apparent in the annual variation of their phase angles.