Table XIII.—Range of the Diurnal Inequality of Declination.Place.Period.Jan.Feb.March.April.May.June.July.Aug.Sept.Oct.Nov.Dec.′′′′′′′′′′′′Pavlovsk1890-1900 a4.936.158.5810.9312.1812.2711.8211.388.706.875.544.63”1890-1900 q2.964.208.7311.2812.8913.2812.3111.709.376.913.952.66Ekatarinburg1890-1900 a3.334.327.6311.1911.8211.5811.0910.458.135.603.733.14Greenwich1865-1896 a5.877.079.4011.4210.5510.9010.8210.939.668.156.415.15Kew1890-1900 a4.926.069.0810.9510.6610.9210.5911.019.497.735.374.46”1890-1900 q4.074.768.8210.5710.9210.6210.1811.019.767.514.753.34Toronto1842-1848 a5.966.059.189.9411.5512.3412.2113.1410.766.966.324.97Manila1890-1900 a1.791.092.133.023.843.944.214.894.531.830.851.33Trivandrum1853-1864 a2.061.480.791.672.903.063.063.643.311.272.142.33Batavia1884-1899 a4.184.643.572.932.382.032.313.163.804.514.504.19St Helena1842-1847 a3.725.194.933.302.643.243.423.592.404.434.053.54Mauritius1876-1890 a5.26.16.34.74.12.93.44.95.05.55.65.1Cape1841-1846 a5.148.217.275.003.913.213.544.984.335.966.365.47Hobart1841-1848 a11.6611.809.507.264.563.704.615.898.2411.0112.0511.81§ 17. Variations of inclination are connected with those of horizontal and vertical force by the relationδI = ½ sin 2I {V−1δV − H−1δH}.Thus in temperate latitudes where V is considerably in excess of H, whilst diurnal changes in V are usually less than those in H, it is the latter which chiefly dominate the diurnal changes in inclination. When the H influence prevails, I has its highest values at hours when H is least. This explains why the dip is above its mean value near midday at stations in Table XI. from Pavlovsk to Parc St Maur. Near the magnetic equator the vertical force has the greater influence. This alone would tend to make a minimum dip in the late forenoon, and this minimum is accentuated owing to the altered type of the horizontal force diurnal variation, whose maximum now coincides closely with the minimum in the vertical force. This accounts for the prominence of the minimum in the diurnal variation of the inclination at Kolaba and Batavia, and the large amplitude of the range. Tiflis shows an intermediate type of diurnal variation; there is a minimum near noon, as in tropical stations, but inclination is also below its mean for some hours near midnight. The type really varies at Tiflis according to the season of the year. In June—as in the mean equality from the whole year—there is a well marked double period; there is a principal minimum at 2 p.m. and a secondary one about 4 a.m.; a principal maximum about 9 a.m. and a secondary one about 6 p.m. In December, however, only a single period is recognizable, with a minimum about 8 a.m. and a maximum about 7 p.m. The type of diurnal inequality seenat the Cape of Good Hope does not differ much from that seen at Batavia. Only a single period is clearly shown. The maximum occurs about 8 or 9 p.m. throughout the year. The time of the minimum is more variable; at midsummer it occurs about 11 a.m., but at midwinter three or four hours later. At Hobart the type varies considerably with the season. In June (midwinter) a double period is visible. The principal minimum occurs about 8 a.m., as at the Cape. But, corresponding to the evening maximum seen at the Cape, there is now only a secondary maximum, the principal maximum occurring about 1 p.m. At midsummer the principal maximum is found—as at Kew or Greenwich—about 10 or 11 a.m., the principal minimum about 4 p.m.§ 18. Even at tropical stations a considerable seasonal change is usually seen in the amplitude of the diurnal inequality in at least one of the magnetic elements. At stations in Europe, and generally in temperate latitudes, the amplitude varies notably in all the elements. Table XIII. gives particulars of the inequality range of declination derived from hourly readings at selected stations, arranged in order of latitude from north to south. The letters “a” and “q” are used in the same sense as before. At temperate stations in either hemisphere—e.g.Pavlovsk, Greenwich or Hobart—the range is conspicuously larger in summer than in winter. In northern temperate stations a decided minimum is usually apparent in December. There is, on the other hand, comparatively little variation in the range from April to August. Sometimes, as at Kew and Greenwich, there is at least a suggestion of a secondary minimum at midsummer. Manila and Trivandrum show a transition from the December minimum, characteristic of the northern stations, to the June minimum characteristic of the southern, there being two conspicuous minima in February or March and in November or October. At St Helena there are two similar minima in May and September, while a third apparently exists in December. It will be noticed that at both Pavlovsk and Kew the annual variation in the range is specially prominent in the quiet day results.Table XIV. gives a smaller number of data analogous to those of Table XIII., comprising inequality ranges for horizontal force, vertical force and inclination. In some cases the number of years from which the data were derived seems hardly sufficient to give a smooth annual variation. It should also be noticed that unless the same group of years is employed the data from two stations are not strictly comparable. The difference between the all and quiet day vertical force data at Pavlovsk is remarkably pronounced. The general tendency in all the elements is to show a reduced range at midwinter; but in some cases there is also a distinct reduction in the range at midsummer. This double annual period is particularly well marked at Batavia.Table XIV.—Ranges in the Diurnal Inequalities.Jan.Feb.March.April.May.June.July.Aug.Sept.Oct.Nov.Dec.H (unit 1γ)Pavlovsk1890-1900a122032464749494439321711””q121731424545424037311710Ekatarinburg”a11152937404039363327139Kew”q151726363839383835272011Toronto1843-1848a232124282929262841252120Batavia1883-1898a494754605148505358524340St Helena1843-1847a434148534640404541404032Mauritius1883-1890a211521232021202220212120Cape of Good Hope1841-1846a131013131516141821141720Hobart1842-1848a424334281917222323353942V (unit 1γ)Pavlovsk1890-1900a152729242620231923201814””q45913131213109754Ekatarinburg”a10151721221920161413119Kew1891-1900q710202531272823201596Toronto1843-1848a121417232614273234251918Batavia1883-1898a424848453131322941504033St Helena1843-1847a161312141311171117111518Mauritius1884-1890a121618151413152120161311Cape of Good Hope1841-1846a294741382112141919353328Hobart1842-1848a252722232421222826222327Inclination′′′′′′′′′′′′Pavlovsk1890-1900a0.971.242.072.792.722.882.852.642.522.181.200.89Ekatarinburg”a0.790.941.702.082.252.192.182.082.001.700.880.69Kew”q0.981.011.381.862.052.022.052.151.981.571.270.63Toronto1843-1848a1.150.941.191.231.311.371.131.261.871.161.091.05Batavia1883-1898a4.885.225.565.624.214.054.244.175.135.584.513.85Cape of Good Hope1842-1846a1.552.292.232.231.601.411.541.701.862.031.552.04Hobart1842-1848a1.952.161.721.621.231.161.281.421.391.752.042.10§ 19. When discussing diurnal inequalities it is sometimes convenient to consider the components of the horizontal force in and perpendicular to the astronomical meridian, rather than the horizontal force and declination. If N and W be the components of H to astronomical north and west, and D the westerly declination, N = H cos D, W = H sin D. Thus corresponding small variations in N, W, H and D are connected by the relations:—δN = cos DδH − H sin DδD, δW = sin DδH + H cos DδD.If δH and δD denote the departures of H and D at any hour of the day from their mean values, then δN and δW represent the corresponding departures of N and W from their mean values. In this way diurnal inequalities may be calculated for N and W when those for H and D are known. The formulae suppose δD to be expressed in absolute measure,i.e.1′ of arc has to be replaced by 0.0002909. If we take as an example a station at which H is .185 then HδD = .0000538 (number of minutes in δD). In other words, employing 1γ as unit of force, one replaces HδD by 5.38δD, where δD represents declination change expressed as usual in minutes of arc. In calculating diurnal inequalities for N and W, one ought, strictly speaking, to assign to H and D the exact mean values belonging to these elements for the month or the year being dealt with. For practical purposes, however, a slight departure from the true mean values is immaterial, and one can make use of a constant value for several successive years without sensible error. As an example, Table XV. gives the mean diurnal inequality for the whole year in N and W at Falmouth, as calculated from the 12 years 1891 to 1902. The unit employed is 1γ.The data in Table XV. are closely similar to corresponding Kew data, and are presumably fairly applicable to the whole south of England for the epoch considered. At Falmouth there is comparatively little seasonal variation in the type of the diurnal variation in either N or W. The amplitude of the diurnal range varies, however, largely with the season, as will appear from Table XVI., which is based on the same 12 years as Table XV.Diurnal inequalities in N and W lend themselves readily to the construction of what are known asvector diagrams. These are curves showing the direction and intensity at each hour of the day of the horizontal component of the disturbing force to which the diurnal inequality may be regarded as due. Figs. 7 and 8, taken from thePhil. Trans.vol. 204A, will serve as examples. They refer to the mean diurnal inequalities for the months stated at Kew (1890 to 1900) and Falmouth (1891 to 1902), thick lines relating to Kew, thin to Falmouth. NS and EW represent the geographical north-south and east-west directions; their intersection answers to the origin (thick lines for Kew, thin for Falmouth). The line from the origin to M represents the magnetic meridian. The line from the origin to any cross—the number indicating the corresponding hour counted from midnight as 0—represents the magnitude and direction at that hour of the horizontal component of the disturbing force to which the diurnal inequality may be assigned. The cross marks the point whose rectangular co-ordinates are the values of δN and δW derived from the diurnal inequalities of these elements. In figs. 7 and 8 the distances of the points N, E, S, W from their corresponding origin represents 10γ. The tendency to form a loop near midnight, seen in the November and Decembercurves, is characteristic of the winter months at Kew and Falmouth. The shape is less variable in summer than in winter; but even in summer the portion answering to the hours 6 p.m. to 6 a.m. varies a good deal. The object of presenting the Kew and Falmouth curves side by side is to emphasize the close resemblance between the magnetic phenomena at places in similar latitudes, though over 200 miles apart and exhibiting widely different ranges for their meteorological elements. With considerable change of latitude however the shape of vector diagrams changes largely.Table XV.—Diurnal Inequalities in N. and W. at Falmouth (unit 1γ).Hour.123456789101112N.a.m.+ 6+ 5+ 5+ 5+ 6+ 6+ 5+ 1− 6−14−20−20p.m.−17−12− 6− 1+ 3+ 6+ 9+ 9+ 9+ 8+ 7+ 7W.a.m.− 2− 2− 3− 4− 6− 9−13−17−19−13− 3+11p.m.+20+22+17+11+ 6+ 4+ 2+ 10− 1− 2− 2§ 20. Any diurnal inequality can be analysed into a series ofFourier Series.harmonic terms whose periods are 24 hours and submultiples thereof. The series may be expressed in either of the equivalent forms:—a1cos t + b1sin t + a2cos 2t + b2sin 2t + ...(i)c1sin (t + α1) + c2sin (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.212330393937373936322415W.202446545555545651392415In 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).The a, b, c, α constants are called sometimes Fourier, sometimes Bessel coefficients.(FromPhil. Trans.)Fig. 7.By taking a sufficient number of terms a series can always be obtained which will represent any set of diurnal inequality figures; but unless one can obtain a close approach to the observational figures from the terms possessing the periods 24, 12, 8 and 6 hours the physical significance and general utility of the analysis is somewhat problematical. In the case of the magnetic elements, the 24 and 12 hour terms are usually much the more important; the 24-hour term is generally, but by no means always, the larger of the two. The c constants give the amplitudes of the harmonic terms or waves, the α constants the phase angles. An advance of 1 hour in the time of occurrence of the first (and subsequent, if any) maximum and minimum answers to anincreaseof 15° in α1of 30° in α2, of 45° in α3, of 60° in α4and so on. In the case of magnetic elements the phase angles not infrequently possess a somewhat large annual variation. It is thus essential for a minute study of the phenomena at any station to carry out the analysis for the different seasons of the year, and preferably for the individual months. If the a and b constants are known for all the individual months of one year, or for all the Januarys of a series of years, we have only to take their arithmetic means to obtain the corresponding constants for the mean diurnal inequality of the year, or for the diurnal inequality of the average January of the series of years. This, however, is obviously not true of the c or α constants, unless the phase angle is absolutely unchanged throughout the contributory months or years. This is a point requiring careful attention, because when giving values of c and α for the whole year some authorities give the arithmetic mean of the c’s and α’s calculated from the diurnal inequalities of the individual months of the year, others give the values obtained for c and α from the mean diurnal inequality of the whole year. The former method inevitably supplies a larger value for c than the latter, supposing α to vary with the season. At some observatories,e.g.Greenwich and Batavia, it has long been customary to publish every year values of the Fourier coefficients for each month, and to include other elements besides the declination. For a thoroughly satisfactory comparison of different stations, it is necessary to have data from one and the same epoch; and preferably that epoch should include at least one 11-year period. There are, however, few stations which can supply the data required for such a comparison and we have to make the best of what is available. Information is naturally most copious for the declination. For this element E. Engelenburg20gives values of C1, C2, C3, C4, and of α1, α2, α3, α4for each month of the year for about 50 stations, ranging from Fort Rae (62° 6′ N. lat.) to Cape Horn (55° 5′ S. lat.). From the results for individual stations, Engelenburg derives a series of means which he regards as representative of 11 different zones of latitude. His data for individual stations refer to different epochs, and some are based on only one year’s observations. The original observations also differ in reliability; thus the results are of somewhat unequal value. The mean results for Engelenburg’s zones must naturally have some of the sources of uncertainty reduced; but then the fundamental idea represented by the arrangement in zones is open to question. The majority of the data in Table XVII. are taken from Engelenburg, but the phase angles have been altered so as to apply to westerly declination. The stations are arranged in order of latitude from north to south; in a few instances results are given for quiet days. The figures represent in all cases arithmetic means derived from the 12 monthly values. In the table, so far as is known, the local mean time of the observatory has been employed. This is a point requiring attention, because most observatories employ Greenwich time, or time based on Greenwich or some other national observatory, and any departure from local time enters into the values of the constants. The data for Victoria Land refer to the “Discovery’s” 1902-1903 winter quarters, where the declination, taken westerly, was about 207°.5.As an example of the significance of the phase angles in Table XVII., take the ordinary day data for Kew. The times of occurrence of the maxima are given by t + 234° = 450° for the 24-hour term, 2t + 39°.7 = 90° or = 450° for the 12-hour term, and so on, taking an hour in t as equivalent to 15°.Thus the times of the maxima are:—24-hour term, 2 h. 24 m. p.m.; 12-hour term, 1 h. 41 m. a.m. and p.m.8-hour term, 4 h. 41 m. a.m., 0 h. 41 m. p.m., and 8 h. 41 m. p.m.6-hour term, 0 h. 33 m. a.m. and p.m., and 6 h. 33 m. a.m. and p.m.The minima, or extreme easterly positions in the waves, lie midway between successive maxima. All four terms, it will be seen, have maxima at some hour between 0h. 30m. and 2h. 30m. p.m. They thus reinforce one another strongly from 1 to 2 p.m., accounting for the prominence of the maximum in the early afternoon.(FromPhil. Trans.)Fig. 8.The utility of a Fourier analysis depends largely on whether the several terms have a definite physical significance. If the 24-hour and 12-hour terms, for instance, represent the action of forces whose distribution over the earth or whose seasonal variation is essentially different, then the analysis helps to distinguish these forces, and may assist in their being tracked to their ultimate source. Suppose, for example, one had reason to think the magnetic diurnal variation due to some meteorological phenomenon,e.g.heating of the earth’s atmosphere, then a comparison of Fourier coefficients, if such existed, for the two sets of phenomena would be a powerful method of investigation.Table XVII.—Amplitudes and Phase Angles for Diurnal Inequality of Declination.Place.Epoch.c1.c2.c3.c4.α1.α2.α3.α4.′′′′°°°°Fort Rae (all)1882-188318.498.221.992.07156.541.9308104Fort Rae (quiet)”9.094.511.320.73166.537.5225350Ekatarinburg1841-18622.571.810.730.22223.37.4204351Potsdam1890-18992.811.900.830.31239.932.623749Kew (ordinary)1890-19002.911.790.790.27234.039.723957Kew (quiet)”2.371.820.900.30227.342.124055Falmouth (quiet)1891-19022.181.820.910.29226.240.523856Parc St Maur1883-18992.701.870.850.30238.632.523595Toronto1842-18482.652.341.000.33213.734.9238350Washington1840-18422.381.860.650.33223.026.622353Manila1890-19000.530.580.430.17266.350.722689Trivandrum1853-18640.540.460.290.10289.049.6114Batavia1883-18990.800.880.430.13332.0163.25236St. Helena1842-18470.680.610.630.34275.8171.427244Mauritius1876-18900.861.110.760.2221.6172.7350161C. of G. Hope1841-18461.151.130.800.35287.7156.0351193Melbourne1858-18632.522.451.230.3527.4176.79193Hobart1841-18482.292.150.870.3233.6170.8349185S. Georgia1882-18832.131.280.760.3130.3185.37180Victoria Land (all)1902-190320.514.811.211.32158.7306.9292303Victoria Land (quieter)”15.344.051.241.18163.8312.9261§ 21. Fourier coefficients of course often vary much with the season of the year. In the case of the declination this is especially true of the phase angles at tropical stations. To enter on details for a number of stations would unduly occupy space. A fair idea of the variability in the case of declination in temperate latitudes may be derived from Table XVIII., which gives monthly values for Kew derived from ordinary days of an 11-year period 1890-1900.Fourier analysis has been applied to the diurnal inequalities of the other magnetic elements, but more sparingly. Such results are illustrated by Table XIX., which contains data derived from quiet days at Kew from 1890 to 1900.Winterincludes November to February,SummerMay to August, andEquinoxthe remaining four months. In this case the data are derived from mean diurnal inequalities for the season specified. In the case of thecor amplitude coefficients the unit is 1′ for I (inclination), and 1γ for H and V (horizontal and vertical force). At Kew the seasonal variation in the amplitude is fairly similar for all the elements. The 24-hour and 12-hour terms tend to be largest near midsummer, and least near midwinter; but the 8-hour and 6-hour terms have two well-marked maxima near the equinoxes, and a clearly marked minimum near midsummer, in addition to one near midwinter. On the other hand, the phase angle phenomena vary much for the different elements. The 24-hour term, for instance, has its maximum earlier in winter than in summer in the case of the declination and vertical force, but the exact reverse holds for the inclination and the horizontal force.Table XVIII.—Kew Declination: Amplitudes and Phase Angles(local mean time).Month.c1.c2.c3.c4.α1.α2.α3.α4.′′′′°°°°January1.790.860.410.27251.229.825464February2.411.110.570.30242.027.723539March3.051.981.110.45233.236.122349April3.352.481.170.39224.839.222861May3.572.380.870.17221.350.824589June3.832.390.740.05212.646.723972July3.722.300.770.11214.648.12338August3.642.431.050.18228.257.224451September3.352.021.040.35236.955.324570October2.691.690.920.48240.135.623565November1.941.060.510.32248.328.324761December1.610.810.350.20255.122.024356§ 22. If secular change proceeded uniformly throughout the year, the value Enof any element at the middle of the nth month of the year would be connected withE, the mean value for the whole year, by the formula En=E+ (2n − 13)s/24,Annual Inequality.where s is the secular change per annum. For the present purpose, difference in the lengths of the months may be neglected. If one applies to En−Ethe correction −(2n − 13)s/24 one eliminates a regularly progressive secular change; what remains is known as theannual inequality. If only a short period of years is dealt with, irregularities in the secular change from year to year, or errors of observation, may obviously simulate the effect of a real annual inequality. Even when a long series of years is included, there is always a possibility of a spurious inequality arising from annual variation in the instruments, or from annual change in the conditions of observation. J. Liznar,21from a study of data from a number of stations, arrived at certain mean results for the annual inequalities in declination and inclination in the northern and southern hemispheres, and J. Hann22has more recently dealt with Liznar’s and newer results. Table XX. gives a variety of data, including the mean results given by Liznar and Hann. In the case of declination + denotes westerly position; in the case of inclination it denotes a larger dip (whether the inclination be north or south). According to Liznar declination in summer is to the west of the normal position in both hemispheres. The phenomena, however, at Parc St Maur are, it will be seen, the exact opposite of what Liznar regards as normal; and whilst the Potsdam results resemble his mean in type, the range of the inequality there, as at Parc St Maur, is relatively small. Of the three sets of data given for Kew the first two are derived in a similar way to those for other stations; the first set are based on quiet days only, the second on all but highly disturbed days. Both these sets of results are fairly similar in type to the Parc St Maur results, but give larger ranges; they are thus even more opposed to Liznar’s normal type. The last set of data for Kew is of a special kind. During the 11 years 1890 to 1900 the Kew declination magnetograph showed to within 1′ the exact secular change as derived from the absolute observations; also, if any annual variation existed in the position of the base lines of the curves it was exceedingly small. Thus the accumulation of the daily non-cyclic changes shown by the curves should closely represent the combined effects of secular change and annual inequality. Eliminating the secular change, we arrive at an annual inequality, based on all days of the year including the highly disturbed. It is this annual inequality which appears under the headings. It is certainly very unlike the annual inequality derived in the usual way. Whether the difference is to be wholly assigned to the fact that highly disturbed days contribute in the one case, but not in the other, is a question for future research.Table XIX.—Kew Diurnal Inequality: Amplitudes and Phase Angles (local mean time).c1.c2.c3.c4.α1.α2.α3.α4.°°°°IWinter0.2400.2220.1040.076250.091.8344194Equinox0.6010.2900.2130.127290.3135.54207Summer0.8010.3220.1720.070312.5155.539238HWinter3.623.861.811.1382.9277.31546Equinox10.975.873.321.84109.6303.516716Summer14.856.232.350.95130.3316.519941VWinter2.461.670.860.42153.9300.8108280Equinox6.154.702.510.94117.2272.399289Summer8.636.452.240.55122.0272.4100285In the case of the inclination, Liznar found that in both hemispheres the dip (north in the northern, south in the southern hemisphere) was larger than the normal when the sun was in perihelion, corresponding to an enhanced value of the horizontal force in summer in the northern hemisphere.In the case of annual inequalities, at least that of the declination, it is a somewhat suggestive fact that the range seems to become less as we pass from older to more recent results, or from shorter to longer periods of years. Thus for Paris from 1821 to 1830 Arago deduced a range of 2′ 9″. Quiet days at Kew from 1890 to 1894 gave a range of 1′.2, while at Potsdam Lüdeling got a range 30% larger than that in Table XX. when considering the shorter period 1891-1899. Up to the present, few individual results, if any, can claim a very high degree of certainty. With improved instruments and methods it may be different in the future.Table XX.—Annual Inequality.Declination.Inclination.Liznar,N. Hemi-sphere.Potsdam,1891-1906.Parc StMaur,1888-1897.Kew (1890-1900).Batavia,1883-1893.Mauritius.Liznar &Hann’smean.Potsdam.Parc St.Maur.Kew.q.o.s.′′′′′′′′′′′′January−0.25+0.04+0.01+0.08+0.03+0.32+0.23+0.06+0.49+0.32+0.44−0.03February−0.54−0.110.00+0.48+0.25−0.20+0.19+0.29+0.39+0.56+0.29−0.07March−0.27+0.04+0.17+0.03+0.05−1.02−0.12+0.27+0.20+0.38+0.13+0.53April−0.03+0.10+0.12−0.31−0.14−0.90−0.11+0.30−0.08−0.02−0.13+0.18May+0.19+0.07−0.11−0.39−0.28+0.29−0.30+0.08−0.43−0.29−0.37−0.15June+0.46+0.13−0.14−0.47−0.39+0.78−0.13−0.19−0.70−0.77−0.59−0.35July+0.48+0.14−0.17−0.30−0.13+0.44−0.08−0.44−0.72−0.67−0.27−0.13August+0.47+0.11+0.01+0.08+0.05+0.52−0.18−0.38−0.47−0.23−0.05−0.19September+0.31+0.010.00+0.29+0.24−0.02+0.06−0.06−0.06+0.16+0.01+0.20October−0.07−0.11+0.09+0.06+0.01−0.26+0.03−0.04+0.31+0.27+0.190.00November−0.30−0.28−0.05+0.17+0.11−0.02+0.08−0.01+0.51+0.30+0.43+0.18December−0.36−0.14+0.05+0.26+0.23+0.05+0.35+0.06+0.55+0.19+0.24−0.29Range1.020.420.340.950.641.800.650.741.271.331.030.88§ 23. The inequalities in Table XX. may be analysed—as has in fact been done by Hann—in a series of Fourier terms, whose periods are the year and its submultiples. Fourier series can also be formed representing the annual variation in theAnnual Variation Fourier Coefficients.amplitudes of the regular diurnal inequality, and its component 24-hour, 12-hour, &c. waves, or of the amplitude of the absolute daily range (§ 24). To secure the highest theoretical accuracy, it would be necessary in calculating the Fourier coefficients to allow for the fact that the “months” from which the observational data are derived are not of uniform length. The mid-times, however, of most months of the year are but slightly displaced from the position they would occupy if the 12 months were exactly equal, and these displacements are usually neglected. The loss of accuracy cannot be but trifling, and the simplification is considerable.The Fourier series may be represented byP1sin (t + θ1) + P2sin (2t + θ2) + ...,where t is time counted from the beginning of the year, one month being taken as the equivalent of 30°, P1, P2represent the amplitudes, and θ1, θ2the phase angles of the first two terms, whose periods are respectively 12 and 6 months. Table XXI. gives the values of these coefficients in the case of the range of the regular diurnal inequality for certain specified elements and periods at Kew23and Falmouth.23aIn the case of P1and P2the unit is 1′ for D and I, and 1γ for H and V. M denotes the mean value of the range for the 12 months. The lettersqandorepresent quiet and ordinary day results. S max. means the years 1892-1895, with a mean sun spot frequency of 75.0. S min. for Kew means the years 1890, 1899 and 1900 with a mean sun spot frequency of 9.6; for Falmouth it means the years 1899-1902 with a mean sun spot frequency of 7.25.Increase in θ1or θ2means an earlier occurrence of the maximum or maxima, 1° answering roughly to one day in the case of the 12-month term, and to half a day in the case of the 6-month term. P1/M and P2/M both increase decidedly as we pass from years of many to years of few sun spots;i.e.relativelyconsidered the range of the regular diurnal inequality is more variable throughout the year when sun spots are few than when they are many.The tendency to an earlier occurrence of the maximum as we pass from quiet days to ordinary days, or from years of sun spot minimum to years of sun spot maximum, which appears in the table, appears also in the case of the horizontal force—at least in the case of the annual term—both at Kew and Falmouth. The phenomena at the two stations show a remarkably close parallelism. At both, and this is true also of the absolute ranges, the maximum of the annual term falls in all cases near midsummer, the minimum near midwinter. The maxima of the 6-month terms fall near the equinoxes.Table XXI.—Annual Variation of Diurnal Inequality Range. Fourier Coefficients.P1.P2.θ1.θ2.P1/M.P2/M.KewDo3.360.94279°280°0.400.111890-1900Dq3.811.22275°273°0.470.15Iq0.670.16264°269°0.420.10Hq13.63.0269°261°0.480.11Vq11.72.2282°242°0.630.12S max.Kew4.501.26277°282°0.470.13DqFalmouth4.101.40277°286°0.430.15S min.Kew3.351.10274°269°0.490.16DqFalmouth3.191.14275°277°0.490.17§ 24. Allusion has already been made in § 14 to one point which requires fuller discussion. If we take a European station such as Kew, the general character of, say, the declination does not vary very much with the season, but still it doesAbsolute Range.vary. The principal minimum of the day, for instance, occurs from one to two hours earlier in summer than in winter. Let us suppose for a moment that all the days of a month are exactly alike, the difference in type between successive months coming inper saltum.Suppose further that having formed twelve diurnal inequalities from the days of the individual months of the year, we deduce a mean diurnal inequality for the whole year by combining these twelve inequalities and taking the mean. The hours of maximum and minimum being different for the twelve constituents, it is obvious that the resulting maximum will normally be less than the arithmetic mean of the twelve maxima, and the resulting minimum (arithmetically) less than the arithmetic mean of the twelve minima. The range—or algebraic excess of the maximum over the minimum—in the mean diurnal inequality for the year is thus normally less than the arithmetic mean of the twelve ranges from diurnal inequalities for the individual months. Further, as we shall see later, there are differences in type not merely between the different months of the year, but even between the same months in different years. Thus the range of the mean diurnal inequality for, say, January based on the combined observations of, say, eleven Januarys may be and generally will be slightly less than the arithmetic mean of the ranges obtained from the Januarys separately. At Kew, for instance, taking the ordinary days of the 11 years 1890-1900, the arithmetic mean of the diurnal inequality ranges of declination from the 132 months treated independently was 8′.52, the mean range from the 12 months of the year (the eleven Januarys being combined into one, and so on) was 8′.44, but the mean range from the whole 4,000 odd days superposed was only 8′.03. Another consideration is this: a diurnal inequality is usually based on hourly readings, and the range deduced is thus an under-estimate unless the absolute maximum and minimum both happen to come exactly at an hour. These considerations would alone suffice to show that theabsolute rangein individual days,i.e.the difference between the algebraically largest and least values of the element found any time during the 24 hours, must on the average exceed therange in the mean diurnal inequality for the year, however this latter is formed. Other causes, moreover, are at work tending in the same direction. Even in central Europe, the magnetic curves for individual days of an ordinary month often differ widely amongst themselves, and show maxima and minima at different times of the day. In high latitudes, the variation from day to day is sometimes so great that mere eye inspection of magnetograph curves may leave one with but little idea as to the probable shape of the resultant diurnal curve for the month. Table XXII. gives the arithmetic mean of the absolute daily ranges from a few stations. The values which it assigns to the year are the arithmetic means of the 12 monthly values. The Mauritius data are for different periods, viz. declination 1875, 1880 and 1883 to 1890, horizontal force 1883 to 1890, vertical force 1884 to 1890. The other data are all for the period 1890 to 1900.
Table XIII.—Range of the Diurnal Inequality of Declination.
§ 17. Variations of inclination are connected with those of horizontal and vertical force by the relation
δI = ½ sin 2I {V−1δV − H−1δH}.
Thus in temperate latitudes where V is considerably in excess of H, whilst diurnal changes in V are usually less than those in H, it is the latter which chiefly dominate the diurnal changes in inclination. When the H influence prevails, I has its highest values at hours when H is least. This explains why the dip is above its mean value near midday at stations in Table XI. from Pavlovsk to Parc St Maur. Near the magnetic equator the vertical force has the greater influence. This alone would tend to make a minimum dip in the late forenoon, and this minimum is accentuated owing to the altered type of the horizontal force diurnal variation, whose maximum now coincides closely with the minimum in the vertical force. This accounts for the prominence of the minimum in the diurnal variation of the inclination at Kolaba and Batavia, and the large amplitude of the range. Tiflis shows an intermediate type of diurnal variation; there is a minimum near noon, as in tropical stations, but inclination is also below its mean for some hours near midnight. The type really varies at Tiflis according to the season of the year. In June—as in the mean equality from the whole year—there is a well marked double period; there is a principal minimum at 2 p.m. and a secondary one about 4 a.m.; a principal maximum about 9 a.m. and a secondary one about 6 p.m. In December, however, only a single period is recognizable, with a minimum about 8 a.m. and a maximum about 7 p.m. The type of diurnal inequality seenat the Cape of Good Hope does not differ much from that seen at Batavia. Only a single period is clearly shown. The maximum occurs about 8 or 9 p.m. throughout the year. The time of the minimum is more variable; at midsummer it occurs about 11 a.m., but at midwinter three or four hours later. At Hobart the type varies considerably with the season. In June (midwinter) a double period is visible. The principal minimum occurs about 8 a.m., as at the Cape. But, corresponding to the evening maximum seen at the Cape, there is now only a secondary maximum, the principal maximum occurring about 1 p.m. At midsummer the principal maximum is found—as at Kew or Greenwich—about 10 or 11 a.m., the principal minimum about 4 p.m.
§ 18. Even at tropical stations a considerable seasonal change is usually seen in the amplitude of the diurnal inequality in at least one of the magnetic elements. At stations in Europe, and generally in temperate latitudes, the amplitude varies notably in all the elements. Table XIII. gives particulars of the inequality range of declination derived from hourly readings at selected stations, arranged in order of latitude from north to south. The letters “a” and “q” are used in the same sense as before. At temperate stations in either hemisphere—e.g.Pavlovsk, Greenwich or Hobart—the range is conspicuously larger in summer than in winter. In northern temperate stations a decided minimum is usually apparent in December. There is, on the other hand, comparatively little variation in the range from April to August. Sometimes, as at Kew and Greenwich, there is at least a suggestion of a secondary minimum at midsummer. Manila and Trivandrum show a transition from the December minimum, characteristic of the northern stations, to the June minimum characteristic of the southern, there being two conspicuous minima in February or March and in November or October. At St Helena there are two similar minima in May and September, while a third apparently exists in December. It will be noticed that at both Pavlovsk and Kew the annual variation in the range is specially prominent in the quiet day results.
Table XIV. gives a smaller number of data analogous to those of Table XIII., comprising inequality ranges for horizontal force, vertical force and inclination. In some cases the number of years from which the data were derived seems hardly sufficient to give a smooth annual variation. It should also be noticed that unless the same group of years is employed the data from two stations are not strictly comparable. The difference between the all and quiet day vertical force data at Pavlovsk is remarkably pronounced. The general tendency in all the elements is to show a reduced range at midwinter; but in some cases there is also a distinct reduction in the range at midsummer. This double annual period is particularly well marked at Batavia.
Table XIV.—Ranges in the Diurnal Inequalities.
§ 19. When discussing diurnal inequalities it is sometimes convenient to consider the components of the horizontal force in and perpendicular to the astronomical meridian, rather than the horizontal force and declination. If N and W be the components of H to astronomical north and west, and D the westerly declination, N = H cos D, W = H sin D. Thus corresponding small variations in N, W, H and D are connected by the relations:—
δN = cos DδH − H sin DδD, δW = sin DδH + H cos DδD.
If δH and δD denote the departures of H and D at any hour of the day from their mean values, then δN and δW represent the corresponding departures of N and W from their mean values. In this way diurnal inequalities may be calculated for N and W when those for H and D are known. The formulae suppose δD to be expressed in absolute measure,i.e.1′ of arc has to be replaced by 0.0002909. If we take as an example a station at which H is .185 then HδD = .0000538 (number of minutes in δD). In other words, employing 1γ as unit of force, one replaces HδD by 5.38δD, where δD represents declination change expressed as usual in minutes of arc. In calculating diurnal inequalities for N and W, one ought, strictly speaking, to assign to H and D the exact mean values belonging to these elements for the month or the year being dealt with. For practical purposes, however, a slight departure from the true mean values is immaterial, and one can make use of a constant value for several successive years without sensible error. As an example, Table XV. gives the mean diurnal inequality for the whole year in N and W at Falmouth, as calculated from the 12 years 1891 to 1902. The unit employed is 1γ.
The data in Table XV. are closely similar to corresponding Kew data, and are presumably fairly applicable to the whole south of England for the epoch considered. At Falmouth there is comparatively little seasonal variation in the type of the diurnal variation in either N or W. The amplitude of the diurnal range varies, however, largely with the season, as will appear from Table XVI., which is based on the same 12 years as Table XV.
Diurnal inequalities in N and W lend themselves readily to the construction of what are known asvector diagrams. These are curves showing the direction and intensity at each hour of the day of the horizontal component of the disturbing force to which the diurnal inequality may be regarded as due. Figs. 7 and 8, taken from thePhil. Trans.vol. 204A, will serve as examples. They refer to the mean diurnal inequalities for the months stated at Kew (1890 to 1900) and Falmouth (1891 to 1902), thick lines relating to Kew, thin to Falmouth. NS and EW represent the geographical north-south and east-west directions; their intersection answers to the origin (thick lines for Kew, thin for Falmouth). The line from the origin to M represents the magnetic meridian. The line from the origin to any cross—the number indicating the corresponding hour counted from midnight as 0—represents the magnitude and direction at that hour of the horizontal component of the disturbing force to which the diurnal inequality may be assigned. The cross marks the point whose rectangular co-ordinates are the values of δN and δW derived from the diurnal inequalities of these elements. In figs. 7 and 8 the distances of the points N, E, S, W from their corresponding origin represents 10γ. The tendency to form a loop near midnight, seen in the November and Decembercurves, is characteristic of the winter months at Kew and Falmouth. The shape is less variable in summer than in winter; but even in summer the portion answering to the hours 6 p.m. to 6 a.m. varies a good deal. The object of presenting the Kew and Falmouth curves side by side is to emphasize the close resemblance between the magnetic phenomena at places in similar latitudes, though over 200 miles apart and exhibiting widely different ranges for their meteorological elements. With considerable change of latitude however the shape of vector diagrams changes largely.
Table XV.—Diurnal Inequalities in N. and W. at Falmouth (unit 1γ).
§ 20. Any diurnal inequality can be analysed into a series ofFourier Series.harmonic terms whose periods are 24 hours and submultiples thereof. The series may be expressed in either of the equivalent forms:—
a1cos t + b1sin t + a2cos 2t + b2sin 2t + ...
(i)
c1sin (t + α1) + c2sin (2t + α2) + ....
(ii)
Table XVI.—Ranges in Diurnal Inequalities at Falmouth (unit 1γ).
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).
The a, b, c, α constants are called sometimes Fourier, sometimes Bessel coefficients.
By taking a sufficient number of terms a series can always be obtained which will represent any set of diurnal inequality figures; but unless one can obtain a close approach to the observational figures from the terms possessing the periods 24, 12, 8 and 6 hours the physical significance and general utility of the analysis is somewhat problematical. In the case of the magnetic elements, the 24 and 12 hour terms are usually much the more important; the 24-hour term is generally, but by no means always, the larger of the two. The c constants give the amplitudes of the harmonic terms or waves, the α constants the phase angles. An advance of 1 hour in the time of occurrence of the first (and subsequent, if any) maximum and minimum answers to anincreaseof 15° in α1of 30° in α2, of 45° in α3, of 60° in α4and so on. In the case of magnetic elements the phase angles not infrequently possess a somewhat large annual variation. It is thus essential for a minute study of the phenomena at any station to carry out the analysis for the different seasons of the year, and preferably for the individual months. If the a and b constants are known for all the individual months of one year, or for all the Januarys of a series of years, we have only to take their arithmetic means to obtain the corresponding constants for the mean diurnal inequality of the year, or for the diurnal inequality of the average January of the series of years. This, however, is obviously not true of the c or α constants, unless the phase angle is absolutely unchanged throughout the contributory months or years. This is a point requiring careful attention, because when giving values of c and α for the whole year some authorities give the arithmetic mean of the c’s and α’s calculated from the diurnal inequalities of the individual months of the year, others give the values obtained for c and α from the mean diurnal inequality of the whole year. The former method inevitably supplies a larger value for c than the latter, supposing α to vary with the season. At some observatories,e.g.Greenwich and Batavia, it has long been customary to publish every year values of the Fourier coefficients for each month, and to include other elements besides the declination. For a thoroughly satisfactory comparison of different stations, it is necessary to have data from one and the same epoch; and preferably that epoch should include at least one 11-year period. There are, however, few stations which can supply the data required for such a comparison and we have to make the best of what is available. Information is naturally most copious for the declination. For this element E. Engelenburg20gives values of C1, C2, C3, C4, and of α1, α2, α3, α4for each month of the year for about 50 stations, ranging from Fort Rae (62° 6′ N. lat.) to Cape Horn (55° 5′ S. lat.). From the results for individual stations, Engelenburg derives a series of means which he regards as representative of 11 different zones of latitude. His data for individual stations refer to different epochs, and some are based on only one year’s observations. The original observations also differ in reliability; thus the results are of somewhat unequal value. The mean results for Engelenburg’s zones must naturally have some of the sources of uncertainty reduced; but then the fundamental idea represented by the arrangement in zones is open to question. The majority of the data in Table XVII. are taken from Engelenburg, but the phase angles have been altered so as to apply to westerly declination. The stations are arranged in order of latitude from north to south; in a few instances results are given for quiet days. The figures represent in all cases arithmetic means derived from the 12 monthly values. In the table, so far as is known, the local mean time of the observatory has been employed. This is a point requiring attention, because most observatories employ Greenwich time, or time based on Greenwich or some other national observatory, and any departure from local time enters into the values of the constants. The data for Victoria Land refer to the “Discovery’s” 1902-1903 winter quarters, where the declination, taken westerly, was about 207°.5.
As an example of the significance of the phase angles in Table XVII., take the ordinary day data for Kew. The times of occurrence of the maxima are given by t + 234° = 450° for the 24-hour term, 2t + 39°.7 = 90° or = 450° for the 12-hour term, and so on, taking an hour in t as equivalent to 15°.
Thus the times of the maxima are:—
24-hour term, 2 h. 24 m. p.m.; 12-hour term, 1 h. 41 m. a.m. and p.m.
8-hour term, 4 h. 41 m. a.m., 0 h. 41 m. p.m., and 8 h. 41 m. p.m.
6-hour term, 0 h. 33 m. a.m. and p.m., and 6 h. 33 m. a.m. and p.m.
The minima, or extreme easterly positions in the waves, lie midway between successive maxima. All four terms, it will be seen, have maxima at some hour between 0h. 30m. and 2h. 30m. p.m. They thus reinforce one another strongly from 1 to 2 p.m., accounting for the prominence of the maximum in the early afternoon.
The utility of a Fourier analysis depends largely on whether the several terms have a definite physical significance. If the 24-hour and 12-hour terms, for instance, represent the action of forces whose distribution over the earth or whose seasonal variation is essentially different, then the analysis helps to distinguish these forces, and may assist in their being tracked to their ultimate source. Suppose, for example, one had reason to think the magnetic diurnal variation due to some meteorological phenomenon,e.g.heating of the earth’s atmosphere, then a comparison of Fourier coefficients, if such existed, for the two sets of phenomena would be a powerful method of investigation.
Table XVII.—Amplitudes and Phase Angles for Diurnal Inequality of Declination.
§ 21. Fourier coefficients of course often vary much with the season of the year. In the case of the declination this is especially true of the phase angles at tropical stations. To enter on details for a number of stations would unduly occupy space. A fair idea of the variability in the case of declination in temperate latitudes may be derived from Table XVIII., which gives monthly values for Kew derived from ordinary days of an 11-year period 1890-1900.
Fourier analysis has been applied to the diurnal inequalities of the other magnetic elements, but more sparingly. Such results are illustrated by Table XIX., which contains data derived from quiet days at Kew from 1890 to 1900.Winterincludes November to February,SummerMay to August, andEquinoxthe remaining four months. In this case the data are derived from mean diurnal inequalities for the season specified. In the case of thecor amplitude coefficients the unit is 1′ for I (inclination), and 1γ for H and V (horizontal and vertical force). At Kew the seasonal variation in the amplitude is fairly similar for all the elements. The 24-hour and 12-hour terms tend to be largest near midsummer, and least near midwinter; but the 8-hour and 6-hour terms have two well-marked maxima near the equinoxes, and a clearly marked minimum near midsummer, in addition to one near midwinter. On the other hand, the phase angle phenomena vary much for the different elements. The 24-hour term, for instance, has its maximum earlier in winter than in summer in the case of the declination and vertical force, but the exact reverse holds for the inclination and the horizontal force.
Table XVIII.—Kew Declination: Amplitudes and Phase Angles(local mean time).
§ 22. If secular change proceeded uniformly throughout the year, the value Enof any element at the middle of the nth month of the year would be connected withE, the mean value for the whole year, by the formula En=E+ (2n − 13)s/24,Annual Inequality.where s is the secular change per annum. For the present purpose, difference in the lengths of the months may be neglected. If one applies to En−Ethe correction −(2n − 13)s/24 one eliminates a regularly progressive secular change; what remains is known as theannual inequality. If only a short period of years is dealt with, irregularities in the secular change from year to year, or errors of observation, may obviously simulate the effect of a real annual inequality. Even when a long series of years is included, there is always a possibility of a spurious inequality arising from annual variation in the instruments, or from annual change in the conditions of observation. J. Liznar,21from a study of data from a number of stations, arrived at certain mean results for the annual inequalities in declination and inclination in the northern and southern hemispheres, and J. Hann22has more recently dealt with Liznar’s and newer results. Table XX. gives a variety of data, including the mean results given by Liznar and Hann. In the case of declination + denotes westerly position; in the case of inclination it denotes a larger dip (whether the inclination be north or south). According to Liznar declination in summer is to the west of the normal position in both hemispheres. The phenomena, however, at Parc St Maur are, it will be seen, the exact opposite of what Liznar regards as normal; and whilst the Potsdam results resemble his mean in type, the range of the inequality there, as at Parc St Maur, is relatively small. Of the three sets of data given for Kew the first two are derived in a similar way to those for other stations; the first set are based on quiet days only, the second on all but highly disturbed days. Both these sets of results are fairly similar in type to the Parc St Maur results, but give larger ranges; they are thus even more opposed to Liznar’s normal type. The last set of data for Kew is of a special kind. During the 11 years 1890 to 1900 the Kew declination magnetograph showed to within 1′ the exact secular change as derived from the absolute observations; also, if any annual variation existed in the position of the base lines of the curves it was exceedingly small. Thus the accumulation of the daily non-cyclic changes shown by the curves should closely represent the combined effects of secular change and annual inequality. Eliminating the secular change, we arrive at an annual inequality, based on all days of the year including the highly disturbed. It is this annual inequality which appears under the headings. It is certainly very unlike the annual inequality derived in the usual way. Whether the difference is to be wholly assigned to the fact that highly disturbed days contribute in the one case, but not in the other, is a question for future research.
Table XIX.—Kew Diurnal Inequality: Amplitudes and Phase Angles (local mean time).
In the case of the inclination, Liznar found that in both hemispheres the dip (north in the northern, south in the southern hemisphere) was larger than the normal when the sun was in perihelion, corresponding to an enhanced value of the horizontal force in summer in the northern hemisphere.
In the case of annual inequalities, at least that of the declination, it is a somewhat suggestive fact that the range seems to become less as we pass from older to more recent results, or from shorter to longer periods of years. Thus for Paris from 1821 to 1830 Arago deduced a range of 2′ 9″. Quiet days at Kew from 1890 to 1894 gave a range of 1′.2, while at Potsdam Lüdeling got a range 30% larger than that in Table XX. when considering the shorter period 1891-1899. Up to the present, few individual results, if any, can claim a very high degree of certainty. With improved instruments and methods it may be different in the future.
Table XX.—Annual Inequality.
§ 23. The inequalities in Table XX. may be analysed—as has in fact been done by Hann—in a series of Fourier terms, whose periods are the year and its submultiples. Fourier series can also be formed representing the annual variation in theAnnual Variation Fourier Coefficients.amplitudes of the regular diurnal inequality, and its component 24-hour, 12-hour, &c. waves, or of the amplitude of the absolute daily range (§ 24). To secure the highest theoretical accuracy, it would be necessary in calculating the Fourier coefficients to allow for the fact that the “months” from which the observational data are derived are not of uniform length. The mid-times, however, of most months of the year are but slightly displaced from the position they would occupy if the 12 months were exactly equal, and these displacements are usually neglected. The loss of accuracy cannot be but trifling, and the simplification is considerable.
The Fourier series may be represented by
P1sin (t + θ1) + P2sin (2t + θ2) + ...,
where t is time counted from the beginning of the year, one month being taken as the equivalent of 30°, P1, P2represent the amplitudes, and θ1, θ2the phase angles of the first two terms, whose periods are respectively 12 and 6 months. Table XXI. gives the values of these coefficients in the case of the range of the regular diurnal inequality for certain specified elements and periods at Kew23and Falmouth.23aIn the case of P1and P2the unit is 1′ for D and I, and 1γ for H and V. M denotes the mean value of the range for the 12 months. The lettersqandorepresent quiet and ordinary day results. S max. means the years 1892-1895, with a mean sun spot frequency of 75.0. S min. for Kew means the years 1890, 1899 and 1900 with a mean sun spot frequency of 9.6; for Falmouth it means the years 1899-1902 with a mean sun spot frequency of 7.25.
Increase in θ1or θ2means an earlier occurrence of the maximum or maxima, 1° answering roughly to one day in the case of the 12-month term, and to half a day in the case of the 6-month term. P1/M and P2/M both increase decidedly as we pass from years of many to years of few sun spots;i.e.relativelyconsidered the range of the regular diurnal inequality is more variable throughout the year when sun spots are few than when they are many.
The tendency to an earlier occurrence of the maximum as we pass from quiet days to ordinary days, or from years of sun spot minimum to years of sun spot maximum, which appears in the table, appears also in the case of the horizontal force—at least in the case of the annual term—both at Kew and Falmouth. The phenomena at the two stations show a remarkably close parallelism. At both, and this is true also of the absolute ranges, the maximum of the annual term falls in all cases near midsummer, the minimum near midwinter. The maxima of the 6-month terms fall near the equinoxes.
Table XXI.—Annual Variation of Diurnal Inequality Range. Fourier Coefficients.
§ 24. Allusion has already been made in § 14 to one point which requires fuller discussion. If we take a European station such as Kew, the general character of, say, the declination does not vary very much with the season, but still it doesAbsolute Range.vary. The principal minimum of the day, for instance, occurs from one to two hours earlier in summer than in winter. Let us suppose for a moment that all the days of a month are exactly alike, the difference in type between successive months coming inper saltum.Suppose further that having formed twelve diurnal inequalities from the days of the individual months of the year, we deduce a mean diurnal inequality for the whole year by combining these twelve inequalities and taking the mean. The hours of maximum and minimum being different for the twelve constituents, it is obvious that the resulting maximum will normally be less than the arithmetic mean of the twelve maxima, and the resulting minimum (arithmetically) less than the arithmetic mean of the twelve minima. The range—or algebraic excess of the maximum over the minimum—in the mean diurnal inequality for the year is thus normally less than the arithmetic mean of the twelve ranges from diurnal inequalities for the individual months. Further, as we shall see later, there are differences in type not merely between the different months of the year, but even between the same months in different years. Thus the range of the mean diurnal inequality for, say, January based on the combined observations of, say, eleven Januarys may be and generally will be slightly less than the arithmetic mean of the ranges obtained from the Januarys separately. At Kew, for instance, taking the ordinary days of the 11 years 1890-1900, the arithmetic mean of the diurnal inequality ranges of declination from the 132 months treated independently was 8′.52, the mean range from the 12 months of the year (the eleven Januarys being combined into one, and so on) was 8′.44, but the mean range from the whole 4,000 odd days superposed was only 8′.03. Another consideration is this: a diurnal inequality is usually based on hourly readings, and the range deduced is thus an under-estimate unless the absolute maximum and minimum both happen to come exactly at an hour. These considerations would alone suffice to show that theabsolute rangein individual days,i.e.the difference between the algebraically largest and least values of the element found any time during the 24 hours, must on the average exceed therange in the mean diurnal inequality for the year, however this latter is formed. Other causes, moreover, are at work tending in the same direction. Even in central Europe, the magnetic curves for individual days of an ordinary month often differ widely amongst themselves, and show maxima and minima at different times of the day. In high latitudes, the variation from day to day is sometimes so great that mere eye inspection of magnetograph curves may leave one with but little idea as to the probable shape of the resultant diurnal curve for the month. Table XXII. gives the arithmetic mean of the absolute daily ranges from a few stations. The values which it assigns to the year are the arithmetic means of the 12 monthly values. The Mauritius data are for different periods, viz. declination 1875, 1880 and 1883 to 1890, horizontal force 1883 to 1890, vertical force 1884 to 1890. The other data are all for the period 1890 to 1900.