Formulae are also wanted to show how the value of an element, or the rate of change of an element, at a particular place has varied throughout a long period. For comparatively short periods it is best to use formulae of the type E = a + bt + ct², where E denotes the value of an element t years subsequent to some convenient epoch; a, b, c are constants to be determined from the observational data. For longer periods formulae of the type E = a + b sin (mt + n), where a, b, m and n are constants, have been used by Schott16and others with considerable success. The following examples, due to G. W. Littlehales,17for the Cape of Good Hope, will suffice for illustration:Declination (West)= 14°.63 + 15°.00 sin {0.61 (t − 1850) + 77°.8}.Inclination (South)= 49°.11 + 8°.75 sin {0.8 (t − 1850) + 34°.3}.Here t denotes the date. It is perhaps hardly necessary to point out that the extension of any of these empirical formulae—whether to places outside the surveyed area, or to times not included in the period of observation—is fraught with danger, which increases rapidly the further the extrapolation is pushed.Table VII.—Inclination (northerly) and Horizontal Force at London.Date.I.Date.I.Date.I.H.Date.I.H.° ′° ′° ′° ′157671 50180170 36.0185768 24.9.17474189167 33.2.18193160072 0182170 3.4186069 19.8.17550189567 25.4.18278167673 30183069 38.0186568 8.7.17662190067 11.8.18428172374 42183869 17.3187067 58.6.17791190567 3.8.18510177372 19185468 31.1187467 50.0.17903190867 0.9.18515178672 9Fig. 5.Bauer has employed a convenient graphical method of illustrating secular change. Radii are drawn from the centre of a sphere parallel to the direction of the freely dipping needle, and are produced to intersect the tangent plane drawn at the point which answers to the mean position of the needle during the epoch under consideration. The curve formed by the points of intersection shows the character of the secular change. Fig. 5 (slightly modified fromNature, vol. 57, p. 181) applies to London. The curve is being described in the clockwise direction. This, according to Bauer’s18own investigation, is the normal mode of description. Schott and Littlehales have found, however, a considerable number of cases where it is difficult to say whether the motion is clockwise or not, while in some stations on both the east and west shores of the Pacific it was clearly anti-clockwise. Fritsche19dealing with the secular changes from 1600 to 1885—as given by his calculated values of the magnetic elements—at 204 points of intersection of equidistant lines of latitude and longitude, found only sixty-three cases in which the motion was unmistakably clockwise, while in twenty-one cases it was clearly the opposite.§ 14. All the magnetic elements at any ordinary station show a regular variation in the solar day. To separate this from the irregular changes, means of the hourly readings must be formed making use of a number of days. The amplitude ofDiurnal Variations.the diurnal change usually varies considerably with the season of the year. Thus a diurnal inequality derived from all the days of the year combined, or from a smaller number of days selected equally from all the months of the year, can give only the average effect throughout the year. Also unless the hours of maxima and minima at a given station are but slightly variable with the season, the result obtained by combining data from all the months of the year may be a hybrid which does not very closely resemble the phenomena in the majority of individual months. This remark applies in particular to the declination at places within the tropics. One consequence is obviously to make the range of a diurnal inequality which answers to the year as a whole less than the arithmetic mean of the twelve ranges obtained for the constituent months. At stations in temperate latitudes, whilst minor differences of type do exist between the diurnal inequalities for different months of the year, the difference is mainly one of amplitude, and the mean diurnal inequality from all the months of the year gives a very fair idea of the nature of the phenomena in any individual month.Table VIII.—Diurnal Inequality of Declination, mean from whole year (+ to West).Station.Jan Mayen.St Petersburgand Pavlovsk.Greenwich.Kew.ParcSt Maur.Tiflis.Kolaba.Batavia.Mauritius.South Vic-toria Land.Latitude.71° 0′ N.59° 41′ N.51° 28′ N.51° 28′ N.48° 49′ N.41° 43′ N.18° 54′ N.6° 11′ S.20° 6′ S.77° 51′ S.Longitude.8° 28′ W.30° 29′ E.0° 0′.0° 19′ W.2° 29′ E.44° 48′ E.72° 49′ E.106° 49′ E.57° 33′ E.166° 45′ E.Period.1882-1883.1873-1885.1890-1900.1890-1900.1883-1897.1888-1898.1894-1901.1883-1894.1876-1890.1902-1903.a.q.a.q.a.a.q.a.a.q.a.a.a.q.Hour.′′′′′′′′′′′′′′1− 6.6−4.2−1.3−0.7−1.4−1.5−0.9−1.4−0.7−0.2+0.1+0.1+ 2.0+ 0.92−10.5−6.4−1.2−0.8−1.3−1.4−0.9−1.2−0.6−0.1−0.1+0.1− 2.1− 1.83−15.2−7.8−1.2−1.0−1.3−1.5−1.0−1.2−0.6−0.1−0.1+0.1− 5.2− 4.54−16.9−8.4−1.4−1.3−1.4−1.7−1.3−1.2−0.5−0.10.0+0.2− 9.4− 6.85−17.0−8.1−1.7−1.8−1.7−2.1−1.8−1.6−0.7−0.10.0+0.3−12.2− 9.06−13.7−7.0−1.9−2.3−2.1−2.4−2.3−1.9−1.2−0.6+0.1+0.4−15.3−11.77− 9.3−5.1−2.2−2.8−2.4−2.7−2.8−2.4−1.9−1.0+0.5+0.6−17.2−15.08− 6.8−3.2−2.5−3.2−2.5−2.8−3.1−2.7−2.4−1.2+1.3+1.1−21.5−17.39− 3.7−0.6−2.3−3.0−1.9−2.1−2.5−2.3−2.3−0.7+1.7+1.8−23.5−18.110− 2.4+2.1−1.0−1.7−0.2−0.3−0.7−0.5−0.90.0+1.5+1.9−21.2−15.811− 0.5+4.6+1.0+0.4+2.1+2.2+1.7+2.0+1.0+0.9+0.9+1.3−15.3− 9.2Noon+ 2.5+6.5+3.1+2.7+4.2+4.3+3.9+4.2+2.6+1.4+0.10.0− 9.8− 4.91+ 3.7+7.3+4.6+4.3+5.1+5.3+4.8+5.3+3.3+1.2−0.6−1.1− 3.2− 0.12+ 6.4+7.1+4.9+4.5+4.7+4.9+4.4+4.9+3.1+0.6−1.1−2.0+ 3.8+ 5.93+ 7.4+5.9+4.1+3.6+3.6+3.7+3.1+3.7+2.3+0.1−1.3−2.3+11.1+ 9.54+ 8.5+4.3+2.7+2.3+2.2+2.4+1.8+2.3+1.3−0.2−1.2−1.8+16.6+12.95+10.6+3.0+1.5+1.3+1.1+1.2+0.7+1.1+0.6−0.1−0.9−0.9+19.9+14.66+14.2+2.3+0.6+0.7+0.3+0.4+0.2+0.2+0.20.0−0.6−0.1+22.0+15.57+15.2+2.20.0+0.4−0.3−0.2−0.1−0.4+0.1+0.1−0.4+0.1+22.0+15.98+15.8+2.6−0.4+0.2−0.9−0.6−0.3−0.9−0.1+0.2−0.2+0.1+19.9+14.69+13.2+2.6−1.00.0−1.2−1.0−0.5−1.3−0.4+0.10.0+0.1+16.0+10.610+ 7.4+2.0−1.4−0.2−1.5−1.3−0.7−1.5−0.60.0+0.1+0.1+11.6+ 7.211+ 1.1+0.5−1.6−0.4−1.6−1.4−0.8−1.6−0.70.0+0.1+0.1+ 7.6+ 4.212− 3.6−1.8−1.5−0.6−1.6−1.5−0.9−1.6−0.8−0.1+0.1+0.1+ 3.3+ 1.9Range32.815.77.47.77.68.17.98.05.72.63.04.245.534.0Tables VIII. to XI. give mean diurnal inequalities derived from all the months of the year combined, the figures representing the algebraic excess of the hourly value over the mean for the twenty-four hours. The + sign denotes in Table VIII. that the north end of the needle is to the west of its mean position for the day; in Tables IX. to XI. it denotes that the element—the dip being the north or south as indicated—is numerically in excess of the twenty-four hour mean. The letter “a” denotes that all days have been included except, as a rule, those characterized by specially large disturbances. The letter “q” denotes that the results are derived from a limited number of days selected as being specially quiet,i.e.free from disturbance. In all cases the aperiodic or non-cyclic element—indicated by a difference between the values found for the first and second midnights of the day—has been eliminated in the usual way,i.e.by treating it as accumulating at a uniform rate throughout the twenty-four hours. The years from which the data were derived are indicated. The algebraically greatest and least of the hourly values are printed in heavy type; the range thence derived is given at the foot of the tables.Table IX.—Diurnal Inequality of Horizontal Force, mean from whole year (Unit 1γ = .00001 C.G.S.)Station.Jan Mayen.St Petersburgand Pavlovsk.Greenwich.Kew.ParcSt Maur.Tiflis.Kolaba.Batavia.Mauritius.S. VictoriaLand.Period.1882-1883.1873-1885.1890-1900.1890-1900.1883-1897.1888-1898.1894-1901.1883-1894.1883-1890.1902-1903.a.q.a.q.a.q.a.a.q.a.a.a.Hour.1−57−22+ 4+ 5+ 4+ 4+ 5+ 3−10−11− 3−122−64−24+ 4+ 4+ 3+ 4+ 5+ 3− 9−10− 1−133−74−25+ 4+ 4+ 3+ 4+ 5+ 3− 9− 8+ 1−144−69−24+ 4+ 4+ 3+ 4+ 5+ 4− 9− 7+ 2−155−60−22+ 5+ 4+ 3+ 4+ 6+ 4− 9− 5+ 3−156−37−19+ 4+ 4+ 1+ 2+ 4+ 4− 7− 1+ 4−127−15−15+ 2+ 2− 3− 1+ 1+ 2− 1+ 5+ 7− 98− 1−13− 3− 4− 9− 7− 5− 3+ 8+14+ 9− 79+ 8−12−10−10−16−13−12− 8−19+24+ 9− 310+17−12−16−16−20−18−17−10+26+31+9+ 311+32−10−19−20−19−18−16− 7+30+35+ 9+ 7Noon+49−4−17−18−13−12−12− 1+26+31+ 8+121+65+ 8−12−13− 7− 7− 7+ 4+19+22+ 7+182+78+22− 6− 6− 1− 2− 4+5+10+10+ 2+203+89+3700+ 2+ 1− 1+ 3+ 2− 1− 2+194+83+43+ 3+ 3+ 5+ 30− 1− 3− 9− 6+185+68+49+ 5+ 5+ 7+ 5+ 2− 4− 7−13− 7+156+37+43+ 6+ 6+ 9+ 7+ 4− 6− 8−14− 7+117+13+30+ 7+ 7+10+8+ 6− 4− 9−15− 7+ 58−11+15+ 8+ 8+10+ 8+7− 1−10−16− 8+ 09−33+ 1+ 9+ 9+ 8+ 7+ 7+ 1−11−16−8− 410−36−10+ 8+ 9+ 7+ 6+ 6+ 2−11−16− 8− 711−40−16+ 7+ 8+ 6+ 6+ 6+ 3−10−15− 7− 912−51−20+ 6+ 6+ 5+ 5+ 6+ 3−10−13− 5−11Range1637428293026241541511735Table X.—Diurnal Inequality of Vertical Force, mean from whole year (Unit 1γ).Station.Jan Mayen.St Petersburgand Pavlovsk.Greenwich.Kew.Parc StMaur.Tiflis.Kolaba.Batavia.Mauritius.South Vic-toria Land.Period.1882-1883.1873-1885.1890-1900.1891-1900.1883-1897.1888-1898.1894-1901.1883-1894.1884-1890.1902-1903.a.q.a.q.a.q.a.a.q.a.a.a.Hour1+65+ 3− 7− 1− 3+ 10+ 2+ 4+ 7+ 2+132+65+ 2−7− 1− 4+ 10+ 2+ 4+ 5+ 2+123+56− 1− 7− 1− 40− 1+ 1+ 3+ 4+ 2+104+37− 5− 60− 300+ 1+ 3+ 3+ 2+ 85+16− 7− 50− 2+ 10+ 2+ 5+ 2+ 2+ 36− 7− 8− 40− 1+ 1+ 1+ 3+7+ 1+ 207−17− 6− 3000+ 1+ 3+ 60+ 308−14− 4− 200− 10+ 30− 3+ 4− 29− 90− 3− 1− 3−4− 4− 1− 8−11+5− 610− 6+ 5− 2− 2− 6− 8− 8− 7−14−20+ 3−1311− 6+10− 3− 4− 9−11−12−11−15−260−17Noon−10+16− 3−5−10−11−12−11−10−27− 4−201−13+21− 1− 4− 6− 8− 9− 9− 3−21− 7−202−24+23+ 2− 10− 3− 3− 5+ 1−13− 9−163−31+20+ 8+ 2+ 5+ 2+ 2− 1+ 4− 4− 8−124−40+13+ 9+ 3+ 8+ 5+ 6+ 1+ 3+ 4− 5− 65−48+ 2+10+ 3+ 9+ 6+ 7+ 30+10− 3− 16−53− 9+10+3+10+7+8+40+130+ 37−47−18+ 9+ 3+ 9+ 6+ 7+ 30+140+ 68−36−20+ 8+ 3+ 7+ 5+ 6+ 3+ 1+14+ 1+ 99− 7−19+ 6+ 2+ 5+ 5+ 5+ 3+ 2+14+ 2+1110+18−13+ 3+ 2+ 3+ 4+ 3+ 3+ 3+13+ 2+1211+42− 5− 200+ 3+ 2+ 3+ 3+11+ 2+1212+540− 5− 1− 2+ 2+ 1+ 2+ 3+ 9+ 2+13Range118431782018201522411433When comparing results from different stations, it must be remembered that the disturbing forces required to cause a change of 1′ in declination and in dip vary directly, the former as the horizontal force, the latter as the total force. Near a magnetic pole the horizontal force is relatively very small, and this accounts, at least partly, for the difference between the declination phenomena at Jan Mayen and South Victoria Land on the one hand and at Kolaba, Batavia and Mauritius on the other. There is, however, another cause, already alluded to, viz. the variability in the type of the diurnal inequality in tropical stations. With a view to illustrating this point Table XII. gives diurnal inequalities of declination for June and December for a number of stations lying between 45° N. and 45° S. latitude. Some of the results are represented graphically in fig. 6, plus ordinates representing westerly deflection. At the northmost station, Toronto, the difference between the two months is mainly a matter of amplitude, the range being much larger at midsummer than at midwinter. The conspicuous phenomenon at both seasons is the rapid swing to the west from 8 or 9 a.m. to1 or 2 p.m. At the extreme southern station, Hobart—at nearly equal latitude—the rapid diurnal movement is to the east, and so in the opposite direction to that in the northern hemisphere, but it again takes place at nearly the same hours in June (midwinter) as in December. If, however, we take a tropical station such as Trivandrum or Kolaba, the phenomena in June and December are widely different in type. At Trivandrum—situated near the magnetic equator in India—we have in June the conspicuous forenoon swing to the west seen at Toronto, occurring it is true slightly earlier in the day; but in December at the corresponding hours the needle is actually swinging to the east, just as it is doing at Hobart. In June the diurnal inequality of declination at tropical stations—whether to the north of the equator like Trivandrum, or to the south of it like Batavia—is on the whole of the general type characteristic of temperate regions in the northern hemisphere; whereas in December the inequality at these stations resembles that of temperate regions in the southern hemisphere. Comparing the inequalities for June in Table XII. amongst themselves, and those for December amongst themselves, one can trace a gradual transformation from the phenomena seen at Toronto to those seen at Hobart. At a tropical station the change from the June to the December type is probably in all cases more or less gradual, but at some stations the transition seems pretty rapid.Table XI.—Diurnal Inequality of Inclination mean from whole year.Station.Jan Mayen.St Petersburgand Pavlovsk.Greenwich.Kew.ParcSt Maur.Tiflis.Kolaba.Batavia.Mauritius.South Vic-toria Land.End DippingNorth.North.North.North.North.North.North.South.South.South.Period.1882-1883.1873-1885.1890-1900.1891-1900.1883-1897.1888-1898.1894-1901.1883-1894.1884-1890.1902-1903.a.q.a.q.a.q.a.a.q.a.a.a.Hour′′′′′′′′′′′′1+4.6+1.5−0.5−0.3−0.4−0.3−0.3−0.1+0.6+0.9+0.3+0.62+5.0+1.6−0.5−0.3−0.3−0.2−0.3−0.1+0.6+0.8+0.2+0.73+5.6+1.6−0.5−0.3−0.3−0.2−0.3−0.1+0.5+0.60.0+0.74+5.0+1.5−0.4−0.3−0.3−0.2−0.4−0.2+0.5+0.5−0.0+0.75+4.2+1.4−0.5−0.3−0.2−0.2−0.4−0.2+0.7+0.3−0.1+0.76+2.4+1.2−0.4−0.3−0.1−0.1−0.3−0.1+0.8+0.1−0.2+0.57+0.7+0.9−0.2−0.1+0.2+0.10.00.0+0.5−0.2−0.3+0.48−0.1+0.8+0.1+0.3+0.6+0.4+0.4+0.3−0.2−0.8−0.4+0.39−0.7+0.8+0.6+0.6+1.0+0.8+0.7+0.5−1.2−1.7−0.4+0.110−1.2+0.9+1.0+1.0+1.1+1.0+0.9+0.3−1.9−2.7−0.5−0.211−2.2+0.8+1.2+1.2+1.0+0.9+0.70.0−2.1−3.3−0.6−0.4Noon−3.4+0.4+1.1+1.1+0.6+0.6+0.4−0.5−1.6−3.1−0.7−0.71−4.5−0.2+0.7+0.7+0.3+0.2+0.2−0.6−0.8−2.4−0.8−0.92−5.6−1.2+0.4+0.4+0.1+0.1+0.2−0.5−0.2−1.3−0.6−1.03−6.3−2.2+0.2+0.10.00.0+0.2−0.3+0.3−0.2−0.3−1.04−6.1−2.90.0−0.1−0.1−0.1+0.2+0.1+0.3+0.7+0.1−0.95−5.1−3.2−0.1−0.3−0.2−0.2+0.1+0.4+0.2+1.3+0.4−0.76−3.1−2.9−0.2−0.3−0.3−0.30.0+0.5+0.2+1.5+0.5−0.57−1.7−2.2−0.3−0.4−0.4−0.4−0.2+0.4+0.3+1.6+0.5−0.28+0.3−1.3−0.3−0.5−0.4−0.4−0.3+0.2+0.4+1.6+0.60.09+2.0−0.3−0.4−0.6−0.4−0.4−0.3+0.1+0.5+1.6+0.6+0.210+2.5+0.5−0.5−0.6−0.4−0.3−0.30.0+0.6+1.5+0.6+0.411+3.0+1.0−0.5−0.6−0.4−0.3−0.30.0+0.6+1.4+0.5+0.512+4.0+1.3−0.5−0.4−0.4−0.3−0.3−0.1+0.6+1.2+0.4+0.6Range11.94.81.71.81.51.41.31.12.94.91.41.7Table XII.—Diurnal Inequality of Declination (+ to West).Station.Toronto.Kolaba.Trivandrum.Batavia.St Helena.Mauritius.Cape.Hobart.Month.June.Dec.June.Dec.June.Dec.June.Dec.June.Dec.June.Dec.June.Dec.June.Dec.Hour′′′′′′′′′′′′′′′′1−0.4−0.1−0.30.0−0.3−0.1+0.1+0.1−0.1−0.40.0+0.1−0.4−0.7+0.8+1.12−0.2+0.4−0.3+0.1−0.4+0.1−0.1+0.1−0.2−0.1−0.2+0.2−0.5−0.4+0.3+1.13−0.2−0.1−0.3+0.1−0.4+0.3−0.2+0.2−0.2+0.1−0.2+0.4−0.7−0.1−0.1+1.04−1.2−0.4−0.3+0.3−0.5+0.5−0.3+0.3−0.3+0.3−0.2+0.7−0.6+0.3−0.1+1.15−2.9−0.6−0.7+0.4−0.7+0.7−0.3+0.5−0.5+0.6−0.3+1.0−0.7+1.00.0+1.76−5.2−0.6−1.6+0.5−1.6+1.1−0.5+1.2−1.0+0.9−0.4+1.7−1.0+2.20.0+2.77−6.2−0.9−2.2+0.7−1.7+1.4−1.1+2.0−2.2+1.9−1.1+2.6−1.6+3.3−0.1+4.48−6.0−1.2−2.1+0.2−1.1+0.9−0.4+2.3−1.5+2.2−1.0+2.4−0.8+3.6+0.1+5.69−4.4−1.8−1.1−0.1−0.2+0.5+0.5+2.0−0.3+1.3+0.2+2.0+0.7+3.1+0.6+5.610−1.5−1.10.0−0.2+0.6+0.3+0.9+1.3+0.3+0.2+1.2+1.1+1.6+1.6+1.2+3.611+2.1+0.6+1.20.0+1.2+0.1+1.0+0.4+0.5−1.0+1.40.0+1.5+0.1+1.0+0.7Noon+4.8+2.2+2.10.0+1.4−0.4+0.7−0.6+0.3−1.4+1.0−1.4+0.8−1.0−0.1−2.61+6.1+3.2+2.0−0.2+1.1−0.8+0.3−1.4+0.3−1.2+0.1−2.2+0.3−1.8−1.4−5.12+6.1+3.2+1.6−0.3+0.7−0.9−0.2−1.8+0.2−0.4−0.9−2.5−0.3−1.9−2.2−6.23+5.2+2.4+0.9−0.3+0.3−0.9−0.7−1.9+0.2+0.4−1.5−2.2−0.3−1.4−2.4−5.84+3.6+1.5+0.2−0.3+0.1−0.8−0.8−1.6+0.7+0.6−1.3−1.6+0.2−0.8−1.6−4.85+1.8+0.50.0−0.20.0−0.4−0.5−1.2+1.1+0.4−0.3−1.0+0.5−0.8−0.7−3.36+0.7−0.1+0.1−0.2+0.2−0.4−0.1−0.7+1.0+0.1+0.5−0.5+0.5−0.6−0.4−1.970.0−0.8+0.3−0.2+0.5−0.4+0.1−0.6+0.6−0.4+0.7−0.3+0.4−0.80.0−1.080.0−1.2+0.4−0.1+0.5−0.3+0.2−0.5+0.5−0.7+0.7−0.3+0.3−0.9+0.5−0.39−0.5−1.4+0.3−0.1+0.4−0.2+0.4−0.3+0.4−0.9+0.6−0.2+0.2−0.9+1.10.010−0.5−1.7+0.10.0+0.2−0.1+0.4−0.1+0.2−1.0+0.4−0.1+0.1−1.0+1.3+0.611−0.7−1.1−0.1−0.10.0−0.1+0.30.0+0.1−0.8+0.30.00.0−1.0+1.3+0.912−0.6−0.7−0.2−0.1−0.2−0.1+0.2+0.1−0.1−0.6+0.1+0.1−0.2−1.0+1.1+1.2Range12.35.04.31.03.12.32.14.23.33.62.95.13.25.53.711.8§ 15. In the case of the horizontal force there are, as Table IX. shows, two markedly different types of diurnal inequality. In the one type, exemplified by Pavlovsk or Greenwich, the force is below its mean value in the middle of the day; it has a principal minimum about 10 or 11 a.m., and morning and evening maxima, the latter usually the largest. In the other type, exemplified by Kolaba or Batavia, the horizontal force is above its mean in the middle of theday, and has a maximum about 11 a.m. The second type may be regarded as the tropical type. At tropical stations, such as Kolaba, Batavia, Manila and St Helena, the type is practically the same in summer as in winter, and is the same whether the station is north or south of the equator. Similarly, what we may call the temperate type is seen—with comparatively slight modifications—both in summer and winter at stations such as Greenwich or Pavlovsk. In winter, it is true, the pronounced daily minimum is a little later and the early morning maximum is relatively more important than in summer. There is not, as in the case of the declination, any essential difference between the phenomena at temperate stations in the northern and southern hemispheres.Fig. 6.With diminishing latitude, there is a gradual transition from the temperate to the tropical type of horizontal force diurnal variation, and at stations whose latitude is under 45° there is a very appreciable variation in type with the season. The mean diurnal variation for the year at Tiflis in Table IX. really represents a struggle between the two types, in which on the whole the temperate type prevails. If we take the diurnal variations at Tiflis for midsummer and midwinter, we find the former essentially of the temperate, the latter essentially of the tropical type. A similar conflict may be seen in the mean diurnal inequality for the year at the Cape of Good Hope, but there the tropical type on the whole predominates, and it prevails more at midwinter than at midsummer. Toronto and Hobart, though similar in latitude to Tiflis, show a closer approach to the temperate type. Still at both stations the hours during which the force is below its mean value tend to extend back towards midnight, especially at midsummer. The amplitude of the horizontal force range appears less at intermediate stations, such as Tiflis, than at stations in either higher or lower latitudes. There is a very great difference in this respect between the north and the south of India.§ 16. In the case of the vertical force in higher temperate latitudes—at Pavlovsk for instance—the diurnal inequalities from “all” and from “quiet” days differ somewhat widely in amplitude and slightly even in type. In mean latitudes,e.g.at Tiflis, there is often a well marked double period in the mean diurnal inequality for the whole year; but even at Tiflis this is hardly, if at all, apparent in the winter months. In the summer months the double period is distinctly seen at Kew and Greenwich, though the evening maximum is always pre-eminent. Speaking generally, the time of the minimum, or principal minimum, varies much less with the season than that of the maximum. At Kew, for instance, on quiet days the minimum falls between 11 a.m. and noon in almost all the months of the year, but the time of the maximum varies from about 4 p.m. in December to 7 p.m. in June. At Kolaba the time of the minimum is nearly independent of the season; but the changes from positive to negative in the forenoon and from negative to positive in the afternoon are some hours later in winter than in summer. At Batavia the diurnal inequality varies very little in type with the season, and there is little evidence of more than one maximum and minimum in the day. At Batavia, as at Kolaba, negative values occur near noon; but it must be remembered that while at Kolaba and more northern stations vertical force urges the north pole of a magnet downwards, the reverse is true of Batavia, as the dip is southerly. At St Helena vertical force is below its mean value in the forenoon, but the change from − to + occurs at noon, or but little later, both in winter and summer. At the Cape of Good Hope the phenomena at midsummer are similar to those at Kolaba, the force being below its mean value from about 9 a.m. to 3 p.m. and above it throughout the rest of the day; but at midwinter there is a conspicuous double period, the force being below its mean from 1 a.m. to 7 a.m. as well as from 11 a.m. to 3 p.m., and thus resembling the all-day annual results at Greenwich. At Hobart vertical force is below its mean value from 1 a.m. to 9 a.m. at midsummer, and from 4 a.m. to noon at midwinter; while the force is above its mean persistently throughout the afternoon both in summer and winter, there is at midwinter a well marked secondary minimum about 6 p.m., almost the same hour as that at which the maximum for the day is observed in summer.
Formulae are also wanted to show how the value of an element, or the rate of change of an element, at a particular place has varied throughout a long period. For comparatively short periods it is best to use formulae of the type E = a + bt + ct², where E denotes the value of an element t years subsequent to some convenient epoch; a, b, c are constants to be determined from the observational data. For longer periods formulae of the type E = a + b sin (mt + n), where a, b, m and n are constants, have been used by Schott16and others with considerable success. The following examples, due to G. W. Littlehales,17for the Cape of Good Hope, will suffice for illustration:
Here t denotes the date. It is perhaps hardly necessary to point out that the extension of any of these empirical formulae—whether to places outside the surveyed area, or to times not included in the period of observation—is fraught with danger, which increases rapidly the further the extrapolation is pushed.
Table VII.—Inclination (northerly) and Horizontal Force at London.
Bauer has employed a convenient graphical method of illustrating secular change. Radii are drawn from the centre of a sphere parallel to the direction of the freely dipping needle, and are produced to intersect the tangent plane drawn at the point which answers to the mean position of the needle during the epoch under consideration. The curve formed by the points of intersection shows the character of the secular change. Fig. 5 (slightly modified fromNature, vol. 57, p. 181) applies to London. The curve is being described in the clockwise direction. This, according to Bauer’s18own investigation, is the normal mode of description. Schott and Littlehales have found, however, a considerable number of cases where it is difficult to say whether the motion is clockwise or not, while in some stations on both the east and west shores of the Pacific it was clearly anti-clockwise. Fritsche19dealing with the secular changes from 1600 to 1885—as given by his calculated values of the magnetic elements—at 204 points of intersection of equidistant lines of latitude and longitude, found only sixty-three cases in which the motion was unmistakably clockwise, while in twenty-one cases it was clearly the opposite.
§ 14. All the magnetic elements at any ordinary station show a regular variation in the solar day. To separate this from the irregular changes, means of the hourly readings must be formed making use of a number of days. The amplitude ofDiurnal Variations.the diurnal change usually varies considerably with the season of the year. Thus a diurnal inequality derived from all the days of the year combined, or from a smaller number of days selected equally from all the months of the year, can give only the average effect throughout the year. Also unless the hours of maxima and minima at a given station are but slightly variable with the season, the result obtained by combining data from all the months of the year may be a hybrid which does not very closely resemble the phenomena in the majority of individual months. This remark applies in particular to the declination at places within the tropics. One consequence is obviously to make the range of a diurnal inequality which answers to the year as a whole less than the arithmetic mean of the twelve ranges obtained for the constituent months. At stations in temperate latitudes, whilst minor differences of type do exist between the diurnal inequalities for different months of the year, the difference is mainly one of amplitude, and the mean diurnal inequality from all the months of the year gives a very fair idea of the nature of the phenomena in any individual month.
Table VIII.—Diurnal Inequality of Declination, mean from whole year (+ to West).
Tables VIII. to XI. give mean diurnal inequalities derived from all the months of the year combined, the figures representing the algebraic excess of the hourly value over the mean for the twenty-four hours. The + sign denotes in Table VIII. that the north end of the needle is to the west of its mean position for the day; in Tables IX. to XI. it denotes that the element—the dip being the north or south as indicated—is numerically in excess of the twenty-four hour mean. The letter “a” denotes that all days have been included except, as a rule, those characterized by specially large disturbances. The letter “q” denotes that the results are derived from a limited number of days selected as being specially quiet,i.e.free from disturbance. In all cases the aperiodic or non-cyclic element—indicated by a difference between the values found for the first and second midnights of the day—has been eliminated in the usual way,i.e.by treating it as accumulating at a uniform rate throughout the twenty-four hours. The years from which the data were derived are indicated. The algebraically greatest and least of the hourly values are printed in heavy type; the range thence derived is given at the foot of the tables.
Table IX.—Diurnal Inequality of Horizontal Force, mean from whole year (Unit 1γ = .00001 C.G.S.)
Table X.—Diurnal Inequality of Vertical Force, mean from whole year (Unit 1γ).
When comparing results from different stations, it must be remembered that the disturbing forces required to cause a change of 1′ in declination and in dip vary directly, the former as the horizontal force, the latter as the total force. Near a magnetic pole the horizontal force is relatively very small, and this accounts, at least partly, for the difference between the declination phenomena at Jan Mayen and South Victoria Land on the one hand and at Kolaba, Batavia and Mauritius on the other. There is, however, another cause, already alluded to, viz. the variability in the type of the diurnal inequality in tropical stations. With a view to illustrating this point Table XII. gives diurnal inequalities of declination for June and December for a number of stations lying between 45° N. and 45° S. latitude. Some of the results are represented graphically in fig. 6, plus ordinates representing westerly deflection. At the northmost station, Toronto, the difference between the two months is mainly a matter of amplitude, the range being much larger at midsummer than at midwinter. The conspicuous phenomenon at both seasons is the rapid swing to the west from 8 or 9 a.m. to1 or 2 p.m. At the extreme southern station, Hobart—at nearly equal latitude—the rapid diurnal movement is to the east, and so in the opposite direction to that in the northern hemisphere, but it again takes place at nearly the same hours in June (midwinter) as in December. If, however, we take a tropical station such as Trivandrum or Kolaba, the phenomena in June and December are widely different in type. At Trivandrum—situated near the magnetic equator in India—we have in June the conspicuous forenoon swing to the west seen at Toronto, occurring it is true slightly earlier in the day; but in December at the corresponding hours the needle is actually swinging to the east, just as it is doing at Hobart. In June the diurnal inequality of declination at tropical stations—whether to the north of the equator like Trivandrum, or to the south of it like Batavia—is on the whole of the general type characteristic of temperate regions in the northern hemisphere; whereas in December the inequality at these stations resembles that of temperate regions in the southern hemisphere. Comparing the inequalities for June in Table XII. amongst themselves, and those for December amongst themselves, one can trace a gradual transformation from the phenomena seen at Toronto to those seen at Hobart. At a tropical station the change from the June to the December type is probably in all cases more or less gradual, but at some stations the transition seems pretty rapid.
Table XI.—Diurnal Inequality of Inclination mean from whole year.
Table XII.—Diurnal Inequality of Declination (+ to West).
§ 15. In the case of the horizontal force there are, as Table IX. shows, two markedly different types of diurnal inequality. In the one type, exemplified by Pavlovsk or Greenwich, the force is below its mean value in the middle of the day; it has a principal minimum about 10 or 11 a.m., and morning and evening maxima, the latter usually the largest. In the other type, exemplified by Kolaba or Batavia, the horizontal force is above its mean in the middle of theday, and has a maximum about 11 a.m. The second type may be regarded as the tropical type. At tropical stations, such as Kolaba, Batavia, Manila and St Helena, the type is practically the same in summer as in winter, and is the same whether the station is north or south of the equator. Similarly, what we may call the temperate type is seen—with comparatively slight modifications—both in summer and winter at stations such as Greenwich or Pavlovsk. In winter, it is true, the pronounced daily minimum is a little later and the early morning maximum is relatively more important than in summer. There is not, as in the case of the declination, any essential difference between the phenomena at temperate stations in the northern and southern hemispheres.
With diminishing latitude, there is a gradual transition from the temperate to the tropical type of horizontal force diurnal variation, and at stations whose latitude is under 45° there is a very appreciable variation in type with the season. The mean diurnal variation for the year at Tiflis in Table IX. really represents a struggle between the two types, in which on the whole the temperate type prevails. If we take the diurnal variations at Tiflis for midsummer and midwinter, we find the former essentially of the temperate, the latter essentially of the tropical type. A similar conflict may be seen in the mean diurnal inequality for the year at the Cape of Good Hope, but there the tropical type on the whole predominates, and it prevails more at midwinter than at midsummer. Toronto and Hobart, though similar in latitude to Tiflis, show a closer approach to the temperate type. Still at both stations the hours during which the force is below its mean value tend to extend back towards midnight, especially at midsummer. The amplitude of the horizontal force range appears less at intermediate stations, such as Tiflis, than at stations in either higher or lower latitudes. There is a very great difference in this respect between the north and the south of India.
§ 16. In the case of the vertical force in higher temperate latitudes—at Pavlovsk for instance—the diurnal inequalities from “all” and from “quiet” days differ somewhat widely in amplitude and slightly even in type. In mean latitudes,e.g.at Tiflis, there is often a well marked double period in the mean diurnal inequality for the whole year; but even at Tiflis this is hardly, if at all, apparent in the winter months. In the summer months the double period is distinctly seen at Kew and Greenwich, though the evening maximum is always pre-eminent. Speaking generally, the time of the minimum, or principal minimum, varies much less with the season than that of the maximum. At Kew, for instance, on quiet days the minimum falls between 11 a.m. and noon in almost all the months of the year, but the time of the maximum varies from about 4 p.m. in December to 7 p.m. in June. At Kolaba the time of the minimum is nearly independent of the season; but the changes from positive to negative in the forenoon and from negative to positive in the afternoon are some hours later in winter than in summer. At Batavia the diurnal inequality varies very little in type with the season, and there is little evidence of more than one maximum and minimum in the day. At Batavia, as at Kolaba, negative values occur near noon; but it must be remembered that while at Kolaba and more northern stations vertical force urges the north pole of a magnet downwards, the reverse is true of Batavia, as the dip is southerly. At St Helena vertical force is below its mean value in the forenoon, but the change from − to + occurs at noon, or but little later, both in winter and summer. At the Cape of Good Hope the phenomena at midsummer are similar to those at Kolaba, the force being below its mean value from about 9 a.m. to 3 p.m. and above it throughout the rest of the day; but at midwinter there is a conspicuous double period, the force being below its mean from 1 a.m. to 7 a.m. as well as from 11 a.m. to 3 p.m., and thus resembling the all-day annual results at Greenwich. At Hobart vertical force is below its mean value from 1 a.m. to 9 a.m. at midsummer, and from 4 a.m. to noon at midwinter; while the force is above its mean persistently throughout the afternoon both in summer and winter, there is at midwinter a well marked secondary minimum about 6 p.m., almost the same hour as that at which the maximum for the day is observed in summer.