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Encyclopedia Britannica - Main :: LUP-MAL |
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MAGNETISM, TERRESTRIAL , the science which has for its province the study of the magnetic phenomena of the earth. 1. Terrestrial magnetism has a long history. Its early growth was slow, and considerable uncertainty prevails as to its earliest developments. The properties of the magnet Historical. (see MAGNETISM) were to some small extent known to the Greeks and Romans before the Christian era, and compasses (see COMPASS) of an elementary character seem to have been employed in Europe at least as early as the 12th century. In China and Japan compasses of a kind seem to have existed at a much earlier date, and it is even claimed that the Chinese were aware of the declination of the compass needle from the true north before the end of the 11th century. Early scientific know-ledge was usually, however, a mixture of facts, very imperfectly ascertained, with philosophical imaginings. When an early writer makes a statement which to a modern reader suggests a knowledge of the declination of the compass, he may have had no such definite idea in his mind. So far as Western civilization is concerned, Columbus is usually credited with the discoveryin 1492 during his first voyage to Americathat the pointing of the compass needle to the true north represents an exceptional state of matters, and that a declination in general exists, varying from place to place. The credit of these discoveries is not, how-ever, universally conceded to Columbus. G. Hellmann 6 * considers it almost certain that the departure of the needle from the true north was known in Europe before the time of Columbus. There is indirect evidence that the declination of the compass was not known in Europe in the early part of the 15th century, through the peculiarities shown by early maps believed to have been drawn solely by regard to the compass. Whether Columbus was the first to observe the declination or not, his date is at least approximately that of its discovery. The next fundamental discovery is usually ascribed to Robert Norman, an English instrument maker. In The Newe Attractive (1581) Norman describes his discovery made some years before of the inclination or dip. The discovery was made more or less by accident, through Norman's noticing that compass needles which were truly balanced so as to be horizontal when unmagnetized, ceased to be so after being stroked with a magnet. Norman devised a form of dip-circle, and found a value for the inclination in London which was at least not very wide of the mark. Another fundamental discovery, that of the secular change of the declination, was made in England by Henry Gellibrand, professor of mathematics at Gresham College, who described it in his Discourse Mathematical on the Variation of the Magneticall Needle together with its Admirable Diminution lately discovered (1635). The history of this discovery affords a curious example of knowledge long delayed. William Borough, in his Discourse on the Variation of the Compas or Magneticall Needle (1581), gave for the declination at Limehouse in October 158o the value II4 E. approximately. Observations were repeated at Lime-house, Gellibrand tells us, in 1622 by his colleague Edmund Gunter, professor of astronomy at Gresham College, who found the much smaller value 6 13'. The difference seems to have been ascribed at first to error on Borough's part, and no suspicion of the truth seems to have been felt until 1633, when some rough observations gave a value still lower than that found by Gunter, * For explanation of these nurnbers, see end of article. It was not until midsummer 1634 that Gellibrand felt sure of his facts, and yet the change of declination since 158o exceeded 7 The delay probably arose from the strength of the preconceived idea, apparently universally held, that the declination was absolutely fixed. This idea, it would appear, derived some of its strength from the positive assertion made on the point by Gilbert of Colchester in his De magnete (1600). A third fundamental discovery, that of the diurnal change in the declination, is usually credited to George Graham (16751751), a London instrument maker. Previous observers, e.g. Gellibrand, had obtained slightly different values for the declination at different hours of the day, but it was natural to assign them to instrumental uncertainties. In those days the usual declination instrument was the compass with pivoted needles, and Graham himself at first assigned the differences he observed to friction. The observations on which he based his conclusions were made in 1722; an account of them was communicated to the Royal Society and published in the Philosophical Transactions for 1724. The movements of the compass needle throughout the average day represent partly a regular diurnal variation, and partly irregular changes in the declination. The distinction, however, was not at first very clearly realized. Between 1756 and 1759 J. Canton observed the declination-changes on some 600 days, and was thus able to deduce their general character. He found that the most prominent part of the regular diurnal change in England consisted of a westerly movement
movement
Another landmark in the history of terrestrial magnetism was the discovery towards the end of the 18th century that the intensity of the resultant magnetic force varies at different parts of the earth. The first observations clearly showing this seem to be those of a Frenchman, Paul de Lamanon, who observed in 17851787 at Teneriffe and Macao, but his results were not published at the time. The first published observations seem to be those made by the great traveller Humboldt in tropical America between 1998 and 1803. The delay in this discovery may again be attributed to instrumental imperfections. The method first devised for comparing the force at different places consisted in taking the time of oscillation of the dipping needle, and even with modern circles this is hardly a method of high precision. Another discovery worth chronicling was made by Arago in 1827. From observations made at Paris he found that the inclination of the dipping needle and the intensity of the horizontal component of the magnetic force both possessed a diurnal variation. 2. Whilst Italy, England and France claim most of the early observational discoveries, Germany deserves a large share of credit for the great improvement in instruments and methods during the first half of the 19th century. Measurements of the intensity of the magnetic force were somewhat crude until Gauss showed how absolute results could be obtained, and not merely relative data based on observations with some particular needle. Gauss also devised the bifilar magnetometer, which is still largely represented in instruments measuring changes of the horizontal force; but much of the practical success attending the application of his ideas to instruments seems due to Johann von Lamont (18051879), a Jesuit of Scottish origin resident in Germany. The institution of special observatories for magnetic work is largely due to Humboldt and Gauss. The latter's observatory at Gottingen, where regular observations began in 1834, was the centre of the Magnetic Union founded by Gauss and Weber for the carrying out of simultaneous magnetic observations and it was long customary to employ Gottingen time in schemes of international co-operation. In the next decade, mainly through the influence of Sir Edward Sabine (17881883), afterwards president of the Royal Society, several magnetic observatories were established in the British colonies, at St Helena, Cape of Good Hope, Hobarton (now Hobart) and Toronto. These, with the exception of Toronto, continued in full action for only a few years; but their recordsfrom their widely distributed positionsthrew much fresh light on the differences between magnetic phenomena in different regions of the globe. The introduction of regular magnetic observatories led ere long to the discovery that there are notable differences between the amplitudes of the regular daily changes and the frequency of magnetic disturbances in different years. The discovery that magnetic phenomena have a period closely similar to, if not absolutely identical with, the " eleven year " period in sunspots, was made independently and nearly simultaneously about the middle of the 19th century by Lamont, Sabine and R. Wolf. The last half of the 19th century showed a large increase in the number of observatories taking magnetic observations. After 1890 there was an increased interest
3. All the more important magnetic observatories are provided with instruments of two kinds. Those of the first kind give the absolute value of the magnetic elements at the time of Observaobservation. The unifilar magnetometer (q.v.), for tionat instance, gives the absolute values of the declination and Methods and horizontal force, whilst the inclinometer (q.v.) or dip Records. circle gives the inclination of the dipping needle. Instruments of the second kind, termed magnetographs (q.v.), are differential and self-recording, and show the changes constantly taking place in the magnetic elements. The ordinary form of magnetograph records photographically. Light reflected from a fixed mirror gives a base line answering to a constant value of the element in question; the light is cut off every hour or second hour so that the base line also serves to make the time. Light reflected from a mirror carried by a magnet gives a curved line answering to the changes in position of the magnet. The length of the ordinate or perpendicular drawn from any point of the curved line on to the base line is proportional to the extent of departure of the magnet from a standard position. If then we know the absolute value of the element which corresponds to the base line, and the equivalent of r cm. of ordinate, we can deduce the absolute value of the element answering to any given instant of time. In the case of the declination the value of 1 cm. of ordinate is usually dependent almost entirely on the distance of the mirror carried by the magnet from the photographic paper, and so remains invariable or very nearly so. In the case of the horizontal force and vertical force magnetographsthese being the two force components usually recordedthe value of 1 cm. of ordinate alters with the strength of the magnet. It has thus to be determined from time to time by observing the deflection shown on the photographic paper when an auxiliary
As an example of how the curves are standardized, suppose that absolute observations of declination are taken four times a month, and that in a given month the mean of the observed values is 16 34'6 W. The curves are measured at the places which correspond to the times of the four observations, and the mean length of the four ordinates is, let us say, 2.52 cros. If cm. answers to 1o', then 2.52 cms. represents 25'2, and thus the value of the base linei.e. the value which the declination would have if the curve came down to the base lineis for the month in question 16 34'6 less 25'2 or 16 9' 4, If now we wish to know the declination at any instant in this particular month all we have to do is to measure the corresponding ordinate and add its value, at the rate of to' per crn., to the base value 16 9'4 just found. Matters are a little more complicated iii the case of the horizontal and vertical force magnetographs. Both instruments usually possess a sensible temperature coefficient, i.e. the position of the magnet is dependent to some extent on the temperature it happens to possess, and allowance has thus to be made for the difference from a standard temperature. In the case of the vertical force an " observed " value is derived by combining the observed value of the inclination with the simultaneous value of the horizontal force derived from the horizontal force magneto-graph after the base value of the latter has been determined. In themselves the results of the absolute observations are of minor interest
4. Partly on account of the uncertainties due to disturbances, and partly with a view to economy of labour, it has been the practice at some observatories to derive diurnal inequalities from a comparatively small number of undisturbed or quiet days. Beginning with 189o, five days a month were selected at Greenwich by the astronomer royal as conspicuously quiet. In the selection regard was paid to the desirability that the arithmetic mean of the five dates should answer to near the middle of the month. In some of the other English observatories the routine measurement of the curves was limited to these selected quiet days. At Greenwich itself diurnal inequalities were derived regularly from the quiet days alone and also from all the days of the month, excluding those of large disturbance. If a quiet day differed from an ordinary day only in that the diurnal variation in the latter was partly obscured by irregular disturbances, then supposing enough days taken to smooth out irregularities, one would get the same diurnal inequality from ordinary and from quiet days. It was found, however, that this was hardly ever the case (see 29 and 30). The quiet day scheme thus failed to secure exactly what was originally aimed at ; on the other hand, it led to the discovery of a number of interesting results calculated to throw valuable sidelights on the phenomena of terrestrial magnetism.The idea of selecting quiet days seems due originally to H. Wild. His selected quiet days for St Petersburg
5. The intensity and direction of the resultant magnetic force at a spoti.e. the force experienced by a unit magnetic poleare known if we know the three components of force parallel to any set of orthogonal axes. It is usual to take for these axes the vertical at the spot and two perpendicular axes in the horizontal plane; the latter are usually taken in and perpendicular to the geographical meridian. The usual notation in mathematical work is X to the north, Y to the west or east, and Z vertically downwards. The international magnetic committee have recommended that Y be taken positive to the east, but the fact that the declination is westerly over most of Europe has often led to the opposite procedure, and writers are not always as careful as they should be in stating their choice. Apart from mathematical calculations, the more usual course is to define the force by its horizontal and vertical components usually termed H and Vand by the declination or angle which the horizontal component makes with the astronomical meridian. The declination is sometimes counted from o to 36o, o answering to the case when the so-called north pole (or north seeking pole) is directed towards geographical north, 9o to the case when it is directed to the east, and so on. It is more usual, however, to reckon declination only from o to 18o, characterizing it as easterly or westerly according as the north pole points to the east or to the west of the geographical meridian. The force is also completely defined by H or V, together with D the declination, and I the inclination to the horizon of the dipping needle. Instead of H and D some writers make use of N the northerly component, and W the westerly (or E the easterly). The resultant force itself is denoted sometimes by R, sometimes by T (total force). The following relationships exist between the symbols X=N, Y=W or E, Z=V, R=T, H V (X2-+2), Rse (X2+Y2+Z2), tan D = Y/X, tan I = V/H. The term magnetic element is applied to R or any of the components, and even to the angles D and I. 6. Declination is the element concerning which our know-ledge is most complete, and most reliable. With a good unifilar magnetometer, at a fixed observatory distant charts. from the magnetic poles, having a fixed mark of known azimuth, the observational uncertainty in a single observation should not exceed o'5 or at most r'.o. It cannot be taken for granted that different unifilars, even by the best makers, will give absolutely identical values for the declination, but as a matter of fact the differences observed are usually very trifling. The chief source of uncertainty in the observation lies in the torsion of the suspension fibre, usually of silk or more rarely of phosphor bronze or other metal. A very stout suspension must be avoided at all cost, but the fibre must not be so thin as to have a considerable risk
The usual method of conveying information as to the value of the declination at different parts of the earth's surface is to draw curves on a mapthe so-called isogonalssuch that at all points on any one curve the declination at a given specified epoch has the same value. The information being of special use to sailors, the preparation of magnetic charts has been largely the work of naval authoritiesmore especially of the hydrographic department of the British admiralty. The object of the admiralty world chartsfour of which are reproduced here, on a reduced scale, by the kind per-mission of the Hydrographeris rather to show the general features boldly than to indicate minute details. Apart from the immediate necessities of the case, this is a counsel of prudence. The observations used have mostly been taken at dates considerably anterior to that to which the chart is intended to apply. What the sailor wants is the declination now or for the next few years, not what it was five, ten or twenty years ago. Reliable secular change data, for reasons already indicated, are mainly obtainable from fixed observatories, and there are enormous areas outside of Europe where no such observatories exist. Again, as we shall see presently, the rate of the secular change sometimes alters greatly in the course of a comparatively few years. Thus, even when the observations themselves are thoroughly reliable, the prognostication made for a future date by even the most experienced of chart makers may be occasionally somewhat wide of the mark. Fig. i is a reduced copy of the British admiralty declination chart for the epoch 1907. It shows the isogonals between 70 N. and 65 S. latitude. Beyond the limits of this chart, the number of exact measurements of declination iswhose centre is the pole. At all points on the circle the positions of the needle will be parallel; but whereas the north pole of the magnet will point exactly towards the centre of the circle at one of the points where the straight line drawn on the ground cuts the circumference, it will at the opposite end of the diameter point exactly away from the centre. The former part is clearly on the isogonal where the declination is 0, the latter on the isogonal where it is 18o. Isogonals will thus radiate out from the north geographical pole (and similarly of course from the south geographical pole) in all directions. If we travel along an isogonal, starting from the north magnetic pole, our course will generally take us, often very circuitously, to the north geographical pole. If, for example, we select the isogonal of 1o E., we at first travel nearly south, but then more and more westerly, then north-westerly across the north-east of Asia; the direction then gets less northerly, and makes a dip to the south before finally making for the north geographical pole. It is possible, however, according to the chart, to travel direct from the north magnetic to the south geographical pole, provided we select an isogonal answering to a small westerly or easterly declination (from about 19 W. to 7 E.). Special interest attaches to the isogonals answering to declination 0. These are termed agonic lines, but sailors often call them lines of no variation, the term variation having at one time been in common use in the sense of declination. If we start from the north magnetic pole the agonic line takes us across Canada, the United States and South America in a fairly straight course to the south geographical pole. A curve continuous with this can be drawn from the south leduW f..t.. S..n. by p~.sl.iw e111,. Ltd. Cgputl.si...n of 3L. 3.4 fr4ty TLv pe.tron .n.,b. IhttLD T.In wd Yr nc~ Ls. Leon .y.dWlf down f.. th,..L.rt ,Emory w.ikr. w. 'FIG. 1.Isogonals, or lines of equal magnetic declination. somewhat limited, but the general nature of the phenomena is easily inferred. The geographical and the magnetic poleswhere the dipping needle is verticalare fundamental points. The north magnetic pole is situated in North America near the edge of the chart. We have no reason to suppose that the magnetic pole is really a fixed point, but for our present purpose we may regard it as such. Let us draw an imaginary circle round it, and let us travel round the circle in the direction, west, north, east, south, starting from a point where the north pole of a magnet (i.e. the pole which in Europe or the United States points to the north) is directed exactly towards the astronomical north. The point we start from is to the geographical south of the magnetic pole. As we go round the circle the needle keeps directed to the magnetic pole, and so points first slightly to the east of geographical north, then more and more to the east, then directly east, then to south of east, then to due south, to west of south, to west, to north-west, and finally when we get round to our original position due north once more. Thus, during our course round the circle the needle will have pointed in all possible directions. In other words, isogonals answering to all possible values of the declination have their origin in the north magnetic pole. The same remark applies of course to the south magnetic pole. Now, suppose ourselves at the north geographical pole of the earth. Neglecting as before diurnal variation and similar temporary changes, and assuming no abnormal local disturbance, the compass needle at and very close to this pole will occupy a fixed direction relative to the ground underneath. Let us draw on the ground through the pole a straight line parallel to the direction taken there by the compass needle, and let us carry a compass needle round a small circlegeographical to the south magnetic pole at every point of which the needle points in the geographical meridian; but here the north pole of the needle is pointing south, not north, so that this portion of curve is really an isogonal of 18o. In continuation of this there emanates from the south magnetic pole a second isogonal of or agonic line, which traverses Australia, Arabia and Russia, and takes us to the north geographical pole. Finally, we have an isogonal of 18o, continuous with this second isogonal of 0 which takes us to the north magnetic pole, from which we started. Throughout the whole area included within these isogonals of 0 and 180excluding locally disturbed areasthe declination is westerly; outside this area the declination is in general easterly. There is, however, as shown in the chart, an isogonal of 0 enclosing an area in eastern Asia inside which the declination is westerly though small. 7. Fig. 2 is a reduced copy of the admiralty chart of inclination or dip for the epoch 1907. The places where the dip has the same value lie on curves called isoclinals. The dip is northerly (north pole dips) or southerly (south pole dips) according as the place is north or south of the isoclinal of 0. At places actually on this isoclinal the dipping needle is horizontal. The isoclinal of o is nowhere very far from the geographical equator , but lies to the north of it in Asia and Africa, and to the south of it in South America. As we travel north from the isoclinal of 0 along the meridian containing the magnetic pole the dipping needle's north pole dips more and more, until when we reach the magnetic pole the needle is vertical. Going still farther north, we have the dip diminishing. The northerly inclination is considerably less in Europe than in the same latitudes of North America; and correspondinglythe southerly inclination is less in South America than in the same force. The total force is least in equatorial regions, where values latitudes of Africa. slightly under 0.4 C.G.S. are encountered. In the northern hemis- Fig. 3 is a reduced copy of the admiralty horizontal force chart phere there are two distinct maxima of total force. One of these for 1907. The curves, called isomagnelics, connect the places where so-called foci is in Canada, the other in the north-east of Siberia, the P. 9, /- it/y%' im, O j ~//, ~' / % j fib.. '''' .4.V 55 4"; / .405 ~{~5% ~~ '260 /// .. ~jryyy ~~ ~ ~ `.. :Xf iEl _ .._/7 1, - ' ` _ ' '' a .y :600 =' :., .'.. d-om. !Y +.~.~-_. the horizontal force has the same value; the force is expressed in C.G.S. units. The horizontal force vanishes of course at the magnetic poles. The chart shows a maximum value of between 0.39 and 0.40 in an oval including the south of Siam and the China Sea. The horizontal force is smaller in North America than in corresponding latitudes in Europe. Charts are sometimes drawn for other magnetic elements, especially vertical force (fig. 4) and total force. The isomagnetic of zero vertical force coincides necessarily with that of zero dip, and there is in general considerable resemblance between the forms of lines of equal vertical force and those of equal dip. The highest values of the vertical force occur in areas surrounding the magnetic poles, and are fully 50% larger than the largest values of the horizontalformer having the higher value of the force. There are, however, higher values of the total force than at either of these foci throughout a considerable area to the south of Australia. In the northern hemisphere the lines of equal total forcecalled isodynamic linesform two sets more or less distinct, consisting of closed ovals, one set surrounding the Canadian the other the Siberian focus. 8. As already explained, magnetic charts for the world or for large areas give only a general idea of the values of the elements. If the region is undisturbed, very fairly approximate values are derivable from the charts, but when the highest accuracy is necessary the only thing to do is to observe at the precise spot. In disturbed areas local values often depart somewhat widely from what one would infer from the chart, and occasionally there are large differences 70 60 48 30 r0 6 ,20 r40 :6o rio 0 4 9 , tic so 9, q b~ I~j // O /tY w.. ~-1f // ~~[ HiY//.~/.IH~ /.b~:~~.m~" ym//. - <'Uwy tZ ~1 _A'"?i : /6V0'a76IDw..do x////.197 `:~D 4kitfri 4n boy -q.-~ _zf.. . um: 1/ 0 Amummmmilmmizoilote.m......mcii.. -3 110r't . N) 4'7 URI 16!L. 1s^ "NIL, 80 W. 6o''11,'. 46w. 60E. 80 E. rro E. r45E. 946E. .66E. :8o 966W. 196'W. :2u W. 1o6 W. 86W. aL.wa fr.. . .4.:9.99 N9...1e..9 w 1,9.1. C...nu.ium... 0.. wan.;..~ir emery Wsikar i4. 70" 50 a4 s 40 SQ 6o between places only a few miles apart. Magnetic observatories usually publish the mean value for the year of their magnetic elements. It has been customary for many years to collect and publish these results in the annual report of the Kew Observatory (Observatory Department of the, National Physical Laboratory). The data in Tables I. and II. are mainly derived from this source. The observatories are arranged in order of latitude, and their geographical co-ordinates are given in Table II., longitude being reckoned from Greenwich. Table I. gives the mean values of the declination, inclination and horizontal force for January I, 1901; they are in the main arithmetic means of the mean annual values for the two years 1900 and 1901. The mean annual secular changes given in this table are derived from a short period of yearsusually 1898 to 1903the centre of which fell east all over Europe, and the rate at which it is moving seems not to vary much throughout the continent. The needle is also moving to the east throughout the western parts of Asia, the north and east of Africa, and the east of North America. It is moving to the west in the west of North America, in South America, and in the south and east of Asia, including Japan, south-east Siberia, eastern China and most of India. 9. The information in figs. I, 2, 3 and 4 and in Tables I. and II. applies only to recent years. Owing to secular change, recent charts differ widely from the earliest ones constructed. The first charts believed to have been constructed were those of Edmund Halley the astronomer. According to L. A. Bauer,' who has made a special study of the subject, Halley issued two declination charts for the epoch 1700; one, published in 1701, was practically confined to the Atlantic Ocean, whilst the second, published in 1702, contained Magnetic Elements and their Secular Change. S ,y /ma y " a! e' ~2 , . a //~ ig O ~L~/PJ.! ~i~~1 `b / / a . // ?Asia "j////// ~,Ej'y%y y ~a j/~~~ ////j / ~ ~/%%%/'/' q ~ )4/ f//~. %//.r///// ^ . ................. /o 4014rivims,zavi,rzArif/A or ~~ i ' ` =~Jl//~//. -~~j ~~'y///~~i-v e/ar0/////'~~ gym- .h` ./ rr .. r' ' JO' - .. .~ ~~ Qi mG lI Y eo ` 6_._ homaahon.al p..o.i..loo . . . lady 4oaaaf.oiaen of Ma dmirJy Smefy mater ay Fin. 4.Isomagnetics, lines of equal vertical force. at the beginning of 1901. Table II. is similar to Table I., but includes vertical force results; it is more extensive and contains more recent data. In it the number of years is specified from which the mean secular change is derived; in all cases the last year of the period employed was that to which the absolute values assigned to the element belong. The great majority of the stations have declination west and inclination north; it has thus been convenient to attach the + sign to increasing westerly (or decreasing easterly) declination and to increasing northerly (or decreasing southerly) inclination. In other words, in the case of the declination + means that the north end of the needle is moving to the west, while in the case of the inclination + means that the north end (whether the dipping end or not) is moving towards the nadir. In the case, however, of the vertical force + means simply numerical increase, irrespective of whether the north or the south pole dips. The unit employed in the horizontal and vertical force secular changes is Iy, i.e. o0000I C.G.S. Even in the declination, at the very best observatories, it is hardly safe to assume that the apparent change from one year to the next is absolutely truthful to nature. This is especially the case if there has been any change of instrument or observer, or if any alteration has been made to buildings in the immediate vicinity. A change of instrument is a much greater source of uncertainty in thecase of horizontal force or dip than in the case of declination, and dip circles and needles are more liable to deterioration than magnetometers. Thus, secular change data for inclination and vertical force are the least reliable. The uncertainties, of course, are much less, from a purely mathematical standpoint, for secular changes representing a mean from five or ten years than for those derived from successive years' values of the elements. The longer, however, the period of years, the greater is the chance that one of the elements may in the course of it have passed through a maximum or minimum value. This possibility should always be borne in mind in cases where a mean secular change appears exceptionally small. As Tables I. and II. show, the declination needle is moving to thealso data for the Indian Ocean and part of the Pacific. These charts showed the isogonic lines, but only over the ocean areas. Though the charts for 1700 were the first published, there are others which apply to earlier epochs. W. van Bemmelen a has published charts for the epochs 1500, 1550, 1600, 165o and 1700, whilst H. Fritsche9 has more recently published charts of declination, inclination and horizontal force for 1600, 1700, 1780, 1842 and 1915. A number of early declination charts were given in Hansteen's Atlas and in G. Hellmann's reprints, Die Altesten Karten der Isogonen, Isoklinen, Isodynamen (Berlin, 1895). The data for the earlier epochs, especially those prior to 1700, are meagre, and in many cases probably of indifferent accuracy, so that the reliability of the charts for these epochs is somewhat open to doubt. If we take either Hansteen's or Fritsche's declination chart for 1600 we notice a profound difference from fig. 1. In 1600 the agonic line starting from the north magnetic pole, after finding its way south to the Gulf of Mexico, doubled back to the north-east, and passed across or near Iceland. After getting well to the north of Iceland it doubled again to the south, passing to the east of the Baltic. The second agonic line which now lies to the west of St Petersburg appears in 1600 to have continued, after traversing Australia, in a nearly northerly direction through the extreme east of China. The nature of the changes in declination in western Europe will be under-stood from Table III., the data from which, though derived from a variety of places in the south-east of England,10 may be regarded as approximately true of London. The earliest result is that obtained by Borough at Limehouse. Those made in the 16th century are due to Gunter, Gellibrand, Henry Bond and Halley. The observations from 1787 to 1805 were due to George Gilpin, who published particulars of his own and the earlier observations in the Phil. Trans. for 1806. The data for 1817 and 1820 were obtained by Col. Mark Beaufoy, at Bushey, Herts. They seem to come precisely at the time when the needle, which had been continuously moving to the west since the earliest observations, began to retrace its steps. The data from 1860 onwards apply to Kew. Place. Absolute Secular change. values. D. I. H. D. I. H. , , Pavlovsk . 0 39.8E 70 36.8N .16553 - 4.1 -0.8 Y + 7 Ekatarinburg 10 6.3E 70 40.5N .17783 - 4.6 +0.5 -13 Copenhagen 10 10.4W 68 38.5N .17525 Stonyhurst . 18 103W 68 480N .17330 - 4.0 +22 Wilhelmshaven. 12 260W 67 397N 18108 4.1 -2.1 +20 Potsdam . 9 54.2W 66 24.5N .18852 - 4.2 -1.6 +16 Irkutsk . . . 2 ioE 70 15.8N 20122 + 0.5 +I.6 -14 de Bilt . . . 13 48.3W 66 55.5N .18516 - 4.4 -2.2 +14 Kew. . . 16 5o8W 67 ,o6N .18440 - 4.2 -2.2 +25 Greenwich . 16 27.5W 67 7.3N .18465 - 4.0 -2.2 +23 Uccle . . 14 iioW 66 8.8N .18954 - 4'2 -2.1 +23 Falmouth . . 18 27.3W 66 44oN 18705 - 3.8 -2.7 +26 Prague 16 581W 65 441N 19956 - 35 -2.7 +20 St Helier . . Pare St Maur . 14 43.4W 64 52.3N .19755 Val Joyeux. . 15 13.7W 65 ooN 1967o - 4'0 -2.2 +23 Munich. . . 10 25.8W 63 18.1N .20629 - 4.8 -2.7 +21 O'Gyalla . 7 26.1W .21164 - 4.8 +13 Pola. . . . 9 22.7W 6o 14.5N 22216 - 4.0 +23 Toulouse . . 14 16.4W 6o 559N .21945 - 3'9 -2.5 +25 Perpignan . . 13 34.7W 59 57.6N 22453 Capo di Monte. 9 8oW 56 22.3N - 5.2 -2.3 Madrid . . . 15 39oW Coimbra . . 17 18.IW 59 22.0N 22786 - 3.7 -4'3 +34 Lisbon . . . 17 15.7W 57 53'oN .23548 Athens . 5 38.2W 52 7.5N 26076 San Fernando . 15 57.5W 55 8.8N '24648 Tokyo . . . 4 34.9W 49 0.3N 29932 Zi-ka-wei . . 2 23.5W 45 43.5N .3z875 + 1.5 -1.5 +37 Helwan . 3 39.7W 40 30.8N 30136 - 7.0 -0.4 - 7 Hong-Kong. . 017.5E 31 22.8N .36753 + 1.8 -4.3 +45 Kolaba . . . 0 23.2E 21 26.5N 37436 + 2.2 +7.0 - 9 Manila . . . 0 52.2E 16 13.5N .38064 + 0I -5.3 +47 Batavia. . . 1 7.3E 30 35.5S 36724 + 3.0 -7.3 -II Mauritius . . 9 25.2W 54 9.4S .23820 - 4'7 +4'6 -39 Rio de Janeiro. 8 2.9W 13 20IS .2501 +10.4 -2.3 Melbourne . . 8 25.6E 67 24.65 .23295 The rate of movement of the needle to the east at London-and throughout Europe generally-fell off markedly subsequent to 1880. The change of declination in fact between 188o and 1895 was only about 75% of that between 1865 and 1880, and the mean annual change from 1895 to 1900 was less than 75% of the mean annual change of the preceding fifteen years. Thus in 1902 it was at least open to doubt whether a change in the sign of the secular change were not in immediate prospect. Subsequent, however, to that date there was little further decline in the rate of secular change, and since 1905 there has been very distinct acceleration. Thus, if we derive a mean value from the eighteen European stations for which declination secular changes are given in Tables I. and II. we find mean value from table I. -4.18 II. -5.21 The epoch to which the data in Table II. refer is somewhat variable, but is in all cases more recent than the epoch, January 1, 1901, for Table I., the mean difference being about 5 years. to. At Paris there seems to have been a maximum of easterly declination (about 9) about 1580; the needle pointed to true north about 1662, and reached its extreme westerly position between 1812 and 1814. The phenomena at Rome resembled those at Paris and London, but the extreme westerly position is believed to have been attained earlier. The rate of change near the turning point seems to have been very slow, and as no fixed observatories existed in those days, the precise time of its occurrence is open to some doubt. Perhaps the most complete observations extant as to the declination phenomena near a turning point relate to Kolaba observatory at Bombay ; they were given originally by N. A. F. Moos,l'- the director of the observatory. Some of the more interesting details are given in Table IV.; here W denotes movement to be west, and so answers to a numerical diminution in the declination, which is easterly. Prior to 188o the secular change at Kolaba was unmistakably to the east, and subsequent to 1883 it was clearly to the west; but between these dates opinions will probably differ as to what actually happened. The fluctuations then apparent in the sign of the annual change may be real, but it is at least conceivable that they are of instrumental origin. From 187o to 1875 the mean annual change was -i'2; from 1885 to 1890 it was +I'5, from 1890 to 1895 it was +2'0, while from 1895 to 1905 it was +2'35, the + sign denoting movement to the west. Thus, in this case the rate of secular change has increased fairly steadily since the turning point was reached. Table V. contains some data for St Helena and the Cape of Good Hope,12 both places having a long magnetic history. The remarkable feature at St Helena is the uniformity in the rate of secular change. The figures for the Cape show a reversal in the direction of the secular change about 1840, but after a few years the arrested movement to the west again became visible. According, however, to J. C. Beattie's Magnetic Survey of South Africa the movement to the west ceased shortly after 1870. A persistent movement to the east then set in, the mean annual change increasing from 1'8 between 1873 and 1890 to 3'.8 between 1890 and 1900. t I. Secular changes of declination have been particularly interesting in the United States, an area about which information is unusually complete, thanks to the labours and publications of the United States Coast and Geodetic Survey.13 At present the agonic line passes in a south-easterly direction from Lake Superior to South Carolina. To the east of the agonic line the declination is westerly, and to the west it is easterly. In 1905 the declination varied from about 21 W. in the extreme north-east to about 24 E. in the extreme north-west. At present the motion of the agonic line seems to be towards the west, but it is very slow. To the east of the agonic line westerly declination is increasing, and to the west of the line, with the exception of a narrow strip immediately adjacent to it, easterly declination is increasing. The phenomena in short suggest a motion southwards in the north magnetic pole. Since 1750 declination has always been westerly in the extreme east of the States, and always easterly in the extreme west, but the position of the agonic line has altered a good deal. It was to the west of Richmond, Virginia, from 1750 to about 1772, then to the east of it until about 1838 when it once more passed to the west ; since that time it has travelled farther to the west. Table VI. is intended to show the nature of the secular change throughout the whole country. As before, + denotes that the north pole of the magnet is moving to the west, -that it is moving to the east. The data in Table VI. represent the mean change of declination per annum, derived from the period (ten years, except for 1900-1905) which ended in the year put at the top of the column. The stations are arranged in four groups, the first group representing the extreme eastern, the last group the extreme western states, the other two groups being intermediate. In each group the stations are arranged, at least approximately, in order of latitude. The data are derived from the values of the declination given in the Geodetic Survey's Report for 1906, appendix 4, and Magnetic Tables and Magnetic Charts by L. A. Bauer, 1908. The values seem, in most cases, based to some extent on calculation, and very probably the secular change was not in reality quite so regular as the figures suggest. For the Western States the earliest data are comparatively recent, but for some of the eastern states data earlier than any in the table appear in the Report of the Coast and Geodetic Survey for 1902. These data indicate that the easterly movement of the magnet, visible in all the earlier figures for the Eastern States in Table VI., existed in all of them at least as far back as 1700. There is not very much evidence as to the secular change between 1700 and 1650, the earliest date to which the Coast and Geodetic Survey's figures refer. The figures show a maximum of westerly declination about 167o in New Jersey and about 1675 in Maryland. They suggest that this maximum was experienced all along the Atlantic border some time in the 17th century, but earlier in the extreme north-east than in New York
Examination of Table VI. shows that the needle continued to move to the east for some time after 1750 even in the Eastern States. But the rate of movement was clearly diminishing, and about 1765 the extreme easterly position was'reached in Eastport, Maine, the needle then beginning to retrace its steps to the west. The phenomena visible at Maine are seen repeating themselves at places more and more to the west, in Boston about 1785, in Albany about 1800, in Washington, D.C., about 1805, in Columbus (Ohio) about 1815, in Montgomery (Alabama) about 1825, in Bloomington (Ill.) about 1830, in Des Moines (Iowa) about 1840, in Santa Rosa (New Mexico) about 186o and in Salt Lake about 1870. In 1885 the needle was moving to the west over the whole United States with the exception of a comparatively narrow strip along the Pacific coast. Even an acute observer would have been tempted to prophesy in 1885 that at no distant date the secular change would be pronouncedly westerly right up to the Pacific. But in a few years a complete change took place. The movement to the east, which had become exceedingly small, if existent, in the Pacific states, began to accelerate; the movement to the west continued in the central, as in the eastern states, but perceptibly slackened. In 1905 the area throughout which the movement to the west still continued had greatly contracted and lay to the east of a line drawn from the west end of Lake Superior to the west of Georgia. If we take a station like Little Rock (Arkansas), we have the secular change to the Geographical position. Absolute Values of Elements. Secular change (mean per annum). Place. Latitude. Longitude. Year. D. I. H. V. Interval D. I. H. V. in years. Pavlovsk . . 59 41N 30 29E 1906 4.2E 70 36.6N .16528 .46963 5 -4.5 +0.1 - 6 -14 Sitka (Alaska) . 57 3N 135 20W 1906 30 3.3E 74 4I.7N .15502 .56646 4 -3.0 -1.6 +18 -38 Ekatarinburg . 56 49N 6o 38E 1906 IO 31.0E 70 49.5N .17664 .50796 5 --4.5 +1'7 -23 +18 Rude Skov (Copenhagen) 55 51N 12 27E 1908 9 43.3W 68 45N .17406 44759 Stonyhurst . . 53 51N 2 28W 1909 17 28.6W 68 42.8N .17424 .44722 5 -5.9 -1.1 + 6 -25 Hamburg . . 53 33N 9 59E 1903 II I0.2W 67 23.5N .18126 .43527 Wilhelmshaven. 53 32N 8 9E 1909 II 46.8W .18129 5 -5.2 - 7 Potsd2m. 52 23N 13 4E 1909 9 Io6W 66 200N .18834 .42971 5 -5.8 +0I - 9 -19 Irkutsk . . 52 16N 104 16E 1905 158.1E 70 25oN 20011 .56250 5 +o6 +2.0 -24 +39 de Bilt . . 52 5N 5 IIE 1907 13 190W 66 49.9N '18559 .43368 5 -4.7 -o6 + 2 -16 Valencia. . . . 51 56N Io T5W 1909 20 503W 68 15IN .17877 .44812 5 -5.0 -1.2 + 7 -25 Kew . . . . 51 28N o 19W 1909 16 Io8W 66 597N .18506 .43588 5 -5.4 -1.1 + 2 -35 Greenwich . . 51 28N o 0 1909 15 476W 66 53.9N .18526 .43432 5 -5.5 -0.7 + I -20 Uccle . . 50 48N 4 21E 1908 13 36.7W . 66 1.6N 19061 .42867 4 -5.3 -o8 - 3 -35 Falmouth . . 50 9N 5 5W 1909 17 48.4W 66 306N 188o2 43266 5 -4.7 -1.4 + 9 -30 Prague . . . 50 5N 14 25E 1908 8 20.9W 5 -6.5 Cracow . . 50 4N 19 58E 1909 5 35'1W 64 18N 3 -7'3 St Helier . . 49 12N 2 5W 1907 16 27.4W 65 34.5N 5 -5'3 -1.2 Val Joyeux . 48 49N 2 IE 1909 14 32'9W 64 43.9N .19727 .41792 5 -5'4 -1'7 + I -51 Vienna . . 48 15N 16 21E 1898 8 241W Munich . . . 48 9N II 37E 1906 9 59'5W 63 IooN .20657 '40835 5 -4'8 -1.3 + 4 -31 O'Gyalla . . 47 53N 18 ,2E 1909 6 43'9W .21094 5 -5.0 -10 Odessa . . 46 26N 30 46E 1899 4 36.7W 62 18.2N 2,869 .41660 Pola . . 44 52N 15 51E 1908 8 432W 6o 6.8N .22207 .38640 5 -5.5 -0.6 - 4 -23 Agincourt (Toronto) 43 47N 79 16W 1906 5 45'3W 74 35.6N 16397 .59502 4 +3'4 +0.9 -23 -24 Nice . 43 43N 7 16E 1899 12 4oW 6o 11.7N .22390 .39087 Toulouse . . 43 37N 128E 1905 13 56.3W 6o 491N .22025 '39439 5 -4'5 -I.5 + 2 - 2 Perpignan . . 42 42N 2 53E 1907 13 4.4W 7 -4.7 Tiflis. . . . 41 43N 44 48E 1905 2 41.6E 56 2.8N .25451 .37799 7 -5.2 +1.7 -26 + 2 Capo di Monte. 40 52N 14 15E 1906 8 403W 56 I3.5N 5 -5.1 -1.5 Madrid . . . 40 25N 3 40W 1901 15 35.6W Coimbra. . . 40 12N 8 25W 1908 16 46.2W 58 573N 22946 .38120 5 -4.6 -2.9 +17 -45 Baldwin (Kansas) . 38 47N 95 IoW 1906 8 3oIE 68 45IN 21807 .56081 4 -1.7 +1.8 -36 - 8 Cheltenham (Maryland) 38 44N 76 50W 1906 5 22.0W 70 27.3N .20035 .56436 4 +3'8 +I.2 -38 -45 Lisbon . . 38 43N 9 9W 1900 17 18oW 57 54.8N .23516 '37484 Athens . 37 58N 21 23E 1908 452.9W 52II.7N .26197 .33613 5 -5'5 San Fernando . 36 28N 6 12W 1908 15 25.6W 54 48.4N .24829 .35206 5 -4.6 -2.8 +26 -24 Tokyo . . . 35 41N 139 45E 1901 4 361W 49 ooN .29954 '34459 Zi-ka-wei . . 31 12N 121 26E 1906 2 320W 45 35.3N .33040 .33726 5 +1.5 -1'3 +30 + 6 Dehra Dun . . 30 19N 78 3E 1907 2 38'3E 43 361N 33324 .31736 4 +o'8 +5.5 -26 +77 Helwan . . . . 29 52N 31 21E 1909 2 492W 40 404N .30031 .25804 5 -5.7 +1.2 - 6 +13 Havana . . . 23 8N 82 25W 1905 2 25.0E 52 574N .30531 .40452 Barrackpore 22 46N 88 22E 1907 1 9.9E 30 30.2N .37288 .21967 3 +4'2 +3'4 +21 +62 Hong-Kong. 22 ,8N 114 ToE 1908 0 3.9E 31 2.5N .37047 .22292 5 +I.9 -1.8 +43 - I Honolulu . . 21 19N 158 4W 1906 9 21.7E 40 I.8N .29220 24545 4 -0'9 -3'2 -19 -62 Kolaba . . . 18 54N 72 49E 1905 0 14.0E 21 58.5N '37382 .15084 5 +2.1 +7.2 -II +86 Alibagh . . i8 39N 72 52E 1909 1 03E 23 29oN 36845 16008 3 +I.7 +6.8 -To +82 Vieques (Porto Rice) 18 9N 65 26W 1906 i 33.2W 49 47.7N .28927 .34224 2 +7.2 +6.8 -49 +66 Manila . 14 35N 120 59E 1904 0 51.4E 16 o2N .38215 .10960 5 +0.1 -3'9 +47 -34 Kodaikanal . . 10 14N 77 28E 1907 0 407W 3 27.2N .37431 .02259 4 +4.3 +5'5 +16 +61 Batavia . . . 6 ITS 106 49E 1906 0 54.1E 30 48.55 .36708 .21889 4 +2.1 -7.7 - 2 +110 Dar es Salaam . 6 49S 39 18E 1903 7 35'2W Mauritius . . 20 6S 57 33E 1908 9 14'3W 53 44.95 .23415 .31932 5 -0.3 +2.9 -53 -131 Rio de Janeiro . 22 55S 43 IIW 1906 8 55'3W 13 571S .24972 .06164 5 +9'1 -6.8 -42 +44 Santiago (Chile) 33 27S 70 42W 1906 14 18.7E 30 11.85 +99 Melbourne 37 50S 144 58E 1901 8 26.7E 67 25.OS .23305 .56024 3 +6 I Christchurch, N.Z. 43 32S 172 37E 1903 16 18.4E 67 42.35 22657 .55259 west lasting for about sixty years. Further west the period shortens. At Pueblo (Colorado) it is about forty years, at Salt Lake under thirty years, at Prescott (Arizona) about twenty years. Considering how fast the area throughout which the secular change is easterly has extended to the east since 1885, one would be tempted to infer that at no distant date it will include the whole of the United States. In the extreme north-east, however, the movement of the needle to the west, which had slackened perceptibly after 186o or 1870, is once more accelerating. Thus the auspices do not all point one way, and the future is as uncertain as it is interesting. 12. Table VII. gives particulars of the secular change of horizontal force and northerly inclination at London. Prior to the middle of the 19th century information as to the value of H is of uncertain value. The earlier inclination data" are due to Norman, Gilbert, Bond, Graham, Heberden and Gilpin. The data from 1857 onwards, both for H and I, refer to Kew. " London " is rather a vague term, but the differences between the values of H and I at Kew and Greenwich-in the extreme west and east-are almost nil. For some time after its discovery by Robert Norman inclination at London increased. The earlier observations are notsufficient to admit of the date of the maximum inclination or its absolute value being determined with precision. Probably the date was near 1723. This view is supported by the fact that at Paris the inclination fell from 720 15' in 1754 to 71 48' in 1780. The Date. Declination. Date. Declination. Date. Declination. / 0 / 0 / 1580 11 15E 1773 21 9W 1860 21 38.9W 1622 6 0 1787 23 19 1865 20 58'7 1634 4 6 1795 23 57 1870 20 18.3 1657 o 0 1802 24 6 1875 19 35'6 1665 I 22W 1805 24 8 188o 18 52.1 1672 2 30 1817 24 36 1885 18 19.2 1692 6 o 1818 24 38 1890 17 50.6 1723 14 17 1819 24 36 1895 17 16.8 1748 17 40 1820 24 34 1900 16 52.7 1905 i6 32'9 earlier observations in London were probably of no very high accuracy, and the rates of secular change deducible from them are correspondingly uncertain. It is not improbable that the average annual change o'8 derived from the thirteen years 1773-1786 is too small, and the value 6'2 derived from the fifteen years 1786-18o1 too large. There is, however, other evidence of unusually Year. Declina- Change since Year. Declina- Change since tion East. previous year. tion East. previous year. O / Y / p O / N / M '1876 0 55 58 0 37 E 1881 0 57 12 0 3 E 1877 56 39 0 41 E 1882 0 56 5o 0 22 W 1878 57 6 0 27 E 1883 57 2 0 12 E 1879 57 30 0 24 E 1884 55 39 123 W 188o 57 9 0 21 W 1885 55 3 0 36 W rapid secular change of inclination towards the end of the 18th century in western Europe; for observations in Paris show a fall of 56' between 178o and 1791, and of 90' between 1791 and 1806. Between 18oI and 1901 inclination in London diminished by 3 26'5, or on the average by 2'I per annum, while between 1857 and 1900 H increased on the average by 22y a 'year. These values differ but little from the secular changes given in Table I. as applying at Kew for the epoch ,Jan. 1, 1901. Since the beginning, however, of the loth century a notable change has set in, which seems shared by the whole of western Europe. This is shown in a striking fashion by contrasting the data from European stations in Tables I. and II. There are fifteen of these stations which give secular change data for H in both tables, while thirteen give secular data for I. The mean values of the secular changes derived from these stations are as follows:- H From Table I. -2'.35 +2I.0'y From Table II. -P12 +1.6y The difference in epoch between the two sets of results is only about 5 years, and yet in that short time the mean rate of annual increase in H fell to a thirteenth of its original value. During 1908-19o9 H diminished throughout all Europe except in the extreme west. Whether we have to do with merely a temporary phase, or whether a general and persistent diminution in the value of H is about to set in over Europe it is yet hardly possible to say. 13. It is often convenient to obtain a formula to express the mean annual change of an element during a given period throughout an area of some size. The usual method is to assume that the change at a place whose latitude is 1 and longitude X is given byan expression of the type c+a(1-l)-l-b(X-A), where a, b, c are constants, l and X. denoting some fixed latitude and longitude which it is convenient to take as point of departure. Supposing observational data available from a series of stations throughout the area, a, b and c can be determined by least squares: As anexample, we may take the following slightly modified formula given by Ad. Schmidt is as applicable to Northern Europe for the period 1890 to 1900. OD, DI and AH represent the mean annual changes during this period in westerly declination, in inclination and in horizontal force:- i AD = -5.24-0.071(1-50)+0.033(A-10), AI = -1.58+0.010(1-50) +0.036(A -10), OH = +23.5-0.59 (1-5o)-0'35 (A-10). Longitude X is here counted positive to the east. The central position assumed here (lat. 50, long. 1o E.) falls in the north of St Helena. Cape of Good Hope. Date. Declination. Date. Declination. 16,o 7 13 E 16o5 0 30 E 1677 0 40 1609 0 12 W 1691 I o W 1675 8 14 1724 7 30 1691 II o 1775 12 18 1775 2I 14 1789 15 30 1792 24 31 1796 15 48 1818 26 31 1806 17 i8 1839 29 9 1839 22 17 1842 29 6 184o 22 53 1846 29 9 1846 23 II 1850 29 19 1890 23 57 1857 29 34 1874 30 4 1890 29 32 1903 28 44 Bavaria. In the case of the horizontal force unity represents 1y. Schmidt found the above formulae to give results in very close agreement with the data at the eight stations which he had employed in determining the constants. These stations ranged from Pavlovsk to Perpignan, and from Stonyhurst to Ekaterinburg in Siberia. Formulae involving the second as well as the first powers of 1-l and A-A have also been used, e.g., by A. Tanakadate in the Magnetic Survey of Japan. Place. Epoch 176o 70 8o 90 1800 lo 20 30 40 50 6o 70 8o 90 1900 50 Eastport, Maine -P2 0.0 +P2 +2.1 +3.2 +4'0 +4'5 +4'9 +5.0 +5.6 +4.5 +3'0 +2.1 +1.0 +1.8 +2.4 Boston, Mass.. -2.7 -P9 -P0 0o +1I +I.9 +21 +3'5 +4'2 +4'4 +4'0 +3'3 +3'1 +3'0 +3'2 +3'4 Albany, New York
Philadelphia, Penn. . -4'6 -4'2 -3'5 -2'3 -1'3 +O.1 +P3 +2.5 +3.4 +4.3 +4'2 +4'6 +4'4 +3'4 +3'5 +3'4 j Baltimore, Maryland . -3'9 -3'4 -2.7 -2.0 -0.9 0.0 +0.9 +2.0 +21 +3'4 +3'9 +4'0 +3'9 +3.6 +3.5 +3.2 Richmond, Virginia . -3.6 -3.2 -2.5 -I.8 -0.9 0.0 +0.9 +1.8 +2.5 +3'1 +3'6 +3'9 +3.8 +3'7 +3'4 +3.2 Columbia, S. Carolina . -3'7 -3'4 -2'9 -2.2 -1'3 -0'5 +0.5 +P3 +2.2 +2.9 +3.4 +3'8 +3'8 +3'8 +3'6 +1.8 L Macon, Georgia . -3.7 -3.6 -3.2 -2.5 -1.8 -0.9 0.0 +0.9 +1.8 +2'5 +3.2 +3'6 +3'9 +3'5 +3'1 +P2 lTampa, Florida . -3.0 -2.5 -2.0 -PI -0.4 +0.4 +1I +2o +2.5 +3'0 +3'2 +3'5 +3'7 +2.8 +2.9 +1.6 `Marquette, Michigan . 0.0 +1.4 +2.6 +31 +4'7 +5'1 +4'9 +3.8 +2.4 Columbus, Ohio . -0.9 0.0 +0.9 +2.0 +2.9 +3'4 +3'6 +3'7 +3'9 +4'o +2.4 Bloomington, Illinois . -2.4 -1.5 -0.4 +0.4 +P5 +2.4 +2.8 +4'2 +3'9 +2.9 +I'o 1 Lexington, Kentucky . -0.9 0.0 +0.9 +1.8 +2'5 +3'2 +3'6 +3'8 +3'8 +3'4 +1.8 Chattanooga,Tennessee -0.9 0.0 +0.9 +1.8 +2.5 +3'2 +3'6 +4'o +3'5 +3.1 +P6 Little Rock, Arkansas -2'3 -I'5 -0.9 +0.1 +0.8 +11 +2.0 +3'6 +31 +2'3 -P2 Montgomery, Alabama -3'6 -3'5 -3'1 -2.8 -2.2 -P5 -o8 +0.1 +0.8 +1.6 +2.2 +2.8 +3'8 +3'9 +2'6 +0.2 Alexandria, Louisiana . -2.1 -i6 -o8 +0.1 +o8 +P6 +2.2 +3'6 +3'3 +2.0 -P4 Northome, Minnesota . -1.7 -o6 +0.6 +1.7 +2.8 +4'2 +4'4 +3'5 0.0 Jamestown, N. Dakota +I.O +1.9 +3'1 +4'8 +P9 -2.2 Des Moines, Iowa. -P5 -o6 +o6 +P5 +2.5 +3'8 +4'5 +2.7 -o6 1 Douglas, Wyoming -o8 0.0 +P2 +2.3 +0.5 -p6 -1 Emporia, Kansas . +0.6 +1.6 +2.7 +3.8 +P7 -I.8 Pueblo, Colorado . -0.3 +0.4 +1.5 +3'I +0.7 -2.2 Okmulgee, Oklahoma . +0.9 +P5 +2.7 +3.9 +P4 -2.4 Santa Rosa, New Mexico -0.4 +0.4 +P4 +2.6 +0.4 -2'4 San Antonio, Texas . -T1 -0.5 -0-5 +I.1 +1.8 +2.7 +0.9 -2'4 (Seattle, Washington . -3.3 -3'5 -3.7 -3.7 -3.5 -3'3 -3'0 -2.6 -2.1 -I.3 -I.9 -2o -3.2 Wilson Creek,Washing- -2.1 -1.5 -0.4 -I.0 -1.6 -3.2 ton . . . . Detroit, Oregon . -3'8 -3.9 -3'9 -3'7 -3'4 -2.9 -2'5 -1.8 -o8 -1.8 -3.8 Salt Lake, Utah . -1.1 -0.4 +PO +PO -o-8 -2.8 Prescott, Arizona -1'4 -0'7 +0.4 +0.4 -P2 -3'2 California -2.6 -2.9 -2.9 -2'9 -2.7 -2.5 -2.3 -2.0 -1.5 -o8 -0.4 -1.9 -3.8 San Jose, C VI .1 I -3'4 -3'4 -3.5 -3'2 -3.0 --2.7 -2.1 -1.6 -1.1 -0.9 -0.3 -I.6 -3.6 Los Angeles, Date. I. Date. I. Date. I. H. Date. I. H. o o o o i 1576 71 50 1801 70 36.0 1857 68 24.9 '17474 1891 67 33.2 .18193 1600 72 O 1821 70 3.4 186o 69 19.8 .17550 1895 67 25.4 .18278 1676 73 30 1830 69 38.0 1865 68 8.7 17662 190067 11.8 18428 1723 74 42 1838 69 17.3 1870 67 58.6 '17791 1905 67 3.8 18510 1773 72 19 1854 68 31.1 1874 67 50.0 .17903 1908 67 0.9 .18515 1786 72 9 Formulae are also wanted to show how the value of an element, described in the clockwise direction. This, according to Bauer's 18 or the rate of change of an element, at a particular place has own investigation, is the normal mode of description. Schott varied throughout a long period. For comparatively short periods and Littlehales have found, however, a considerable number it is best to use formulae of the type E_~ bt I ct2, where E of cases where it is difficult to say whether the motion is clockwise denotes the value of an element t years subsequent to some or not, while in some stations on both the east and west shores of convenient epoch; , b, care constants to be determined from the Pacific it was clearly anti-clockwise. Fritsche 19 dealing with the observational data. For longer periods formulae of the type the secular changes from 1600 to 1885-as given by his calculated E _ + b sin (mt fin), where a, b, m and n are constants, have values of the magnetic elements-at 204 points of intersection of been used by Schott 16 and others with considerable success. The equidistant lines of latitude and longitude, found only sixty-three following examples, due to G. W. Littlehales, 17 for the Cape of Good cases in which the motion was unmistakably clockwise, while in Hope, will suffice for illustration: twenty-one cases it was clearly the opposite. Declination (West) =14 63 } 150o sin {o61(t-1850) X77 8) 14. All the magnetic elements at any ordinary station show a Inclination (South)=49 II+ 8 75 sin {0 8 (t-1850)+34'3} irregular changes, lmeans oflthe hourly Toreadings tmust sbe from the formed Here t denotes the date. It is perhaps hardly necessary to point making use of a number of days. The ampl tude of the diurnal change usually varies considerably with the Damao Diurnal out that the extension of any of these empirical formulae-whether to places outside the surveyed area, or to times not included in the season of the year. Thus a diurnal inequality derived period of observation-is fraught with danger, which increases from all the days of the year combined, or from a smaller rapidly the further the extra-potation is pushed. number of days selected equally from all the months of the Table V.I I.-Inclination (northerly) and Horizontal Force at London. year, can give only the average effect through-out 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 Bauer has employed a convenient graphical method of illustrating I to the year as a whole less than the arithmetic mean of the twelve secular change. Radii are drawn from the centre of a sphere 1 ranges obtained for the constituent months. At stations in tem- parallel to the direction of perate latitudes, whilst minor differences of type do exist between 16 ~s s a o 4 the freely dipping needle, the diurnal inequalities for different months of the year, the difference s ~~^~.a and are produced tp in- tersect the tanggent plane ~~II II ~ drawn at the point which ea ansswe ers to the mea mean po posi- tion of the needle during ' the epoch under consider- '..., bon. Thn curve formed ,/ I? byy the points of intersec- tion shows a character o of thhe e sa ecuular ch change. . Fig. 5 (slightly modified py g from Nature, vol. 57, p. 181) applies to London. The curve i s being from a limited number of days selected as being specially quiet, Table VIII.-Diurnal Inequality of Declination, mean from whole year (+ to West). 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. Tables VIII. to XI. give mean diurnal inequalities derived from' aIl the months of tho year combined, the figures representing tho 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 Station. Jan Mayen. St Petersburg Greenwich. Kew. St Maurc . Pars. Kolaba. Batavia. Mauritius. South Vic- Tiflis. and Pavlovsk. Land. Latitude. 71 o' N. 59 41' N 51 28' N. 51 28' N. 48 49' N.41 43' N. 18 54' N. 6 II'S.2o 6' S. 77 51' S. Longitude. 8 28' W. 3o 29' E. 0 o'. 0 19'W. 2 29' E.44 48'E. 72 49' E. Io6 49'E. 57 33' E. 166 45' E. Period. 1882-1883. 1873-1885. 1890-1900. 1890-1900. 1883-1897.1888-1898 .1894-1901.1883-1894.1876-189o. 1902-1903. a. q. a. q. a. a. q. a. a. q. a. a. a. q. Hour. r r r r r r r r -6.6 -4.2 -1.3 -0.7 -1'4 -1.5 -0.9 -P4 -0.7 -0.2 +0.1 +0.1 +2.0 +0.9 2 -10.5 -6.4 -P2 -o8 -P3 -1.4 -0.9 -1.2 -0.6 -oi -oi -{-OI -2I -I.8 3 -15.2 -7.8 -1.2 -I.0 -P3 -1.5 -1.0 -P2 -0.6 -o I -0.1 +o.1 -5.2 -4.5 4 -16.9 -8.4 -P4 -1.3 -i4 -1.7 -1.3 -1.2 -0.5 0.0 +0.2 -9.4 -6.8 5 -17.0 -8.1 -1.7 -I.8 -I.7 -2I -1.8 -I.6 -O.7 -oI 0.0 +0.3 -12.2 -9.0 6 -13.7 -7.0 -1'9 -2'3 -2.1 -2.4 -2.3 -1.9 -1.2 -o6 +0.1 +0.4 -15'3 -11.7 7 -9.3 -5.1 -2.2 -2.8 -2.4 -2.7 -2.8 -2.4 -P9 -1o +0.5 +0.6 -17.2 -15.0 8 -6.8 -3.2 -2.5 -3.2 -2.5 -2.8 -3.1 -2.7 -2.4 -1.2 +1.3 +II -2P5 -17.3 9 -3.7. -o6 -2.3 -3.0 -1.9 -2.1 -2.5 -2.3 . -2.3 -0.7 +1.7 +1.8 -23.5 -18.1 10 -2.4 +2.1 -1.0 -1.7 -0.2 -0.3 -0.7 -0.5 -0.9 0.0 +P5 +1'9 -2P2 -15.8 11 -0.5 +4.6 +1.0 +0.4 +2.1 +2.2 +P7 +2.0 +1o +0.9 +0.9 +P3 -15.3 -9.2 Noon +2.5 +6.5 +3.1 +2'7 +4'2 +4'3 +3'9 +4'2 +2.6 +1.4 +0.1 0.0 -9.8 -4.9 I +3'7 +7.3 +4'6 +4'3 +5.1 +5.3 +4.8 +5.3 +3.3 +1.2 -o6 -1I -3'2 -0.1 2 +6.4 +7.1 +4.9 +4.5 +4'7 +4'9. +4.4 +4'9 +3.1 +o6 -PI -zo +3'8 +5.9 3 +7-4 +5'9 +4'1 ' +3'6 +3'6 +3'7 +3.1 +3.7 +2.3 +oI -1.3 -2.3 +11I +9'5 4 +8.5 +4'3 +2'7 +2'3 +2.2 +2.4 +1.8 +2.3 +1.3 -0.2 -1.2 -I.8 +16.6 +I2.9 5 +Io6 +3'0 +P5 +1.3 +I.1 +1.2 +0.7 +I.1 +0.6 -oI -0.9 -0.9 +19.9 +14.6 6 +14.2 +2.3 +0.6 +0.7 +0.3 +0'4 +0.2 +0.2 +0.2. 0.0 -0.6 -0.1 +22.0 +15.5 7 +15.2 +2.2 0.0 +0.4 -0.3 -0.2 -0.1 -0.4 +0.1 +0I -0.4 +o1 +22.0 +15.9 8 +15.8 +2.6 -0.4 +0.2 -o.9 -o.6 -0.3 -0.9 -0.1 +0.2 -O.2 +0.1 +19.9 +14.6 9 +13.2 +2.6 -1o 0.0 -1.2 -1.0 -0.5 -1'3 -0.4 +oI oo +oI +,6o +Io6 10 +7.4 +2.0 -I.4 -O.2 -I.5 -I.3 -0'7 -1.5 -0.6 0.0 +0.1 +O.1 +1I.6 +7.2 II +1i +0.5 -i.6 -0.4 -i.6 -1.4 -o8 -I.6 -0.7 0.0 +0.1 +0.1 +7.6 +4.2 12 -3.6 -1.8 -P5 -o6 -P6 -1.5 -0.9 -I.6 -o8 -0I +0.1 +0.1 +3'3 +P9 Range 32.8 15.7 7.4 7.7 . 7'6 8I 7.9 8.0 5'7 2.6 3'0 4'2 45'5 34'0 St Petersburg Parc S. Victoria Station. Jan Mayen. and Pavlovsk. Greenwich. Kew. St Maur. Tiflis. Kolaba. Batavia. Mauritius. Land. (Period. 18821883. 18731885. 18901900. 18901900. 18831897. 18881898. 18941901. 18831894. 18831890. 19021903. a. q. a. q. a. q. a. a. q. a. a. a. Hour. I -57 -22 + 4 + 5 + 4 + 4 + 5 + 3 -to -II - 3 -I2 2 -64 -24 + 4 + 4 + 3 + 4 + 5 + 3 9 to t -13 3 -74 -25 +4 +4 +3 +4 +5 +3 -9 8 +1 -14 4 -69 -24 +4 +4 +3 +4 +5 +4 -9 -7 +2 -15 5 -60 -22 +5 +4 +3 +4 +6 +4 -9 5 +3 -15 6 -37 -19 + 4 + 4 + 1 + 2 + 4 + 4 7 I + 4 -12 7 -15 -15 + 2 + 2 3 I + 1 + 2 I + 5 + 7 9 8 -1 -13 -3 -4 -9 -7 -5 3 +8 +14 +9 -7 9 + 8 -12 -I0 -I0 -16 -13 -12 - 8 +19 +24 + 9 3 10 +17 -I2 -16 -16 -20 -18 -17 -10 +26 +31 + 9 + 3 I I +32 -10 -19 -20 -19 -18 -16 7 +30 +35 + 9 + 7 Noon +49 4 -17 -18 -13 -12 -12 I +26 +31 + 8 +12 t +65 + 8 -12 -13 7 7 7 + 4 +19 +22 + 7 +18 2 +78 +22 -6 -6 I -2 -4 + 5 +to +to + 2 +20 3 +89 +37 0 0 + 2 + I - I + 3 + 2 1 2 +19 4 +83 +43 + 3 + 3 + 5 + 3 0 I 3 9 6 +18 5 +68 +49 + 5 + 5 + 7 + 5 + 2 4 7 -13 7 +15 6 +37 +43 + 6 + 6 + 9 + 7 + 4 6 8 -14 7 +11 7 +13 +30 + 7 + 7 +10 + 8 + 6 -4 -9 -15 7 + 5 8 -11 +t5 '+ 8 + 8 +10 + 8 + 7 - I -I0 -16 - 8 + 0 9 -33 +1 + 9 + 9 + 8 + 7 + 7 +1 -11 -16 -8 -4 to -36 to + 8 + 9 + 7 + 6 + 6 + 2 -tt -16 - 8 - 7 II -40 -16 + 7 + 8 + 6 + 6 + 6 + 3 to -15 7 9 12 -51 -20 + 6 + 6 + 5 + 5 + 6 + 3 -10 -13 5 II Range 163 74 28 29 30 26 24 15 41 51 17 35 i.e. free from disturbance. In all cases the aperiodic or non-cyclic elementindicated by a difference between the values found for the first and second midnights of the dayhas 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. When comparing results from different stations, it must be remembered that the disturbing forces required to cause a change of t' 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 et 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 o a.m. to St Petersburg Parc St South Vic- Station. Jan Mayen. and Pavlovsk. Greenwich. Kew. Maur. Tiflis. Kolaba. Batavia. Mauritius. toria Land. Period. 18821883. 18731885. 18901900. 18911900. 18831897. 18881898. 18941901. 18831894. 1884189o. 19021903. a. q. a. q. a. q. a. a. q. a. a. a. Hour I +65 +3 7 1 3 + 1 0 +2 +4 +7 +2 +13 2 +65 +2 7 I 4 + 1 0 +2 +4 +5 +2 +12 3 +56 1 -7 I 4 0 I +1 +3 +4 +2 +to 4 +37 -5 -6 0 -3 O o +1 +3 +3 +2 +8 5 +16 7 5 0 2 +1 0 +2 +5 +2 +2 +3 6 7 8 4 0 I + 1 + 1 + 3 + 7 + 1 + 2 0 7 -17 -6 -3 o 0 0 + 1 +3 +6 0 +3 0 8 -14 -4 -2 0 0 I 0 +3 0 -3 +4 -2 9 -9 0 -3 1 -3 -4 -4 t -8 II +5 -6 to -6 +5 -2 -2 -6 -8 -8 -7 -14 -20 +3 -13 II - 6 +to 3 4 9 II -12 It -15 -26 0 -17 Noon to +16 3 5 -10 -11 -12 -11 to -27 4 -20 t -13 +21 I -4 -6 -8 ,9 -9 -3 -21 -7 -20 2 -24 +23 +2 I o -3 3 -5 +1 -13 -9 -16 3 -31 +20 +8 +2 +5 +2 +2 I +4 -4 8 -12 4 -40 +13 +9 +3 +8 +5 +6 +1 +3 +4 5 -6 5 + 2 +10 + 3 + 9 + 6 + 7 + 3 O +10 - 3 - I 6 48 -9 +10 +3 +10 +7 +8 +4 0 +13 0 +3 7 -47 -18 +9 +3 +9 +6 +7 +3 0 +14 0 -{-6 8 -36 -20 +8 +3 +7 +5 +6 +3 +1 +14 + 1 +9 9 7 -19 +6 +2 +5 +5 +5 +3 +2 +14 +2 +II to +18 -13 +3 +2 +3 +4 +3 +3 +3 +13 +2 +12 II +42 5 2 0 0 +3 +2 +3 +3 +11 +2 +12 12 +54 0 5 I 2 +2 + 1 +2 +3 +9 +2 +13 Range 118 43 17 8 20 18 20 15 22 41 14 33 . St. Petersburg Pare South Vic- Station. Jan Mayen. and Pavlovsk. Greenwich. Kew. St Maur. Tiflis. Kolaba. Batavia. Mauritius. toria Land. End Dipping. North. 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 / / / / / / / / / / / / i +4.6 +1.5 -0.5 -0.3 -0.4 -0.3 -0.3 -0.1 +0.6 +9 +0.3 +0.6 2 +5.0 +1.6 -0.5 -0.3 -0.3 -0.2 -O.3. -O.1 +0.6 +0.8 +0.2 +0.7 3 +5.6 +1.6 -o.5 -0.3 -0.3 -0.2 -0.3 -0.1 .+0.5 +06 0.0 +0.7 4 +5.0 +1.5 -0.4 -0.3 -0.3 -0.2 -0.4 -0.2 +0.5 +0.5 o0 +0.7 5 +4.2 +1.4 -0.5 -0.3 -0.2 -0.2 -4 -0.2 +0.7 +0.3 -0I +0.7 6 +2.4 +1.2 -0.4 -0.3 -0.1 -0I -0.3 -0.I +0.8 +0.1 -0'2 +0.5 7 +0.7 +0.9 -0.2 -0I +0.2 +0I 0o 0.0 +0.5 -0.2 -0.3 +0.4 8 -0.1 +o8 +0.1 +3 +0.6 +o4 +0.4 +0.3 -0.2 -o8 -0.4 +0.3 9 -0.7 +0.8 +0.6 +o6 +1o +0.8 +0.7 +0.5 -I.2 -I.7 -0.4 +OI lo -1.2 +0.9 +1.0 +I.0 +1.1 +1.0 +0.9 +0.3 -1.9 -2.7 -0.5 -0.2 II -2.2 +0.8 +1.2 +1.2 +J0 +0.9 +0.7 0.0 -2.1 -3.3 -o6 -0.4 Noon -3.4 +0.4 +II +1.1 +o6 +o6 +0.4 -0.5 -1.6 -3.1 -0.7 -0.7 -4.5 -0.2 +01 +01 +0.3 +0.2 +0.2 -0.6 -08 -2.4 -0.8 -0.9 2 -5.6 -1.2 +0'4 +0.4 +0I +0I +0.2 -0.5 -0.2 -1.3 -o6 -1.0 3 -6.3 -2.2 +0'2 +0.1 0.0 0.0 +0.2 -0.3 +0.3 -0.2 -0.3 -10 4 -6.1 -2.9 0.0 -0I -OI -0I +O.2 +0.1 +0.3 +0.7 +0I -0.9 5 -5'I -3.2 -0.1 -0.3 -0.2 +0.1 +0.4 +0.2 +P3 +O.4 -0.7 6 -3.1 -2.9 -0.2 -0.3 -0.3 -0'3 0.0 +0.5 +0.2 +P5 +5 -0.5 7 -1.7 -2.2 -0.3 -0.4 -0.4 -0.4 -0.2 +0.4 +0.3 +1.6 +0.5 -0.2 8 +0.3 -0.3 -0.5 -0.4 -0.4 -0.3 +0.2 +0.4 +1.6 +o6 0.0 9 +2.0 -0.3 -0.4 -o.6 -0.4 -0.4 -0'3 +0.1 +0.5 +1.6 +0.6 +0.2 to +2.5 +0.5 -0.5 -0.6 -0.4 -0.3 -0.3 0.0 +0.6 +1.5 +o6 +0.4 II +3.0 +1.0 -0.5 -0.6 -0.4 -0.3 -0.3 0.O +0.6 +1'4 +0.5 +0.5 12 +4.0 +1.3 -0.5 -0.4 -0.4 -0.3 -0.3 -0.1 +o6 +1.2 +0.4 +o6 Range 11.9 4.8 1.7 1.8 I.5 P4 1.3 1.1 2.9 4'9 1.4 1.7 I 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 northernhemisphere; 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 them-selves, 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. 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 lo or II 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 the 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 -O.4 -0 I -0.3 0.0 -0.3 -0.1 +0 I +0 I -O I -O.4 O.0 +0 I -O.4 -O.7 +0.8 +1.1 I 2 -0'2 +0.4 -O'3 +0.1 -0.4 +o1 -0.1 +OI -0.2 -0.1 -O.2 +0.2 -O.5 -O'4 +O'3 +II 3 -0.2 -OI -0.3 +0I -0.4 +0.3 -0.2 +O.2 -O.2 +0I -0.2 +0.4 -0.7 -0'1 -O.1 +1.0 4 -1.2 -0.4 -0.3 +0.3 -0.5 +0.5 -0.3 +'3 -0 3 +0.3 -0.2 +0.7 -o6 +0.3 -0.1 +II 5 -2.9 -o6 -O'7 +0.4 -0.7 +01 -0.3 +0.5 -0.5 +0.6 -0.3 +PO -0.7 +1.0 0.0 +11 6 -5.2 -o6 -1.6 +0.5 -1.6 +I.I -O.5 +1.2 -I.O +0.9 -0.4 +I.7 -PO +2.2 O.0 +21 7 -6.2 -0.9 -2.2 +0.7 -1.7 +1.4 -1.1 +2o -2.2 +I.9 -I I +2.6 -1.6 +3'3 -0'1 +4'4 8 -6o -1.2 -2.1 +0.2 -II +0.9 -0.4 +2.3 -1.5 +2.2 -1o +2.4 -o8 +3.6 +o1 +5.6 9 -4.4- -1.8 -1.1 -0.1 -0.2 +-5 +0.5 +2.0 -0.3 +1.3 +0.2 +2.0 +01 +3'1 +o6 +5.6 10 -I.5 -1I 0.0 -0.2 +o6 +0.3 +0.9 +1.3 +0.3 +0.2 +1.2 +II +1.6 +1.6 +1.2 +3'6 II +2I +0.6 +I.2 0.0 +I.2 +0I +1.0 +0.4 +0.5 -I.O +1.4 O.O +P5 +0I +I.0 +01 Noon +4.8 +2.2 +2.1 0.0 +1.4 -0.4 +01 -o6 +0.3 -1.4 +Io -1.4 +0.8 -I.0 -0I -2.6 +6I +3.2 +2.0 -0.2 +II -0.8 +0.3 -1.4 +0.3 -1.2 +0.1 -2.2 +0.3 -1.8 -I.4 -5.1 2 +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 -6.2 3 +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 -5.8 4 +3'6 +1.5 +0.2 -0.3 +OI -0.8 -0.8 -I.6 +0.7 +o6 -1.3 -1.6 +0.2 -o8 -1.6 -4.8 5 +1.8 +0.5 0'O -O.2 0.O -O.4 -O.5 -1'2 +1.1 +0.4 -0.3 -1.O +0.5 -O.8 -0.7 -3'3 6 +0 7 -0.1 +0.1 -O.2 +0.2 -0.4 -0.1 -0.7 +I.O +0.1 +0.5 -0.5 +0.5 -o6 -0.4 -P9 7 0.0 -o8 +0.3 -0.2 +0.5 -0.4 +0.1 - 0.6 +o6 -0.4 +01 -0'3 +0.4 -o'8 0.0 - Io 8 0.0 - I.2 +0.4 -0.1 +0.5 -0.3 +0.2 -0.5 +0.5 -O.7 +0.7 -O.3 +0.3 -0.9 +5 -- O'3 9 -0.5 -1.4 +0.3 -0.1 +0.4 -0'2 +4 -O'3 +0.4 -0.9 +O.6 -0.2 +O.2 -0.9 +1.1 0.0 10 -0.5 -1.7 +0I 0.0 +0.2 -0.1 +0'4 -OI +0.2 -PO +0.4 -0I +0I -I.0 +I.3 +0.6 II -O.7 -1.1 -0'I -0I O.0 -0I +0.3 0.O +OI -O.8 +0'3 0.O 0.O -I.0 +1.3 +0.9 12 -0.6 -0.7 -0.2 -0I -O.2 -0.1 +0.2 +0I -0.1 -O.6 +OI +O.1 -0'2 -PO +II +P2 Range 12.3 5.0 4.3 1.0 ~ 3.1 2.3 I 2.1 4.2 3.3 3.6 2.9 5.1 3.2 5.5 3.7 11.8 day, and has a maximum about II 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. +6 +4 +2 0 +2 0 +2 0 +2' 0 0 +4 +2 0 Midt. 6 Noon 6 .MIdt. Midt...6 Noon 6 Midt. 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 mid-winter, 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, b'ut 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 mid-winter there is a conspicuous double period, the force being below its mean from 1 a.m. to 7 a.m. as well as from II 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 I 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. 17. Variations of inclination are connected with those of horizontal and vertical force by the relation SI=a sin 2IIV-i SV-H-i all. 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 seen JUNE DECEMBER 0 Place. Period. an. Feb. March. April. May. June. July. Aug. Sept. Oct. Nov. Dec. Pavlovsk 1890-1900 a 4.93 6.15 8.58 10.93 12.18 12.27 II.82 11.38 8.70 6.87 5.54 4.63 q 2.96 4.20 8.73 1I.28 12.89 13.28 12.31 11.70 9.37 6.91 3.95 2.66 Ekatarinburg 1890 o-1900 a 3.33 4.32 7.63 I I.19 I I'82 11.58 I I.09 10.45 8.13 5.6o 3.73 3.14 Greenwich 1865-1896 a 5.87 7.07 9.40 1 1.42 10.55 10'90 Io'82 10.93 9'66 8.15 6.41 5.15 Kew 1890-1900a 4.92 6o6 9.08 19.95 Io66 10'92 10.59 I1.01 9'49 7.73 5.37 4'46 q 4.07 4.76 8.82 10.57 10.92 Io62 ,o18 11.01 9.76 7.51 4'75 3'34 Toronto . . 1842-1848 a 5.96 6.05 9.18 9'94 11.55 12.34 12.21 13.14 10.76 6.96 6.32 4'97 Manila _ x890-1900 a 1.79 I.09 2.13 3.02 3.84 3'94 4'21 4.89 4'53 I.83 0.85 I.33 Trivandrum 1853-1864 a 2.06 1.48 0.79 I.67 2.90 3.06 3.06 3.64 3'31 1'27 '2.14 2.33 Batavia . . 1884-1899 a 4.18 4.64 3'57 2.93 2.38 2.03 2.31 3'16 3.80 4'51 4.50 4'19 St Helena 1842-1847 a 3'72 5'19 4'93 3.30 2.64 3.24 3.42 3'59 2'40 4.43 4.05 3'54 Mauritius 1876-1890 a 5.2 6.1 6.3 4'7 4'1 2'9 3'4 4'9 5'0 5.5 5.6 5'1 Cape . 1841-1846a 5.14 8'21 7'27 5.00 3'91 3.21 3.54 4.98 4.33 5.96 6.36 5'47 Hobart 1841-1848 a 11.66 11.8o 9.50 7'26 4.56 3.70 4.61 5'89 8.24 II.0I 12.05 II8I at the Cape of Good Hope does not differ much from that seen at N =H cos D, W =H sin D. Thus corresponding small Batavia. Only a single period is clearly shown. The maximum in N, W, H and D are connected by the relations:occurs about 8 or 9 p.m. throughout the year. The time of the minimum is more variable; at midsummer it occurs about II 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 I p.m. At midsummer the principal maximum is foundas at Kew or Greenwichabout to 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 hemispheree.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. 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, an. Feb. March. April. May. June. July. Aug. Sept. Oct. Nov. Dec. H (unit 17) 12 20 32 46 47 49 49 44 39 32 17 II Pavlovsk . . . 1890-1900 a q 12 17 31 42 45 45 42 40 37 31 17 to Ekatarinburg . . a iI 15 29 37 40 40 39 36 33 27 13 9 Kew q 15 17 26 36 38 39 38 38 35 27 20 II Toronto . 1843-1848 a 23 21 24 28 29 29 26 28 41 25 21 20 Batavia . . 1883-1898 a 49 47 54 6o 51 48 50 53 58 52 43 40 St Helena . . 1843-1847 a 43 41 48 53 46 40 40 45 41 40 40 32 Mauritius . 1883-1890 a 21 15 21 23 20 21 20 22 20 21 21 20 Cape of Good Hope 1841-1846 a 13 to 13 13 15 16 14 18 21 14 17 20 Hobart . 1842-1848 a 42 43 34 28 19 17 22 23 23 35 39 42 V (unit Iy) 15 27 29 24 26 20 23 19 23 20 18 14 Pavlovsk . . . 189o-1900 a , q 4 5 9 13 13 12 13 10 9 7 5 4 Ekatarinburg . .. a 10 15 17 21 22 19 20 i6 14 13 II 9 Kew . . . 1891-1900 q 7 10 20 25 31 27 28 23 20 15 9 6 Toronto . . . 1843-1848 a 12 14 17 23 26 14 27 32 34 25 19 18 Batavia . . . 1883-1898 a 42 48 48 45 31 31 32 29 41 50 40 33 St Helena . . 1843-1847 a 16 13 12 14 13 II 17 II 17 II 15 18 Mauritius . 1884-1890 a 12 16 18 15 14 13 15 21 20 16 13 11 Cape of Good Hope 1841-1846 a 29 47 41 38 21 12 14 19 19 35 33 28 Hobart . . . . 1842-1848 a 25 27 22 23 24 21 22 28 26 22 23 27 Inclination ' ' ' ' ' ' ' Pavlovsk . . . 18901900 a 0.97 1.24 2.07 2.79 2'72 2.88 215 2.64 2.52 2.18 1'20 0'89 Ekatarinburg . . a 0.79 0'94 I'70 2'08 2'25 2'19 2'18 2'08 2'00 I.70 o'88 o'69 Kew . . . q 0.98 1'ot r38 1'86 2.05 2'02 2'05 2'15 F98 1.57 1'27 9'63 Toronto . 1843-1848 a I'15 0'94 1'19 1.43 1'31 1'37 1'13 1'26 r87 1.16 1'09 1'05 Batavia . 1883-1898 a 4'88 5'22 5.56 5'62 4'21 4'05 4'24 4'17 5'13 5.58 4'51 3'85 Cape of Good Hope 1842-1846 a 1'55 2'29 2'23 2'23 1'6o 1'41 r54 1'70 1'86 2.03 1'55 2'04 Hobart . 1842-1848 a 1'95 2'16. 1'72 I'62 1'23 1'16 r28 1'42 1'39 1.75 2'04 2 10 variations SN= cosDSHH sinDSD, SW sinDSH+H cosDSD. If SH and SD denote the departures of H and D at any hour of the day from their mean values, then ON and SW 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 SD 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 HSD=0000538(number of minutes in SD). In other words, employing IT as unit of force, one replaces HSD by 5'38SD, where SD 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 IT. 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 as vector 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 the Phil. 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 crossthe number indicating the corresponding hour counted from midnight as orepresents 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 ON and SW 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 by. The tendency to form a loop near midnight, seen in the November and December curves, 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. 20. Any diurnal inequality can be analysed into a series of Fourier
Series. multiples thereof. The series may be expressed in either of the equivalent forms: al cos t+bl sin t+a2 cos 2t+b2 sin 2t+ .. . c1 sin (t+al)+c2 sin (21+a2)+ ... . 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, a, . . . constants are at once derivable from the formulae: tan an = an/bn; cn = an/sin an = bn/cos an = ^I (ant+bn2). The a, b, c, a constants are called sometimes Fourier
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 June.TERRESTRIAL 367 months of one year, or for all the Januarys of a seriee 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 in- equality of the average January of the series of years. This, however, is obviously not true of the c or a con- stants, 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 a for the whole year some authorities give the arithmetic mean of the c's and a's calculated from the diurnal inequalities of the individual months of the year, others give the values obtained for c and a from the mean diurnal inequality of the whole year. The former method inevitably supplies a larger value for c than the latter, supposing a 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 it-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. Engelenburg 20 gives values of c1, c2, c3, c4, and of al, as, as, a4 for each month of the year for about 5o stations, ranging from Fort Rae (62 6' N. lat.) to Cape Horn (55 5' S. lat.). From the results for individual sta- tions, Engelenburg derives a series of means which he regards as representative of 1i differ- ent 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 July. (i) (ii) Table XV.Diurnal Inequalities in N. and W. at Falmouth (unit Iy). Hour. 1 2 3 4 5 6 7 8 9 10 I I 12 ,vSa.m. + 6 + 5 + 5 + 5 + 6 + 6 + 5 +I 6 -14 -20 -20 jp.m. -17 -12 6 1 + 3 + 6 + 9 + 9 + 9 + 8 + 7 + 7 W~a.m. 2 2 3 4 -6 -9 -13 -17 -19 -13 -3 +II (p.m. +20 +22 +17 +II + 6 + 4 + 2 + I 0 I 2 2 Table XVI.Ranges in Diurnal Inequalities at Falmouth (unit 1y). Jan. Feb. Mar. April. May. June. July. Aug. Sept. Oct. Nov. Dec. N. 21 23 30 39 39 37 37 39 36 32 24 15 W. 20 24 46 54 55 55 54 56 51 39 24 I5 figures from the terms possessing the periods 24, I2, 8 and 6 hours the physical significance and general utility of the analysis is some-what 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 a constants the phase angles. An advance of i hour in the time of occurrence of the first (and subsequent, if any) maximum and minimum answers to an increase of 15 in al of 30 in as, of 45 in as, of 60 in a4 and 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 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" 19021903 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 1+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 150 Thus the times of the maxima are: 24-hour term, 2 h. 24 M. p.m.; 12-hour term, I h. 41 M. a.m. and p.m. 8-hour term, 4 h. 41 M. a.m., o h. 41 M. p.m., and 8 h. 41 M. Q.M. 6-hour term, o h. 33 M. a.m. and p.m., and 6 h. 33 M. a.m. and p.m. 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. (local mean time). Month. c1. C2. C3. C4 al. as. as. a4. o o 0 0 January . . 1.79 0.86 0.41 0.27 251.2 29'8 254 64 February . 2.41 1.11 0.59 0.30 242.0 27.7 235 39 March . . 3.05 1.98 1.11 0.45 233'2 36'1 223 49 April . . . 3.35 2.48 1.17 0'39 224'8 39.2 228 61 May . . . 3.59 2.38 0.87 0.17 221.3 50'8 245 89 June . . . 3.83 2.39 0.74 0.05 212.6 46'7 239 72 July . . . 3.72 2.30 0'77 0.11 214.6 48.1 233 8 August . . 3.64 2.43 1.05 0.18 228.2 57'2 244 51 September . 3.35 2.02 1.04 0'35 236'9 55'3 245 70 October . . 2.69 1.69 0'92 0.48 240.1 35'6 235 65 November . 1.94 1.06 0.51 0.32 248.3 28'3 247 61 December 1.61 0.81 0.35 O.20 255.1 22'0 243 56 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 oh. Som. and 2h. 3om. p.m. They thus reinforce one another strongly from I to 2 p.m., accounting for the prominence of the maximum in the early afternoon. 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. Winter includes November to February, Summer May to August, and Equinox the remaining four months. In this case the data are derived from mean diurnal inequalities for the season specified. In the case of the c or amplitude coefficients the unit is 1` for I (inclination), and I-y 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 22. If secular change proceeded uniformly throughout the year, the value E of any element at the middle of the nth month of the year would be connected with E, the mean value for the Annual whole year, by the formula E, =E+(2n-13)s/24, /:equality. where s is the secular change per annum. For the pre-sent purpose, difference in the lengths of the months may be neglected. If one applies toE-E the correction -(2n-13)s/24 one eliminates a regularly progressive secular change; what remains is known as the annual 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 in-equality. 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, 2i from 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. Hann 22 has 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 Pare 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 in-equality 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 II 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 November. (From Phil. 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. Declination. Place. Epoch. C1. C2. C2. C4. a1. as. as. a4 - o o 0 0 Fort Rae (all) . . 1882-1883 18.49 8.22 1.99 2'07 156.5 41.9 308 104 (quiet) 9'09 4'51 1.32 0'73 166'5 37.5 225 350 Ekatarinburg 1841'1862 2.57 1.81 0.93 0'22 223.3 7.4 204 351 Potsdam . 1890-1899 2.81 1.90 0.83 0.31 239.9 32'6 237 49 Kew (ordinary) . 1890-1900 2.91 1.79 0.79 0.27 234.0 39'7 239 57 Kew (quiet) . . 00 2.37 1.82 0.90 0.30 227.3 42.1 240 55 Falmouth (quiet) 1891-1902 2.18 1.82 0.91 0.29 226.2 40.5 238 56 Parc St Maur . . 1883-1899 2.70 1.87 0.85 0.30 238.6 32.5 235 95 Toronto 1842-1848 2.65 2.34 1.00 0.33 213'7 34'9 238 350 Washington 1840-1842 2.38 1.86 0.65 0'33 223'0 26.6 223 53. Manila . 1890-1900 0.53 0.58 0.43 0.17 266.3 50.7 226 89 Trivandrum 1853-1864 0.54 0.46 0.29 0.10 289.0 49.6 114 Batavia . . . 1883-1899 0.80 0.88 0.43 0'13 332.0 163.2 5 236 St. Helena . . 1842-1847 0.68 0.61 0.63 0'34 275'8 171'4 27 244 Mauritius . . 1876-1890 0.86 1.11 0.76 0.22 21.6 172.7 350 161 C. of G. Hope . 1841-1846 1.15 1.13 0.8o 0.35 287'7 156'0 351 193 Melbourne . . 1858-1863 2.52 2.45 1.23 0.35 27.4 176.7 9 193 Hobart . 1841-1848 2.29 2.15 0.87 0'32 33.6 170.8 349 185 S. Georgia . 1882-1883 2.13 1.28 0.76 0.31 30.3 185.3 7. 180 Victoria Land (all). 1902-1903 20.51 4.81 1.21 I.32 158.7 306.9 292 303 (quieter). 00 15'34 4.05 1.24 1.18 163-8 312.9 261 CI. C2. C3. C4. a1. a2. a3. a4. 0 0 0 0 Winter . 0.240 0.222 0.104 0.076 250.0 91.8 344 194 I Equinox 0.601 0.290 0.213 0.127 290.3 135.5 4 207 Summer 0.801 0.322 0.172 0.070 312.5 155.5 39 238 Winter . 3.62 3.86 1.81 1.13 82.9 277'3 154 6 H- Equinox 10.97 5.87 3.32 1.84 109.6 303.5 167 16 Summer 14.85 6.23 2.35 0'95 130.3 316.5 199 41 Winter . 2.46 1.67 0.86 0.42 153'9 300.8 108 280 V Equinox 6.15 4.70 2'51 0'94 117'2 272'3 99 289 Summer 8.63 6'45 2'24 0'55 122'0 272.4 100 285- 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 in-equality which appears under the heading s. 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. 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,369 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 mid-winter. The maxima of the 6-month terms fall near the equinoxes. 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 Absolute not vary very much with the season, but still it does 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 in per TERRESTRIAL Declination. Inclination. Liznar, Potsdam, Parc St Kew (1890-1900). Batavia, Liznar & Parc St. LiHarm's N. Hemi- Maur, 1883.1893 Mauritius. Potsdam. Maur. Kew. sphere. 1891 1906 1888-1897. q. o. S. mean. I January -0.25 +0.04 +0.01 +0.08 +0.03 +0.32 +0'23 +0.06 +0'49 +0.32 +0'44 -0.03 February -0.54 -01I 0.00 +0.48 +0.25 -0.20 +0.19 +0.29 +0.39 +0.56 +0.29 -0.07 March -0.27 +0.04 +0.17 +0.03 +0.05 - 1.02 -0.12 +0.27 +0.20 +0.38 +0.13 +0'53 April -0.03 +0.10 +0.12 -0.31 -0.14 -0.90 -0.11 +0.30 -0.08 -0.02 -0.13 +0.18 May +0.19 +0.07 -0.11 -0.39 -0.28 +0.29 -0.30 +0.08 -0'43 -0.29 -0.37 -0.15 June +0.46 +0.13 -0.14 -0.47 -0.39 +0.78 -0.13 -0.19 -0.70 -0.77 -0.59 -0.35 July +0.48 +0.14 -0.17 -0.30 -0.13 +0.44 -0.08 -0.44 -0.72 -0.67 -0.27 -0.13 August +0.47 +0.11 +0.01 +0.08 +0.05 +0.52 -0.18 -0.38 -0.47 -0.23 -0.05 -0.19 September +0.31 +0.01 0.00 +0.29 +0.24 -0.02 +0.06 -0.06 -0.06 +0.16 +0.01 +0.20 October -0.07 -0.11 +0.09 +0.06 +ooI -0.26 +0.03 -004 +0.31 +0.27 +0.19 0.00 November -0.30 -0.28 -0.05 +0.17 +01I -0.02 +0.08 -0.01 +0.51 +o30 +0.43 +0.18 December -0.36 -0.14 +0.05 +0.26 +0.23 +0.05 +0'35 +0.06 +0'55 +0.19 +0.24 '-0.29 Range 1.02 0.42 0.34 0.95 0.64 I.8o 0.65 0.74 1.27 1'33 1.03 o88' 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 183o Arago deduced a range of 2' 9". Quiet days at Kew from 1890 to 1894 gave a range of 1'2, while at Potsdam Ludeling 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. 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 Annual also be formed representing the annual variation in the variation amplitudes of the regular diurnal inequality, and its Fourier C0-component 24-hour, 12-hour, &c. waves, or of the etficients. 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 P1 sin (t+B1)+P2 sin (2t+02)+ ... , where t is time counted from the beginning of the year, one month being taken as the equivalent of 30, PI, P2 represent the amplitudes, and B1, 02 the 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 cert |
