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A TEEATISE 
 
 MAGNETISM. 
 
Camfcrttrge : 
 
 PRINTED BY C. J. CLAY, M.A. 
 AT THE UNIVERSITY PRESS. 
 
A TREATISE 
 
 ON 
 
 MAGNETISM 
 
 for tije use of jetutients in tl)c 
 
 BY 
 
 GEORGE BIDDELL AIRY, M.A. LL.D. D.C.L. 
 
 FORMERLY FELLOW, NOW HONORARY FELLOW, OF TRINITY COLLEGE; 
 
 LATE LUCASIAN PROFESSOR OF MATHEMATICS, AFTERWARDS PLUMIAN PROFESSOR 
 
 OF ASTRONOMY AND EXPERIMENTAL PHILOSOPHY, 
 
 IN THE UNIVERSITY OF CAMBRIDGE ; 
 
 ASTRONOMER ROYAL. 
 
 MACMILLAN AND CO. 
 
 1870. 
 
 [All Rights reserved.} 
 
/ 
 
 PBINTED BY C. J. CLAY, M.A. 
 AT THE UNIVERSITY PEESS. 
 
IN ENDEAVOURING TO INTRODUCE INTO 
 
 THE UNIVERSITY OF CAMBRIDGE 
 A PHYSICAL SUBJECT, OF MATHEMATICAL CHARACTER, 
 
 HITHERTO UNRECOGNIZED 
 IN ITS ACADEMICAL COURSE ; 
 I VENTURE TO INSCRIBE THIS WORK 
 TO MY HONOURED FRIEND 
 
 SIR JOHN FREDERICK HERSCHEL, BARONET, K.IL, 
 
 ONE OF A SMALL BAND 
 WHO BY THEIR PRIVATE EFFORTS 
 ESTABLISHED IN THE UNIVERSITY 
 
 THE FORM OF MATHEMATICS 
 THEN AND NOW ACCEPTED IN THE SCIENTIFIC WORLD. 
 
 G. B. AIRY. 
 
 1870, November. 
 
ADVERTISEMENT. 
 
 IN the spring of 1864 I was honoured with a request 
 from the Vice-Chancellor of the University of Cam- 
 bridge to deliver the Lecture on Sir Robert Rede's 
 foundation : and in my Address in the Senate House 
 on 1864 May 10, in speaking of the advantages which 
 might be expected to follow the establishment of that 
 Lecture, I took occasion to point out what appeared to 
 be defects in the system of education in the University 
 as connected with Mathematical Physics. I followed 
 up this oral remark by a letter to the Vice-Chancellor ; 
 and the subject by degrees attracted the attention of 
 the University. 
 
 Remarking that, in addition to excellent works 
 on Spherical and Gravitational Astronomy, General 
 Mechanics, Hydrostatics, Pneumatics, and common 
 Optics, a treatise on Physical Optics existed in the 
 University; it appeared desirable to provide for the 
 subjects of Tides, Waves, Sound, Electricity, and Mag- 
 netism: as well as for some of the modifications of 
 
VI ADVERTISEMENT. 
 
 Pure Mathematics specially applicable to the Observing 
 Sciences. The foundations for treatises on Electricity, 
 Tides, and Waves, exist in articles in the Encyclopaedia 
 Metropolitana ; and I trust that some Resident Mem- 
 ber of the University may be induced to exhibit these 
 branches of science in a form adapted to University 
 Education. In redemption of the engagement into 
 which I had virtually entered to place the other sub- 
 jects before the University, I have published works 
 on Probabilities and on Partial Differential Equations 
 (both with express reference to Mathematical Physics), 
 and on Sound. I now close my part of the undertaking 
 by this Treatise on Magnetism. 
 
 I am indebted to James Glaisher, Esq. F.R.S. and 
 F.R.A.S., and to James Carpenter, Esq. F.R.A.S., of 
 the Royal Observatory, for much assistance in the 
 preparation of the diagrams inserted in the pages of 
 this work. 
 
 G. B. AIRY, 
 
 ROYAL OBSEKVATOBY, GREENWICH;,. 
 1870, November. 
 
CONTENTS. 
 
 SECTION I. 
 
 PHYSICAL EXTENSION OF MAGNETISM, AND LIMITATION OF ITS 
 TREATMENT IN THE PRESENT WORK. 
 
 Articles 1, 2. Pages 1, 2. 
 
 ART. P.AGB 
 
 1. DISSEMINATION of magnetism through many components of 
 
 the Earth, and probable cosmical extension of magnetism . ] 
 
 2. Our accurate knowledge of magnetism is limited to the 
 
 magnetism of iron, steel, and the Earth, to which this 
 work will be confined: with allusion finally to galvanism 
 and thermo-electricity ....... 2 
 
 SECTION II. 
 
 PROPERTIES OF STEEL MAGNETS. 
 Articles 313. Pages 323. 
 
 3. The steel magnets will be supposed to be slender bars ; 
 
 usually, they will be supposed to be straight . . . 
 
 4. Definition of a steel magnet ; the definition sometimes 
 
 applies to iron bars 
 
 5. Size of magnets most convenient for experiments ... 
 
 6. Mounting of magnets for experiments 
 
 7. The opposite ends of a magnet have different properties. 
 
 Explanation of the terms "red" and " blue " magnetism, 
 and of the symbols employed to represent them. Allusion 
 to horse-shoe magnets 
 
Vlll CONTENTS. 
 
 ART. PAOB 
 
 8. Method of magnetizing steel bars, and of preserving their 
 
 magnetic power 6 
 
 9. The terrestrial force upon a magnet is a Couple: the red 
 
 end is drawn towards the north, the blue end towards the 
 south, with equal forces. First Law of Magnetism: the 
 Duality of Powers 9 
 
 10. Action of one magnet upon another. Second Law of 
 
 Magnetism. There is attraction between dissimilar ends, 
 repulsion between similar ends. This is exhibited in 
 various ways. Idea of poles. When one magnet dis- 
 turbs a compass, another magnet may be so placed as to 
 neutralize the disturbance. Poles of a horse-shoe magnet. 10 
 
 11. The 'action of the Earth is exactly the same as that of a 
 
 large magnet whose red end is on the south side and 
 whose blue end is on the north side. For experimental 
 purposes, the Earth's action may be neutralized over a 
 large space. Or its action on a special magnet may be 
 rendered insensible by use of the astatic needle . . .13 
 
 12. Experimental examination of the action of a large magnet on 
 
 a small needle. Third Law of Magnetism: magnetism 
 collected in or near each pole of a magnet acts (as to 
 sense) equally in all directions 16 
 
 13. Experimental proof of the Fourth Law of Magnetism, that 
 
 the attraction or repulsion (as the case may be) between 
 two masses of magnetism, estimated as a statical force, is 
 proportional to the product of their inagnetical energies. 
 Definition of the units of the elements used in succeeding 
 investigations ; novel unit of statical pressure . . .19 
 
 SECTION III. 
 
 ALGEBRAICAL INVESTIGATIONS OF THE ACTION OF ONE MAGNET UPON 
 ANOTHER, THE MAGNETS BEING IN THE SAME PLANE, AND THE 
 FORCE OF ATTRACTION OR REPULSION VARYING AS A POWER OF 
 THE DISTANCE. 
 
 Articles 1421. Pages 2441 . 
 
 14. The disturbing magnet presents the center of one side at 
 
 right angles to the disturbed magnet (a position which we 
 shall hereafter term "broadside-on"), and the disturbed 
 magnet presents one end to the center of the disturbing 
 
CONTENTS. ix 
 
 ART. PAGJK 
 
 magnet (a position which we shall call "end-on"): the 
 magnetic energies are supposed to be collected in the 
 poles; and the attractive or repulsive force to vary 
 inversely as the mth power of the distance: to find the 
 angular momentum impressed on the disturbed magnet. 
 Attractions will be represented in the diagrams by con- 
 tinuous lines, repulsions by interrupted lines . . .24 
 
 15, In the same arrangement, to find the tendency of the 
 
 action of A to produce a motion of translation of B .27 
 
 1G. The disturbing magnet is end-on towards the center of the 
 disturbed magnet, which is broadside-on ; to find the 
 angular momentum impressed ...... 28 
 
 17. In the same arrangement, to find the tendency of the action 
 
 of A to produce a motion of translation of B . . .31 
 
 18, To find the tendency of the action of A upon B in the 
 
 simplest cases of parallelism of the two magnets . . .31 
 
 19. Remarks on the computation of these quantities by the 
 
 method of Potentials . , . . . . .33 
 
 20, Investigation of the cases of Articles 14, 15, 16, 17, 18, on 
 
 the supposition that magnetism is not confined to two 
 poles of each magnet, but is disseminated according to 
 some law through its whole length; being entirely red 
 magnetism on one side of the center, and entirely blue on 
 the other side, with similar laws of distribution. Defini- 
 tion of the "magnet-power" of a magnet . . .36 
 
 21, Investigation of the action of a magnet, whose poles are very 
 
 widely separated, upon a needle, whose center is in the line 
 
 of those poles, and whose axis is inclined to that line . . 39 
 
 SECTION IV. 
 
 ON TERRESTRIAL MAGNETISM, AS ACTING IN THE HORIZONTAL PLANE 
 AT EACH PLACE OF OBSERYATJON. 
 
 Articles 2234. Pages 4276. 
 
 22. Definition of Local Magnetic Meridian, and Magnetic Vari- 
 
 ation or Declination; instruments and methods for ascer- 
 taining roughly the Declination : Azimuth Compass : obser- 
 vation of Sun's azimuth: Variation Theodolite: Declination- 
 Charts H 
 
X CONTENTS. 
 
 AP.T. PA6B 
 
 23. More accurate methods of determining the Local Magnetic 
 
 Meridian ; Reversed Telescope carried by Magnet ; elimination 
 of corrections for position of Magnetic Axis, and for torsion 
 power of suspension-thread .46 
 
 24. Terrestrial Magnetic Meridians; Historical Physical Changes 
 
 in the system of Magnetic Meridians ..... 50 
 
 25. Imperfect Method of measuring the horizontal directive force 
 
 of terrestrial magnetism at any locality, by vibrations of a 
 magnetic needle. Correction for the torsion-power of the 
 suspension-thread ........ 53 
 
 23. Experimental determination of the proportion of the magnet- 
 power of a magnet to the Earth's horizontal magnet-power 
 by the method of deflexions. First process, the disturbing 
 needle applied broadside-on . .... 56 
 
 27. Experimental determination continued. Second process, the 
 
 disturbing magnet applied end-on . . . . . 61 
 
 28. Experimental determination continued. Each of the preceding 
 
 operations performed with the needles at different distances. 
 Reference to numerical observations. The discussion of the 
 first process alone, the discussion of the second process alone, 
 and the comparison of the two processes, shew independently 
 that the attraction or repulsion of magnetic masses is in- 
 versely as the square of the distance between them : Final 
 Law of Magnetism . . . .. .. . . .62 
 
 ^9. Inference as to the numerical value of the proposfcion of A to 
 
 E. Remark on the unit of measure for A and E . .Co 
 
 30. Investigation of the most advantageous proportion of the 
 
 two distances of the disturbing magnet from the disturbed 
 needle 67 
 
 31. Incidental inference as to the effect of temperature upon the 
 
 magnet-power of the disturbing magnet, and necessity for 
 correction for temperature in various experiments . .68 
 
 32. Accurate determination of the Earth's local horizontal magnet- 
 
 power, founded on the method of deflexions, used in com- 
 bination with the measure of the Earth's horizontal action 
 upon the disturbing magnet 70 
 
 33. Investigation of the proportion in which the numerical value 
 
 for E will be altered, when, instead of using the foot and 
 the grain as units, we use other units, as the millimetre 
 and milligramme 73 
 
CONTENTS. XI 
 
 ART. PAOI: 
 
 3i. Special values of E\ historical physical change in the value ; 
 
 lines upon the Earth's surface, passing through points of 
 equal horizontal force . 74 
 
 SECTION V. 
 
 ON TERRESTRIAL MAGNETISM, AS ACTING IN THE VERTICAL AT EACH 
 PLACE OF OBSERVATION: AND ON THE COMBINATION OF THE 
 HORIZONTAL AND VERTICAL FORCES, AND THE TOTAL TERRESTRIAL 
 MAGNETIC TORCE, AT EACH PLACE OF OBSERVATION. 
 Articles 3542. Pages 7799. 
 
 35. First evidence of the existence of a vertical magnetic force . 77 
 
 36. Description of the Dipping Needle 78 
 
 37- Manipulation of the Dipping Needle; reversion of its pivot- 
 bearing ; rotation round its vertical axis ; reversal of its 
 magnetic poles ......... 81 
 
 38. Mathematical Theory of the Dipping Needle : first, on the 
 
 supposition that the magnetic intensity after reversion is 
 equal io that before reversion; simplification when the 
 needle- is very nearly balanced 83 
 
 39. Mathematical theory of the Dipping Needle continued; 
 
 secondly, on the supposition that the intensity is not the 
 same after reversion, and that the needle is not nearly 
 balanced .......... 86 
 
 40. Theory of the Dipping Needle, when the dips are observed 
 
 in different vertical planes, inclined to the plane of the 
 magnetic meridian 88 
 
 41. Determination of the Total Terrestrial Magnetic Force at 
 
 any locality: lines upon the Earth's surface passing through 
 points of equal dip, and lines passing through points of 
 equal total force ; historical changes . . . . .90 
 
 42. Reference to the points of principal interest in Figures 20, 21, 
 
 28, 29, 35, 36; secular change in the place of North 
 Magnetic Pole 94 
 
 SECTION VI. 
 
 THEORIES 'ON THE PHYSICAL CAUSE OR REPRESENTATION 
 
 OF TERRESTRIAL MAGNETISM. 
 Articles 4349. Pages 100121. 
 
 43". Reasons for believing that Terrestrial Magnetism is not pro- 
 duced, in any important degree, by magnetic forces external 
 to the Earth . 100 
 
Xll CONTENTS. 
 
 ABT. PACK 
 
 44. Reasons for believing that Terrestrial Magnetism does not 
 
 reside, in any important degree, in the Earth's surface . 101 
 
 45. Attempt to explain Terrestrial Magnetism by the action of 
 
 a magnet of small dimensions but of very great power, near 
 the center of the Earth 103 
 
 46. Attempt to explain Terrestrial Magnetism by the action of 
 
 two magnets within the Earth 106 
 
 47- Gauss's more general explanation of Terrestrial Magnetism 
 by supposing that the red and blue magnetisms are dis- 
 tributed irregularly through the Earth .... 107 
 
 48. Incidental introduction of Laplace's Coefficients (not further 
 
 used in this Treatise) . . . . . . . .115 
 
 49. Continuation . of Gauss's investigation; application in a 
 
 numerical form , . .117 
 
 SECTION yn. 
 
 DISTURBING FORCE PRODUCED ON 4 SMALL COMPASS-NEEDLE BY A 
 LARGE MAGNET, IN VARIOUS POSITIONS: AND COMPOSITION OP 
 THIS DISTURBING FORCE WITH TERRESTRIAL HORIZONTAL FORCE. 
 Articles 5055. Pages 122129. 
 
 50. The disturbing magnet is horizontal; its center is broadside- 
 
 on to the center of the compass ; to find its effect at different 
 distances and elevations 122 
 
 51. The disturbing magnet is end-on to the compass; first, in the 
 
 horizontal plane ; secondly, in an inclined plane, the axis of 
 the magnet still directed to the compass . . , . }23 
 
 52. The disturbing magnet is horizontal; it is directed end-on 
 
 to the vertical axis of the conipass, and is not necessarily at 
 the same elevation as the compass . . , . . 124 
 
 53. The disturbing magnet is vertical . . . . . .126 
 
 54. The disturbing magnet is in the horizontal plane which passes 
 
 through the compass, but is inclined at any angle to the line 
 joining the centers of the magnets 127 
 
 55. Composition of the disturbing force in the horizontal plane 
 
 with the terrestrial horizontal force 128 
 
 SECTION VIII, 
 
 ON TRANSIENT INDUCED MAGNETISM IN SOFT IRON AND IN MAGNETS. 
 Articles 5674, Pages 130167. 
 
 56. Definition of Soft Iron, and criterion of the magnetic differ- 
 
 ence between Soft Iron and Magnetized Steel . . . 130 
 
CONTENTS. xiii 
 
 ART. PAGE 
 
 57. Experiments on the induction of magnetism in soft iron by 
 
 the action of a steel magnet 131 
 
 58. Explanation of the attraction of soft iron by either pole of a 
 
 steel magnet, as an effect of induction .... 133 
 
 59. Rapid diminution, with increase of distance, of the attraction 
 
 between a magnet and soft iron .*.... 134 
 
 60. Induction of magnetism in soft iron, produced by terrestrial 
 
 magnetism < 135 
 
 61. Effect of the terrestrially-induced magnetism in a mass of soft 
 
 iron which is carried round a compass, at the same level as 
 the compass, and with the same part of the ma&s always 
 directed to the compass-center 137 
 
 62. Effect of the combination of two masses of iron, in opposite 
 
 azimuths : and of two masses of iron, in azimuths differing 90 139 
 
 63. Simplest form of theory for explanation of the phenomena of 
 
 induction . . 139 
 
 C4. The inductive energy may be resolted in different directions, 
 
 in the same manner as statical forces ..... 141 
 
 65. A mass of iron, symmetrical with respect to the plane directed 
 
 to the axis of a compass and with respect to the horizontal 
 plane, and with its center at the same height as the compass, 
 is subject to terrestrial induction : theoretical investigation 
 of its deviating energy on the compass ; it follows the law 
 of sine 2 azimuth . . . . . . . .142 
 
 66. Simpler investigation when the mass is spherical, with its 
 
 center at the same height as the compass . . . .144 
 
 67. In these cases, the magnitude of the deviation produced in 
 
 the compass is independent of the magnitude of the terres- 
 trial horizontal force * . .146 
 
 68. General investigation of the disturbing force produced by 
 
 a mass of iron, symmetrical with respect to a vertical 
 plane passing through the compass-axis (as an iron-built 
 ship), subject to terrestrial induction 147 
 
 69. Examination of the physical meaning of the different terms 
 
 of the disturbing force 152 
 
 70. Defect of this theory; sketch of Poisson's more complete 
 
 theory ..*....... 154 
 
 71. Inadmissibility of Poisson's fundamental suppositions, and 
 
 indication of the wants of a new theory .... 156 
 
 72. Complexity introduced, by induction, into the actions of 
 
 magnets upon each other . . . . . . . 160 
 
xiv CONTENTS. 
 
 ART. P*G3 
 
 73. Method of measuring the amount of magnetism produced in a 
 
 steel magnet by terrestrial induction 162 
 
 74. Correction of the formulae, used in the determination of the 
 
 Earth's horizontal magnet-power, for effects of induction . 165 
 
 SECTION IX. 
 
 ON SUBPERMANENT MAGNETISM IN IRON SUBJECTED TO MECHANICAL 
 VIOLENCE. 
 
 Articles 7577. Pages 168172. 
 
 75. Primary experiment on Subpermanent Magnetism; a long 
 
 plate or slender bar of iron is placed on a firm frame 
 (sometimes called the "Magnetic Anvil") with its length 
 parallel to the local dip, and is struck repeatedly with a 
 hammer; it becomes a magnet, with red magnetism in the 
 end which dipped (in northern magnetic latitudes), and this 
 magnetism does not change with change of the magnet's 
 position 168 
 
 76. Variations of the experiment. All lead to the supposition 
 
 that iron, in a state of tremor or jar, is peculiarly able to 
 .receive induced magnetism, and .to retain it firmly . . 1G9 
 
 77. Eeversion or destruction of the magnetism. Origin of the 
 
 term "Subpermanent." , 171 
 
 SECTION X. 
 
 ON THE MAGNETISM OF IRON SHIPS, AS AFFECTING THEIR COMPASSES. 
 Articles 7882. Pages 173188. 
 
 78. Philosophical and Commercial Importance of this subject. 
 
 Complication of the Magnetic considerations involved in it . 173 
 
 79. Brief History of the first steps in this science . . . 174 
 
 80. Reference to the causes of partial failure in the correction of 
 
 the compass 178 
 
 81. Continuation of the history. Investigation of the effect of the 
 
 ship's heeling 181 
 
 82. Examination of the heeling-disturbance, and remarks on the 
 
 possibility of correcting it ....... 186 
 
CONTENTS. xv 
 
 SECTION XI. 
 
 ON THE CONTINUOUS REGISTRATION OF SMALL CHANGES IN 
 TERRESTRIAL MAGNETISM. 
 
 Articles 8387. Pages 189-206. 
 
 ART. PACK 
 
 8J. General principle of photographic self- registration, now usually 
 adopted. Distinction of the magnetic elements which are 
 to be registered, and appropriate positions of the recording 
 apparatus 189 
 
 8 t. Kecord of the small changes of magnetic declination, and 
 
 evaluation of their scale 191 
 
 85. Bifilar magnetometer for record of the small changes of mag- 
 
 netic horizontal force, and evaluation of their scale . .192 
 
 86. Balance-magnetometer for record of the small changes of mag- 
 
 netic vertical force, and evaluation of their scale . . .198 
 
 87. Results obtained from the continuous registers of small changes 
 
 in terrestrial magnetism ....... 202 
 
 SECTION XII. 
 
 ON THE RELATION BETWEEN GALVANIC CURRENTS AND MAGNETIC 
 FORCES; AND ON THE REGISTER OF TERRESTRIAL GALVANIC 
 CURRENTS, WITH SPECIAL REFERENCE TO DISTURBANCES OF TER- 
 RESTRIAL MAGNETISM. 
 
 Articles 88- 91. Pages 207 220, 
 
 88. Fundamental principles of the creation of a galvanic current, 
 
 and of its magnetic action ; application to the galvanometer 
 and to the speaking-telegraph ...... 207 
 
 89. Inductive magnetic power of the galvanic current; its action 
 
 on steel and on iron : formation of transient magnets ; regis- 
 tering-telegraphs . . . .211 
 
 90. Spontaneous terrestrial galvanic currents.; investigation of the 
 
 magnetic effects due to them, and comparison of those mag- 
 netic effects with the magnetic disturbances recorded by tho 
 self- registering magnetometers 214 
 
 91. Note on thermo-electric or thermo-galvanic currents, and on 
 
 their possible connexion with terrestrial magnetism . . 218 
 
ERRATA. 
 
 73> line n, for 83, read 85. 
 73 5 > 1 6, /or 72, read 74. 
 
ON MAGNETISM. 
 
 SECTION I. 
 
 PHYSICAL EXTENSION OF MAGNETISM, AND LIMITA- 
 TION OF ITS TREATMENT IN THE PRESENT WORK. 
 
 1. Dissemination of magnetism through many com- 
 ponents of the earth, and probable cosmical extension of 
 magnetism. 
 
 In ordinary observation, magnetism is scarcely 
 known except as existing in iron and especially in 
 steel, and as related in some obscure manner to the 
 earth. But there is reason to believe that it is one 
 of the most extensively diffused agents in nature. It 
 can be traced not only in iron but also in every sub- 
 stance into which iron (one of the most widely spread 
 substances in nature) enters into composition. It is 
 found in nickel and other substances, and even in some 
 gases. Wherever a galvanic current exists in nature, 
 whether produced by chemical action or appearing in 
 the thermo-electric form as originating from the effects 
 
 1 
 
2 ON MAGNETISM. 
 
 of heat at the place of union of different substances, 
 magnetic effects can be elicited. On the larger scale, 
 it is certain that the whole Earth acts as a combination 
 of magnets, and there is reason to think that the Sun 
 and the Moon also act as magnets. 
 
 2. Our accurate knowledge of magnetism is limited 
 to the magnetism of iron, steel, and the Earth, to which 
 this work will be confined: with allusion finally to 
 galvanism and thermo-electricity. 
 
 The laws of magnetic force, however, have been 
 experimentally examined with philosophical accuracy, 
 only in its connexion with iron and steel; and, by 
 inferences bearing considerable probability, in the in- 
 fluences excited by the Earth as a whole. The accurate 
 portions of the following work will therefore be con- 
 fined to the investigations connected with these metals 
 and the Earth. But it will be advantageous, before 
 terminating the treatise, to allude in a more general 
 way to the laws of the connexion between magnetism 
 on the one hand and galvanism and thermo-electricity 
 on the other hand. 
 
PROPERTIES OF STEEL MAGNETS. 
 
 SECTION II. 
 
 PROPERTIES OF STEEL MAGNETS. 
 
 3. The steel magnets will be supposed to be slender 
 bars : usually they will be supposed to be straight. 
 
 As a general rule, it is found impracticable to give 
 magnetism, admitting of careful experimental investiga- 
 tion, to a mass of steel of any form except that of a 
 long bar, straight or bent. (Detached bar-magnets are 
 usually made of uniform breadth throughout : compass- 
 needles, and other mounted magnets, are frequently 
 made with pointed ends, as having smaller moment of 
 inertia in proportion to the energy of their magnetism.) 
 The mathematical investigations which follow will be 
 confined to the case of straight bars, in which the length 
 greatly exceeds the breadth. General reference will 
 however be made to the horse-shoe magnet. 
 
 4. Definition of a steel magnet : the definition some- 
 times applies to iron bars. 
 
 The practical definition of a magnet is, "a bar of 
 steel which, when so suspended or so mounted on a fine 
 point that it can vibrate freely in the horizontal plane, 
 will take a definite direction ; and, if disturbed from 
 
 12 
 
4 ON MAGNETISM. 
 
 that direction, will return to it by a series of vibrations, 
 gradually diminishing in extent, from the effect of 
 atmospheric resistance, &c." As a general rule, the 
 material possessing this property to a degree admitting 
 of experimental examination must be steel. In some 
 exceptional cases however (to be hereafter mentioned) 
 the same properties may be given in a minor degree 
 to bars of iron. 
 
 5. Size of magnets most convenient for experiments. 
 
 For a few important experiments, to be mentioned 
 feelow, it is desirable to be provided with a large magnet, 
 perhaps one foot or two feet in length. But, generally, 
 the best magnets for experiment are small compass- 
 needles, mounted and unmounted. These are capable 
 of possessing a great magnetic power in proportion 
 to their weight, and they can be procured at small 
 expense. 
 
 6. Mounting of magnets for experiments. 
 
 In experiments where the position taken by the 
 magnet, or its vibration, or its displacement by the 
 action of an external magnetic substance, is to be ob- 
 served, it is desirable that the magnet (and, if suspended, 
 its suspending apparatus) should be inclosed in a glass 
 case. For many ordinary experiments, the support of 
 the magnet upon a fine point, as in the common com- 
 pass, is sufficiently delicate ; especially if the point be 
 made of the hard iridium-ore, now universally employ- 
 ed for the compasses of the Royal Navy. But for 
 
PROPERTIES OF STEEL MAGNETS. 5 
 
 delicate experiments suspension is far superior. A 
 very small magnet may be carried by a single fibre as 
 spun by .the silk-worm : a larger magnet may be sup- 
 ported by a manufacturer's silk thread, formed by the 
 union of six or seven of the silk -worm's threads : and 
 the largest may be suspended by a skein consisting of 
 a number of these threads in parallel lines. In all these 
 cases of suspension, the torsion-power of the support is 
 very small, and there is an almost total absence of fric- 
 tion properly so called. 
 
 7. The opposite ends of a magnet have different 
 properties. Explanation of the terms "red" and "blue" 
 magnetism, and of the symbols employed to represent 
 them. Allusion to horse-shoe magnets. 
 
 Understanding then that one end of the magnet 
 thus freely suspended will point to the direction called 
 "Magnetic North" (not generally coinciding with Astro- 
 nomical or Geographical North, but at the present time, 
 in Greenwich, about 20 west of North), and that the 
 other end will point to the "Magnetic South," and that 
 if the magnet be disturbed in direction it immediately 
 returns to its first position, it is evident that the oppo- 
 site ends of the magnet possess different properties. 
 The magnetism of the end of the magnet which points 
 nearly to the geographical north will be called red mag- 
 netism, and that of the opposite end will be called blue 
 magnetism. The student is particularly requested to 
 observe that these words "red" and "blue" have here no 
 meaning whatever, except as distinguishing the two 
 
ON MAGNETISM. 
 
 ends. (The words, from their brevity, and their appli- 
 cability to the colour of the paint put on magnets, are 
 convenient : it has long been the custom of tradesmen 
 to paint with red the north-seeking end of a magnet.) 
 In the diagrams, the red end will be distinguished by a 
 cross-hatching and the blue end by a longitudinal 
 hatching. (It is usual for tradesmen to mark the north- 
 seeking end by a transversal file-mark.) 
 
 Introductory Diagram explaining the representations 
 of Magnets. 
 
 Poles of Ked Magnetism, seeking the North. 
 
 Poles of Blue Magnetism, seeking the South. 
 
 A horse-shoe magnet is merely a straight magnet 
 bent into the horse-shoe form : it will be shewn here- 
 after that the properties of the two ends differ in the 
 same manner as those of the ends of a straight magnet. 
 
 8. Method of magnetizing steel bars, and of preserv- 
 ing their magnetic power. 
 
 It is not easy to say how artificial magnets were 
 formed in the first instance. Probably they may have 
 been derived from the natural magnet ; whose attrac- 
 tive properties, and whose power of producing temporary 
 
MAGNETIZATION OF STEEL BARS. 7 
 
 magnetism in iron, have been known from a very dis- 
 tant age. But, magnets having been once formed, there 
 is little difficulty in forming other magnets from them. 
 The most convenient process for magnetizing a steel bar 
 or compass-needle, &c. is that known by the name 
 "double touch." It requires the use of two magnets. 
 The bar which is to be magnetized being laid horizon- 
 tally, with some slight band to prevent it from moving, 
 the operator takes one magnet in his right hand with 
 (say) the red end downwards, and one in his left hand 
 with the blue end downwards (or both in the opposite 
 positions, according to the nature of the magnetism 
 which he wishes to impart), and, touching the bar with 
 the ends of the magnets near the middle of its length 
 
 Fig. 1. 
 
 (see Figure 1), he draws the two magnets simultaneously 
 to the two ends of the bar (constantly maintaining the 
 contact) till they slip off. He raises the magnets, again 
 places them in contact with the middle of the bar, and 
 again slides them to the ends : and repeats this opera- 
 tion : the motion, while in contact, being always from 
 the middle to the ends. The bar is thus converted into 
 a magnet : the end of the bar which was touched by 
 the red end of the magnet employed becomes a blue 
 end; and vice versa. The magnetizing magnets, in 
 
8 ON MAGNETISM. 
 
 general, suffer no deterioration from this employment. 
 The new steel magnet will retain its magnetism through 
 an indefinitely long period : its permanency depending 
 greatly on the quality of the steel. The steel best 
 adapted for large magnets is that which is best for fine 
 cutlery : and the steel should be perfectly hard through 
 the whole length of the bar. For compass-needles, the 
 same steel at spring-temper is found more advan- 
 tageous. 
 
 It is possible, by various contrivances, as for instance 
 by holding both the hand-magnets with the red ends 
 downwards or both with the blue ends downwards, to 
 create a magnet with a concentration of magnetism in 
 the middle of its length. Such magnets however are 
 useless, and we shall not further notice them. 
 
 Some artists prefer the following method of magnet- 
 izing simultaneously three or more bars. The bars are 
 laid so as to form a closed circuit, and a powerful horse- 
 shoe magnet is placed upon any one bar with both its 
 ends in contact with the bar, and is carefully carried 
 thus round the whole circuit of bars, always in contact, 
 and with the same end of the horse-shoe magnet always 
 preceding. (See Figure 2.) The circuit is repeated 
 
 Fig. 2. 
 
DUALITY OF MAGNETIC POWERS. 9 
 
 several times without lifting the horse-shoe magnet. 
 The red- and blue ends of the resulting magnets are 
 respectively opposed in position to those of the horse- 
 shoe magnet. (On the mode of distinguishing the ends 
 of a horse-shoe magnet, we shall speak shortly.) 
 
 In either process, after a time, a limit to the inten- 
 sity of the communicated magnetism is reached. This 
 is usually expressed by the phrase "magnetized to 
 saturation." 
 
 The steel which is the most valuable for retention 
 of magnetism is also the most favourable for reception 
 of a strong dose of magnetism. 
 
 For preserving the magnets with full magnetic 
 intensity, it is found prudent to place them side by side 
 with their red and blue ends in opposite positions, and 
 to connect the opposite ends (the red end of each with 
 the blue end of the other) by pieces of iron in contact 
 with both. 
 
 Valuable information connected with this subject 
 will be found in the Encyclopaedia Metropolitana, article 
 Magnetism. 
 
 9. The terrestrial force upon a magnet is a Couple : 
 the red end is drawn towards the north, the blue end to- 
 wards the south, with equal forces. First Law of Mag- 
 netism, the Duality of Powers. 
 
 If a magnet, on which the Earth's directive power is 
 strong, be suspended by a very long suspension-thread, 
 and the position of the thread be noted ; if then the 
 magnet be removed and a lump of lead of equal weight 
 
10 ON MAGNETISM. 
 
 be substituted for it ; the suspension thread takes 
 exactly the same position as before. This shews that, 
 upon the whole, there is no horizontal force tending to 
 produce a motion of translation of the magnet ; and 
 therefore, if there is one force tending to draw the red 
 end towards the north, there is an equal force tending 
 to draw the blue end towards the south. 
 
 If a small magnet, as a compass-needle, be supported 
 by two small pieces of cork and floated on water, it 
 speedily takes the north-and-south position, but it shews 
 no disposition to approach any side of the basin. 
 
 These experiments are very important. They shew 
 either that there are different attractions from different 
 parts of the Earth upon different parts of the magnet, or 
 that attraction of the Earth on one part of the magnet 
 is accompanied with equal repulsion on another part. 
 We shall find that, without negativing the first of these 
 suppositions, other experiments shew that the second is 
 universally true : that attraction on one part of a mag- 
 net is universally accompanied with repulsion on 
 another part. And thus we arrive at the first import- 
 ant law of magnetism, the " Duality of Powers." It is 
 this duality which essentially distinguishes the force of 
 magnetism from that of gravitation : in other respects, 
 it will be seen, there is much similarity of their laws. 
 
 10. Action of one magnet upon another. Second 
 Law of Magnetism. There is attraction between dis- 
 similar ends, repulsion between similar ends. This is 
 exhibited in various ways. Idea of poles. When one 
 
ATTRACTION AND REPULSION OF MAGNETS. 11 
 
 magnet disturbs a compass, another magnet may be so 
 placed as to neutralize the disturbance. Poles of a 
 horse-shoe magnet. 
 
 The experiments proving these general laws are the 
 easiest of all. Turn the red end of a magnet held in 
 the hand towards the red end of a suspended needle 
 or compass-needle, and it repels the needle's red end. 
 In like manner, the blue end of the hand-magnet 
 repels the blue end of the needle. On the contrary, 
 turn the red end of the hand-magnet towards the blue 
 end of the needle, and it attracts the needle's blue end ; 
 and in like manner, the blue end of the hand-magnet 
 attracts the red end of the needle. 
 
 The same principle may be exhibited in various 
 forms. If the red end of the hand-magnet be pointed, 
 from a distance, at right angles towards the middle of 
 the needle, it attracts the blue end and repels the red 
 end ; shewing (in addition to the law which we have 
 before us) that the ends of a magnet can act obliquely: 
 an important remark on which we will speak further. 
 If the hand-magnet be placed, at a distance, with its 
 center in the line of the needle produced, and its 
 direction transversal to that of the needle, it disturbs 
 the needle according to the same law. If the hand- 
 magnet be placed with its center vertically above or 
 vertically below the center of the needle, and its 
 direction transversal to that of the needle, the same 
 remark holds. All these experiments lead to the 
 Second Law of Magnetism ; that there is repulsion 
 
12 OX MAGNETISM. 
 
 between magnetisms of similar character and attraction 
 between magnetisms of dissimilar character. 
 
 An additional result, of some importance, is gained 
 by holding the hand-magnet in a vertical direction and 
 bringing it sideways towards one end of the needle. 
 It will be found that the energy of the attraction (or 
 repulsion, as the case may be) varies as the hand- 
 magnet is moved up and down; and that it is greatest 
 when a part of the hand -magnet near to its end but 
 not at its end (distant from it perhaps by -^ of the 
 magnet's whole length) is nearest to the needle. This 
 suggests the idea that the whole of the magnetism 
 peculiar to that end of the magnet is collected into 
 that one point : and that point is called a " Pole." But 
 in fact it is found that, in varying the experiment, no 
 point can be fixed on as strictly corresponding to this 
 idea of a pole ; still the language and the idea are so 
 convenient that we shall make use of them, in general 
 description, and even in some investigations. 
 
 It is easily found that the effect of one magnet may 
 be neutralized by that of another magnet. Thus, if one 
 magnet be below the needle, a similar magnet above 
 the needle with its poles in opposite positions will 
 neutralize it. The reader will have no difficulty in 
 varying this experiment, so as to make it applicable 
 to the other cases of magnetic disturbance. 
 
 If a horse-shoe magnet be held in a vertical position, 
 and if its ends be separately presented sideways to a 
 suspended magnet, it will be found that they possess 
 
QUALITY OF THE EAKTH's MAGNETISM. 
 
 13 
 
 respectively red and blue poles, exactly similar to those 
 of a straight bar-magnet. 
 
 11. The action of the Earth is exactly the same as 
 that of a large magnet, whose red end is on the south 
 side and whose lime end is on the north side. For 
 experimental purposes, the Earth's action may be neu- 
 tralized over a large space. Or its action on a special 
 magnet may be rendered insensible by use of the astatic 
 
 needle. 
 
 Fig. 3. 
 
 JUT 
 
 In Figure 3, suppose the needle A to be turning 
 freely on a fine point and the magnet B to be delicately 
 suspended above it, both magnets taking the position 
 given to them by the Earth's magnetic power, and 
 therefore parallel, with their red ends pointing to the 
 north. In this state, the needle A is maintained vigor- 
 
14 
 
 ON MAGNETISM. 
 
 ously in its position ; and, if it is drawn aside for a 
 moment, it returns rapidly to that position. Lower 1$ 
 gradually : at a certain elevation of B, the needle A 
 will become indifferent to position, and if drawn aside 
 will not return to its former direction. Lower B still 
 more, and A will reverse its position, its red end 
 pointing to the south, as in Figure 4. 
 
 Fig. 4. 
 
 A 
 
 O- 
 
 It is evident here that, at the second or inter- 
 mediate position of B, the action of B is sensibly 
 neutralized by the Earth's action. But, as we have 
 remarked in the last article, the action of B may be 
 neutralized by that of another magnet, at a proper 
 distance, with its red pole to the south. Consequently, 
 the Earth's action is exactly the same as that of a 
 magnet whose red pole is south, and for magnetic 
 
QUALITY OF THE EARTH'S MAGNETISM, 
 
 15 
 
 purposes the Earth may be represented by such a 
 magnet. 
 
 The importance of this inference for theories of 
 magnetism cannot be over-estimated. It shews, not 
 only that the Earth's red and blue poles must be 
 considered to be on the south and north sides, but also 
 that the quality of the Earth's magnetism is the same 
 as that of a steel magnet. 
 
 Advantage may be taken of this principle, in ex- 
 periments, for removing terrestrial influence. If, as 
 in Figure 5, a large magnet be placed at a proper 
 distance below a table, magnetic experiments may be 
 performed upon that table without disturbance by 
 terrestrial magnetism. 
 
 Fig. 5. 
 
 There is however another way of neutralizing the 
 Earth's action, by use of the " astatic needle." This 
 instrument, represented in Figure 6, consists of two 
 needles of equal magnetic power, attached to the same 
 central axis, with their poles in opposite positions. In 
 this state, the action of the Earth on one needle is 
 
16 ON MAGNETISM. 
 
 balanced by its action on the other, and the united 
 frame is indifferent to terrestrial magnetism. But one 
 
 Fig. 6. 
 
 of the needles may be brought so near to the magnet 
 whose force is to be tried that the comparative in- 
 fluence on the more distant needle may sometimes 
 be neglected ; and the experiments on the action of the 
 magnet on the nearer needle will not differ much from 
 what they would have been if terrestrial magnetism 
 did not exist. 
 
 12. Experimental examination of the action of a 
 large magnet on a small needle. Third Law of Mag- 
 netism ; the magnetism collected in or near each pole of 
 a magnet acts (as to sense) equally in all directions. 
 
 Underneath a table, let a large magnet be placed 
 with its red pole north, at such a distance (determined 
 by trial with a small needle on the table) that on the 
 surface of the table the Earth's magnetism is sensibly 
 neutralized. Place in that region a magnet of mode- 
 rate size, carry round it a small compass, and register 
 the positions of its needle. A ^ ;ries of directions is 
 
POLES OF MAGNETS. 
 
 17 
 
 obtained similar to those in Figure 7 (which is drawn 
 
 Fig. 7. 
 
 I I 
 
 I \\ 
 
 from actual experiment; . It will be evident here 
 
 2 
 
18 ON MAGNETISM. 
 
 that the direction in which the red pole (for instance) 
 of the needle is drawn is everywhere determined by 
 the composition of two forces, namely, attraction to the 
 blue pole of the magnet and repulsion from the red 
 pole: the influence of the more distant pole (which- 
 ever it may be) diminishing very rapidly with the 
 increase of distance. Thus, in the neighbourhood of 
 each pole of the magnet, the attractive force on one 
 pole of the needle and repulsive force on the other 
 sensibly draw the needle into the same position as if 
 the distant pole of the magnet did not exist ; opposite 
 the middle of the magnet's length, the distances of the 
 needle from the two poles of the magnet are equal, 
 the attraction of the needle's red pole to the magnet's 
 blue pole and its repulsion from the red pole (and the 
 opposite for the needle's blue pole) are sensibly equal, 
 and the needle lies parallel to the magnet but in the 
 opposite direction. Thus it is seen that the action 
 of a pole of the magnet is not limited to the direction 
 longitudinal from the pole or even transversal from the 
 pole, but that it is equally distinct in a direction nearly 
 backwards from the pole. It is not so easy to judge of 
 the magnitude of the force which one pole exerts in 
 different directions, because it is soon complicated by 
 the effect of the other pole : but, on trying it at small 
 distances by the time of vibration of the needle, there 
 appears to be good reason for thinking that the force 
 when the needle's center is at a transversal separation 
 from the magnet's pole is exactly the same as the force 
 when the needle's center is at a longitudinal separation 
 
PRODUCT OF MAGNETIC ENERGIES. 19 
 
 from the magnet's pole. The experiment of vibration 
 may be extended to a position of the needle much 
 nearer to the center of the magnet than is the magnet's 
 pole. And thus we arrive at the Third Law of 
 Magnetism, that the magnetism collected in or near 
 the pole of a magnet acts equally, as to sense, in all 
 directions. In this respect, magnetism resembles gravi- 
 tation. (The law of force, as depending on the distance, 
 will be a subject of future inquiry.) 
 -. 
 
 13. Experimental proof of the Fourth Law of 
 Magnetism, that the attraction or repulsion (as the case 
 may be) between two masses of magnetism, estimated as 
 a statical force, is proportional to the product of their 
 magnetic energies. Definition of the units of the ele- 
 ments used in succeeding investigations ; novel unit of 
 statical pressure. 
 
 Without at present giving an algebraical or nu- 
 merical definition of magnetic energy, it may be under- 
 stood as being, in needles of similar form, proportional 
 to the force by which, under the action of terrestrial 
 magnetism, the red end is drawn to the north and the 
 blue end to the south. The successive steps of experi- 
 ment bearing upon the law now under consideration 
 will be the following : 
 
 (a) Provide a light axis capable of receiving, at 
 pleasure, one, two, three, or more needles, made as 
 similar as possible, and charged as nearly as possible 
 with the same amount of magnetism. (This is easily 
 
 22 
 
20 ON MAGNETISM. 
 
 verified, by their power of deflecting a compass-needle.) 
 The apparatus is represented in Figure 8. Suspend the 
 
 Fig. 8. 
 
 axis delicately; load it with each of the needles in 
 succession, one at a time ; observe the time of vibration 
 as produced by terrestrial magnetism ; and, if they 
 differ slightly, take the mean. Then place all the 
 needles on the axis, and it will be found that the time 
 of vibration is the same as that mean. This shews 
 that the terrestrial magnetic statical force on the 
 assemblage of needles bears the same relation to the 
 moment of inertia of the assemblage as that which 
 existed for a single one : and therefore that the ter- 
 restrial magnetic statical force is proportional to. the 
 number of similar magnets on which it acts. 
 
 (6) The direction in which terrestrial magnetism 
 acts being known, and a line being drawn through 
 the center of the axis at right angles to that direction, 
 place in that line a magnet (either similar to one of 
 the needles, or of any other form and magnitude) which 
 
PRODUCT OF MAGNETIC ENERGIES. 21 
 
 will deflect the needles mounted on the axis. As every 
 one of the actions between the respective poles is a 
 statical action, and as the mean of the actions on the 
 nearer pole and the further pole of the needle will be 
 sensibly the same as if each was at the needle's center, 
 the trigonometrical tangent of deflection will be the 
 proportion of the statical force exerted by the magnet 
 to the statical force exerted by the Earth. Now the 
 fact of experiment is, that the deflection produced by 
 the external magnet in the assemblage of needles is 
 exactly the same as the deflection produced in a single 
 needle. And therefore the proportion of the statical 
 force exerted by the external magnet to the statical 
 force exerted by the Earth is the same in both cases. 
 But, as we have seen, the statical force exerted by the 
 Earth is proportioned to the number of needles or to 
 the sum of magnetic energies of the needles ; and 
 therefore the statical force exerted by the external 
 magnet is proportional to the sum of magnetic energies 
 of the needles. And the algebraical expression for 
 that statical force must contain that sum of magnetic 
 energies of the disturbed needles as factor. The same 
 rule holds good with regard to gravitation. 
 
 It may at first appear strange that the pull exerted 
 by a magnet upon several needles is greater than the 
 pull exerted upon a single needle, and that in fact a 
 new equal pull is ready to act upon every new equal 
 needle exposed to it. But the fact is so; and it is ana- 
 logous to the gravitation-attraction exercised by a 
 planet upon several satellites, in which the force upon 
 
22 ON MAGNETISM. 
 
 one satellite is not diminished by the circumstance that 
 the same planet is acting also upon another satellite. 
 
 (c) Use the apparatus of Figure 8 as a deflecting 
 apparatus, to deflect from its ordinary position a com- 
 pass-needle. Place the axis of Figure 8 in the direction 
 magnetic E. or- W. from the center of the compass: and 
 mount successively upon it one needle, two needles, 
 three needles. If the single needles continue in their 
 combined state each to exercise the same action as 
 when it is alone, so that the whole statical pull on the 
 compass-needle is successively represented by 1, 2, 3, 
 then the trigonometrical tangents of the angles of 
 deflection of the compass-needle will be in the succes- 
 sive proportions of 1, 2, 3. And the fact, in experiment, 
 is so. It follows from this that the statical force exerted 
 by the assemblage of needles is proportional to the sum 
 of the statical forces exerted by each single needle : that 
 is, it is proportional to the sum of the magnetic energies 
 of these needles. And therefore, the expression for the 
 statical force exerted must contain the sum of the 
 energies of the disturbing needles as factor. 
 
 (This might have been inferred from the conclusion 
 of (6), or vice versd, by assuming the equality of statical 
 action and reaction. But in a matter of such funda- 
 mental importance, it appears well to establish each 
 proportion by independent experiment.) 
 
 (d) Combining the results of (6) and (c), it will be 
 seen that the algebraical expression for the statical 
 
PRODUCT OF MAGNETIC ENERGIES. 23 
 
 force exerted between the two magnetic systems must 
 contain as factor the product of the energies of the two 
 systems. 
 
 The experiments cited in this Article have been 
 carefully verified by the writer of this Treatise. 
 
 It is necessary now to fix with precision the units of 
 the different elements which we have to employ. For 
 the unit of time, 1 second of mean solar time is uni- 
 versally adopted: for the unit of measure of length, 
 1 foot is commonly used in England, and 1 millimetre 
 by the nations which adopt the Metrical system : for 
 the measure of mass, reference is made to weight, and 
 the received units are, 1 grain in England, and 1 milli-" 
 gramme in the Metrical system. For the measure of 
 statical force, it is found convenient to depart from the 
 custom usually followed in mechanical investigations 
 (in which the unit of pressure is considered to be the 
 pressure produced by a unit of mass under the action 
 of terrestrial gravity), and to adopt, instead, that pres- 
 sure which, acting through the time 1 upon the mass 
 1, would produce in it the velocity 1. (This unit, in 
 
 English experiments, is about -^- of the ordinary unit 
 
 O*u A. 
 
 of pressure.) This selection of unit of pressure amounts 
 to the same as saying that the unit of accelerative force 
 will be that which produces the velocity 1 in the 
 time 1. 
 
24 ON MAGNETISM. 
 
 SECTION III. 
 
 ALGEBRAICAL INVESTIGATIONS OF THE ACTION OP 
 
 ONE MAGNET UPON ANOTHER ; THE MAGNETS BEING 
 IN THE SAME PLANE, AND THE FORCE OF ATTRAC- 
 TION OR REPULSION VARYING AS A POWER OF THE 
 DISTANCE. 
 
 14. The disturbing magnet presents the center of one 
 side at right angles to the disturbed magnet (a position 
 which we shall hereafter term "broadside-on"), and the 
 disturbed magnet presents one end to the center of the 
 disturbing magnet (a position which we shall call "end- 
 on") ; the magnetical energies are supposed to be collected 
 in the poles, and the attractive or repulsive force to vary 
 inversely as the mth power of the distance : to find the 
 angular momentum impressed on the disturbed magnet 
 Attractions will be represented in the diagrams by 
 continuous lines, repulsions by interrupted lines. 
 
 In Figure 9, suppose that we require the angular 
 
DISTURBING MAGNET BROADSIDE-ON. 
 
 25 
 
 momentum which A produces on B. The continuous 
 ^S m 9< lines denote attraction ; the inter- 
 
 rupted lines denote repulsion. Let 
 2a and 26 be the lengths of the two 
 magnets as measured from pole to 
 pole : a and /3 the magnetic energies 
 at the poles (meaning by this that 
 the attraction or repulsion will be 
 expressed by a/3 x (distance )~ m ) ; c 
 the distance between the centers of 
 the magnets, which is supposed to 
 be considerably greater than a or 
 6. Then the distance from the 
 blue pole of A to the red pole of 
 B is 
 
 the attractive force is 
 
 the resolved part of this, drawing the red pole of B to 
 the right, is 
 
 a similar term is obtained from the repulsion of the red 
 pole of A on the red pole of B : and the whole angular 
 momentum which they impress on B is 
 
26 ON MAGNETISM. 
 
 tending to turn it in the direction opposite to the mo- 
 tion of watch-hands. Similarly, the momentum obtain- 
 ed from the action of the two poles of A upon the blue 
 pole of B is 
 
 -m-l 
 
 in the same direction as the former. And the whole 
 angular momentum, opposite to watch-hands, is 
 
 -m-l 
 
 It is a great convenience, connected with the assump- 
 tion that c is large in proportion to a or b, that we can 
 at once proceed to expand these expressions in terms 
 with progressive powers of c in the denominator, stop- 
 ping at a definite power of c. It will be found sufficient, 
 for future use, to stop with c" 2 in the expansions of the 
 brackets. Thus we shall have 
 
 24 
 
DISTURBING MAGNET BROADSIDE-ON. 27 
 
 m 4- 1 . m + 3 46 2 
 2.4 ~? 
 
 sum= 2 - (m + 1) . ^^+ (m + 1) (m + 3) ^ . 
 
 The whole angular momentum 
 = 2aj&xa6x [2C-"- 1 + (m + l).^- 3 { - a 2 + (m + 2) . 6 2 )], 
 
 or = 4a/3 x a6 x [c^ 1 + ^ . c 3 { - a 2 + (m + 2) . 
 
 15. /ft ^A^ same arrangement, to find the tendency of 
 the action of A. to produce a motion of 'translation of 'B. 
 
 It is obvious that there is no tendency to carry B to 
 or from A. But there is a tendency to carry it to the 
 right. The forces on the two poles of B are the same 
 as those just found, but that on the blue pole must be 
 subtracted from that on the red pole. The efficient 
 force therefore is 
 
 Expanding as before, this becomes 
 
 2 J c/3xa.cf fn - 1 x2(m + l)- 
 
 c 
 
28 
 
 ON MAGNETISM. 
 
 It is important to observe here that the negative 
 power of c is greater than in the expression found in 
 Article 14. The force which would produce the first or 
 principal term in the expression for angular momentum 
 is 4a/3 x a x c"" 1 " 1 . The proportion of the force of trans- 
 
 lation now found to that force is - . 
 
 c 
 
 If then c be much 
 
 larger than b, the force tending to produce translation 
 is much smaller than the force producing angular mo- 
 mentum. 
 
 16. The disturbing magnet is end- on towards the 
 center of the disturbed magnet, which is broadside-on : 
 to find the angular momentum impressed. 
 
 Fig. 10. 
 
 The notation for Figure 10 being the same as for 
 
DISTURBING MAGNET END-ON. 29 
 
 Figure 9, it will be seen that the resolved part of the 
 force which the red pole of A impresses on the red pole 
 of B tending to turn it against watch-hands is 
 
 -m-l 
 
 a/3x (c-a)x (c 2 - 2ca + a 2 + 6 2 ) 2 , 
 
 -m-l 
 
 which produces the angular momentum 
 
 -m-l 
 
 o 7 /-, o\ /- 2a a? + V\ 2 
 
 o.B x b . c~ m x [ 1 -- x 1 -- + 2 
 \ cj \ c c 2 / 
 
 A similar term is produced by the attraction of the red 
 pole of A on the blue pole of B : thus the whole angular 
 momentum opposite watch-hands produced by the red 
 pole of A is 
 
 -m-l 
 
 2 
 
 The momentum produced by the blue pole of A is found 
 in like manner to be 
 
 and therefore the entire momentum opposite watch 
 hands is 
 
30 ON MAGNETISM. 
 
 In expanding these brackets, it will quickly be found 
 that, in order to secure the same accuracy as in the de- 
 velopment of Article 14, it is necessary to use the 
 binomial theorem one step further : 
 
 therefore ( 1 --- 1 -- 5 1 
 
 \ c c J 
 
 ra + 1/ 2a a 2 +& 2 \ ra+l.m+3/ 2a 2 +&V 
 = 1- x --- h 2 M --- cr~A - --- 1" 2" 
 2 \ c c J 2.4 \ c c 2 J 
 
 m + l.m43.m + o/ 2q q 2 + & 2 \ 3 
 2.4.6 ~\~ c ?~~ J 
 
 TO + 1 . m + 3 
 . __ .. 
 
 Multiply this by ( 1 -- ) , and we obtain 
 \ o/ 
 
 a m.ra+1 a 2 m + 1 b* m.m 
 U ~* 
 
 2.3 V 
 
 m . r?i + 1 cib* 
 ~~2 ? 
 
 Similarly the second bracket is 
 
 a m.ra + 1 a 2 m+1 6 2 m.m+l.m + 2 a 3 
 
 c 2 * 2V 23 V 
 
 m . m + 1 
 
 and the whole large bracket is 
 a m . m + 1 . m + 2 a 3 
 2w c + - 3 -- ^ 
 
DISTURBING MAGNET END-ON. 31 
 
 and the entire angular momentum opposite watch- 
 hands is 
 
 or 
 
 x ab x 
 
 L . c + ^^ . c-f^ '- 6=)l . 
 I * \ 6 J) 
 
 On comparing this result with that of Article 14, it 
 will be seen that the first or principal term here has the 
 factor m, while that in Article 14 has the factor 1. 
 
 17. In the same arrangement, to find the tendency 
 of the action of A to produce a motion of translation of B. 
 
 Here also it is seen that there is no tendency to carry 
 B to or from A 9 but there is a tendency to carry it to 
 the right. Both actions of the red pole of A tend to 
 carry B to the right, and both actions of the blue pole 
 of A tend to carry B to the left. The result is the 
 same as that of Article 15, of which the circumstances 
 are exactly reversed in this problem. 
 
 18. To find the tendency of the action of A upon B 
 in the simplest cases of parallelism of the two magnets. 
 
 The cases to be considered are those represented in 
 Figures 11, 12, 13, 14. In all these, there is no tendency 
 to produce angular motion, or to produce motion of 
 translation to the right or left. But there is tendency 
 to produce motion of B to or from A, The student will 
 easily verify the following results. 
 
32 
 
 ON MAGNETISM. 
 
 In Fig. 11 there is force, tending to produce a 
 motion of translation of B from A, represented by 
 
 4 (w + 1) a/3 x ab . c"" 1 " 2 . 
 
 Fig. 13. 
 fl 
 
 Fig. 14. 
 
 ft\ 
 
 In Fig. 12, there is an equal force tending to produce 
 motion of B towards A. 
 
 In Fig. 13, there is force tending to produce motion 
 of B towards A, represented by 
 
 4m . (m + 1) aft x ab . c~ m ~ z . 
 
 In Fig. 14, there is an equal force tending to produce 
 motion of B from A. 
 
 In all these cases, the index of c is - m 2, and the 
 
INVESTIGATION BY POTENTIALS. 
 
 33 
 
 result is subject to the same remark as those of Articles 
 15 and 17. 
 
 19. Remarks on the computation of these quantities 
 by the method of Potentials. 
 
 The method of Potentials depends on the algebraical 
 fact, representing a mechanical law like that of Virtual 
 Velocity, that when a point x'y'z attracts a point xyz 
 with a force R which is a function of the distance r be- 
 tween the points, the force in the direction x, or 
 
 R . - , can be expressed as - R . -j- : and therefore, 
 v ax 
 
 f T\ Ct/ O -. * d O \AJt \A/*^J -1 
 
 if R = , it can be expressed as -7- . or : and 
 
 dS dr dS 
 
 T ~J~ or T~ 
 
 dr dx dx 
 so in the directions of the other co- 
 ordinates. (Repulsion must be con- 
 sidered as negative attraction.) Here 
 S is the Potential. 
 
 In the case of a needle B, let x 
 be measured upwards on the paper 
 and y to the right hand : if its semi- 
 length is b inclined at an angle to 
 the axis of x, and its center has for 
 co-ordinates c in x and e in y, x will 
 = c -f b cos 6, y = e + b sin 6. These 
 apply to the pole which is on the 
 right hand of the diagram Fig. 15 : 
 for the opposite pole, b is negative. 
 The form of the general theorem can 
 then be conveniently altered, thus : 
 
 3 
 
34 ON MAGNETISM. 
 
 (a) The force on the right-hand pole of B tending 
 to increase x is 
 
 dS c + b ,cos 6 x' 
 dr' r 
 
 dr 
 
 But since r 2 = (c + b cos x') z -f (e + & sin 6 y)*, 
 
 ac 
 
 .,, C+&COS0 X , , - , r ,. 
 
 will = - , and therefore the force tending 
 
 d8 dr 
 
 to increase x will = -j- . -j- 
 dr dc 
 
 __ 
 dc' 
 
 To estimate the whole force on the needle in the di- 
 rection of x, the aggregate of the functions S for all the 
 various attractions and repulsions on the two poles must 
 be taken, and developed in the most convenient form 
 (that proceeding by inverse powers of c will always be 
 most convenient), and then it is only necessary to differ- 
 entiate with respect to c. 
 
 (6) In like manner, the force tending to carry the 
 needle to the right is -y- . But if we suppose the 
 
 ae 
 
 center of the needle B to be really on the axis of x, it 
 is only necessary to retain e for the purpose of differen- 
 tiation, and then, after the differentiation, e may be 
 made = 0. It is evident that, in this case, it is only 
 necessary to develope S as far as the first power of e. 
 
 (c) The force at right arg^s to the length of B, 
 tending to produce motion opposite to watch-hands, is 
 
 force in x x sin 9 force in y x cos 0, 
 
INVESTIGATION BY POTENTIALS. 35 
 
 7O -t 
 
 or -j- . - {(c -f b cos 6 x') sin - (e+6sin 0-y') cos 0J : 
 CUT T 
 
 dS 1 dr Id8 
 
 or T-.T -f~T^) ory- 7/1 j 
 dr &r d<9 & d0 
 
 and the angular momentum opposite to watch-hands 
 
 _dS 
 
 ~ d6' 
 
 (d) Thus we have all the forces that we require, 
 expressed in terms of a single function S m , which we 
 must now proceed to develope. If the force of attrac- 
 tion between a pole of magnet A and a pole of needle 
 
 B = a/3 x r~, which must be taken for one part of -j , 
 
 ar 
 
 then the corresponding part of 8= . -. x r"" 1 " 1 " 1 . Let 
 
 the disturbing magnet A be inclined at the angle <f> to 
 the axis of x\ then for the right-hand pole, x = a cos <, 
 y = a sin <. And, supposing the similar poles of the 
 two magnets to be on the same side, the complete 
 value of S will be 
 
 m 1 
 
 -m+l 
 
 VI * 
 
 {(cbcos0-acos(j)y+(e-bsm6asin<f)) 
 
 (e) For applying this method to specific cases, 
 such as those of Articles 14, 15, 16, 17, 18, it is neces- 
 
 32 
 
36 ON MAGNETISM. 
 
 sary to begin by taking the ordinates c, e, in all their 
 generality; and, after differentiation, to substitute the 
 specific values of c, e, 6. The special investigations 
 will be found to be easier. 
 
 20. Investigation of the cases of Articles 14, 15, 16, 
 17, 18, on the supposition that magnetism is not confined 
 to two poles of each magnet but is disseminated accord- 
 ing to some law through its whole length, being entirely 
 red magnetism on one side of the center and entirely blue 
 on the other side, with similar laws of distribution. 
 Definition of the " magnet-power " of a magnet. 
 
 Let x be measured from the center of A towards 
 the right-hand pole or upper pole (in the preceding 
 diagrams), and let the amount of magnetic energy 
 there, for the length Sx, be ae. And let y be similarly 
 measured for B, and let the amount of magnetic energy 
 for the length By be (3y. To the action of these two 
 masses of magnetism, the investigations of Articles 14 
 to 19 will apply. But in the investigation for the 
 action of a single pole of A on a single pole of B, 
 certain terms appeared which disappeared in the 
 aggregate. We will therefore adopt the results of the 
 investigations as if those terms had never appeared at 
 all. 
 
 (a) From Article 14, with the disturbing magnet 
 broadside-on and the disturbed needle end-on, the 
 angular momentum produced by these two masses is 
 
 xxyx 
 
MAGNETISM SUPPOSED NOT COLLECTED IN POLES. 37 
 or c~ m ~ 1 .zx$x.{3ySy 
 
 / 
 ( 
 
 This is to be integrated through the whole length 
 of A, from x a to x = + a, and through the whole 
 length of B, from y = btoy = + b. Now, remarking 
 that x and y are absolutely independent, and that the 
 limits of the integrations are absolutely independent, it 
 will be seen at once that the double integral of each 
 term is simply the product of the two single integrals 
 related to the two variables x and y in that term, and 
 thus the double integral becomes 
 
 We shall call the definite integral / ax "the magnet- 
 
 J x 
 
 power of A," and shall put the symbol A for it : and 
 similarly, we shall call the definite integral / /3y "the 
 
 J y 
 
 magnet-power of B" and shall use the symbol B for it. 
 For the whole of the bracket, which is a function of 
 definite integrals, we shall put K. (It is to be re- 
 marked that every one of these terms has a real valu, 
 inasmuch as, when the sign of x is changed, that is 
 when we consider the other half of the magnet, the 
 quality of the magnetism is changed, and therefore the 
 sign of a is changed : and therefore the sign of ax or of 
 
38 ON MAGNETISM. 
 
 ax 3 is unaltered, and the integral throughout is a con- 
 tinual aggregation of quantities with the same sign: 
 and so for j3y and /3y 3 .) Then the expression for the 
 angular momentum is 
 
 in which, when c is large, the second term is much 
 smaller than the first. Succeeding terms would be 
 multiplied by c~ m ~ 5 &c., and, by taking c large enough, 
 may certainly be made insensible. 
 
 (b) Treating the result of Article 16 in the same 
 way, and putting L for the bracket which then presents 
 itself, we find that, with the disturbing magnet end-on 
 and the disturbed needle broadside-on, the expression 
 for the angular momentum is 
 
 in which, when c is large, the second term is much 
 smaller than the first. 
 
 (c) The reader's attention is particularly called to 
 the circumstance that the principal term in the second 
 case (disturbing magnet end -on) is m times as great as 
 that in the first case (disturbing magnet broadside-on) : 
 and that in both the successive powers of c are m 1, 
 m 3. Thus, if the law of attraction and repulsion 
 of magnetic masses be the inverse square of the dis- 
 tance, or m = 2, then the principal term in the second 
 case is double the principal term in the first case, and 
 the successive indexes of c are 3, 5. It will be 
 found that these remarks are of the utmost importance 
 
EAKTH'S ACTION ON A NEEDLE. 
 
 39 
 
 in determining the law of magnetic attraction and 
 repulsion. 
 
 21. Investigation of the action of a magnet, whose 
 poles are very widely separated, upon a needle whose 
 center is in the line of these poles and whose axis is 
 inclined to that line. 
 
 Fig. 16. 
 
 In Figure 16, the repulsion of the magnetic red 
 mass S upon the red mass fiby at the point y of the 
 needle is 
 
 #./% (c 2 + 2cy cos + y z ) "^; 
 
 which being resolved into a part acting endways of the 
 needle and a part parallel to c gives for the latter part 
 
 -m-l 
 
 (c 2 + 
 
40 ON MAGNETISM. 
 
 which produces the angular momentum opposite watch- 
 hands 
 
 y. sin . 8. /%.c (c 2 + 2c#.cos + 2 / 2 )~ T ~. 
 
 Now when c is very large, it will suffice to retain the 
 first efficient term in the expansion, which gives 
 
 or 
 
 m 0.8 < 
 
 The integral of this, through the whole needle, will be 
 
 In like manner, the attraction of the mass of N of blue 
 magnetism will produce an angular momentum in the 
 same direction represented by 
 
 and the total angular momentum will be 
 
 Let the quantity in the bracket be called E: then the 
 total angular momentum = 
 
 EB. sin 6. 
 
 It is evident that, under the action of this force, 
 the needle B will vibrate according to the same law as 
 a common pendulum. 
 
 It will be remarked that the conditions here sup- 
 posed correspond to those of the Earth's horizontal 
 magnetic influence as observable at any one place. 
 We know nothing of the precise action of the Earth's 
 magnetism, except that, for any one place, it may be 
 
EAKTH'S ACTION ON A NEEDLE. 41 
 
 represented by the action of a blue mass on the north 
 side and a red mass on the south side, or by one of 
 these alone. The result at which we have arrived is 
 analogous in form to that which we have obtained 
 in other magnetical investigations; and, to preserve 
 the analogy of language, we shall call E the Earth's 
 magnet-power. It is important to remark that, on 
 account of the assumed magnitude of c and c', the 
 angular momentum here is expressed by a single term, 
 a multiple of B, without any of the additional terms 
 which occur in Article 20. 
 
42 ON MAGNETISM. 
 
 SECTION IV. 
 
 ON TERRESTRIAL MAGNETISM, AS ACTING IN THE HORI- 
 ZONTAL PLANE AT EACH PLACE OF OBSERVATION. 
 
 22. Definition of Local Magnetic Meridian and 
 Magnetic Variation or Declination; instruments and 
 methods for ascertaining roughly the Declination: 
 Azimuth Compass; observation of Suns Azimuth, Va- 
 riation Theodolite; Declination Charts. 
 
 The ' Local Magnetic Meridian ' is the direction, on 
 the horizontal plane of any place of observation, which 
 is taken by the compass-needle. The ' Magnetic Varia- 
 tion' (a very bad term, still commonly employed by 
 nautical men, but for which, among men of science, the 
 term ' Magnetic Declination ' is usually substituted) is 
 the angle made by the Local Magnetic Meridian with 
 the Astronomical Meridian : it requires to have append- 
 ed to it the word " east " or " west " as applied to the 
 north point of the needle. Thus the Magnetic Decli- 
 nation at Greenwich at the present time is about 20 
 West: meaning that the north point of the magnetic 
 needle points west of the astronomical north meridian 
 by 20 nearly. 
 
LOCAL MAGNETIC MERIDIAN. 
 
 43 
 
 The general direction of the Local Magnetic Meri- 
 dian may be obtained by merely observing the direc- 
 tion of the needle : but the Astronomical Meridian is 
 not so obviously visible : and therefore the Declination 
 cannot be easily found. Navigators however require 
 the Declination in order to enable them to adapt their 
 compass-steering to ordinary maps and charts. The 
 
 Fig. 17. 
 
 Fig. 17*. 
 
 element required was obtained by the use of an 
 ' azimuth compass' represented in Figures 17 and 17*. 
 
44 ON MAGNETISM. 
 
 (The jimbal-rings, by which the compass-card retains 
 its horizon tali ty in all motions of the ship, are omitted 
 in these Figures.) It must be remarked that in ships' 
 compasses the compass-needle is never exhibited naked, 
 but is inclosed in the thickness of a * compass-card/ a 
 circular card on which points of the compass and degrees 
 of azimuth are engraved, and which, being firmly con- 
 nected with the compass-needle, is so directed by the 
 magnetic power of the earth acting on the compass-needle 
 that the N, E, S, W on the card point truly to magnetic 
 north, east, south, west. In the compass, nothing touches 
 the circumference of this card, but there are, rising from 
 the compass-box, two small frames carrying vertical 
 wires; by directing the eye-view along the two wires 
 and turning the box till that eye-view sees a distant 
 object in the same line as the two wires, the line of 
 wires is made to coincide with the direction of the 
 object : and then the observer must read the gradua- 
 tions of the card which correspond to points in the 
 box below the two wires. In this way he obtains the 
 azimuth of the object as referred to the Local Magnetic 
 Meridian. 
 
 In the best modern instruments, a horizontal ring 
 is expressly provided to carry the vertical wire-frame: 
 and, instead of having a wire next to the eye, a glass 
 prism acting by internal reflection is placed there; so 
 arranged, that one half of the pupil of the eye can 
 observe the wire on the further side of the horizontal 
 ring, and the distant object; and the other half of the 
 pupil can see the graduations of the compass-card by 
 
AZIMUTH COMPASS. 45 
 
 internal reflection in the prism. (Small instruments 
 of this kind, called "prismatic compass," are to be 
 obtained at little expence, and are very convenient.) 
 
 The azimuth of an object being obtained as refer- 
 red to the Local Magnetic Meridian, the next point is 
 to find its azimuth as referred to the Astronomical 
 Meridian. The power of doing this depends on the 
 choice of the object. Navigators almost invariably 
 choose the rising or setting Sun. The latitude of the 
 ship being known with sufficient accuracy for this 
 purpose, and the Sun's declination and consequent 
 distance from the celestial pole being taken from the 
 Nautical Almanac, the solution of a quadrantal equa- 
 tion gives the Sun's azimuth at rising or setting as 
 referred to the Astronomical Meridian. The difference 
 between this and the azimuth as referred to the Local 
 Magnetic Meridian gives the Magnetic Declination. 
 (The poetic reader will find much of this operation 
 correctly described in Falconer's " Shipwreck.") 
 
 For determinations on shore, a " Variation-theodo- 
 lite" was sometimes used, consisting of a common 
 theodolite adapted to the measure of horizontal angles, 
 the axis of its telescope being so much raised that the 
 telescope could be pointed to the pole-star, by which 
 the reading of the horizontal circle for astronomical 
 meridian could therefore be found; and that the tele- 
 scope could also be pointed down to a compass-needle 
 immediately below the theodolite-frame, to the ends of 
 which the telescope-object-glass could be directed, and 
 which ends were made distinctly visible by affixing 
 
46 ON MAGNETISM. 
 
 another lens on the telescope-object-glass; by which 
 process the horizontal reading for magnetic meridian 
 was found. 
 
 By such methods as these, "Variation Charts" or 
 more properly "Declination Charts" have been pre- 
 pared for the use of mariners, exhibiting the Compass- 
 Declination on every part of our globe. And it is 
 mainly by the use of these, in connexion with the 
 compass, that ships are steered. 
 
 23. More accurate method of determining the Local 
 Magnetic Meridian; Reversed Telescope carried by Mag- 
 net; elimination of corrections for position of Magnetic 
 Axis, and for torsion-power of suspension-thread. 
 
 The most accurate apparatus for determining the 
 Declination is however that represented in Figure 18, 
 which is nearly copied from the instrument used at the 
 Royal Observatory, Greenwich. It is founded essen- 
 tially on the use of the " Reversed Telescope " or (as 
 it is frequently but inappropriately called) the u Colli- 
 mator." It is an ordinary optical theorem that, if 
 parallel rays fall upon an object-glass, they will be 
 made to converge to a point: and conversely, if rays 
 diverge from a point at the focus of an object-glass and 
 fall on that object-glass, they will emerge parallel. 
 This quality of parallelism of rays possesses two im- 
 portant advantages. First, that the small object at 
 the focus (which object will usually be a cross of wires) 
 can be distinctly seen by use of a telescope, adapted to 
 receive parallel rays, or rays coming from a distant 
 
MAGNET WITH REVERSED TELESCOPE. 
 
 47 
 
 object, as a star. Second, that if that telescope be 
 shifted laterally so as to receive the rays from one part 
 of the parallel pencil instead of another, it still receives 
 them in the same direction: so that minute adjustment 
 of the lateral position of that telescope is not necessary. 
 
 Fig. 18. 
 
 In Figure 18 therefore B represents a magnet 
 suspended by a suspension-thread : C a small frame 
 firmly attached to it, carrying a fine wire, or two wires 
 crossing each other, and thus producing a fine point for 
 observation : D another frame clamped to the magnet 
 and carrying an object-glass whose focus is exactly at 
 the place of the wires carried by C. The rays of light 
 
48 ON MAGNETISM. 
 
 diverging from the wire or mark in C and falling on 
 the object-glass in D will emerge from it in a parallel 
 pencil, every ray being parallel to that which passes 
 from the mark G through the center of the object-glass 
 in D: or parallel to the optical axis of the Reversed 
 Telescope CD. A theodolite E must be placed at 
 any convenient distance so that the object-glass of its 
 telescope shall receive the whole or a part of the pencil 
 of parallel rays coming from D : the eye, applied to the 
 theodolite-telescope, will see the mark C distinctly : the 
 theodolite may be slightly turned till the wire in its 
 field of view is seen to coincide with the image of the 
 mark C: and then the graduations of the horizontal 
 circle of the theodolite indicated by the pointer or 
 vernier of the rotatory part of the theodolite (which 
 carries its telescope) are to be read. After this, the 
 theodolite is to be turned with its telescope elevated 
 at the proper angle, as at E', to view the pole-star or 
 other circumpolar stars; a meridional direction being 
 thus obtained, the horizontal circle of the theodolite is 
 to be read as before: and the difference between this 
 reading and that formerly taken when observing the 
 mark C is the magnetic declination. 
 
 But there are some delicate points requiring atten- 
 tion, which it may be well to describe here. First: the 
 line along which our theodolite-telescope is directed is 
 the optical axis of the Reversed Telescope CD. But 
 the line which takes the direction of Terrestrial Mag- 
 netism is an undiscoverable line in the interior of the 
 magnet, called its "magnetic axis." How can we be 
 
MAGNET WITH REVERSED TELESCOPE. 
 
 49 
 
 certain that this magnetic axis is parallel to the optical 
 axis of GDI The only thing that we can do is, so to 
 arrange observations in a different state of the magnet 
 that whatever error is produced by want of parallelism 
 in the first case shall be exactly reversed in the second 
 case ; and the mean of results will then be free -from 
 error. It is merely necessary to invert the magnet in 
 respect of the side on which it carries CD, as shewn in 
 Figure 19; if the optical axis of CD pointed too much 
 
 Fig. 19. 
 
 to the west of magnetic north in the first case, it now 
 points too much to the east of magnetic north in the 
 second case, or vice versd. 
 
50 ON MAGNETISM. 
 
 In practice there is another cause of error, from the 
 torsion-power of the suspension-thread. If, in the 
 observations above described, the thread has any 
 tendency to turn the magnet horizontally, the results 
 are certainly erroneous. To give the means of applying 
 a mechanical correction, the suspension-thread must not 
 be tied at its top to a fixed bar, but must be fastened 
 to or hung on a hook or ring attached to a small frame 
 which can be turned round horizontally. (Sometimes 
 this means of rotation is given at the lower end of the 
 thread.) And to discover whether correction is wanted, 
 the magnet must be taken out of the frame which 
 carries it, and a brass bar of equal weight and furnished 
 with a similar Reversed Telescope must be hung in its 
 place. If, on viewing with the theodolite, it is found 
 that the mark in the Reversed Telescope of the brass 
 bar occupies the same position as in that of the magnet, 
 no correction is necessary. If it does not occupy the 
 same position, the rotating frame carrying the hook 
 must be turned till the position becomes the same. 
 When this adjustment is completed, the determination 
 of the Local Magnetic Meridian will be extremely 
 accurate. 
 
 In some instruments used for this purpose, the 
 magnet is made in the form of a hollow steel tube 
 (which can be magnetized perfectly well), and the mark 
 C and lens I) are fixed in its interior, forming a verit- 
 able telescope. 
 
 24. Terrestrial Magnetic Meridians; Historical 
 physical changes in the system of Magnetic Meridians. 
 
TERRESTRIAL MAGNETIC MERIDIANS. 
 
 51 
 
 The directions of the Local Magnetic Meridian 
 being found at numerous points of the Earth, it is not 
 difficult to trace through them a curve starting from 
 any assumed point, and so drawn that in every part of 
 its course its direction represents the direction of the 
 Local Magnetic Meridian at that point. Such a curve 
 may properly be called a Terrestrial Magnetic Meridian : 
 and a number of these, at convenient intervals of 
 
 (A is the North Magnetic Pole. See Article 42.) 
 
 geographical longitude, may be advantageously used 
 as a system of Terrestrial Magnetic Meridians. Such 
 
 42 
 
52 ON MAGNETISM. 
 
 a system is represented by the diverging strong lines in 
 
 Figures 20 and 21, which are maps on the stereographic 
 
 Fig. 21. 
 
 (A is the South Magnetic Pole. See Article 42.) 
 
 projection. (Although this system is essentially found- 
 ed on observations, collected about forty years ago, 
 some portions of it in inaccessible regions of the Earth 
 are supplied from a theory of Gauss's, to which we shall 
 soon allude: some small alterations would be required 
 to make it quite correct for the present date.) The 
 forms of these Magnetic Meridians are very remarkable. 
 None of them appears to be exactly a great circle. 
 
COMPARISON OF TERRESTRIAL DIRECTIVE FORCES. 53 
 
 They converge to a north pole, north of Hudson's Bay, 
 and a south pole, in South Victoria: but these poles 
 are not opposite. From E. longitude 70 to 150, the 
 northern parts agree nearly with Geographical Meri- 
 dians: the same remark also applies to a large portion 
 of the southern part in longitude 150. 
 
 The system of Magnetic Meridians has undergone 
 considerable changes in the times of modern accurate 
 science. The southern point of Africa received from 
 the Portuguese voyagers in the fifteenth century the name 
 of L'Agulhas (the needle), because the direction of the 
 compass-needle, or the Local Magnetic Meridian, coin- 
 cided there with the Geographical Meridian : it now 
 makes with it an angle of about 30. In the sixteenth 
 century, the compass-needle in Britain pointed east of 
 north : it now points from 20 to 30 (in different parts 
 of the British isles) west of north. At the present time, 
 a change of the opposite character is going on : in 1819 
 the westerly declination at Greenwich was about 24 23', 
 which was probably its maximum ; in the last thirty 
 years it has diminished from 23J to 20, nearly. It is 
 believed that the magnetic poles are rotating round the 
 geographical poles from East to West. 
 
 25. Imperfect method of measuring the horizontal 
 directive force of terrestrial magnetism at any locality, 
 by vibrations of a magnetic needle. Correction for the 
 torsion-power of the suspending-thread. 
 
 In Article 21, it was found that the angular momen- 
 tum, which the Earth's horizontal magnetic action im- 
 
54 ON MAGNETISM. 
 
 presses on a suspended needle whose axis makes the 
 angle 6 with the Local Magnetic Meridian, is EB sin 6 : 
 and if M be the moment of inertia of the needle, the 
 equation which determines its angular motion will be 
 
 72/1 77' 73 
 
 -T72 = yp sin 6. If 6 be very small, or if proper cor- 
 
 CLt Jj-L 
 
 rections be applied to the observed time of vibration (as 
 for an ordinary pendulum) so as to reduce the time of 
 vibration to what it would have been if the arc of vibra- 
 tion had been indefinitely small, we may use the equa- 
 
 tion -j 2 = -- iTf 6 : of which the solution is 
 at M. 
 
 and the time T of a complete double vibration is 
 
 / M 
 2-7T./ - . This is a single equation containing two 
 
 unknown quantities E and B, and neither of them can 
 be determined from it. 
 
 If we repeat the experiment at a different time at 
 the same station, or at any time at another station, 
 where it may be presumed that the Earth's magnet- 
 
 / M 
 power is different, we shall have T' = 2?r /-rrp* which 
 
 still gives no information. And if there is suspicion 
 that the magnet-power B may have changed, there is 
 no hope of arriving at any conclusion. 
 
 There is, however, one way of using the vibrating 
 needle, from which an imperfect result may be obtained. 
 If the vibrations be taken at a standard place, (as 
 
COMPARISON OF TERRESTRIAL DIRECTIVE FORCES. 55 
 
 Greenwich) ; and if the needle be carried to other 
 stations in rapid course, so that there is little reason to 
 fear any change in its magnet-power ; and if, for con- 
 firmation, the needle be again brought to the standard 
 place : then we may obtain a certain result thus. Di- 
 vide the square of the first equation above by the square 
 of the second, and we have 
 
 _T _.#' 
 T'*~ E' 
 
 Thus we obtain the proportion of the Earth's magnet- 
 power at one station to its magnet-power at another 
 station, at the same time. But we get no positive in- 
 formation on the measure of the Earth's magnet-power 
 at either station. And as we cannot suppose that the 
 magnet-power of the needle will be unaltered through 
 an unlimited time, we cannot use the experiment to 
 determine whether the Earth's magnet-power at any 
 station has altered with the lapse of time. 
 
 The principal value of this method is for very re- 
 stricted local experiments. 
 
 In all applications of the method, we ought in strict- 
 ness to take account of the torsion-power of the suspen- 
 sion-thread : as, on changing the suspension-thread, or 
 in comparing observations where the difference of 
 external magnetic action is great, the omission of that 
 consideration may introduce important error. The 
 torsion-power will be measured thus. Suppose the 
 suspension-piece to be furnished with apparatus which 
 admits of being turned round horizontally (as described 
 
56 ON MAGNETISM. 
 
 in Article 23). Note the position taken by the needle. 
 Then turn the suspension-apparatus through a mea- 
 surable angle, say 100, and again note the position 
 taken by the needle. If this position differs from the 
 former, say by 1, it shews that the torsion-power pro- 
 duced by a torsion of 100 is able to neutralize the 
 Earth's action by the quantity corresponding to an incli- 
 nation 1 ; and therefore that the terrestrial action on 
 the needle at any inclination 6 is augmented by the 
 power necessary to overcome -^ part of terrestrial 
 action at that angle ; and that in fact the power 
 which we measure is f of the terrestrial power. 
 
 In this imperfect state the determination of the 
 Earth's magnet-power remained till about the year 1835. 
 And it does appear, at first view, impossible to obtain a 
 numerical value, entirely freed from all dependence on 
 weight of "needle, quality of its steel, intensity of its 
 magnetism, &c., for such an unprehensible element as 
 terrestrial magnetism. A method however was intro- 
 duced (first suggested, the writer believes, in an imprac- 
 ticable form by Poisson, and subsequently changed to 
 an easy and accurate shape by Gauss) which solves the 
 problem perfectly, and is now in daily use : it may well 
 be considered as one of the most beautiful processes in 
 experimental philosophy. To this process, after some 
 preliminary experiments, and a determination of the 
 last Law of Magnetism, we shall turn our attention. 
 
 26. Experimental determination of the proportion of 
 the magnet-power of a magnet to the Earth's horizontal 
 
MEASURE OF MAGNETIC DEFLEXIONS. 
 
 57 
 
 magnet-power, by the method of deflexions. First process, 
 the disturbing magnet applied broadside-on. 
 
 Fig. 22. 
 
 'G 
 
 N 
 
 In Figure 22 is represented a side view of the deflex- 
 ion-apparatus, and in Figures 23, 24, 25, horizontal plans 
 of it under different circumstances. E is a graduated 
 circle firmly fixed, preserving absolutely the same posi- 
 tion in Figures 23, 24, 25. jP is a bar turning horizon- 
 tally round the centre of E, and having pointers 
 or verniers f, f, for reading the graduations of E. 
 F supports all the rest of the apparatus. From F 
 there rise two uprights 6r, 6r, with a cross-bar at 
 the top, to which is attached a suspension-thread 
 
58 
 
 ON MAGNETISM. 
 
 carrying the needle B. To B are attached a frame C 
 with cross- wires and a frame D with object-glass, as in 
 
 Fig. 24. 
 H 
 
 Article 23, forming a Reversed Telescope, .fi" is a tele- 
 scope firmly fixed to F, at the same height as CD, which 
 can therefore see distinctly the mark G. A is the 
 magnet for which the proportion of magnet-power to the 
 Earth's horizontal magnet-power is to be found : it may 
 sometimes be placed in the position A': it is so sup- 
 ported that the view of the telescope H is not interrupt- 
 ed. In the Figure, the telescope is supposed to point 
 to the north, but it may equally well point to the south. 
 In the case now under consideration A is presented 
 broadside-on to B. The observation begins without A, 
 as in Figure 23 ; F is turned to such a position that, 
 
MEASURE OF 'MAGNETIC DEFLEXIONS. 
 
 59 
 
 while B takes its direction under the action of terres- 
 trial magnetism only, the mark C is viewed with the 
 telescope H and its image is made to coincide with the 
 wire in H\ in this state, the graduations of Sunder the 
 
 pointers/,/, are read. A is then mounted in its place, 
 with its red end to the east, suppose, as in Figure 24 : 
 the north end of B immediately turns to the west : the 
 bar F is turned carefully in the same direction, till the 
 mark G is again seen by the telescope H, and its 
 image coincides with the wire in H: then the gradua- 
 tions of E under / /, are read. The difference between 
 this reading of the graduations of E and the former 
 reading is the angle through which B has been deflect- 
 
60 ON MAGNETISM. 
 
 ed by the action of A in the circumstances considered 
 in the case (a) of Article 20, A being presented broad- 
 side-on to B which is end-on. It is to be remarked that 
 no new torsion of the suspension-thread is introduced 
 by the rotation of B, because the suspension-bar, which 
 is carried by the uprights on F, is made to rotate 
 through the same angle as B. 
 
 The experiment can now be varied by reversing 
 A, end for end, and B will be deflected the opposite 
 way, as in Figure 25 : the mean of the two deflexions 
 will be free from any error in the reading of Figure 
 23 : and, generally, information will be obtained which 
 may assist to remove accidental errors. Also the mag- 
 net may be placed in the position A', whose distance 
 from A can be measured with great accuracy : if in 
 one case the center of B was too near to A, in the other 
 case it will be equally too far from A ; and the mean 
 of the results will be free from the effect of error in the 
 position of the center of B. 
 
 From the mean of these deflexions, the true deflex- 
 ion 6 will be found with great accuracy. And now we 
 remark that the needle B is kept in a definite angular 
 position by the opposition of two forces whose angular 
 moments must be equal. One is, the angular moment 
 produced by A in the case () of Article 20 : its value is 
 
 The other is, the angular moment produced by the 
 Earth, as in Article 21 : its value is 
 
EQUATION GIVEN BY DEFLEXIONS. 61 
 
 The comparison of these (0 having such a sign that the 
 forces are opposed) gives the equation 
 
 c^.AB + c-*. K=EB. sin0 ; 
 
 in which it is to be remembered that the second term 
 on the left side is small. 
 
 27. Experimental determination continued. Second 
 process, the disturbing magnet applied end-on. 
 
 Here the form of the apparatus must be slightly 
 varied. The reversed Telescope CD, and the telescope 
 H, must occupy the same position as before : but the 
 magnet A must be carried upon an arm F at right angles 
 to F\ see Figures 26, 27. For the same reasons as in 
 
 Fig. 26. 
 
 last Article, the magnet A will be used in the two 
 positions shewn in these Figures, and also in two corre- 
 sponding positions on the opposite half of the transversal 
 
62 ON MAGNETISM. 
 
 bar. Let <f> be the mean of the angles thus obtained. 
 Then, referring to the formulae in case (6) of Article 20 
 
 Fig. 27. 
 
 and in Article 21, we shall, as in last Article, obtain the 
 equation 
 
 To this point, the observations, to which this Article 
 and the last refer, appear to throw little light on the 
 subject : but a variation of the experiment, described in 
 the next Article, will give very great information. 
 
 28. Experimental determination continued. Each 
 of the preceding operations performed with the needles at 
 different distances. Reference to numerical observations. 
 The discussion of the first process alone, the discussion of 
 tJie second process alone, and the comparison of the two 
 processes, shew independently that the attraction or repul- 
 
LAW OF FORCE INFERRED FROM DEFLEXIONS. 63 
 
 sion of magnetic masses is inversely as the square of the 
 distance between them ; Final Law of Magnetism. 
 
 Let the' experiment of Article 26 be performed with 
 two values of c (the distance between the centers of the 
 magnets) which we shall call c x and c 2 . Let the cor- 
 responding deflexions be t and # 2 . Then we have the 
 two equations, 
 
 sn 2 ; 
 
 and nearly similar equations from Article 27, in each of 
 which the second term on the left hand is to be small. 
 And the question now to be considered is, What 
 numerical value for m will enable us to satisfy that 
 condition ? 
 
 We shall approximately represent the state of the 
 case, by neglecting the small terms, and thus we get 
 the following approximate equations : 
 
 c- m ~ l . AB = EB . sin t : c 
 c^ .mAB = EB. sin & : c~ m ~ l .mAB=EB. sin fa 
 
 We must now appeal to actual experiment. A few 
 observations bearing on these points were made by 
 Gauss, but in the Greenwich Observations there are to 
 be found about one hundred observations in which the 
 disturbing needle was applied both broadside-on and 
 end-on, and a far greater number in which it was 
 applied end-on alone. We will take the observations 
 of 1860, January 16. 
 
64 ON MAGNETISM. 
 
 Disturbing needle broadside-on. 
 
 l'O foot. 
 
 4. 29'. 20". 
 
 c = 1*5 foot. 
 
 Disturbing needle end-on. 
 
 = 
 
 I'D foot. 
 
 '2 
 
 < 2 = 2.29'.52". 
 It is to be remarked that 
 
 c =r5foot. 
 
 3.375 . ? = 5-0625. 
 
 Now first, dividing the first equation above by the 
 second, we get the approximate equation 
 
 Substituting the numerical values of sin d l and sin 9 V 
 this becomes 3*4490 = (l'5) m+1 . Looking at the powers 
 of 1*5 placed above, it is impossible that m + 1 can be 
 2 or 4, but the equation is so nearly true when m+l=3, 
 (that is, 3*4490 approaches so near to 3'375) as to afford 
 strong presumption that m + 1 = 3. 
 
 Secondly, we obtain in the same way -: -j 1 ap- 
 
 sin < 2 
 
 (0 \m+l . 
 J . Substituting the numerical va- 
 lues of < x and < 2 , this becomes 3*4115 = (l*5) m+1 . This 
 is very nearly correct if m + 1 = 3. 
 
 Thirdly, if we divide the equation for ^ by that for 
 
 0, we obtain . 7 = m. Substituting the numerical 
 
 1 ein H w 
 
UNIT OF MEASURE FOE A AND E. 65 
 
 values of the sines, this becomes 1*8996 m. This 
 affords a strong presumption that m = 2, agreeing with 
 the deductions above. 
 
 Fourthly, if in like manner we divide the equation 
 for < 2 by that for 2 , the numerical equation becomes 
 1-9205 = . 
 
 It is impossible that any integer but 2 can represent 
 the value of m : and, remarking that, in the omission of 
 small terms, the equations above are not rigorously 
 correct, the value m = 2 is at once assumed for further 
 investigations ; and this gives the Final Law of Magnet- 
 ism, that attractions and repulsions of magnetic masses 
 are inversely as the square of their distance. 
 
 29. Inference as to the numerical value of ike pro- 
 portion of A to E. Remark on the unit of measure for 
 A and E. 
 
 Considering the law of inverse square of distance to 
 be established, unless following phenomena should con- 
 tradict or modify it (none of which do so), it has been 
 usual for experimenters to restrict themselves, for finding 
 the ratio of AB to EB, to observations with disturbing 
 needle end-on : partly because the apparatus is more 
 convenient, partly because the deflexion with a given 
 separation of needles is greater. The observations of 
 deflexion being then taken with the needles at the two 
 distances c t and c 2 , we have the two following equations, 
 which possess all the accuracy that is required for com- 
 parison with observation ; 
 
 5 
 
66 ON MAGNETISM. 
 
 cf 3 . 2AB + c," 5 . L = EB . sin & ; 
 ,~ 5 . L = J&?Z? . sin (f> . 
 
 Substituting the numerical values of c x , C 2 , <,, and 
 <k,, we have here two simple equations for determination 
 of the two unknown quantities AB and L. Both will be 
 expressed as multiples of EB. The quantity L is of no 
 further use to us. But AB is now expressed by such a 
 formula as AB = Jc x EB : and therefore A = kxE: k 
 being a number given by the solution of the equations. 
 Thus we have the magnet-power of the Earth expressed 
 accurately as a multiple of the magnet-power of A. 
 
 Having thus acquired the power of separating the 
 first term from the rest, in the expression for the angular 
 momentum produced by the action of one magnet upon 
 another, we may now conceive that first term only to be 
 retained, in an action of which the idea will be service- 
 able to us. If c is 1 (that is, the unit of measure, what- 
 ever it may be), then the angular momentum, caused 
 in B by the action of A broadside-on, is AB. Suppose 
 now that we have two magnets S and S', exactly similar 
 and equal, such that, using the letters for the magnet- 
 powers as defined in Article 20, paragraph (a), S= 1, 
 8' = 1 : and suppose the distance of their centers to be 
 = 1 ; then the angular momentum produced in S' by 
 S, broadside-on, as depending on the first term of the 
 formula, = 1. Such a magnet may well be conceived as 
 a standard magnet. And when we obtain a numerical 
 value A for the magnet-power of the needle A, it means 
 that its action on S' at distance 1 is A times as great 
 
BEST PROPORTION FOR DISTANCES OF MAGNETS. 67 
 
 as that of S at distance 1 : and when we find that the 
 Earth's magnet-power is , x that of A, it means that 
 
 A 
 the Earth's action on 8' is j- x the action of S upon S' 
 
 at distance 1. 
 
 30. Investigation of the most advantageous propor- 
 tion of the two distances of the disturbing magnet from 
 the disturbed needle. 
 
 If c 2 were very nearly = c t , the two equations of last 
 Article would be so nearly alike that the divisors giving 
 the results of the equations would be small, and the 
 possible errors in the measures of the angles 6 or <j> 
 would produce enormous errors in the values found for 
 the two unknown quantities. If c 2 were very large, the 
 corresponding deviation would be very small, and the 
 possible errors would bear a large proportion to the 
 deviation. There is therefore a more advantageous pro- 
 portion of c 2 to c v which it is our object now to ascertain. 
 
 The equations being of this form, 
 
 where the actual error of D l may be E^ and the actual 
 error of D 2 may be E 2 ; we find, by elimination of #, 
 
 (r ~ 3 r ~ 5 r ~ s r ~ 3> l r c ~ 5 D c ~ 5 D 
 ( c i -C 2 ' ~ | <? 2 ; x C 2 **% c i L'zy 
 
 and 
 
 subject to the actual error, 
 
 x c ~ 5 E (c ~ 3 r ~ 5 r " 5 r "" 3> > x c ~ 
 xc 2 .&! (c t ,c u c x .c 2 , x c t 
 
 52 
 
 (c ~ 3 c ~ 5 r ~ s r 
 i .c c .c 
 
68 ON MAGNETISM. 
 
 Let the probable error of D x be e v and the probable error 
 of Z> 2 be e z (see the Author's Treatise on 'Errors of 
 Observations/ Article 28) : then the square of the pro- 
 bable error of the expression for # will be (see Articles 
 44 and 50 of the same treatise) 
 
 (c;*v-v.o^x C 2 " 10 x e>(cf X~ 5 -^~WTX~ 10 >< & 
 
 and if e l = e 2 = e, the square of the probable error of x 
 
 = (of 3 - <** - or*- <vT x (cf 10 + o x e. 
 
 A value of c 2 is to be found which will make this 
 minimum. Differentiating with respect to c 2 and making 
 
 s\ 
 
 the differential coefficient = 0, and putting z for -, this 
 
 c i 
 equation is obtained, 
 
 53 10 -3-2 12 + 23 2 = 0; 
 of which the solution is 
 
 st 
 
 z or = 1*32 very nearly. 
 c i 
 
 It is usual in the best modern instruments to make 
 C 2 = Cj x 1-3. 
 
 31. Incidental inference as to the effect of tempera- 
 ture upon the magnet-power of the disturbing magnet, 
 and necessity for correction for temperature in various 
 experiments. 
 
 In considering the deflexion-experiments made 
 under different atmospheric temperatures, it is found 
 that the proportion of magnet-power of the disturbing 
 magnet to that of the earth is smallest at the highest 
 temperatures. To measure the amount of change for 
 
EFFECT OF TEMPERATURE. 69 
 
 1 of temperature, without depending on the assump- 
 tion of uniformity of the Earth's magnet-power, the 
 disturbing needle is inclosed in a copper box; and 
 water, at different temperatures for the different 
 observations, is poured into the box. At each of 
 these temperatures, the deflexion produced on B is 
 observed, and the thermometer is read. Thus we 
 
 A 
 obtain the values of -~ at different temperatures of A ; 
 
 and, as the experiment can be completed in so short a 
 time that we may presume on the invariability of E 
 during the observations, we find in fact the propor- 
 tionate change of A for a given change of temperature. 
 
 In some instances, the temperature has been 
 changed by heating the air of the room. 
 
 The ratio of change thus found is very different for 
 different magnets. It probably depends on the quality 
 of the steel; or possibly on the mode of magnetization: 
 but on this point nothing is known with certainty. In 
 all instances, it is believed, the magnet-power dimi- 
 nishes when the temperature is raised. In the 
 " Horizontal Force Magnet " of the Royal Observatory, 
 Greenwich, the loss of power for a rise of 1 Fahrenheit 
 is = magnet-power x 0-0000809: in the magnet lately 
 used as " Vertical Force Magnet," the loss of power 
 for 1 is = magnet-power x 0'00013845. 
 
 In using a needle in the manner described in 
 Article 25, - for comparing the terrestrial horizontal 
 force at different localities, the result ought always to 
 
70 ON MAGNETISM. 
 
 be reduced, by applying the correction for temperature, 
 to what it would have been at a fixed temperature of 
 the magnet, as 32. 
 
 32. Accurate determination of the Earths local hori- 
 zontal magnet-power, founded on the method of deflex- 
 ions, used in combination with the measure of the 
 Earth's horizontal action upon the disturbing magnet. 
 
 In the experiment of Article 26, or that of Article 
 27, treated in the manner described in Article 29, we 
 have found A=kxE:k being a number determined 
 by the solution of certain equations given by the 
 observations. In those observations it was necessary 
 to use a needle B: but we have no further occasion to 
 employ that needle. The operation now to be per- 
 formed consists in suspending the magnet A delicately, 
 and observing the time of its vibration under the action 
 of the Earth's horizontal force. The theory of this has 
 been treated in Article 25 ; and the result, substituting 
 A for B, and preserving T to denote the time of a 
 complete double vibration, and M for the moment of 
 inertia of A, is 
 
 T- / M 
 
 ^V EA : 
 
 whence EA = -^ x M. 
 
 It will be remarked that the unit, by which E and 
 A are measured, is the same as that described at the 
 end of Article 29, namely, the action of the standard 
 magnet S upon S' at distance 1. 
 
EARTH'S LOCAL HORIZONTAL MAGNET POWER FOUND. 71 
 Multiplying together the two equations 
 
 E 1 47T 8 
 
 and extracting the square root, 
 
 *? I M - 
 T\' k' 
 
 a result which depends upon no property whatever of 
 the magnets employed, except M the moment of inertia 
 of A. To the determination of the numerical value 
 of M we will now give our attention. 
 
 If, as is usually the case with large magnets, the 
 form of the magnet be a parallelepiped, and its structure 
 
 , ,, , f ,.. .,, , mass in grains 
 
 homogeneous, the value of M will be - x 
 
 \.2i 
 
 {(length in feet) 2 + (breadth in feet) 2 }. This calcula- 
 tion is easily made with accuracy. It is necessary to 
 add the moment of inertia of the stirrup or other hook 
 which supports the magnet and vibrates with it : this 
 in general is a very small quantity, and can be obtained 
 with sufficient accuracy by weighing that apparatus 
 and estimating its radius of gyration. 
 
 If the form of the magnet be not so simple, or if 
 there be any grounds for suspicion of the accuracy of 
 this process, the device proposed by Weber may be 
 adopted. The time of vibration of the magnet having 
 been observed, as above mentioned, the two ends of the 
 magnet are then loaded with brass weights, very care- 
 fully weighed, which rest upon the magnet by sharp 
 points, so that the weights do not partake of the cir- 
 
72 ON MAGNETISM. 
 
 cular movement of the magnet; and the distance be- 
 tween these points is measured. In that state, the 
 time of vibration is again observed. < As the force which 
 causes the vibration is the same in both cases (namely, 
 the action of terrestrial horizontal magnetism upon the 
 magnet), the moments of inertia will be proportional to 
 the squares of the two times of vibration. But the 
 difference between the two moments of inertia is 
 merely the moment of inertia of the two brass weights, 
 each being supposed collected at its sharp supporting- 
 point, and admits of being accurately computed. Then 
 the difference of the moments of inertia of the magnet 
 in its two states being known, and their proportion 
 being known, each of them is determined accurately. 
 
 Whichever method is used, the numerical value of 
 M is found and substituted in the expression for E, and 
 the numerical value of E is obtained. 
 
 In the observation of deflexion described in Articles 
 25 to 29, it is evident that the comparison of the mag- 
 net-powers E and A implies that their numerical values 
 are referred to the same unit. And in the investiga- 
 tion, in the present Article, of the measure of the 
 Earth's action upon the magnet, we have used exactly 
 the same formula as in Article 21, which is founded on 
 the methods of preceding Articles, in which all are 
 referred to the same unit. It follows that the numeri- 
 cal value which we have found for E is referred to the 
 same unit : namely (see Article 29) to the magnet-power, 
 or to the magnetic action at distance 1, of the standard 
 magnet S or its equal $', which are such that, when 
 
EARTH'S POWER EXPRESSED BY DIFFERENT UNITS. 73 
 
 the distance of their centers is 1, the angular momen- 
 tum produced in S' by S broadside-on (using the first 
 term only of the formula of force), is 1. 
 
 In general, the two operations (deflexion and 
 vibration) can be performed in so short a time that the 
 effects of change of temperature and change in the 
 Earth's force may be neglected. If it is thought 
 necessary to recognise them, reference must be made, for 
 temperature-correction, to the experiments of Article 31, 
 and for the terrestrial change, to observations of minute 
 changes (to be mentioned hereafter, Article ^) Cor- 
 rections for torsion-force of the suspension-thread ought 
 to be applied on the principles of Article 25. There is 
 also another small correction, for magnetism induced in 
 the needle A by the Earth's action : for this we must 
 refer to a succeeding section, Article j^ *7 LL 
 
 33. Investigation of the proportion in which the 
 numerical value for E will be altered, when, instead of 
 using the foot and the grain as units, we use other units, 
 as the millimetre and milligramme. 
 
 Let the foot p millimetres, and the grain =q 
 milligrammes. Suppose the observations of Articles 
 28 and 31 adopted without any modification of chv 
 cumstances, and let us examine how the resulting 
 formulae will be modified by use of the new units. 
 The experiment of Article 27 practically gives 2A.c~ 3 
 
 E 2 
 
 = E. sin <f>; or . = - - c~ 3 . With the new units, sin 6 
 A sm$ 
 
 will not be altered, but c (numerically) will be p times 
 
74 ON MAGNETISM. 
 
 r* 
 
 as great as before, and -^ will be (numerically) p~* times 
 
 jL 
 
 as great as before. The experiment of Article 32 gives 
 
 t 2 
 
 'JE4 = -fj^- M; T is not altered by the new units ; but M y 
 
 which depends on product of mass by square of distance 
 from center of angular motion, will be 
 
 TfJ 
 
 times as great as before. The product of -r by EA will 
 
 A. 
 
 therefore be (numerically) - times as great as before : 
 
 and the numerical expression will be A/ times as 
 
 great as before. '* . 
 
 The value of p is 3047941: that of q is 647989$^; 
 therefore the new numerical expression for E on the 
 Metric system will be formed by multiplying that on 
 
 the English system by or 0-46108,4 ^ 
 
 The same numerical result would have been obtain- 
 ed if the units employed, in the Metric system, had 
 been the metre and gramme. 
 
 34.. Special values of E : historical physical change 
 in the value : lines on the Earths surface passing through 
 points of equal horizontal force. 
 
 The mean value of E found at Greenwich in the 
 year 1867 was 3*851 in English measure, or 1776 in 
 Metric measure. In 1848 its value at Greenwich, in 
 English measure, was 3722. The increase in 19 years 
 
LINES OF EQUAL HORIZONTAL FORCE. 7v> 
 
 is in the proportion of 29 to 30: its rate of increase in 
 successive years is sensibly uniform. We believe that 
 this is the longest series of accurate determinations of 
 horizontal force made in one place. 
 
 From all the comparative observations of horizontal 
 force, made by the methods of Article 25, which could 
 be collected about forty years ago, combined by the 
 aid of a theory to which allusion is made in Article 
 24, (to be fully explained in Articles 47 and 49), Gauss 
 formed a series of maps of lines of equal horizontal 
 
 Fig. 28. 
 
76 ON MAGNETISM. 
 
 magnetic intensity, which are copied, without essential 
 change, in Figures 28 and 29* These maps are on the 
 
 Fig. 29. 
 
 stereographic projection. The numbers upon the lines 
 give the value of E in Metric measure. Remarks on 
 the peculiarities of these curves will be given below, in 
 Article 42. 
 
EVIDENCE OF VERTICAL MAGNETIC FORCE. 77 
 
 SECTION V. 
 
 ON TERRESTRIAL MAGNETISM, AS ACTING IN THE 
 VERTICAL AT EACH PLACE OF OBSERVATION; AND 
 ON THE COMBINATION OF THE HORIZONTAL AND 
 VERTICAL FORCES, AND THE TOTAL TERRESTRIAL 
 MAGNETIC FORCE AT EACH PLACE OF OBSERVATION. 
 
 35. First evidence of the existence of a vertical 
 magnetic force. 
 
 When a needle is prepared, in the unmagnetized 
 state, for mounting in a compass, with its center of 
 gravity very little below its point of support, and is 
 adjusted to horizontally ; on being magnetized, its red 
 end (in northern latitudes) dips considerably. This 
 proves that (in northern latitudes) the terrestrial hori- 
 zontal magnetic force towards the north is accompaDied 
 with a vertical force downwards, and the terrestrial 
 horizontal force towards the south is accompanied with 
 a vertical force upwards. 
 
 . When the same compass is carried into southern 
 latitudes, the blue end dips. This proves that, while 
 the sign of the terrestrial .horizontal force in the north 
 direction or in the south direction has not changed, the 
 
78 ON MAGNETISM. 
 
 sign of the vertical force has changed. This is so well 
 known that, in the best compasses, a sliding weight is 
 provided, which in north latitudes can be applied to the 
 blue end of the needle, and in south latitudes can be 
 applied to the red end of the needle. 
 
 The instrument with which this vertical force is 
 most conspicuously exhibited and most accurately ex- 
 amined will be described in the next article. 
 
 36. Description of the Dipping Needle. 
 
 The function of this instrument is limited strictly to 
 the determination of the direction which a needle will 
 take under the action of the total terrestrial magnetic 
 force, when it is constrained to move in an arbitrary 
 vertical plane. This limitation permits the construction 
 of an instrument possessing great simplicity, and, in 
 consequence (viewing the nature of its action) great 
 accuracy. 
 
 The needle must be carried by a horizontal axis, 
 passing as nearly as practicable through its center of 
 gravity. This condition, though convenient, is not neces- 
 sary : for, as will be shewn in the next Article, we can so 
 arrange the observations as perfectly to eliminate the 
 effects of error of position of the axis : and indeed, for 
 some observations, the place of the center of gravity is 
 purposely moved to a sensible distance from the axis. 
 The axis must terminate in two delicate pivots ; and it 
 is mainly in the formation of these that the utmost 
 skill of the artist is required. It is very difficult so to 
 arrange the observations that the injurious effect of an 
 
THE DIPPING NEEDLE. 79 
 
 oval or otherwise ill-formed pivot can be entirely re- 
 moved. To make these pivots turn with the least pos- 
 sible friction is of the utmost importance : and for this 
 object, the pivots must not turn in Ys like those of a 
 transit-instrument, but mu^t roll upon two edges of a 
 very hard substance, usually agate. In the direction 
 parallel to the plane in which the needle moves, these 
 edges must be straight and perfectly horizontal ; in the 
 vertical section at right angles to that plane, or in the 
 direction of the needle-axis, the section of each edge is 
 rounded ; a form very desirable for permitting the es- 
 cape of particles of dust, &c. Great attention is required 
 for the satisfactory polishing of the edges. When due 
 care is given to these preparations, the friction is ex- 
 tremely small. 
 
 It is necessary now to describe the method of ob- 
 serving the position of the needle. 
 
 The needles employed are always pointed : and, till 
 within a few years past, the needle was allowed to swing 
 within a graduated ring of brass, and the divisions op- 
 posite to the points of the needle were read. The read- 
 ing was very rough, and there was risk of error from 
 the close proximity of the needle to the brass, which is 
 seldom perfectly free from iron. Lately, a far superior 
 form has been introduced, known as the Kew pattern 
 (from the circumstance that it was invented and first 
 used at the Kew Observatory). A view of that in- 
 strument is given in Figure 30. Several auxiliary parts, 
 unimportant to the general principle, are omitted in 
 this drawing. There is no metal near the needle : the 
 
80; 
 
 ON MAGNETISM. 
 
 points of the needle are observed by means of micro- 
 scopes which are attached to a revolving frame that 
 carries verniers by which the graduations of an external 
 
 Fig. 30. 
 
 vertical circle are read. The true position of the needle 
 (including all effects of friction, uncertainty of reading, 
 &c.) is rarely doubtful to the extent of 2'. A modified 
 form of the instrument, adapted to the use of needles 
 of different lengths, and with other fittings, is mounted 
 as a permanent instrument at the Royal Observatory, 
 Greenwich. 
 
 In. all cases, the instrument is so mounted that it 
 
MANIPULATION OF THE DIPPING-NEEDLE. 81 
 
 can rotate round an axis, which must be made accurately 
 vertical. The divisions and 180 of the circle as read 
 by the microscopes ought then to correspond to vertical 
 position of the needle. In general, it is sufficient to 
 trust to the artist for this adjustment : but at the Royal 
 Observatory, Greenwich, a loaded brass needle has been 
 introduced, whose position is read in the same manner 
 as that of the magnetic needles ; and by use of this the 
 accuracy of the divisions and 180 can be verified. 
 
 37. Manipulation of the Dipping -Needle : reversion 
 of its pivot-bearing : rotation round its vertical axis : re- 
 versal of its magnetic poles. 
 
 The point to which the theoretical considerations, 
 employed in the use of the dipping-needle, are particu- 
 larly addressed, is the elimination of errors produced by 
 the non-coincidence of the axis of rotation of the needle 
 with its center of gravity. Measuring from the axis 
 of rotation, the center of gravity may have an ordinate 
 of sensible value in the direction longitudinal to the 
 needle, and one in the direction transversal to the 
 needle. It will be easily conceived that the effect of 
 the latter ordinate may be eliminated by reversing the 
 pivots upon the agate edges, so as to present that edge 
 upwards which was downwards (see the difference be- 
 tween Figures 31 and 32, or between Figures 33 and 34). 
 The same effect may be produced by rotating the entire 
 supporting frame with the needle which it carries, round 
 a vertical axis, through 180 : for, as the vertical force 
 of magnetism tends to depress the same end as before, 
 
 6 
 
82 
 
 ON MAGNETISM. 
 
 while the horizontal force, always drawing that end to 
 the magnetic north, now draws it to a part of the 
 supporting frame opposite to the former, the edge of 
 
 Fig. 31. Fig. 32. Fig. 33. Fig. 34. 
 
 the needle which was below will now be above. It is 
 evident here that we shall have the means of eliminating 
 the effect of that ordinate of the centre of gravity which 
 is transversal to the needle. 
 
 But for eliminating the effect of that ordinate which 
 is longitudinal to the needle, we must have some method 
 of altering the direction of magnetism with respect to 
 that ordinate. And there is but one way of doing this ; 
 namely, by reversing the poles of the magnet. This, 
 which may be done by the power of a galvanic current, 
 or in other ways, is done in practice most conveniently 
 by the method of double-touch, described in Article 8. 
 Inasmuch as the application of a pair of magnets will 
 produce certain magnetism in a needle, it is easily con- 
 ceivable, and is accurate in fact, that the use of the 
 opposite ends of those magnets (which possess magnet- 
 
THEORY OF THE DIPPING-NEEDLE. 83 
 
 ism of the kind opposite to that of the ends first used) 
 will first destroy the magnetism planted in the needle, 
 and will then plant in it new magnetism of the opposite 
 kind. It is only necessary to caution the operator that 
 the touch-magnets used must have much greater mag- 
 netic power than the needle : otherwise it might happen 
 that the needle, fully charged with magnetic power, 
 would reverse the poles of the touch-magnets. With 
 touch-magnets of adequate power, this never happens : 
 the poles of the needle are reversed, without injury to 
 the powers of these magnets : and the magnetic power 
 of the needle in its state of reversed magnetism is 
 sensibly equal to that before reversion, as is ascertained 
 by calculations to be mentioned below. 
 
 38. Mathematical theory of the Dipping-Needle : 
 first, on the supposition that the magnetic intensity after 
 reversion is equal to that before reversion ; simplification 
 when the needle is very nearly balanced. - 
 
 In Figures 31, 32, 33, 34, the same part of the edge 
 of the dipping-needle is represented by the strong line 
 in a portion of the outline. The magnetic north is 
 supposed to be to the right. Commencing with Figure 
 31, the needle is so turned in Figure 32 that the edge 
 which was downwards is now upwards, but no change is 
 made in the magnetism of the needle. After this, the 
 magnetism is reversed ; and, as is seen in the shading 
 of the Figure, the end, which was charged with blue 
 magnetism and was uppermost, is now charged with red 
 magnetism and is lowest. Between Figures 33 and 34 
 
 62 
 
84 ON MAGNETISM. 
 
 the needle is so turned that the edge of the needle is 
 reversed in regard to up and down. G is the place of 
 the center of gravity, whose longitudinal and transversal 
 ordinates will be called I and t. Put W for the weight of 
 the needle. Let H and V be the horizontal and vertical 
 terrestrial magnetic forces which act on each pole of the 
 needle before reversion of its magnetism : pulling the 
 red or lower pole, horizontally in the plane of vibration, 
 and downwards, respectively ; and pulling the blue or 
 upper end in the opposite directions : and let nH and 
 n V be the similar forces after the reversion of the poles. 
 (This amounts to the same as supposing that the mag- 
 netic intensity after reversion is to that before reversion 
 as n to 1.) Let a be the distance of each of the poles 
 from the center. Our object now is to find the propor- 
 tion of H to F. 
 
 In Figures 31, 32, 33, 34, let V O y 6 9 0^ be the 
 angles made with the horizon by the line joining the 
 poles of the needles, which angle in each case may be 
 called the apparent dip. 
 
 Then in Figure 31 the equation of equilibrium 
 will be 
 
 TFx (Zcosflj + Jsintfj) + 2 Va cos X - 2Ha sin l = ; 
 or TFx (Zcot0 1 + ) + 2Facot0 1 -25a = 0. 
 
 Similarly, 
 
 inFigure32, TFx (7cot 2 -Q + 2Facot 2 - 2Ta=0; 
 in Figure 33, TFx ( - 1 cot 3 -t) + 2n Facot 8 - 2nHa=Q ; 
 in Figure 34, TFx ( - 1 cot 4 +*) + 2n Fa cot 4 -2ri5a=0. 
 
THEORY OF THE DIPPING-NEEDLE. 85 
 
 The simplest case of these equations will be that given 
 by the usual assumption, that the intensity of. magnet- 
 ism is the same after reversion, or that n=l. The 
 four equations then become 
 
 W x (I cot 6 l + t)+2Va cot O l - 2 Ha = ; 
 
 Adding the first pair, and dividing by cot 6 l + cot 2 , 
 
 Wl + ZVa -- ,,=0. 
 
 cot e l + cot 2 
 
 Adding the second pair, and dividing by cot 3 + cot 4 , 
 
 -TFZ + 2Fa 
 Adding these equations, 
 
 cot 0. cot 0. 
 
 Now, considering the vertical force F and the hori- 
 
 zontal force H as resolved parts of the total terrestrial 
 
 7. 
 force acting on the needle, it is seen that -^ is the tri- 
 
 gonometrical tangent of the angle which the direction 
 of the total force makes with the horizon, or is the 
 tan. Dip. Thus we obtain 
 
86 ON MAGNETISM. 
 
 If the needle is very nearly balanced on its pivots 
 (a condition which the artist always endeavours to 
 secure), so that the four angles V 2 , 8 , 6^ are nearly 
 equal : then we have, without perceptible error, 
 
 cot X + cot 2 = 2 cot * 2 ; cot 8 + cot 4 = 2 cot 3 4 ; 
 
 I, . 
 
 and tan . Dip = - \ tan -*-= 2 + tan 
 
 0, 4- 2 + 19, -f 
 = tan - 1 - ^ 5 
 
 (by the same reasoning as that just employed) : and 
 consequently 
 
 39. Mathematical theory of the Dipping -Needle 
 continued : secondly, on the supposition that the intensity 
 is not the same after reversion, and that the needle is not 
 nearly balanced. 
 
 We must now use the accurate equations near the 
 beginning of the last Article : and first, to find the 
 value of n. 
 
 Multiply the first equation by tan t and the second 
 by tan a , and subtract ; 
 
 Wt (tan t + tan 2 ) - ZHa (tan 0^- tan 2 ) = 0. 
 
 Multiply the third equation by tan 3 and the fourth 
 by tan 4 and subtract ; 
 
 Wt (tan 4 + tan 8 ) - 2nHa (tan 4 - tan 8 ) = 0. 
 
THEORY OF THE DIPPING-NEEDLE. 87 
 
 Eliminating at the same time Wt and Ha, 
 
 _ (tan 4 + tan a ) (tan t - tan 2 ) 
 ~ (tan 64 tan 3 ) (tan O l + tan # 2 ) 
 
 _ (cot fl 3 + cot 4 ) (cot # 2 - cot t ) 
 ~ (cot 3 cot 4 ) (cot 2 + cot 0J ' 
 
 This expression can be used with accuracy when the 
 needle is greatly out of balance, so that l and 2 differ 
 considerably and 3 and 4 differ considerably : but it is 
 not accurate when th'ese angles approach to equality, 
 because the unavoidable errors of observation then 
 greatly affect the proportionate accuracy of cot 2 - cot t 
 and of cot 3 cot 4 . 
 
 On expanding the numerator and denominator, it 
 will be found that the supposition n = 1 corresponds to 
 this very simple equation : 
 
 tan 0j . tan 3 = tan 2 . tan 4 ; 
 or cot 9 l . cot 3 = cot # 2 . cot 4 . 
 
 But this equation, which is founded on the last, cannot 
 be used with safety when the needle is very nearly 
 balanced. 
 
 From the first pair of the original equations, and 
 from the second pair, we find 
 
 
88 ON MAGNETISM. 
 
 of which the sum is 
 
 - + a + 4 
 
 cot l + cot 2 cot 3 + cot 6 
 
 whence 
 
 TV F_ 2 / 1 
 Jlp ~ H~ 1 + 
 
 If n has been numerically calculated, from the for- 
 mula above, the computation of tan . Dip may most 
 readily be made by substituting the numerical value in 
 this expression. Or, if the symbolical expression for n 
 be substituted, 
 
 tan . Dip 
 
 __ 2(cot0 8 -cot0 1 +cot0 a -cot0 4 ) _ 
 
 In ordinary observations with the dipping-needle, 
 these formulae are not required. But they are required 
 in the following case. The adjustment of the instru- 
 ment which it is peculiarly beyond the power of the 
 observer to verify, is the circularity of the pivots. Some 
 observers therefore have thought it desirable that the 
 needle should be so unequally loaded as to be sensibly 
 out of balance, thus making the apparent dips t , 2 , 6 y 
 4 very unequal, and bringing different sides of the 
 pivot into bearing upon its agate edges. With that 
 arrangement, the formulas above investigated are ne- 
 cessary. 
 
 40. Theory of the Dipping-Needle when the dips 
 are observed in different vertical planes inclined to the 
 plane of the magnetic meridian. 
 
DIP IN EXTRAMERIDIONAL PLANE. 89 
 
 The investigations above apply to the case of the 
 needle vibrating in the magnetic meridian : and they 
 then give for result the true magnetic dip. It is only 
 necessary that the direction of the magnetic meridian 
 be nearly known : a small error is unimportant, and the 
 determination of the meridian by a common compass is 
 abundantly accurate for this purpose. 
 
 But the investigations also apply when the needle 
 vibrates in any other vertical plane. They do not then, 
 however, give the true dip ; they give only an apparent 
 dip, corresponding to the proportion of the vertical 
 magnetic force to the resolved part of the horizontal 
 magnetic force in that plane. (For it is obvious that 
 the part of the horizontal magnetic force which is per- 
 pendicular to that plane, being parallel to the needle's 
 axis of rotation, can have no effect on its rotation or 
 vibration.) If ^ and < 2 be the azimuths from the mag- 
 netic meridian, measured in the same direction, of two 
 vertical planes in which the dip is observed ; d t and d 2 
 the apparent dips observed in those planes : then the 
 resolved horizontal forces in these planes will be 
 -Hcos</>, and ZTcos </> 2 , and the equations between the 
 apparent dips and the azimuths will be, 
 
 , H . cos <f>. , H . cos <> 
 cotan. d l = ^- ; cotan. d a = ^ 2 . 
 
 i 
 
 TT 
 
 But -y cotan. True Dip ; therefore 
 
 cotan. d t = cotan. True Dip x cos <, ; 
 cotan. c? 2 = cotan. True Dip x cos < 2 . 
 
90 ON MAGNETISM. 
 
 There is only one case of these equations which 
 offers any interest. If (f> 2 =(f> 1 + 9Q () ) then cos <f> 2 = sin ^ ; 
 and the sum of the squares of the two equations becomes 
 
 (cotan. d$* + (cotan. d^) z = (cotan. True Dip) 2 , 
 from which the angle <, has disappeared. 
 
 Thus it appears that the True Dip can be obtained 
 from observation of the apparent dips in two planes, 
 with no condition as to the position of these planes 
 except that their azimuths differ by 90. This condition 
 can always be secured by means of the azimuthal circle 
 on which the dip-apparatus is mounted. 
 
 41. Determination of the Total Terrestrial Magnetic 
 Force at any locality : lines upon the Earth's surface 
 passing through points of equal dip, and lines passing 
 through points of equal Total Force : historical changes. 
 
 By the investigations extending from Articles 26 to 
 34, the terrestrial horizontal magnetic force is measured. 
 And by those from Articles 38 to 40, the dip is measured. 
 It is plain that the Total Force = Horizontal Force x se- 
 cant of True Dip : and thus the total terrestrial hori- 
 zontal magnetic force is ascertained, without risk of in- 
 accuracy, except at points where the dip is nearly ver- 
 tical (that is, at points near the magnetic poles). 
 
 We confine our attention to this method, because it 
 is the only one which does not rely on the constancy of 
 a needle's magnetism, and because it is very accurate 
 except close to the magnetic poles, where the value of 
 the total force can be inferred from those around it by 
 the laws of continuity. Determinations have been made, 
 
LINES OF EQUAL DIP AND EQUAL TOTAL FORCE. 91 
 
 however, by observing the time of vibration of a dipping- 
 needle in the magnetic meridian : or by observing the 
 extent to which the the needle is displaced by a given 
 weight attached to a thread which is wrapped round the 
 axis of the needle. The theory of these is so simple 
 that there is no need to delay on them. 
 
 The ring-shaped lines in Figures 20 and 21 represent 
 
 Fig. 35. 
 
 -08T 
 
 (B is the primary and A the secondary northern pole of greatest magnetic intensity.) 
 
 the lines of equal dip over the surface of the earth : and 
 the lines in Figures 35 and 36 represent the lines of 
 
92 ON MAGNETISM. 
 
 equal total magnetic force. The numbers upon the 
 latter system of lines shew the value of the total force 
 
 Fig. 36. 
 
 (A is the southern pole of greatest magnetic intensity, and B the primary pole of small 
 intensity.) 
 
 in Metric measure. To present more vividly to the eye 
 the general facts of dip and total force over the earth, 
 Figure 37 is drawn, exhibiting the directions of dip and 
 the magnitude of total force along a meridian of the 
 earth. The magnitude of force is shewn rudely by the 
 lengths of the symbolical needles at the different points 
 of the meridian. The map is on the orthographic pro- 
 
LINES OF EQUAL DIP AND EQUAL TOTAL FOECE. 93 
 
 jection. It will be remarked that there is a little in- 
 accuracy near the south pole, arising from the circum- 
 
 Fig. 37. 
 
 BlU > OU ^ 
 
 stance that it is impossible to include the north and the 
 south magnetic poles in the same geographical meridian. 
 The Dip and the Total Terrestrial Magnetic Force 
 at any place, like the elements of which we have treated 
 in Articles 24 and 34, are slowly changing. In 1843 
 the dip at Greenwich was about 69 1' ; it has diminish- 
 ed, with a rate continually accelerating, till in 1868 it 
 
94 ON MAGNETISM. 
 
 was 67 56'. Adopting as elements of calculation that 
 in 1848 the dip and horizontal force were 68 47' and 
 376, and, in 1866, 68 V and 3'85 : the total force was, 
 in 1848, 10*39, and, in 1866, 10'28 (in English units), 
 or 4'791 and 4'740 (in Metrical units). 
 
 42. Reference to the points of principal interest in 
 Figures 20, 21, 28, 29, 35, 36: secular change in the place 
 of North Magnetic Pole. 
 
 Before entering upon the consideration of the dia- 
 grams, we will allude to some general points regarding 
 the connexion of the magnetic meridians with the curves 
 of equal dip and of equal horizontal force. 
 
 Adopting, as the most convenient definition of 
 " Magnetic Pole," (when not qualified by any other 
 words), " the point where the dip is vertical," there is 
 no reason in nature why there should not be more than 
 one magnetic pole in the north (or in the south). If, at 
 one of these proximate places, the red pole dips, and at 
 the other the blue pole dips, there must be between 
 them a place of no dip (in the same manner as, in 
 Figure 37, there is a place of no dip between the north 
 and south poles of vertical dip with opposite poles of 
 the needle). But, if the red pole dips at both, there is 
 some complication introduced into the forms of the 
 equal-dip curves. We will however commend these to 
 the examination of the speculative student : remarking 
 that we have no reason to think that there is more than 
 one Magnetic Pole or place of vertical dip either in the 
 north or in the south. 
 
TERRESTRIAL MAGNETIC POLES. 95 
 
 Now in progressing along one of the magnetic meri- 
 dians in Figure 20, the observer who follows the direc- 
 tion of the horizontal needle is in fact continually pro- 
 ceeding in the plane of dip. And, if he finds the dip 
 continually increasing (that is, if he advances towards 
 the smaller circles of Figure 20), he will at last arrive 
 at the place of vertical dip. Or, conversely, if he starts 
 from the place of vertical dip, and continues in the 
 course defined by any one of the directions of the hori- 
 zontal needle into which he will immediately fall, he 
 will pass away from the place of vertical dip in the 
 plane of dip, and will therefore, for a time at least, have 
 dip continually diminishing. From all this it appears 
 that the pole of no dip must be the same as the pole 
 common to every magnetic meridian ; that is, the pole 
 to which all magnetic meridians converge. 
 
 The pole to which the lines of equal horizontal force 
 are related, that is, the point where horizontal force 
 vanishes, is evidently the pole of vertical dip. 
 
 Thus the Magnetic Pole is a common pole for the 
 convergence of magnetic meridians, for the verticality of 
 dip, and for the evanescence of horizontal force. 
 
 But the pole of greatest total force is entirely differ- 
 ent in its properties from these. It has not necessarily 
 any connexion with them. There may be any number 
 of points where the total force is maximum (in compari- 
 son with the points that surround them, to a considerable 
 distance). The number of such points in the north may 
 be different from that in the south. 
 
 We will now proceed with the diagrams. And first 
 
96 ON MAGNETISM. 
 
 it must be stated that these diagrams are not, and could 
 not be, drawn simply from observations. They are 
 drawn from a theory (to be explained hereafter) founded 
 upon all the observations which could be collected be- 
 fore the preparation of the charts (published in 1840). 
 In general they represent very accurately the facts of 
 observation : but in later years some sensible but not 
 important inaccuracies have been discovered in the 
 southern hemisphere. 
 
 In Figures 20 and 21, it is to be remarked that the 
 magnetic meridians might have been drawn through 
 any arbitrary points of the geographical equator; they 
 are in fact drawn through the points of east longitude 
 8, 18, 28, &c.: 8 and 188 being the points at which 
 the Magnetic Equator or line of no dip crosses the 
 geographical equator. The magnetic meridians cannot 
 generally be great circles of the sphere, because the 
 two Magnetic Poles through which all must pass are 
 not exactly opposite : they have moreover other irregu- 
 larities of form, which do not depend on the character 
 of the stereographic projection, but are equally con- 
 spicuous when the curves are traced on a globe. There 
 is no trace of more than one pole either in the north or 
 south. 
 
 The form of the lines of equal dip is remarkable. 
 Commencing with the line of no dip or Magnetic 
 Equator, it is easily seen that it is not a great circle : its 
 greatest northerly distance from the geographical equa- 
 tor occurs at about 55 east longitude, or 47 from 
 its node (instead of 90), and its greatest southerly 
 
CURVES OF EQUAL DIP AND EQUAL HORIZONTAL FORCE. 97 
 
 distance at about 318 east longitude, or 50 from 
 the same node. The inclination of the Magnetic 
 Equator to the geographical equator is greater near 
 the west coast of Africa than in any other part. Pro- 
 ceeding to the neighbouring lines, it will be seen that 
 the increase of dip is nearly double of the increase 
 of latitude; and upon this circumstance was founded 
 the conjectural law, tan. Dip = 2 tan. distance from 
 magnetic equator; to which we shall advert in the 
 next Section. Nearer to the Magnetic Poles the 
 curves are oval, or rather pear-shaped; but the major 
 axes of the northern curves, and those of the southern 
 curves, are not in the same direction. The North 
 Magnetic Pole (in longitude 265) and the South 
 Magnetic Pole (in longitude 152) are not opposite 
 each other. These remarks show that the Earth's 
 magnetism cannot be represented as the power of one 
 magnet, and that the distribution of magnetism about 
 the Earth is unsymmetrical. 
 
 The curves of equal Horizontal Force, Figures 28 
 and 29, are still more strange. The greatest values 
 are 3733 at the geographical equator in longitude 
 259, and 3'673 in 14 north latitude, longitude 103. 
 Proceeding along an equatoreal belt, one minimum 
 of 3'039 is reached in longitude 345, north latitude 3, 
 and another 3'408, in longitude 156, south latitude 
 13. Proceeding in either direction, north or south, 
 from this equatoreal belt, the Horizontal Force gradu- 
 ally diminishes to at each pole. The north pole is in 
 the north of Baffin's Bay: the south in South Victoria. 
 
 7 
 
98 ON MAGNETISM. 
 
 But the forms of the southern curves only seem to 
 indicate the existence of two poles of magnetic force. 
 This indication differs remarkably from that which is 
 founded upon the system of curves to be mentioned 
 next. 
 
 The curves of equal Total Magnetic Force present 
 us with the singular phenomenon of two poles of 
 maximum force in the north, and only one in the south. 
 The numerical values of the forces at the former are, 
 6*160 west of Hudson's Bay, and 5 '911 in Siberia: that 
 of the latter 7'898 in South Victoria. Proceeding from 
 these towards the equatoreal belt, the equatoreal max- 
 ima 3'6S6 and 3'649 are reached in longitude 252, 
 south latitude 7, and in longitude 110, north latitude 
 6, and the equatoreal minima 2'828 and 3'248, near 
 St. Helena in longitude 355, latitude 16 south, and 
 in longitude 179, latitude 6 north. There is a rude 
 approach to the law, that the Total Force at the 
 Magnetic Poles is double that at the Magnetic Equator. 
 
 The theoretical connexion of these facts will be 
 treated in the next Section. 
 
 At the end of Article 41 it was remarked that, 
 at Greenwich, the Dip and Total Force are diminishing. 
 Interpreting these by the remarks above, it would 
 seem that the Magnetic Equator is approaching to 
 Greenwich, or the North Magnetic Pole is receding 
 from Greenwich. And remarking also the westerly 
 change in direction of north magnetic meridian, from 
 the sixteenth century to the year 1824, and its subse- 
 quent easterly motion (Article 24), it would seem that 
 
TERRESTRIAL CURVES OF EQUAL- TOTAL FORCE. 99 
 
 the north magnetic pole has rotated round the ter- 
 restrial pole in a small circle from east to west, and 
 having passed the point where its westerly azimuth 
 as viewed from Greenwich is maximum, it is still 
 continuing its course in that circle. It seems probable 
 that in the fifteenth or sixteenth century it was situ- 
 ated between North Cape and Spitzbergen: it is now 
 north-west of Hudson's Bay. 
 
 Valuable information on these changes, from the 
 earliest period to the years about 1830, will be found 
 in the work "Terrestrial and Cosmical Magnetism, 
 the Adams Prize Essay for 1865, by Edward Walker, 
 M.A." (Deightons, Cambridge). There is also some 
 accurate information applying to later years, but not 
 possessing all the completeness which might have been 
 obtained from published records. 
 
 7-2 
 
100 ON MAGNETISM. 
 
 SECTION VI. 
 
 THEORIES ON THE PHYSICAL CAUSE OR REPRESENTA- 
 TION OF TERRESTRIAL MAGNETISM. 
 
 43. Eeasons for believing that Terrestrial Mag- 
 netism is not produced, in any important degree, by 
 magnetic forces external to the earth. 
 
 If there were an external cause for magnetism, it 
 seems scarcely conceivable that some large part of it 
 would not act in planes parallel to the geographical 
 equator: and, if so, its effects at any one place would 
 undergo very great changes in the earth's diurnal 
 revolution; every part of the earth being presented, 
 in the course of a day, in different aspects towards forces 
 so acting. Now the fact is that the diurnal changes are 
 very small, perhaps at Greenwich F J 7 part of the whole 
 horizontal force. It would seem therefore certain that 
 external bodies or space do not produce any sensible 
 part of the magnetism in the planes to which the earth's 
 axis is normal. And this carries with it a very strong 
 improbability that they produce any sensible magnetic 
 forces in the direction of the earth's axis also. 
 
TERRESTRIAL MAGNETISM IS NOT SUPERFICIAL. 101 
 
 44*. Reasons for believing that Terrestrial Mag- 
 netism does not reside, in any important degree, in the 
 earth's surface. 
 
 The first class of reasons are those general ones 
 which are founded on ordinary observation, of the 
 materials of which the earth's surface is composed, and 
 of their non-magnetic property: and upon the general 
 absence of any perceptible change in magnetism de- 
 pending on the change of soil. The materials of a 
 clay-field are not sensibly magnetic, nor are those of 
 a sand-field, nor is there any change of the general 
 terrestrial magnetism in going from one to the other; 
 nor are the granite rocks in one district, or the lime- 
 stone rocks in another, sensibly magnetic. In some 
 places there are ferruginous rocks, specimens of which 
 when brought near to a delicate compass are found 
 to produce sensible disturbance: but the great masses 
 of those rocks on the earth's surface, when examined 
 (by examination of the declination, dip, and horizontal 
 intensity) at corresponding distances in their neighbour- 
 hood, produce no sensible disturbance. 
 
 The second class of reasons consists of those founded 
 on measures of the magnetic elements at different 
 elevations above the earth's surface. One series in- 
 cludes the observations taken on mountain-heights : of 
 these the most valuable are those of Professor James 
 Forbes (Edinburgh Transactions, vol. xiv.), from which 
 it appears that, for a height of 100 feet, horizontal 
 magnetic force is diminished, in Europe, by 3o ^ 0o part, 
 
102 ON MAGNETISM. 
 
 and dip is increased by 5". Both these would corre- 
 spond to the supposition that the magnetic power 
 is sensibly below the earth's surface. As the observer 
 was not actually separated from the earth, the validity 
 of- inference from these may be disputed. Another 
 series is that of observations in balloons, which are 
 free from every objection of that kind, but which are 
 not quite so accurate; and which are necessarily al- 
 most limited to observations of horizontal intensity, 
 as found by vibrations (Article 25). The following are 
 the results of these observations : 
 
 Gay Lussac, 1803, at the height 4000 metres found 
 no sensible diminution of magnetic force' (Annales de 
 Chimie, vol. 52). 
 
 Gay Lussac, 1804, at the height 6900 metres found 
 an apparent very small increase : but this was probably 
 caused by the low temperature of the needle, for 
 which no correction was applied. The dip, imperfectly 
 observed, was not sensibly altered. (A. de C. vol. 52.) 
 
 Glaisher, 1862, found at the height 20200 feet 
 a diminution of power : but in other observations at 
 5300, 11000, and 3800 feet, found the same as on the 
 earth (Report of British Association 1862). 
 
 Glaisher, 1864, found a diminution of about f part 
 at the height 14000 feet. (R. of B. A. 1864.) 
 
 Glaisher and Evans, 1864, found an even larger 
 diminution at height 3600 to 5000 feet. (R. of 
 B. A. 1865.) 
 
 It would appear generally from these observations, 
 that there is a sensible diminution of magnetic hori- 
 
TERRESTRIAL MAGNETISM IS DEEP IN THE EARTH. 103 
 
 zontal force at a great elevation. But the last set 
 of observations casts much doubt on this conclusion. 
 It is to be remarked that all the balloon-observations at 
 great height were compared with observations on the 
 earth. It might have been safer to compare them with 
 balloon-observations at small elevations. Now the last 
 set of observations seems to shew that an apparent 
 large diminution arises simply from the effect of locali- 
 zation in the balloon-car ; and, if this be accepted, there 
 is scarcely any sensible effect to be ascribed to the 
 great elevations. 
 
 Now, remarking how rapidly magnetic power di- 
 minishes with increase of proportion of distance from 
 the magnetic poles, it follows from the observations 
 above that the height of three or four miles must bear 
 a small proportion to the distance of the magnet which 
 produces the magnetic power observed at the earth's 
 surface, and therefore the source of magnetism must 
 be deep. 
 
 45. Attempt to explain Terrestrial Magnetism by 
 the action of a magnet of small dimensions but of very 
 great power, near the center of the earth. 
 
 About the middle of the last century it was sug- 
 gested by Mayer, and in the present century the same 
 idea was independently adopted by Humboldt and Biot 
 (Biot, Traiti de Physique, 181 6r vol. iii. page 139), that 
 the principal phenomena of Terrestrial Magnetism 
 could be explained by the action of a powerful magnet, 
 of limited dimensions, near the center of the earth. Its 
 
104 ON MAGNETISM. 
 
 theory is as follows. In Figure 38, let the magnetic 
 
 Fig. 38. 
 
 pole be defined by prolonging the axis of the magnet 
 till it cuts the earth's surface ; 6 will be the comple- 
 ment of magnetic latitude. The action of the northern 
 or blue pole upon the red end of a needle at P will be 
 represented by 
 
 B (a 2 + Z> 2 - 2ab cos ff)' 1 : 
 
 its resolved part in the horizontal plane at P, towards 
 the pole, will be 
 
 b . sin . (a* + 6 2 - 2ab cos 0)'*; 
 
 the action of the red pole in the same direction will be 
 
 Bb . sin . (a 2 + 6 2 + 2ab cos 0)~* : 
 
 the total horizontal force will be the sum of these two 
 quantities : which, retaining only the first term in the 
 expression when 6 is considered a small quantity, is 
 
 2Bb . sin . a" 2 . 
 
 A similar expression with opposite sign gives the action 
 on the blue pole (the needle being considered to be 
 
MAGNET NEAR THE EARTH'S CENTER. 105 
 
 small) ; and the algebraical difference or numerical sum 
 of these gives the whole horizontal directive force 
 
 = ^Bb . a~ 3 . sin 6. 
 
 The resolved part of the action of the blue pole 
 upon the red end of the needle, in the direction of the 
 vertical at P, is 
 
 B . (a- b cos 0) . (a 2 + b* - 2 ab cos 0)~ : 
 which, expanded to the first power of b, gives 
 B . a~ 3 . (a + 26 cos (9). 
 
 The action of the red pole upon the same red end 
 of the needle is 
 
 -B. o7\ (a -2b cos 0). 
 
 The sum of these gives for the total vertical force 
 downwards upon the red end, 
 
 4>Bb . a\ cos 0. 
 
 As above, there is an opposite force, numerically ad- 
 ditive, upon the blue end : and the whole vertical 
 directive force is 
 
 SBb . oT\ cos B. 
 
 Hence the tangent of dip at P 
 
 vertical force SBb . a~ z . cos 6 
 
 i: T-Tr -- = T~DZ: =3 : ~a = 2 cotan 6 
 horizontal lorce 4tno . a . sin u 
 
 = 2 tan magnetic latitude of P. 
 And the total force at P = 
 
 {(hor. force) 2 + (vert, force) 2 }^ 
 . a' 3 , (sin 2 + 4 cos 2 0)* . 
 
106 ON MAGNETISM. 
 
 At the magnetic equator, = 90, and total force 
 
 At the magnetic pole, 6 = 0, and total force 
 
 or double that at the equator. 
 
 These three results, for horizontal force, for dip, and 
 for total force, are not materially disturbed if we 
 conceive the magnet to be excentric, provided that 
 magnetic latitude is always referred to its center. 
 
 It was soon found that this elegant theory, though 
 well representing the broad facts of terrestrial mag- 
 netism, failed in accuracy when applied to many special 
 cases. Such curves, for instance, as those of equal dip, 
 Figures 20 and 21, could not possibly be explained by 
 it. It was modified by supposing the axis of the mag- 
 net to be distant from the earth's center by one-seventh 
 part of the earth's radius; but it could not then be 
 sufficiently reconciled with observations. 
 
 46. Attempt to explain Terrestrial Magnetism by 
 the action of two magnets within the Earth. 
 
 A celebrated Norwegian magnetical observer, Han- 
 steen, remarking the tendency to the exhibition of 
 two poles in the north and two poles in the south 
 which we have indicated as appearing in some of the 
 diagrams, Figures 20, 21, 28, 29, 35, 36, undertook the 
 task of investigating the effects of two large magnets 
 within the earth, both magnets being excentric, and 
 inclined to the Earth's equator in different planes. 
 
TWO MAGNETS WITHIN THE EARTH. 107 
 
 The investigations are contained in a work entitled 
 Magnetismus der Erde' It will readily be conceived 
 that this is a problem of great complexity. A great 
 number of positions of the magnets were tried, but 
 no one of them was quite satisfactory, though the 
 results were superior to those derived from a single 
 magnet. 
 
 As nothing has really resulted from this theory, 
 it does not appear desirable to load the present 
 Treatise with its laborious investigations. We may 
 however remark that the known phgenomena of ob- 
 servation amply justified the undertaking; and that, 
 if it had not been made, we should often have felt 
 that one possible opportunity of explaining Terrestrial 
 Magnetism had been rejected. 
 
 47. Gauss s more general explanation of Terrestrial 
 Magnetism by supposing that the red and blue magnet- 
 isms are distributed irregularly through the earth. 
 
 The investigation of this theory is given by Gauss 
 in the Resultate &c. des Magnetischen Vereins for the 
 year 1838; and a complete English translation of it 
 is published in Taylor's Scientific Memoirs, volume ii. 
 We shall not attempt here to explain all the generali- 
 ties of this most elegant treatise. It will be sufficient 
 to point out those parts which lead ultimately to the 
 comparison of the results of theory with observation 
 of the most extensive and most accurate kind. 
 
 It is supposed, as a law to which we are led by 
 previous magnetic investigations, that the quantities 
 
108 OX MAGNETISM. 
 
 of red and of blue magnetism in the Earth are equal. 
 And it is supposed that the attraction or repulsion is 
 inversely as the square of the distance. The magnet- 
 ism of every point of the Earth will be supposed, in the 
 algebraical investigation, to be red : blue magnetism 
 being included in the same investigation by conceiving 
 its sign to be negative. As regards the experimental 
 magnet or compass-needle, whose dimensions are ex- 
 ceedingly small in proportion to the distance of the 
 magnetic parts of the Earth, it will be sufficient to 
 consider the terrestrial action upon its red end only. 
 
 Let a, b, c be the coordinates of an attracting point : 
 BJJL the amount of magnetism there (its unit being 
 that quantity of red magnetism which at distance 1 
 exercises on a similar mass the moving force represent- 
 ed by 1) : and let x, y, z be the coordinates of the red 
 end of the needle. The magnetic force on the end 
 
 r\ 
 
 of the needle is 7, in the direction of the line join- 
 
 ing the attracting and attracted points, wherep = 
 *j{x a) 2 + (y 6) 2 + (z c) 2 J. Resolving this in the 
 directions of x, y, z, the several forces are 
 
 Bfi . (x a) Sfj, . (y b) B/J, . (z c) 
 
 ~T~ ~?~ ~^~ 
 
 It is easily seen that those forces are the same as 
 
 A similar system of formulae applies to the effects of 
 
MAGNETISM DISTRIBUTED THROUGH THE EARTH. 109 
 
 every magnetic particle. For summing the effects of 
 
 the whole, let F= Su. -: then the total forces 
 
 J P 
 upon the particle x, y, z in the directions x, y, z, which 
 
 we may call X, Y, Z, are respectively 
 
 dV dV dV 
 dx' dy' da' 
 
 The symbol V will be recognized here as denoting the 
 Potential of the forces acting on the particle x, y, z, 
 affected with the negative sign : it is a physical quan- 
 tity whose numerical value is independent of the 
 directions- of the ordinates x, y, z t provided they are 
 rectangular. 
 
 Now instead of denning the place of the experi- 
 mental needle by a?, y, and z, it is convenient to define 
 it by u the colatitude of the place or its angular 
 distance from the terrestrial pole, A, the longitude of 
 the place as measured from a fixed meridian towards 
 the east, and r the distance of the place from the 
 Earth's center. And it is convenient to estimate the 
 magnetic forces in directions opposite to the directions 
 of those coordinates as they are seen at the locality: 
 namely as a force N towards the north, as a force W 
 towards the west, and a force C in the vertical along 
 the radius towards the Earth's center (the Earth being 
 considered spherical). These three directions are at 
 right angles to each other; and therefore they can be 
 Considered as the x } y, z of the last paragraph, and the 
 
 dV 
 
 expressions -r- &c. can be employed, provided that we use 
 dx 
 
110 ON MAGNETISM. 
 
 proper caution in interpreting these values in reference 
 to our new polar coordinates. Now, considering the 
 fictitious x as in the horizontal plane and towards the 
 
 west, the value &x (by which -KT- and -j- are formed ), is 
 \ * ox ax J 
 
 dV 
 
 r. sin u . SA. ; and therefore the westerly force =- 
 
 ax 
 
 1 dV 
 
 will be -- : .-7-. Considering the fictitious y as 
 r . sin u d\ 
 
 in the horizontal plane and towards the north, By is 
 
 dV 
 
 r.Bu; and therefore the northerly force -=- will be 
 
 -- .-7 And considering the fictitious z as vertical 
 
 at the place, Bz is Br ; and therefore the vertical force 
 
 dV. dV _, 
 
 T~ is -- r~- Inus we have 
 
 dz dr 
 
 .j , 
 
 r . sin u d\ 
 
 The algebraist may perhaps prefer a more rigorous 
 investigation, of the following form. 
 
 Conceiving the place of observation on the globe as 
 turned in some measure towards the spectator, the 
 origin of longitude being to the extreme left hand or 
 west; let x be measured from the Earth's center in the 
 plane of equator towards the left; y in the plane of 
 
GAUSS'S THEORY OF TERRESTRIAL MAGNETISM. Ill 
 
 equator towards the spectator: and z towards the north 
 
 pole. Then 
 
 x = r . sin u . cos X, 
 
 y = r . sin u . sin X, 
 z = r . cos u. 
 For changing our coordinates, we must put 
 
 dV = dVdu dV d\ dVdr 
 dx ~ du ' dx d\ ' dx dr'dx' 
 
 and similar equations for y and z\ where u, X, and r, are 
 supposed to be explicitly expressed (as was V) in terms 
 
 /v. 2 I aj% 
 
 of a?, y, z. Now tan 2 u = - f ; from which, after due 
 
 reductions, 
 
 du 1 
 
 = -cosw.cosX, 
 dx r 
 
 du 1 
 
 -y- = - cos u . sin X, 
 dy r 
 
 du 1 . 
 -r --- sin u. 
 dz r 
 
 And tan X = - ; from which 
 x 
 
 d\ 1 sin X 
 dx r ' sin u' 
 d\ _ 1 cos X 
 dy r ' sin u' 
 
 And r 2 = a; 2 + 2/ 2 + ^ 2 ; whence 
 
 <2r . 
 
 T- = sm it . cos 
 
112 ON MAGNETISM. 
 
 dr . 
 
 = sin u . sin X, 
 
 dr 
 
 -r = COS U. 
 
 dz 
 
 And thus 
 
 ,. dV dVl , dVl sinX dV . 
 
 X= 7 -=+, -- eosw.cosX PT-' -- I- -7-. sum cos X, 
 ax du r aX r smu dr 
 
 . dV dV 1 dVl cosX dV . 
 
 Y=-j- =H ^-.-cosw.sinXH r .-. -: -- h -j- .smit.smX, 
 ay du r d\ r sin?* dr 
 
 dV dV 1 , dV 
 
 Z^- ^ .-smw + -r- .COSM. 
 2 aw r dr 
 
 Now W= JTsinX Fcos X. Also the force in the 
 direction of radius of the .parallel passing through the 
 point of observation = J5fcosX+ Fsin X; from which, 
 combined with the force Z, 
 
 N= Zsm u (JTcos X + Fsin X) cos w, 
 C Zcos u + ( X cos X + i^sin X) sin u. 
 Substituting the values of X, Y, Z t 
 
 - 
 
 JLf ' 7 ? 
 
 r du 
 vr l dV 
 
 W = -- : - .-7T-, 
 
 r sin w X 
 
 rfF 
 ~*- ; 
 
 the same as the values found before. 
 
 Every thing now depends on the function V: and 
 
GAUSS'S THEOKY OF TERRESTKIAL MAGNETISM. 113 
 
 this depends on - or {(x a) 2 +(y &) 2 + (z c) 2 }"*. For 
 
 x, y, z y the ordinates of the experimental magnet, put 
 their values (already used) r . sin u . cos X, r sin u . sin X, 
 r cos u. And for a, b, c, the coordinates of a disturb- 
 ing particle of magnetism, put similar coordinates, 
 a = r Q . sin U Q . cos X , b r Q . sin U Q . sin X , c = r . cos w . 
 (If the experimental magnet be on the earth's surface, 
 and the disturbing magnetism be within the earth, r Q 
 
 is always less than r.) The value of - now becomes 
 
 [r 2 2r r {sin u. sin u . cos (X X ) + cos u . cos u ] + rf]~* ; 
 which can be expanded in a converging series 
 
 where T Q = 1, and T v T z &c., are functions only of u, u , 
 and X X . Put R for the earth's radius (the symbol r 
 being still reserved for the radius at the place of ob- 
 servation, in order to preserve the generality which 
 admits of differentiation with respect to r). Then V 
 
 or 
 
 ISp . - may be put in the form 
 
 where tfP 9 = - f T . $p, 
 f T^ r 2 Sfi, &c. The general term will be 
 B n+ *P n 
 
 r n+l > 
 
 where RP n = -/T n .r n . 8/*. 
 
ON MAGNETISM. 
 
 Now forming the values of N, W, C, and remarking 
 that (as the integral with respect to fyt applies only to 
 elements entirely independent of u, X, and r,} the dif- 
 ferentiations with respect to u and X can be performed 
 under the integral sign, and the differentiation with 
 respect to r will be entirely external to the integral 
 
 1 dV 
 
 sign; the general term of N or . -*- will be 
 
 T Ct/U 
 
 B n+ * dP^ 
 r^'~du> 
 
 1 fJV 
 
 that of TFor = . will be 
 
 r sin u d\ 
 
 1 fi* 1 * dP n 
 
 sin u ' r n ** ' d\ ' 
 
 that of (7 or -j- will be 
 dr 
 
 Also it is to be remarked that T is 1, and therefore 
 / T 9 S/i is (because the total amounts of red and of 
 blue magnetism are supposed to be equal), and there- 
 fore P l is 0. And, if our needle be on the earth's 
 surface, r = R. Thus we obtain 
 
 du 
 
 sin 
 
 C=+ 2P X + 3P, + &c. + (n + 1) P n + &c. 
 where r w+2 .P = -r.r n . 
 
LAPLACE'S COEFFICIENTS. 115 
 
 and T n is the coefficient (in terms of u, U Q , X,X ) of f J 
 
 in the development of -. 
 
 48. Incidental introduction of Laplace's Coefficients 
 (not further used in this Treatise). 
 
 If we differentiate twice the expression 
 
 with respect to x, also with respect to y, and with 
 respect to z, we find 
 
 i 
 
 And since F= jfyt-, and since the application and 
 
 limits of this integration do not depend on x, y y z, it 
 follows that 
 
 _ 
 
 ' 
 
 da? ' dy 2 ' dz* 
 
 Now, by the same principle which we have used in the 
 last Article, for F, and which we shall here use succes- 
 
 , , dV dV , dV 
 
 sively for -=- , -j- , and -j- , 
 
 dx dy dz' 
 
 (dV\ j (^\ fJ i^ 7 \ 
 
 <7 2 F V dx ) du \dxj d\ \dx) dr ^ 
 
 dx* ~ du ' dx d\ ' dx dr ' dx' 
 
 82 
 
116 
 
 <f F J du t d \djjj d\ t 
 
 ' dy d\, dy dr ' dy ' 
 
 dz* du ' dz d\ ' dz dr ' dz' 
 
 The expressions for all the quantities on the right hand 
 
 dV 
 
 are to be found in the last Article : those for -4- &c. re- 
 
 dx 
 
 quiring complete differentiation, with special notice 
 that -5- &c. are themselves subject to differentiation, 
 
 as well as the more explicit quantities cos u, &c. Per- 
 forming all operations, this result is obtained: 
 
 d 2 (rV) 1 cost* 
 
 dr 2 r ' sin u ' du r' da 2 r sin 2 u 
 
 7->+2 75 
 
 Applying this to the general term ^-^ where P n is 
 
 independent of r, 
 
 n(n+I).P n 
 
 _ 
 
 '' du 
 
 CORW 
 
 or 
 
 smw 
 
 from which it is possible to find a general expression 
 for P n . The terms thus found are Laplace's Coefficients. 
 ]n the physical investigation now before us, we shall 
 not have occasion to use the general term. 
 
GAUSS'S THEORY OF TERRESTRIAL MAGNETISM. 117 
 
 49. Continuation of Gauss's investigation : applica- 
 tion in a numerical form. 
 
 Put p for sin u . sin u . cos (X X ) + cos u . cos w , and 
 expand the fraction for - . In the paucity of well-de- 
 
 termined elements, and in the complexity of expressions, 
 Gauss thought it sufficient to develope this to the 4th 
 
 power of r . This gives for -, 
 
 S -"'+"/) 
 
 = 
 r 
 
 cosw.cosX cosw.sinX. ft 
 
 ~i - ( r cosw .smX ) 
 
 sin u 
 
 The corresponding term of V will be 
 
 cos u. cos X fcx 
 --- 2 -- I O/A . r . cos w . cos X 
 
 . sinw 
 
 Each of these integrals is an unknown constant. Call- 
 ing them i v i y i# the term of V will be 
 
 cos u . cos X . cos u . sin X . sin u . 
 
118 ON MAGNETISM. 
 
 where for any special locality on the earth, u, X, and r, 
 must have the proper numerical values, but i v i v i z 
 must for the present be left in a symbolical form. The 
 expansions of p*, p* } &c., will introduce other integrals 
 or unknown constants z 4 , i 5> i 6 , &c. multiplied by other 
 functions of u, \, and r. And thus the forces N, W, C, 
 can be exhibited for every locality, in expressions which 
 involve these unknown constants: then the westerly 
 
 W 
 declination, whose tangent = j=, can be so expressed: 
 
 the total horizontal magnetic force = J(N* + TF 2 ) can 
 be so expressed: and the angle of dip, whose tangent 
 
 ' can be so ex P ressed - 
 
 The number of integrals or undetermined constants 
 thus introduced is large. Limiting the order (as above 
 mentioned) to P 4 or to the fourth power of p, 24 con- 
 stants are required. In order to obtain these numeri- 
 cally, 24 observations of some kind are necessary. Any 
 determinations of magnetic elements will suffice : for 
 instance, determinations of western declination, horizon- 
 tal force, and dip, at each of eight stations. Gauss, refer- 
 ring generally to Sabine's map of Total Intensity in the 
 Seventh Report of the British Association, and to Barlow's 
 map of Declination, Phil. Trans. 1833, and to Homer's 
 map, Physikalisches Worterbuch, Band VI, but without 
 giving numerical details of his process, has obtained the 
 
 y 
 following value for ^ . It is to be remarked that the 
 
 numbers have all been adapted to give horizontal force 
 
GAUSS'S THEORY OF TERRESTRIAL MAGNETISM. 119 
 
 at London = 1732 (it having been customary in former 
 times to call that force 1*732). The first or constant 
 term, which does not appear in our formula for V, pro- 
 bably arises from the conversion of powers of cosines &c. 
 into cosines of multiple arcs. The letter e stands for 
 cos u and f for sin u. 
 
 -= - 1-977 + 937103e+71'245 e 2 - 
 
 + (64-437-79-518 e + 122-936 e*+ 152-589 e 3 )/cos \ 
 + ( - 188-303 -33-507 e + 47794 e* + 64-112 e 3 )/sin \ 
 + (7-035 - 73-193 e - 45791 e 2 )/ 2 cos 2X 
 + ( - 45-092 - 22-766 e - 42-573 e 2 )/ 2 sin 2X 
 
 + (l-396 + 19774e)/ 3 cos3X+ (-18750 -0-l78e)/ 3 sin3X 
 + 4-127/ 4 cos 4X + 3-175/ 4 sin 4X. 
 
 From these, by the formulae in Article 47, are formed 
 numerical values of N, W, and C, for numerous latitudes 
 and longitudes : and from them are derived numerical 
 values of Declination, Horizontal Force, and Dip. By 
 means of the values of Declination, curves of Places of 
 Equal Declination were laid down by Gauss upon a map, 
 from which the writer of this Treatise has formed the 
 Magnetic Meridians in Figures 20 and 21. The Mag- 
 netic Meridians may also be traced by the following 
 process. Conceive fictitious ordinates x, y, z, as in 
 Article 47, where x and y are on the tangent-plane 
 of any point on the earth's surface, and x is in the 
 direction in which V does not alter, that is, in the direc- 
 tion of a curve of Equal Values of V. The general 
 
120 ON MAGNETISM. 
 
 dV 
 expression for force in the direction x is -j- . But in 
 
 this instance, V does not alter with alteration of x ; 
 
 dV 
 
 therefore -7- is 0, and there is no force in the direction 
 dx 
 
 of x. Consequently the whole horizontal force is in the 
 direction of y, or perpendicular to the curve of Equal 
 Values of F Now Gauss has prepared curves of Equal 
 Values of F(not copied in this Treatise), and therefore 
 it is only necessary to draw trajectory-curves cutting all 
 the Equal- F-curves at right angles, and the lines so 
 drawn will be the lines of direction of total horizontal 
 force, or the Magnetic Meridians. 
 
 The curves of Equal Horizontal Force in Figures 28 
 and 29, those of Equal Total Intensity in Figures 35 
 and 36, and those of Equal Dip in Figures 20 and 21, 
 are copied immediately from Gauss. 
 
 The elements from which these curves have been 
 formed having been deduced from 24? magnetic measures 
 made at different places, those measures are necessarily 
 exhibited correctly in the curves. And now the ques- 
 tion arises, whether all other measures made since that 
 time are exhibited accurately by the curves. And the 
 answer is, that they are exhibited so accurately as to 
 leave no doubt on the fundamental correctness of the 
 theory, and yet with small discordances which render it 
 desirable that the formulae should be extended and 
 
 compared with a greater number of measures for nu- 
 
 y 
 
 merical determination of the constants in ~ I* 1 the 
 
 M 
 
GAUSS'S THEOEY OF TERRESTRIAL MAGNETISM. 121 
 
 Astronomische Nachrichten, Nos. 1792 and 1793, 
 H. Peterson has given the results derived from 610 
 measures : but the symbolical development is still limit- 
 ed (as above) to the 4th order. The results are not yet 
 exhibited in a form easily understood by the eye. 
 
 We cannot terminate this Section of our work with- 
 out earnestly inviting the attention of our readers to the 
 whole of Gauss's investigation : one of the most beauti- 
 ful and the most important that has appeared for many 
 years in Physical Mathematics. 
 
122 ON MAGNETISM. 
 
 SECTION VIL 
 
 DISTURBING FORCE PRODUCED ON A SMALL COMPASS- 
 NEEDLE BY A LARGE MAGNET, IN VARIOUS POSITIONS; 
 AND COMPOSITION OF THIS DISTURBING FORCE WITH 
 TERRESTRIAL HORIZONTAL FORCE. 
 
 50. The disturbing magnet is horizontal : its center 
 is broadside-on to the center of the compass : to jind its 
 effect at different distances and elevations. 
 
 In this and subsequent investigations of this Section, 
 we shall consider the dimensions of the compass-needle 
 to be so small, in comparison with other measures, that 
 we may use the lengths of lines measured to the center 
 of the compass-needle instead of those measured to its 
 poles : and we shall investigate the action upon the red 
 end only of the compass-needle, inasmuch as the action 
 on the blue end will be sensibly equal but in the oppo- 
 site direction, and the impressed moment of rotation 
 will be merely doubled. 4 
 
 In Figure 39, the attraction of the blue pole of A 
 
 and the repulsion of the red pole produce in the direc- 
 
 ff ' 
 
ACTION OF A MAGNET ON A COMPASS NEEDLE. 123 
 
 Pig. 39. 
 
 tion of c the respective forces 
 
 which destroy each other : but they 
 produce in the direction towards the 
 right the forces 
 
 which are to be added together, pro- 
 ducing 2 a . a(a z + c 2 )~^ 
 
 acting towards the right. This force, it is to be observed, 
 is parallel to the length of A or perpendicular to the 
 length of c, whatever be the plane containing the center 
 of the compass and the axis of A. The plane aaB may 
 be horizontal, or vertical, or inclined, but the expression 
 found above applies to every one of these cases. 
 
 51. The disturbing magnet is end-on to the compass: 
 first, in the horizontal plane: secondly, in an inclined 
 plane, the axis of the magnet still directed to the compass. 
 
 In Figure 40, the attraction of the blue 
 end of the magnet on the red end of the 
 
 compass is represented by -5 , and the 
 
 repulsion of the red end by , 2 ; the dif- 
 
 (c + a) 
 
 ference is 
 
 ca 
 
124 
 
 ON MAGNETISM. 
 
 Fig. 41. 
 
 This, in every case of the magnetic end directed towards 
 the compass, represents truly the entire action, tending 
 (with the poles as supposed in the figure) to draw the 
 red end of the compass towards A. And, when A is 
 contained in a horizontal plane passing through B, the 
 expression gives the force which tends to disturb the 
 red end of the compass in the horizontal plane. 
 
 But when the direction of A 
 is inclined to the horizon, take the 
 vertical plane passing through A as 
 in Figure 41 ; the force which A 
 exerts on the red end of the compass 
 is that just found, but it acts in 
 the direction BA : and the hori- 
 zontal part of it will be obtained 
 by multiplying by sin <, or will be 
 
 ca 
 
 . sin <f>. 
 
 52. The disturbing magnet is horizontal; it is 
 directed end-on to the vertical axis of the compass, and 
 is not necessarily at the same elevation as the compass. 
 
 This state of things is represented in Figure 42. 
 
 The attraction of the blue end is - 5 -. 
 
 c + a 2ca sm <f> 
 
 the horizontal part of this is 
 
 a(c sin <j> - a) (c 2 + a 2 - 2ca sin 0)~^ 
 The horizontal repulsion of the red end is 
 
 a(c sin <f> + a) (c 2 + a 2 + 2ca sin c)~i 
 
 : and 
 
ACTION OF A MAGNET ON A COMPASS NEEDLE. 125 
 
 Fig. 42. 
 
 The effective attraction in the horizontal plane is the 
 excess of the former over the latter. 
 
 If the magnet be at a considerable distance, or if a 
 be much smaller than c, so that it will suffice to include 
 the first power of a, an approximate value will be 
 
 - 5 (c sin <f> a) (c 2 + 3ca sin <f>) -j (c sin + a) (c 2 3ca sin 0) ; 
 or, nearly, 
 
 -5 (c 3 sin (j> + 3c 2 a sin 2 <f) c 2 a) - s (c 3 sin < 3c 2 a sin 2 ^ + c 2 a) ; 
 c c 
 
 Consequently, when sin = * /- , or ^> = 35 16', there 
 
 is no horizontal action : when < is less than this angle, 
 the action is of the opposite character. 
 
126 
 
 ON MAGNETISM. 
 
 53. The disturbing magnet is vertical. 
 
 In Figure 43, the horizontal part of the attraction 
 
 Fig. 43. 
 
 of the blue pole is a . c . sin < . (c 2 + a 2 2ca cos $)"-, 
 and that of the repulsion of the red pole is 
 a . c sin </> (c 2 + a 2 4- 2ca cos <)~^. 
 
 If these be expanded to the first power of a, the result- 
 ant attraction is found to be 
 
 . 
 3- . sin <j) . cos <f>. 
 
 With a given value of c, it is therefore greatest 
 when (j> = 45. 
 
 With a given horizontal distance h, which makes 
 
 sn 
 
 ~ , the force is 
 
 - . air < . cos 
 
ACTION OF A MAGNET ON A COMPASS NEEDLE. 127 
 
 this is greatest when tan < = 2, which gives the depres- 
 sion of the center of the magnet below the compass 
 h 
 
 54. The disturbing magnet is in the horizontal plane 
 which passes through the compass, but is inclined at any 
 angle to the line joining the centers of the magnets. 
 
 In Figure 44, it will readily be 
 seen that the resolved part of the 
 force in the direction of c is 
 
 a . (c a cos 6) (c 2 2ca cos 6 + a 2 )"^ 
 - a . (c + a cos 6) (c 2 + 2ca cos 6 + a 2 )"^ : 
 and the resolved part perpendicular 
 to c towards the right is 
 
 a . a sin 0(c 2 - 2ca cos + a 2 )" 1 
 
 + a . a sin 0(c 2 + 2ca cos 6 + a 2 )'^. 
 
 If we expand these to the third power of - , we find, 
 
 c 
 
 Force in the direction of c 
 
 = ^cos 6 jl + ^ (5 cos 2 6 - 3) i ; 
 
 Force transversal to c 
 
 a 2 / 15 
 7 2 V2 
 
 The square root of the sum of the squares of these will 
 give the whole force on the red end of the compass- 
 
128 
 
 ON MAGNETISM. 
 
 needle, and the quotient of the second by the first will 
 give the tangent of the inclination of the whole force 
 to c. 
 
 55. Composition of the disturbing force in the hori- 
 zontal plane with the terrestrial horizontal force. 
 
 In the case of Figure 39, the horizontal force pro- 
 duced by A is in the direction at right angles to c: in 
 Figures 40, 41, 42, 43, it is in the vertical plane which 
 contains c ; and in Figure 44, it makes a definite angle 
 with c, depending only on the magnitude and distance 
 of A and its inclination to c. If the disturbing magnet 
 A rotate in the horizontal plane (as for instance when 
 it is part of a ship revolving in azimuth, the compass in 
 these figures being the ship's compass), in Figure 45, 
 
 let BF with length f represent the force which, as found 
 above, is produced by the magnet A acting on the red 
 
COMPOUNDED FORCES OF EARTH AND MAGNET. 129 
 
 end of B ; and let EB with length h, as measured from 
 E to B, represent the earth's horizontal magnetic force 
 acting on the same red end of B. Then EF will repre- 
 sent in magnitude and direction the total composite 
 horizontal force acting on the red end of the compass- 
 needle. As the ship rotates in azimuth, the line / will 
 assume the different positions BF' y BF", &c., the points 
 F'y F", &c. all lying in a circle of which B is the center : 
 and in these different positions, the total horizontal 
 force acting on the needle will be represented in mag- 
 nitude and direction by EF' , EF", &c. It is seen here 
 that when the ship is in such a position that the devia- 
 tion of the compass (which is the same as the angle 
 BEF) = 0, the force is either the greatest possible = h+f, 
 or the least possible = h f. It is also seen that -in the 
 entire revolution of the ship, the compass-needle deviates 
 during half of the revolution to the right and during the 
 other half to the left. The positions of the ship at 
 maximum deviation to the right and maximum devia- 
 tion to the left are not exactly half-way between the 
 positions of no deviation. 
 
 If / be greater than h, the circle will include the 
 point B : and, as the ship revolves uniformly, the com- 
 pass-needle will turn entirely round, but not with 
 uniform angular velocity. 
 
130 ON MAGNETISM. 
 
 SECTION VIII. 
 
 ON TEANSIENT INDUCED MAGNETISM IN SOFT IRON. 
 
 56. Definition of Soft Iron, and criterion of the 
 magnetic difference between Soft Iron and Magnetized 
 Steel 
 
 Under the term Soft Iron may be understood, either 
 Malleable Iron which has not been hammered or sub- 
 jected to any violence when cold, or Cast Iron. (We 
 shall in the next Section discuss the properties of 
 Malleable Iron when subjected in the cold state to 
 violence.) And the best practical criterion by which a 
 bar of Soft Iron is distinguished from a Steel Magnet is 
 this. We have found in Articles 16, 27, and other 
 places, that if, in the horizontal plane, a steel magnet is 
 applied end-on towards the center of a suspended hori- 
 zontal magnet, it tends to produce a deviation in the 
 position of the suspended magnet. Now if a bar of soft 
 iron be substituted for the steel magnet, the suspended 
 magnet will not be disturbed at all. In some positions, 
 if the suspended magnet be constrained by external 
 
INDUCED MAGNETISM IN SOFT IRON. 131 
 
 force to take a position other than north and south (as 
 for instance, if suspended by two threads as in the ap- 
 paratus for measure of small changes of horizontal force, 
 Article 85), the presentation to it of a bar of soft iron 
 end-on to the center will slightly disturb it : but to a 
 degree very much less than that of which we shall speak 
 in the next article. 
 
 57. Experiments on the induction of magnetism in 
 Soft Iron by the action of a Steel Magnet. 
 
 In Figures 46 and 48, suppose that A is a steel magnet 
 Fig. 46. m a vertical position (it matters little whether 
 the red end is upwards or downwards : in the 
 diagram it is supposed that the red end is up- 
 wards). In Figure 46 suppose that G is a small 
 bar of soft iron (as a small nail) lying on a table 
 so far below A that the action of A will not 
 . c . sensibly disturb G. Suppose that B in Figure 47 
 Fio . 47 is a bar of soft iron (as a larger nail) which, alone, 
 would not disturb G. Now let the bar B be 
 placed under A as in Figure 48 (in which case 
 the magnet A if sufficiently powerful will sup- 
 port By the reason of which we shall hereafter 
 1 explain), and B will immediately lift the small 
 
 bar G. If the bar B be held in the left hand, and A 
 in the right, then, upon detaching A from B, G will 
 immediately drop off. On the other hand, if the con- 
 nexion of A, B, and G, be maintained, G will support a 
 piece of iron wire D, as in Figure 48. And this series 
 may sometimes be continued through several steps. 
 
 92 
 
132 
 
 ON MAGNETISM. 
 
 48. It is evident here that B is converted 
 
 into a magnet as long as it is under the 
 influence of A, and no longer. And this 
 A is the characteristic of Transient Induced 
 Magnetism. If the quality of the magnet- 
 ism of the lower end of B be examined 
 B by the disturbance which it produces in a 
 compass-needle, it is found to be the same 
 as that of the lower end of A (blue magnet- 
 ism, in the diagram). This leads to the 
 presumption, in analogy with other phseno- 
 mena of magnetism, that the magnetism of 
 the upper end of B is of the kind opposite to that 
 of the lower end of A: a, presumption which we shall 
 find to be supported in the case which we can examine 
 more perfectly, that of transient induction produced by 
 the earth's action. 
 
 Fig- 49- The same conclusions will 
 
 W 
 
 be arrived at by examination 
 ^^ of the deviation produced in a 
 suspended magnet or compass- 
 needle ; as in Figure 49. If 
 the magnet A has produced 
 deviation of B' to the position 
 shewn in the diagram, and the 
 bar of soft iron B be inserted 
 (under circumstances where, if 
 alone, its effect on B would be 
 imperceptible), it greatly in- 
 creases the deviation of B'. 
 
ATTRACTION OF IRON BY A STEEL MAGNET. 133 
 
 The effect is considerable if B does not touch A t but 
 much larger if B touches A. It is certain here that 
 the nature of the magnetism in the advanced end of 
 B is the same as that in the advanced end of A. 
 
 Or, if the magnet A be held vertically above the 
 center of a small compass (in which state it will not 
 disturb the compass) ; and if the upper end of B touch 
 the magnet, and its lower end be carried conically round 
 the compass : it will disturb it in a manner which shews 
 that the lower end of B has the same kind of magnetism 
 as the lower end of A. 
 
 58. Explanation of the attraction of soft iron by 
 either pole of a steel magnet, as an effect of induction. 
 
 We are now in a position to explain the ordinary 
 phenomenon, (perhaps the best known of all magnetic 
 phenomena), of attraction of soft iron by either pole of 
 a magnet. In Figure 48, B is, for the time, a magnet 
 as well as A ; and the two poles (that of A and that of 
 B) which are in contact, have, one blue magnetism, the 
 other red. Therefore there is attraction. It is seen 
 that it is indifferent which pole of A is presented to B : 
 a blue pole of A produces an adjacent red pole in B, or 
 a red pole of A produces an adjacent blue pole in B : 
 and in both cases there is attraction. 
 
 We see also that the phenomenon is entirely in ac- 
 cordance with that of the magnetization of steel by 
 double-touch, Article 8. It appeared there that the 
 blue magnetism of one end of the dominant magnet 
 
134 ON MAGNETISM. 
 
 dragged the red magnetism of the affected magnet to 
 one end and there left it fixed : here it seems that it 
 draws the red magnetism of the iron bar (or a portion 
 of it) to one end, but cannot leave it fixed there : that 
 in the instance of iron, as distinguished from steel, the 
 separate kinds of magnetism take the earliest oppor- 
 tunity of returning to their original seats and producing 
 neutral magnetism in every part. 
 
 Fig 50 We a ^ so see ^ ne reason wnv a horse-shoe 
 
 magnet so energetically attracts a piece of 
 iron touching both its poles, as in Figure 
 50. Each pole of the horse-shoe converts 
 the corresponding part of the iron into a 
 pole of opposite quality, and the existence 
 of each impressed pole at one end of the 
 iron seems to have a tendency to intensify 
 the opposite pole at the other end, and thus the 
 iron is in the state of a powerful magnet attracted 
 by another powerful magnet, and the attraction (pro- 
 portional to the product of the powers) is very 
 energetic. 
 
 59. Rapid diminution, with increase of distance, of 
 the attraction between a magnet and soft iron. 
 
 The magnetic power of the permanent magnetism 
 in one pole of the magnet varies, as has been demon- 
 strated, inversely as the square of the distance of the 
 magnetic body on which it acts. It appears reasonable 
 to suppose that its influence in inducing magnetism 
 
INDUCTION FKOM TERRESTRIAL MAGNETISM. 135 
 
 follows the same law, and therefore that the energy of 
 the induced magnetism is inversely as the square of the 
 distance. Consequently, the attraction between the two 
 magnetisms (the permanent magnetism of the magnet 
 and the induced magnetism of the iron), which is as the 
 product of these magnetisms directly and as the square 
 of the distance inversely, will be inversely as the fourth 
 power of the distance. With increase of distance there- 
 fore the attraction diminishes very rapidly. 
 
 When the distance is so far increased that the effect 
 of the farther pole of the magnet, though diminished, is 
 less diminished than that of the nearer pole, and be- 
 comes comparatively sensible, it tends still more to 
 diminish the attraction. And, on the whole, the at- 
 traction diminishes with extreme rapidity, and is sensi- 
 ble only at very small distances. 
 
 60. Induction of magnetism in soft iron, produced 
 by terrestrial magnetism. 
 
 Take a bar of soft iron, which for convenience of 
 language we will suppose to have one end painted white 
 and the other end black : hold it vertical, with the 
 black end downwards. Upon applying any of the ordi- 
 nary tests, it will instantly be found that the bar in this 
 position is a genuine magnet, and that its black end is 
 charged with red magnetism and its white end with 
 blue magnetism. The easiest proof will be, holding it 
 parallel to itself, to carry it round a small compass : if 
 the black or lower end is at the level of the compass, it 
 
136 ON MAGNETISM. 
 
 attracts the blue end of the compass-needle : if the white 
 or upper end is at the level of the compass, it attracts 
 the red end of the needle : if the middle of its length is 
 at the level of the compass, it produces no sensible 
 disturbance. 
 
 Yet this magnetism does not imply any permanent 
 modification in the state of the iron bar. For, invert 
 the bar, so that the white end is downwards, and apply 
 it in the same way to the experimental compass. Now, 
 the white end of the bar attracts the blue end of the 
 needle (instead of attracting the red end as it did before) 
 and the black end of the bar attracts the red end of the 
 needle (instead of attracting the blue end as it did be- 
 fore). The iron bar is for the time a magnet, but its 
 poles are in the direction opposite, as regards the struc- 
 ture of the iron, to that in which they were before. 
 
 But they are in the same direction as regards up 
 and down. The upper end (whether white or black) is 
 always a blue pole, and the lower end, (whether black 
 or white), is always a red pole. 
 
 These experiments are described as they are seen in 
 the northern magnetic latitudes of the earth. In the 
 southern magnetic latitudes, the lower end of the bar 
 has blue magnetism. At the magnetic equator, the ex- 
 periment fails in this form ; but a slight variation in the 
 form of the experiments, applicable in every place, ex- 
 hibits the induced magnetism in the greatest possible 
 intensity ; the variation is merely the following : 
 
 Instead of holding the bar in the vertical position, 
 hold it in the direction of the local dip. Then it will 
 
QUADRANTAL EFFECT OF INDUCED MAGNETISM. 137 
 
 be found that the quality of the magnetism of the bar 
 is always the same as that of the dipping-needle. 
 
 Now vary the experiment by holding the bar so that 
 its length is contained in the plane to which the dip- 
 direction is normal. Its disturbing power ceases en- 
 tirely : it has no sensible charge of magnetism. 
 
 All these phenomena are exactly similar to those 
 described in Article 57, conceiving the earth's action to 
 be similar to that of a steel magnet : and the explana- 
 tion is the same as that in Article 58, that the attraction 
 of the earth's red magnetism draws towards itself the 
 blue magnetism which is in the particles of the iron : 
 and similarly for the attraction of the blue on the red. 
 
 61. Effect of the terrestrially-induced magnetism in 
 a mass of soft iron which is carried round a compass, at 
 the same level as the compass, and with the same part of 
 the mass always directed to the compass-center. 
 
 This case is one which theoretically deserves atten- 
 tion, and which in practical application is very impor- 
 tant, inasmuch as in iron-built and other ships it re- 
 presents the state of things where, partly from the iron 
 of the ship-construction and partly from the iron intro- 
 duced for corrective purposes, there is much iron ad- 
 mitting of induction from terrestrial magnetism, at 
 nearly the same elevation as the compass, and revolv- 
 ing round it as the ship swings round, always presenting 
 the same part to the compass. 
 
138 ON MAGNETISM. 
 
 Upon carrying the mass of iron round the compass 
 in the manner described, the phenomena are these : 
 When the central point of the mass (if symmetrically 
 shaped), or a certain central point (in general) is on the 
 N. or S. or E. or W. side of the compass-center, it pro- 
 duces no disturbance in the compass-needle. When 
 the direction of that central point is between N. and E., 
 it turns the N. end of the needle to the E. : when be- 
 tween E. and S., it turns it to the W. : when the central 
 point of the mass is between S. and W., it turns the 
 N. end of the needle to the E. : when between W. and 
 N., it turns the N. end to the W. On comparing these 
 with the deviations produced by a magnet which is 
 carried round the compass in the same manner, as de- 
 scribed in Article 55, it is seen that there is a striking 
 difference ; in the case of a complete revolution of the 
 magnet, the needle is made to deviate once to the right 
 and once to the left ; but in the case of a complete re- 
 volution of the soft iron, the needle deviates twice to 
 the right and twice to the left. If the azimuth of the 
 disturbing mass, as viewed from the center of the com- 
 pass, and measured from N. towards E. be called 0, the 
 amount of deviation produced in the needle from N. 
 towards E. is exactly or approximately proportional to 
 sin 20 : vanishing when is 0, 90, 180, 270, and be- 
 coming negative when is > 90 < 180, or > 270 < 360. 
 The law of disturbance may be represented (for memory 
 only) by this rule : the mass attracts that pole of the 
 needle which is nearest to it. 
 
QUADKANTAL EFFECT OF INDUCED MAGNETISM. 139 
 
 62. Effect of the combination of two masses of iron, 
 in opposite azimuths : and of two masses of iron, in 
 azimuths differing 90. 
 
 One curious consequence of this law, easily verified 
 in experiment, is, that if a mass similar to the first mass 
 be placed on the opposite side of the compass, carried 
 by the same frame so that in revolution it is always 
 opposite to the first mass, it doubles the disturbance ; 
 but if it is placed afc 90 either to the right or to the left 
 of the original mass, always retaining that relative posi- 
 tion, it neutralizes the disturbance. For, the original 
 disturbance being a sin 20, that of an opposite mass will 
 be a sin 2 (0 + 180) 
 
 = a sin 20, 
 
 the addition of which doubles the first : but the disturb- 
 ance produced by a mass 90 to right or left will be 
 a sin 2 (0 + 90) 
 
 = a sin 20, 
 the addition of which neutralizes the first. 
 
 63. Simplest form of theory for explanation of the 
 phenomena of induction. 
 
 In Figure 51 conceive the first line of circles to 
 represent particles of a mass of iron, or at least so 
 many of the particles as contain united portions 
 of red magnetism and blue magnetism, in a line 
 extending through a mass of iron. And conceive 
 
140 
 
 ON MAGNETISM. 
 
 Fig. 51. 
 
 O 
 O 
 O 
 O 
 O 
 O 
 O 
 O 
 O 
 O 
 
 the second line to represent the 
 state of their magnetisms as affect- 
 ed by the induction of the great 
 masses of blue and red magnetism 
 external to them. (The effect of 
 one of these masses alone is pre- 
 cisely similar in kind to that of 
 the two masses.) Then, in analogy 
 with everything that we have seen of 
 magnetization of steel magnets and 
 of iron bars, we may conceive the 
 blue magnetism of each circle to be 
 drawn towards the external red mass, 
 and the red magnetism of each circle 
 to be drawn towards the external 
 blue mass, as shewn in the figure. 
 The effect of this will be that, 
 through all the intermediate parts 
 of the series, the blue and red alternate in such a 
 way that we cannot perceive any clear tendency 
 in them to produce magnetic effect on an external 
 body: but there is certainly a red pole at one end 
 and certainly a blue pole at the other. When we 
 conceive a system of parallel lines of the same kind 
 passing through a mass of iron, we find that the whole 
 exterior surface which is turned towards the great red 
 mass is clothed with blue magnetism, and that the 
 whole which is turned towards the great blue mass is 
 clothed with red magnetism : and the mass resembles 
 to some extent a steel magnet. 
 
THEORETICAL EXPLANATION OF INDUCTION. 141 
 
 64. TIie.inductive energy may "be resolved in different 
 directions, in the same manner as statical forces. 
 
 Fig. 52. In Figure 52, let a and b represent 
 
 the separated masses of magnetism of 
 equal intensity produced by one of the 
 small circles in Figure 51. It seems 
 reasonable to suppose that the extent of 
 ^ their separation will be proportional to 
 the external magnetic energy. Take the 
 positions b'a (coincident in space) for 
 two masses of opposite magnetisms, each 
 equal to a or b. These two masses, 
 while coexisting, neutralize each other. 
 But we may conceive b' associated with 
 a and a associated with Z>; and we may 
 consider the pair ab r as the effect of one 
 inducing magnetism in the direction ab', 
 gmv and the pair a'b as the effect of another 
 inducing magnetism in the direction a'b ; 
 and the magnitudes, of the two inducing magnetisms 
 must (by the general assumption mentioned above) 
 be considered proportional to the lengths ab', ab. 
 It is seen here that we have in fact resolved the 
 primary inducing energy into two, according to the 
 laws of resolution of statical forces : and if, in any pro- 
 posed problem, it can be shewn that one of these is 
 inefficient, we may confine our attention to the 
 other : or if the effects of the resolved inductions 
 can be computed more easily than that of the 
 
142 
 
 ON MAGNETISM. 
 
 original induction we may use them instead of the 
 original induction. 
 
 65. A mass of iron, symmetrical with respect to the 
 plane directed to the axis of a compass and with respect 
 to the horizontal plane, and with its center at the same 
 height as the compass, is subject to terrestrial induction : 
 theoretical investigation of its deviating energy on the 
 compass: it follows the law of sine 2 azimuth. 
 
 In each of the diagrams of Figure 53, the curve 
 
 Fig. 53. 
 
 represents the outline of the mass, and the magnetized 
 needle at which it points is the compass-needle. 
 The terrestrial energy is in the direction of the local 
 dip, and the whole inductive energy will be in that 
 
QUADRANT AL EFFECT EXPLAINED. 143 
 
 direction. Resolve this into horizontal and vertical 
 directions. The effect of the vertical part will be, 
 to produce a series of vertical linear magnets, each of 
 which has its center at the same height as the compass- 
 needle ; and these produce no effect on the compass. 
 The horizontal part remains, which is in the direction 
 of the magnetic meridian, and is proportional to the 
 horizontal force. 
 
 Now this horizontal induction does really produce a 
 series of linear meridional magnets, as shewn in the first 
 diagram : and the clothing of the surface will really be 
 such as is shewn there. But we may resolve the induc- 
 tion into two, one parallel to the length of the mass as 
 in the second diagram, and one transversal to that length 
 as in the third diagram : and their energies will be 
 respectively proportional to cosine azimuth of axis of 
 mass, and sine of the same azimuth. The linear mag- 
 netic needles which they will produce, and the magnetic 
 clothings, are shewn in the second and third diagrams. 
 The aggregate of actions in the second diagram will be 
 represented by that of one magnet, radial to the compass, 
 whose entire action (as already said) is proportional to 
 cosine azimuth : but the resolved part of this tending 
 to give rotation to the needle receives the factor sine 
 azimuth, so that its force tending to deflect the needle 
 may be represented by 
 
 A x cosine azimuth x sine azimuth. 
 The aggregate of actions in the third diagram will be 
 
144 ON MAGNETISM. 
 
 represented by that of one magnet transversal to the 
 radius, whose entire action is proportional to sine 
 azimuth: but the resolved part of this tending to give 
 rotation to the needle receives the factor cosine azimuth j 
 so that its force tending to deflect the needle may be 
 represented by 
 
 B x sine azimuth x cosine azimuth. 
 
 The total deflecting force is therefore 
 A x cosine azimuth x sine azimuth 
 
 + B x sin azimuth x cosine azimuth 
 
 A + B . . 
 = - x sine 2 azimuth. 
 
 It is easy to see that the effects of the two parts of 
 induction which we have considered have the same sign, 
 and that the deflection produced in the compass-needle 
 is such as to bring towards the mass of iron that pole 
 which is nearest to the mass of iron. 
 
 This result agrees with the experiment which is 
 described in Article 61. 
 
 66. Simpler investigation when the mass is spherical 
 with its center at the same height as the compass. 
 
 In Figure 54, the sphere is represented in eight 
 different positions, with the clothing of magnetism 
 which is produced by the induction. In the northern 
 and southern positions, the magnetism of that surface 
 
QUADRANTAL EFFECT OF AN IRON SPHERE. 145 
 
 of the sphere which is nearest to the compass-needle is 
 of the kind opposite to that of the near pole of the 
 
 1 
 
 needle, and there is attraction between them : but this 
 produces no deviation, because it is in the direction of 
 the needle's length. In the east and west positions 
 of the sphere, the magnetism of the north part of the 
 sphere repels that of the north end of the needle, and 
 the magnetism of the south fart of the sphere repels 
 
 10 
 
146 ON MAGNETISM. 
 
 that of the south end of the needle, with equal forces, 
 which balance : and in like manner there is equilibrium 
 between the attraction of the north part of the sphere 
 on the south end of the needle and that of the south 
 part of the sphere on the north end of the needle : and 
 the needle is not disturbed. But in all the other posi- 
 tions, the magnetism with which the nearest part of the 
 sphere is charged is of such a quality that it attracts the 
 nearest pole of the needle : and, when the sphere is in 
 north-east or south-west position, the north end of the 
 needle is made to deviate to the east : and, when the 
 sphere is in north-west or south-east position, the north 
 end of the needle is made to deviate to the west. 
 
 67. In these cases, the magnitude of the deviation 
 produced in the compass is independent of the magnitude 
 of the terrestrial horizontal force. 
 
 In order to judge of the law of compass-deviation in 
 this case and in the case of the last article, as depending 
 on the geographical position of the compass, that is, as 
 depending on the magnitude of the terrestrial horizon- 
 tal force (the only geographical element which affects 
 this problem), it is necessary to observe that, the needle 
 being directed in the magnetical meridian by the terres- 
 trial horizontal force, and being made to deviate by a 
 deviating force, the amount of deviation produced will 
 depend upon the value of the fraction 
 
 deviating force 
 terrestrial horizontal force ' 
 
 But, in a given position of a mass of iron, the deviat- 
 
QUADRANTAL EFFECT IS INDEPENDENT OF LOCALITY.147 
 
 ing force depends only on the amount of magnetism 
 produced by induction: and the amount of induction 
 depends only on the terrestrial horizontal force which 
 produces it : and therefore the deviating force is pro- 
 portional to the terrestrial horizontal force : and there- 
 fore the fraction exhibited above is independent of the 
 terrestrial horizontal force : and, in a given position of 
 a mass of iron with respect to the compass, the deviation 
 produced is the same in all parts of the earth. 
 
 68. General investigation of the disturbance produced 
 by a mass of iron symmetrical with respect to a vertical 
 plane passing through the compass-axis (as an iron-built 
 ship) subject to terrestrial induction. 
 
 It is supposed here that, by the action of terres- 
 trial magnetism, every particle of iron is converted 
 into a small magnet whose direction is parallel to the 
 local direction of the dipping-needle, and whose in- 
 tensity is proportional to the local total intensity of 
 terrestrial magnetism ; the poles of the small magnet 
 being in the same positions as those of the dipping- 
 needle, or opposite to those of a magnet representing 
 local terrestrial action. For convenience of language, 
 we shall use terms applicable to a ship : but the results 
 apply equally to any other masses of iron possessing 
 the symmetry above-mentioned. 
 
 Let the center of the compass be the origin of 
 co-ordinates ; let A be the azimuth of the ship's head, 
 measured from the magnetic north towards the east ; a 
 the azimuth of any particle measured from the ship's 
 
 102 
 
148 ON MAGNETISM. 
 
 head : so that A + a is the azimuth of that particle from 
 the north. Let b be the angular depression of the 
 particle. Then if r be the distance of the particle from 
 the compass ; x, y, z, the ordinates towards the north, 
 towards the east, and vertically downwards ; we have 
 
 x = r . cos b . cos (A + a) , 
 
 y = r . cos b . sin (A + a), 
 
 z = T . sin b. 
 
 Let / represent the local intensity of terrestrial 
 magnetism ; B the local dip, estimated positive for the 
 northern hemisphere ; m a constant for any particle 
 under consideration, representing its susceptibility of 
 inductive magnetization ; 21 the length of the small 
 magnet into which it is changed. Then the ordinates 
 of the blue end of the small magnet are 
 
 x I cos B, y, z l sin B. 
 
 Its distance, or the square root of the sum of the squares 
 of these quantities, omitting P, I s , &c, is 
 
 r (x cos B 4- z sin B). 
 
 The resolved part of its attraction on the red end of 
 the compass -needle in the direction of a? is 
 
 ( I }~ 5 
 
 I'm (x I cos B) \r - (x cos B + z sin B ) Y 
 
 x ( 1 , cos B ~jX cos B + z sin B 
 
 I ^3 j 1 ~ *~^ \-ol~ p 
 
 Similarly, the attraction in the direction of y is 
 
MAGNETIC EFFECT OF SYMMETRICAL IRON. 149 
 And the attraction in the direction of z is 
 
 z( 7 sinS , x cos S 4- z sin 
 
 j m 1_Z_ _ + 
 
 ft* I 
 
 (It is supposed here that the compass is so small that 
 no sensible error will be produced in the small terms of 
 these expressions, by adopting for the red end of the 
 needle the values of x, y, z, which correctly apply to its 
 center.) 
 
 The repulsions of the red end of the small magnet 
 on the red end of the compass are the same, with no 
 change but in the sign of I. 
 
 The true forces upon' the red end of the needle, or 
 the excesses of the attractions over the repulsions, 
 putting H for terrestrial horizontal force or /cos 8, and 
 V for terrestrial vertical force or / sin S, are 
 
 6lm. xz 
 mx 
 
 my 
 
 These are the forces produced by a single particle 
 upon the red pole of the compass-needle. To find the 
 forces which all the iron of the entire ship produces 
 upon that pole, we must take the sum of each of the 
 factors of H or V through the whole ship. And for this 
 
150 ON MAGNETISM. 
 
 purpose we must so express these factors as to shew how 
 much depends on the position of the ship's keel and how 
 much on the position of the particle in the ship. 
 
 Now # 2 = r 2 . cos 2 1 . cos 2 (A + a) 
 
 = - r 2 cos 2 b . [1 + cos 2 A . cos 2a sin 2 A . sin 2a}. 
 
 But, as a is the azimuth of the particle measured from 
 the ship's head or from the line of the keel, there will 
 be as many particles with a positive as with equal a 
 negative. The term sin 2a will therefore vanish : and 
 
 the sum of all the terms 5 will be (putting 2 to 
 express the summation) 
 
 ^ Sim . cos 2 b ^ A ^ 3m cos2 & cos 2o> 
 
 2 + cos 2 A . 2 - o -- . 
 
 r r 
 
 Then xz = r 2 . sin b . cos b . cos (A + a) 
 = r 2 sin b . cos b . (cos A . cos a sin ^4 . sin a) : 
 
 . . , . ,. r ,, Glmxz 
 
 which in the same manner gives for the sum of 
 
 6lm . sin b . cos & . cos a 
 
 r 2 
 
 And xy = ~ cos 2 Z> . sin (2^4 + 2a) 
 25 
 
 r 
 
 = ^ cos 8 6 . (sin 2 J. . cos 2a -f cos 2^4 . sin 2a) : 
 
 . , Slmxy 
 
 which gives for the sum of ~ , 
 
 o A \> . cos 8 b . cos 2a 
 sm 2A . 2 -- o -- 
 r 
 
MAGNETIC EFFECT OF SYMMETRICAL IRON. 151 
 
 Also yz = r 2 . sin b . cos b . sin ( A + a) 
 = r 2 . sin b . cos 6 . (sin -4 > cos a -f- cos ^4 . sin a) : 
 
 7 
 
 which gives for the sum of ~ , 
 
 . , ^ Qlm . sin . cos b . cos a 
 sin -4 . 2, - 3 - . 
 / 
 
 Finally, s 2 = r 2 . sin 2 b, 
 
 , ,, , . 
 
 and the term g produces the sum 
 
 r 3 
 Now assume the following notation ; 
 
 2lm v 31m . cos 2 b ^ 21m /., 3 
 
 ^ 2lm (^ 3 B 
 , or 2 -p- f 1 - g cos 8 
 
 6?wi . sin 5 . cos b . cos a 
 - 3 - 
 
 3lm . cos 2 b . cos 2a ,. 
 
 . sin 2 b ^ , , . ., 
 -^-- p or S - (l-3sm 2 = . 
 
 These four quantities Jf, JV, P, ft do not depend on the 
 terrestrial force or on the position of the ship, but are 
 truly constants of the ship, depending only on its con- 
 struction and its susceptibility of magnetism. Then the 
 disturbing forces are, 
 
152 ON MAGNETISM. 
 
 In x or towards the magnetic north, 
 
 - HM+ HP . cos 2 A + VN. cos A. 
 In y or towards the magnetic east, 
 
 + HP . sin 2A + VN. sin A. 
 In z or vertically downwards, 
 -VQ+HN.cosA. 
 
 These are the forces which act on the red end of the 
 compass-needle. Those which act on the blue end are 
 of the same magnitude but opposite signs, and therefore 
 merely double the power which produces deviation of 
 the needle. 
 
 69. Examination of the physical meaning of the 
 different terms of this disturbing force. 
 
 First. If we compound together the terms VN. cos A 
 towards the north, and VN. sin A towards the east, we 
 find that they produce a term VN directed in the 
 azimuth A, that is, directed to the head of the ship. 
 This term therefore resembles in all respects a permanent 
 magnetism of the ship, so long as the ship remains in 
 one place. But it vanishes when V vanishes (that is, at 
 the magnetic equator) : and it changes sign when V 
 changes sign (that is, in the south magnetic hemisphere): 
 and this circumstance will give facility for determining 
 the influence of this term, and correcting it by a magnet 
 at each place of the ship. It will be seen from, the ex- 
 pression for N that if the whole mass of iron is either 
 
MAGNETIC EFFECT OF SYMMETRICAL IRON. 153 
 
 at the same level as the compass (making sin b = 0), or 
 below the compass (making cos 5 = 0), the expression 
 for VN vanishes. 
 
 Second. The terms - HM + HP . cos 2 A shew that 
 there is a term of fluctuating value in the meridional 
 direction ; if however P vanishes, that is if all the iron 
 be below the compass, the fluctuation with change of 
 azimuth vanishes. In any case, the force towards the 
 north is affected on the whole by HM : and when on 
 
 /2 
 
 the whole cos b is < , / - , M is positive, and the ter- 
 restrial horizontal force is on the whole diminished. 
 
 Third. The term HP. sin 2 A indicates a force 
 changing its sign in every quadrant, which produces the 
 quadrantal deviation described in Articles 61, 65, 66. 
 It has no existence if the iron is entirely below the 
 compass. It changes sign when cos 2<z changes its sign : 
 that is, when the iron is mainly at the sides of the 
 compass. 
 
 The tangent of deviation actually produced will be 
 
 deviating force , TT1 T7 - ,. 
 
 Y^ . When V occurs as factor in the de- 
 
 _Q 
 
 y 
 
 viating force, the quotient ~ is the same as tangent dip. 
 
 When H occurs as factor, the deviation is independent 
 of terrestrial force or dip. 
 
 The terms affecting the vertical force are of little 
 interest in general. They would be comparable with 
 observations only when the vertical force is accurately 
 examined (as for instance by dip-observations). 
 
154 ON MAGNETISM. 
 
 70. Defect of this theory : sketch of Poissons more 
 complete theory. 
 
 We abstain, in this article, from expressing any 
 doubt of the correctness of the assumption from which 
 the preceding investigation starts, namely, that by the 
 action of external magnetic force, every particle of the 
 iron, or a limited proportion of the iron in every small 
 space, is converted into a small magnet, whose axis is 
 parallel to the axis of the external magnetism. The 
 effect of such external magnetic force has been duly 
 taken into account. But there is another disturbing 
 magnetism not taken into account, namely, that every 
 small magnet thus produced in the mass of iron pro- 
 duces a disturbing effect on every other small magnet. 
 
 We have reason to think, from the magnitude of 
 that phenomenon of induction whose effects are most 
 accurately known, namely the quadrantal disturbance, 
 that the total effect of the internal action of the minute 
 magnets is small in comparison with the effect of the 
 terrestrial force : in some instances perhaps one-fifteenth 
 part (with wide uncertainty). If, however, this be 
 considered as a fair representation of its magnitude, 
 then none of the preceding conclusions can be very 
 wrong, and the theory of the last article may be accepted 
 as sufficient for all practical cases. And it has this 
 merit, that it shews clearly the dependence of the mag- 
 nitude and sign of each force upon the distribution of 
 the masses of iron. 
 
 Poisson undertook, in the Memoires de VInstitut de 
 
POISSON's THEORY OF INDUCED MAGNETISM. 155 
 
 France, Tome V., the investigation of this theory in its 
 most general form. He supposes that portions of the 
 iron, occupying a space equal to k x entire volume of 
 the iron (k being a fraction) are magnetic. In each of 
 these spaces the two magnetic fluids can be separated 
 to a constant distance in the direction in which the 
 aggregate of external magnetism (both that of a distant 
 body and that of the surrounding iron) acts on them, 
 the amount of fluid so carried over to opposite sides 
 being proportional to the intensity of the magnetism : 
 this supposition amounts to exactly the same as that of 
 the small magnets in the last articles. He then inves- 
 tigates symbolically (in a triple integral) the aggregate 
 effect of all these little magnets upon any one : and, 
 adding that aggregate to the external force, the total 
 force upon that magnet and the direction of that total 
 force are symbolically expressed. These must be the 
 same as the force and direction assumed for it in the 
 investigation ; and the equations expressing this identity 
 are the equations on the solution of which the solution 
 of the problem depends. 
 
 It will be seen at once that this is a problem of very 
 great complexity. In the general case, no advance 
 whatever can be made to a solution of it. Only in the 
 case of a sphere (solid or hollow), and a spheroid, can a 
 result be obtained. One inference is, that the external 
 attraction of the sphere is not much diminished by the 
 hollow, until the hollow becomes so large as to leave a 
 comparatively thin shell. For other purposes, it is 
 conceived that a very oblate spheroid may in some 
 
156 ON MAGNETISM. 
 
 degree represent a sheet of metal, and that a very pro- 
 late spheriod may represent a rod. And these are the 
 nearest approaches to practical application made by 
 Poisson's theory. 
 
 The final statement by Poisson of general theoretical 
 results is merely the following: If a, /9, 7, are three 
 rectangular components of terrestrial magnetic force, 
 and if a system of iron masses receive magnetic induc- 
 tion from their action, the forces which they will exert 
 on the pole of a compass-needle, in the direction of the 
 three axes, will be 
 
 POL + Q0 + Ey, 
 POL + Q'{3 
 
 P, Q, &c., being constants peculiar to the masses and 
 positions of the iron. The reader who has seen in the 
 preceding investigations how the resolved parts of ter- 
 restrial induction produce in the first instance three 
 systems of molecular magnets, and who remarks that 
 the mutual action of those in each system will still pro- 
 duce a series of magnets peculiar to that system, and 
 who further remarks how the magnitude and direction 
 of the resulting forces depend on the distance and posi- 
 tion of each molecular magnet, and how the forces can 
 be resolved into rectangular directions, will perceive 
 that such a form of result is necessarily obtained without 
 any abstruse investigation whatever. 
 
 71. Inadmissibility of Poisson s fundamental sup- 
 
DEFECT OF POISSON'S THEORY. 157 
 
 positions, and indication of the wants of a new 
 theory. 
 
 It will be remarked in Poisson's theory (and also in 
 the theory of Article 63) that it is assumed that the 
 magnetism of both kinds originally attached to one 
 molecule is never moved beyond the region of that 
 molecule ; and it is also implied in Poisson's assump- 
 tions, that the disturbance of this molecular magnetism 
 is produced by the actions of other molecules or masses, 
 without regard to the question whether those molecules 
 or masses are in the same continuous substance. It 
 follows from this that, according to the fundamental 
 suppositions of both theories, if a given mass of iron be 
 divided into any number of parts, the state of its mag- 
 netism and its action upon a compass-needle will not 
 be altered. At the dividing planes there may be 
 (theoretically) on one surface a peculia state of mag- 
 netism, but this will be accompanied by equal magnet- 
 ism of the opposite kind on the other surface, and the 
 proposition will still hold. The truth of the theories is 
 easily tested by such experiments as the following : 
 
 Provide a bar of iron, 6 inches long, and also four 
 bars whose section is the same, but each 1J inch long. 
 Ascertain (by applying each end of each bar centrally 
 to the E or W side of the compass) that they possess no 
 permanent magnetism. Now apply the long bar 
 endways at azimuth 45, at any distance at which 
 it will produce a quadrantal deviation of several 
 degrees. Remove the long bar, and put the four 
 
158 ON MAGNETISM. 
 
 short bars in its place, just touching or merely separated 
 by bits of thin paper : and the quadrantal deviation 
 will be reduced to four-fifths of its previous value, or 
 less. Or, apply the long bar sideways to the compass 
 with its advanced end abreast of the center, and note 
 the deviation; substitute the four short bars for it, 
 and the deviation will be reduced to about three-fifths. 
 The author has repeatedly tried these experiments, 
 with bars of different lengths, and has always arrived 
 at the same result. 
 
 The following observation is precisely similar, but 
 on a much larger scale. It has been found by Capt. 
 Evans, R.N., that when a ship is built of plates of iron 
 very closely riveted together in every part, the quad- 
 rantal deviation of the compass is considerable. But, 
 when a wood-built ship is covered with heavy iron 
 armour, in plates which (though thick, and screwed to 
 the wood, and perhaps lightly touching each other) are 
 not riveted together, the quadrantal deviation is small. 
 In both these classes of experiment it is evident 
 that Poisson's fundamental suppositions are at fault. 
 It would seem that magnetism can flow through the 
 unbroken bar or the closely-riveted iron nearly as 
 through the steel of a magnet, but is not permanently 
 retained as in a steel magnet. And it appears that ; 
 instead of a thin skin of magnetism on each side of the 
 mass of iron, as in Article 65, Figure 53, where the 
 thickness of the skin is determinate without reference 
 to the depth of the iron ; there will be a dense collection 
 of magnetism of one or the other kind, brought from 
 
DEFECT IN THE THEORIES OF MAGNETISM. 159 
 
 the whole depth of the iron, and in some degree pro- 
 portional to that depth. The results derived in Article 
 68 from the suppositions in Article 63, and the imme- 
 diate results of Poisson's theory, are therefore both 
 erroneous, in regard to the magnitude of the coeffi- 
 cients of the terms representing compass-disturbance. 
 And we are left in complete doubt whether there is 
 or is not a relation or system of relations between 
 the coefficients in the general equations of Article 70. 
 If, however, the inductional effects of external magnetic 
 forces acting in the direction of different co-ordinates 
 admit of being combined by the same law as the super- 
 positions of small displacements, it would appear that 
 formulae similar to Poisson's will hold (with the doubt 
 on relation of coefficients above mentioned), and that 
 the law of quadrantal disturbance in Article 65 and of 
 other disturbances in Article 68, will be unchanged. 
 
 These remarks lead us to a consideration of the 
 really important defect in the present theory of magnet- 
 ism of steel and iron. We possess no information, and 
 no plausible theory, on the permanent distribution of 
 magnetism in a steel magnet, or on the temporary dis- 
 tribution of magnetism in an iron bar affected by 
 external magnetic action. It seems not unlikely that 
 it may be subject to laws something like those of 
 induced electricity. Many of the most important 
 deductions can, however, be securely established with- 
 out that knowledge ; but, till it is obtained, we cannot 
 regard magnetism as possessing the highest claims to 
 regard as a Physical Science. 
 
160 ON MAGNETISM. 
 
 In the mean time, we prefer the theory of Article 
 63 to Poisson's theory ; inasmuch as it gives distinct 
 indications of the connexion between the arrangement 
 of the masses of iron and the laws (irrespective of mag- 
 nitude) of the compass- disturbance ; depending so 
 clearly on general principles of the translation of mag- 
 netism by the influence of external magnetic forces 
 that they will never be materially modified. On these 
 points, Poisson's generalities give no assistance. 
 
 72. Complexity introduced, by induction, into the 
 laws of the action of magnets upon each other. 
 
 We may regard a magnet as consisting throughout 
 of two materials : one the ferruginous part, of which the 
 magnetism is liable to be shifted by the action of an 
 external magnet : the other the magnetizable steel, 
 properly so called, in which the magnetism is fixed. 
 This produces (among other complications) a difference 
 between the attractive and the repulsive powers. For, 
 if the red pole of the first magnet is presented to the 
 blue pole of the second, and strong magnetic attraction 
 takes place, the red pole induces blue magnetism in the 
 adjacent ferruginous part of the second magnet, and the 
 blue pole induces red magnetism in the adjacent ferru- 
 ginous part of the first magnet, and the total attraction 
 is increased. But if the red pole of the first is presented 
 to the red pole of the second, and there is consequent 
 repulsion ; each red pole induces blue magnetism in the 
 adjacent ferruginous part of the other magnet, and 
 
INDUCTION ON A STEEL MAGNET. 
 
 161 
 
 there is attraction, opposed to the magnetic repulsion : 
 and the total repulsion is diminished. The same holds 
 if the blue pole of the first is presented to the blue pole 
 of the second. 
 
 The most accurate information which we possess on 
 the subject is given by Mr W. Ellis, Assistant of the 
 Royal Observatory, Greenwich, in a paper published in 
 The Philosophical Magazine, May 1863, page 325. A 
 magnet 5 J inches long was attached to a clock-pendulum, 
 and a similar magnet was so fixed in the clock-case that 
 one pole of the swinging magnet passed over one pole 
 of the fixed magnet : when there was attraction, the 
 clock was accelerated ; when there was repulsion, the 
 clock was retarded ; and both effects could be measured 
 with extreme accuracy. The following results (ex- 
 tracted from a series) will shew the difference of the 
 effects : 
 
 Distance 
 between 
 poles of 
 magnets 
 in inches. 
 
 Acceleration 
 of clock 
 in seconds per day 
 when dissimilar 
 poles 
 were near. 
 
 Retardation 
 of clock 
 in seconds per day 
 when similar 
 poles 
 were near. 
 
 0-05 
 
 + 5-90 
 
 -2-24 
 
 010 
 
 + 410 
 
 -212 
 
 0-20 
 
 4-3-03 
 
 -1-88 
 
 0-40 
 
 + 2-05 
 
 -1-42 
 
 0-80 
 
 + 1-07 
 
 -0-85 
 
 1-60 
 
 + 0-34 
 
 -0-29 
 
 and these may be thus divided ; 
 
 11 
 
162 
 
 ON MAGNETISM. 
 
 Distance 
 ween poles. 
 
 Part due 
 
 to permanent magnetism. 
 
 Part due 
 
 to induced magne 
 
 0-05 
 
 4-07 
 
 + 1-83 
 
 o- o 
 
 311 
 
 + 0-99 
 
 0-20 
 
 2-455 
 
 + 0-575 
 
 0-40 
 
 1735 
 
 + 0-315 
 
 0-80 
 
 0-96 
 
 + 011 
 
 1-60 
 
 0-315 
 
 + 0-025 
 
 It will be remarked" that the part due to induced 
 magnetism diminishes much more rapidly than that 
 due to permanent magnetism. 
 
 73. Method of measuring the amount of magnetism 
 produced in a magnet by terrestrial induction. 
 
 It is assumed in the following process that the 
 law of distribution of induced magnetism, as regards 
 the length of the magnet, is sensibly the same as 
 that of permanent magnetism : and in fact the law 
 at which we have arrived theoretically of induced 
 magnetism, that the magnetism is exhibited as a coat- 
 ing at the end, is sensibly the same as that found 
 experimentally for permanent magnetism, Article 12. 
 The method adopted at the Kew Observatory for mea- 
 sure of the induced magnetism makes use of that part 
 induced by the terrestrial vertical force, which in this 
 magnetic latitude is large. As in Article 26, a deflexion- 
 apparatus is to be used, in which it is made certain, in all 
 stages of the operation, that the deflected needle (carry- 
 ing a reversed telescope) occupies the same position with 
 
EFFECT OF INDUCTION MEASUKED. 163 
 
 regard to the viewing telescope ; the whole apparatus 
 being turned horizontally round a vertical axis till that 
 condition is obtained, and the graduated horizontal 
 circle which registers its rotation being then read. 
 But there is this difference in adjustment from that of 
 Article 26, that the magnet is placed in a vertical posi- 
 tion, with a definite point near one pole exactly in the 
 horizontal plane of the disturbed needle. 
 
 Suppose now that the red pole of the magnet is 
 downwards, a mark near the red pole being at the same 
 level as the needle, and the blue pole projecting far 
 above the level of the needle. The effect of induc- 
 tion by the earth's vertical force is to add to the red 
 power of the lower end and to the blue power of the 
 upper end, and in fact to make the magnet more power- 
 ful. Now invert the magnet, so that the mark near the 
 red pole is still at the same level as the needle, but the 
 blue pole projects far below the level of the needle. As 
 regards the action of the magnet upon the needle, the 
 force exercised is the same as before. But to the red 
 magnetism at the upper end of the magnet there is now 
 added blue magnetism produced by the earth's vertical 
 induction, and to the blue magnetism at the lower end 
 there is added red magnetism produced by induction, 
 and the power of the magnet is diminished. And 
 these vertical magnetisms are not affected by the hori- 
 zontal rotation of the apparatus round the needle. It is 
 evident here that we have the means of determining the 
 propartion of the induced part to the permanent part of 
 
 magnetism. 
 
 112 
 
164 ON MAGNETISM. 
 
 In order to eliminate any conceivable excentricity 
 in the location of the permanent magnetism, the opera- 
 tion may be repeated, using a mark near the blue pole 
 of the magnet : and the mean of the two results may be 
 taken. For simplicity of reduction, suppose that all 
 observations are made with the separation between the 
 centers of the magnet and the needle equal to the unit 
 of measure, or (in England) 1 foot. As in Article 29, 
 let A be the magnet-power of the magnet, and E the 
 local horizontal magnet-power of the earth. Also let / 
 be the magnet-power induced by the action of the 
 terrestrial vertical force. In one position of the mag- 
 net, its power is A\-b I, and in the other position it is 
 A L On account of the peculiarity of the magnet's 
 position (which is different from those in Articles 26, &c.) 
 these are not the powers which act in the present ex- 
 periment : the real acting powers (see Article 53) will 
 be found by multiplying those by an unknown constant 
 e: so that the real acting powers in the experiment are 
 e (A + P) and e (A I). Let O t and 2 be the deviations 
 in the two experiments. Then as in the last sentence 
 of Article 26, making c = 1 and neglecting K (which 
 which will have the same proportion in the two experi- 
 ments), 
 
 A 4- / sin l 
 whence ~~ A r^ 1 ~~- T\ 
 
CORRECTIONS FOR THE EFFECT OF INDUCTION. 165 
 
 d / _sin0 1 -sm0 1== tan&(0 t --0J 
 
 A ~" sin t + sm0, ~~ tan & (6 1 + 2 ) ' 
 
 The value of I thus obtained is the amount of induced 
 magnetic force produced by the local terrestrial vertical 
 force. The local terrestrial horizontal force is found by 
 multiplying the local vertical force by cotan. local dip : 
 hence the induced magnet power produced by local 
 terrestrial horizontal force E will be 
 
 A x cotan. local dip x 
 
 It is most carefully to be observed that the terrestrial 
 force E and the dip here spoken of are those peculiar to 
 the place of this experiment ; that d l and 2 are angles 
 peculiar to this experiment; and that A is constant so 
 long as the power of the magnet is not changed. 
 
 74. Correction of the formula}, used in the deter- 
 mination of the Earths horizontal magnet-power, for 
 effects of induction. 
 
 In examining the process in Articles 26 to 28, it 
 will be seen that we have to consider the effect of a 
 magnet A in disturbing a needle when A is inclined to 
 the meridian : and in Article 32, we have to treat of the 
 earth's horizontal force upon A when A is in the meri- 
 dian. These must be considered separately. Let E' be 
 the earth's horizontal power at a second station. 
 
 First, for the disturbance of a needle by A. If $ be 
 
166 ON MAGNETISM. 
 
 the angle of deviation of the needle in a deflexion-ex- 
 periment at the second station, the direction of the 
 magnet A will be such, that its blue end will deviate 
 by 90 (j> from the north meridian. The horizontal 
 force which produces induction in A is E x sin <p. 
 Now we found that when the inducing force was E, 
 the magnet-power produced by the induction was 
 
 tan 1(0 -<9 2 ) 
 A x cotan. original local dip x - i ta . Q~\ 
 
 Hence at the second station the induced magnet-power 
 in the experiment of deviating the needle will be 
 
 jE"sin( A 
 
 ~X4 x cotan. original local dip x 
 
 or the efficient magnet-power will be 
 
 But (see Article 27) E' sin (f> is sensibly equal to 2 A. 
 Also, if be the deviation observed in the same manner 
 
 at the original station, E sin 6 2A . Hence - ' ^ 
 = sin 0. And the efficient magnet power is 
 
 A x \ 1 sin 6 x cotan. original local dip x ^\\Q~ Q\ \ '> 
 
 where, after once making the necessary experiments, the 
 quantity within the bracket is constant for all stations. 
 This formula, it will be remarked, applies to the 
 
CORRECTIONS FOR THE EFFECT OF INDUCTION. 167 
 
 efficient magnet-power of A at the second station 
 when its distance from the needle is 1. At any other 
 distance c, let the angle of deviation be <f>'. Then 
 the induced magnet-power will be 
 
 E'.&ind) tan J (0-0,) 
 
 -p xAx cotan. original local dip x - ^~ --' . 
 
 But (see Article 27), putting 2 for ra, E' for E, and <fi 
 for 0, 
 
 , ., 2A , E' sin $ 1 2A 1 
 E . sin 6 =5- , and -- p--*- = - 3 . -= = -3 sin : and 
 c j& c Jii c 
 
 the total effective magnet-power of A will be 
 
 l 3 sin x cotan. original local dip x 7^ 7ff ^ 2 \ ' 
 
 Second, for the force producing the vibrations of A. 
 The red pole of A will be turned to the north, and 
 therefore the inducing force will be -f E r ; the induced 
 
 Tf" 
 
 magnet power will be -f- -^ . A x cotan original local dip 
 
 x i rr^ *\ ' an ^ the real power of the magnet 
 tan (, + 6/J 
 
 will be 
 
 A ! 1 + -4 x cotan. orisrinal local dip x - f-7/r ^ 
 1 E tanH^ + ^ 2 
 
 Here the factor of A does vary from station to station. 
 
 The reader will have little difficulty in investigating 
 the corrections which these considerations shew to be 
 necessary in the formulae terminating in Article 32. 
 
168 ON MAGNETISM. 
 
 SECTION IX. 
 
 ON SUBPERMANENT MAGNETISM IN IRON SUBJECTED 
 TO MECHANICAL VIOLENCE. 
 
 75. Primary experiment on Subpermanent Magnet- 
 ism: a long plate or slender bar of iron is placed on a 
 firm frame (sometimes called the 'Magnetic Anvil), with 
 its length parallel to the local dip, and is struck re- 
 peatedly with a hammer: it becomes a magnet, with red 
 magnetism in the end which dipped (in northern mag- 
 netic latitudes} ; and this magnetism does not change with 
 change of the magnet's position. 
 
 It will be convenient to prepare a small frame, as 
 represented in Figure 55, of which the essential parts 
 
 Fig. 55. 
 
 are, one surface transversal to the direction of local dip, 
 
PRODUCTION OF SUBPERMANENT MAGNETISM. 169 
 
 and one surface containing the direction of local dip ; 
 and to fix it to the floor. Lay a piece of iron plate or 
 iron bar, in contact with the dip plane, and with its 
 length approximately in the direction of dip : and 
 strike it repeatedly with an iron hammer. On re- 
 moving it, it will be found to be a true magnet, the 
 end which was lowest being charged with red mag- 
 netism: and this magnetism is not transient like the 
 induced magnetism of soft iron, changing its place in 
 the bar with every change in the position of the bar 
 (see Article 60) ; but is constant like that of a steel 
 magnet, retaining the same magnetism whatever be 
 the position of the bar. An iron bar which has not 
 been struck, if applied in the horizontal position end- 
 on towards the center of a compass, does not disturb 
 the needle at all; but the same iron bar when it has 
 been struck in the manner described, if applied end-on 
 to the center of a compass, disturbs it powerfully : one 
 end deflecting the needle in one direction, and the 
 other end deflecting the needle in the opposite direc- 
 tion, exactly as a steel magnet would do it: and in 
 all respects comporting itself as a steel magnet. 
 
 76. Variations of the experiment. All lead to the 
 supposition that iron, in a state of tremor or jar, is 
 peculiarly able to receive induced magnetism and to 
 retain it firmly. 
 
 The circumstances of the last experiment, in which 
 the receipt of magnetism depends evidently on the 
 
170 ON MAGNETISM. 
 
 placing the iron in that position in which the earth's 
 power acts most strongly on it, suggest the trial of a 
 steel magnet for the same purpose. Lay a steel mag- 
 net E and W on a table, and bring a nail near it in 
 the same direction, not touching the magnet. In this 
 state, the earth's magnetism has no effect: and the 
 nail receives from the magnet no permanent mag- 
 netism (as is easily verified by applying it end-on to 
 a compass). Now place the nail in its former position, 
 and strike it with a hammer; upon withdrawing it, it 
 will be found that it has become a magnet in all re- 
 spects like a steel magnet, the pole which was nearest 
 to the large magnet having magnetism opposite to that 
 of the pole which it approached. 
 
 If a bar of iron be dropped endwise upon a stone 
 pavement, it immediately acquires polar magnetism 
 of permanent character, like that of a steel magnet. 
 But if it be dropped on a carpeted floor, it scarcely 
 receives any sensible magnetism. This experiment 
 shews that a state of sharp tremor or violent jar among 
 the particles of the iron is necessary to enable it to 
 receive this magnetism. 
 
 As matter of familiar observation it may be men- 
 tioned that a common fire-place poker, of which the 
 same end is usually downwards and is frequently 
 struck upon the hard floor, is almost always well 
 charged with magnetism, its red end being the 
 lower. 
 
DESTRUCTION OF SUBPERMANENT MAGNETISM. 171 
 
 77. Reversion or destruction of the magnetism. 
 Origin of the term l Subpermanent.' 
 
 Suppose that the lower end of the bar in the ex- 
 periment of Article 75 is distinguished by being 
 painted white. This white end, after the bar has 
 been placed with its white end in the direction of 
 dip and has been struck, is found to be charged with 
 red magnetism. Now reverse the bar upon the dip 
 plane with its white end upwards, and strike it; it 
 will be found that its white end is charged with 
 blue magnetism. The magnetism has been reversed. 
 Undoubtedly, in order to arrive at this state, it has 
 gone through the stage of being destroyed or rendered 
 undiscoverable by instruments. 
 
 But the pure destruction may be visibly effected 
 in the following manner. Lay the bar upon that 
 surface of the magnetic anvil which is normal to the 
 direction of dip, and strike it with a few blows of the 
 hammer. On removing the bar, and testing its state 
 by means of a compass, it will be found that all trace 
 of magnetism has disappeared. The bar is now in the 
 same state as before the experiment of Article 75. 
 
 If, however, the magnetized bar be subject to no 
 such violence, but be suffered to rest quietly, or be 
 moved gently into different positions, it will slowly 
 lose a large proportion of its magnetism. And it is 
 this peculiar character which necessitated the intro- 
 duction of a new name. The magnetism of a struck 
 iron bar resembles the magnetism of a permanent 
 
172 ON MAGNETISM. 
 
 steel magnet in all respects but this, that, while 
 perhaps no change can be remarked in hours or days, 
 it infallibly diminishes in a long time. To express 
 this partially permanent character, the term 'Sub- 
 permanent Magnetism ' has been adopted. 
 
 In single bars, the subpermanent magnetism di- 
 minishes sensibly in a few hours, and is lost in a few 
 days. In some large iron ships, a portion of it has 
 remained unaltered for many years. It would seem 
 that where the operation of magnetizing by hammer- 
 blows has been rapid, the magnetism is not very 
 firmly fixed: but where the violence has been long 
 continued, the magnetism is so firmly established as 
 to become an immovable quality of the iron. 
 
MAGNETISM OF IRON SHIPS. 173 
 
 SECTION X. 
 
 ON THE MAGNETISM OF IRON SHIPS, AS AFFECTING 
 THEIR COMPASSES. 
 
 78. Philosophical and Commercial Importance of 
 this subject. Complication of the Magnetic considera- 
 tions involved in it. 
 
 It is unnecessary to remark on the extent to which 
 at the present time, when so large a portion even of 
 the mercantile navies is built of iron, the interests 
 of commerce are involved in the investigations which 
 alone can make the ship-compasses available. But it 
 may be desirable to point out to what an extent 
 Science may benefit from it. It will be shewn that 
 the principal agent in the disturbance of the compass 
 is subpermanerit magnetism, an element little known 
 before the introduction of iron ships, and whose laws 
 have principally been derived from the examination 
 of iron ships. But another element, whose effects are 
 sensible in all, and very important in some, is transient 
 
174 OX MAGNETISM. 
 
 induced magnetism : and the study which in late years 
 has been given to this subject has been stimulated 
 almost entirely by its application to iron ships. 
 
 In applying the science deductively to the control 
 of ships' compasses, every part of the theories treated 
 in the earlier sections is brought into play. Magnetic 
 declination is obviously necessary: terrestrial horizon- 
 tal force enters into every formula of disturbance (see 
 Article 55): both horizontal force and vertical force, 
 or dip, occur in the formula? for induction disturbance : 
 the laws of action of magnet-power enter both into the 
 effects of subpermanent magnetism of the ship and into 
 those of the permanent magnets employed in correction: 
 and the theory of induced magnetism, and especially of 
 quadrantal deviation, presents itself in the correction 
 of the effects of the iron masses. 
 
 79. Brief history of the first steps in this science. 
 
 The fi$st real step appears to have been made by 
 Captain Flinders, about 1803, who remarked that the 
 disturbances of his compass were such as would be 
 produced by the attraction of iron charged with mag- 
 netism; blue for northern latitudes and red for southern 
 latitudes, in the direction of the ship's head ; and sug- 
 gested the use of a vertical bar to be placed aft of the 
 compass, whose upper end having similar magnetism 
 would tend to correct the other. At a later time, 1820 
 to 1833, numerous experiments were made by Professors 
 Barlow and Christie, illustrating the action of induced 
 
MAGNETISM OF IKON SHIPS. 175 
 
 magnetism. In regard to subpermanent magnetism, 
 the first experiments on iron bars, &c., were made by 
 Mr Scoresby, about 1821 ; and the first virtual observa- 
 tion in ships was made by General Sabine in discussing 
 the compass-deviations in Sir James Boss's voyage, 
 1839 to 1843, in which he remarked that the pecu- 
 liarities in the disturbances of the compass lasted for 
 a short time after the ship had left the region in which 
 the terrestrial forces were such as would tend to ex- 
 plain the disturbances. These observations were made 
 in wood-built ships having many accidental masses of 
 iron. 
 
 The first explanation of the character of the com- 
 pass-disturbance produced by iron ships was given by 
 the writer of this Treatise in the Phil. Trans., 1839, 
 as resulting principally from examination of the iron 
 steamer, Rainbow, in 1838. The disturbing forces on 
 the steering-compass of that ship were so great that 
 in one position of the ship the north end of the needle 
 was deflected more than 50 to the east, and in another 
 position it was more than 50 to the west. The first 
 light that was thrown upon the causes of these de- 
 viations was obtained by placing the ship with her 
 head exactly north (which can be done in various 
 ways, one of the most convenient being to use an 
 azimuth-compass on shore, and to adjust the ship 
 by signal till her masts, as seen by the shore-compass, 
 are all in the magnetic meridian), then observing the 
 deviation of the compass, and, replacing the compass 
 by a vibrating needle whose time of vibration on shore 
 
176 ON MAGNETISM. 
 
 had been found, ascertaining the effective total hori- 
 zontal force acting on the compass needle; this total 
 force, which is really in the deviated direction of the 
 compass, was resolved into a~N. direction and an E. 
 direction, and gave the whole force to the N. and that 
 to the E.; subtracting the terrestrial force from the 
 former, the ship's disturbing forces to the N. and to 
 the E. were found. Similar operations were performed 
 with the ship's head E., S., and W.; and in each, the 
 amount of disturbing force in the direction of ship's 
 head and ship's starboard-side were found. On ex- 
 amining these it was at once seen that the four values 
 for force directed to the ship's head agreed so nearly, 
 and the four values to the ship's side agreed so nearly, 
 as to leave no doubt that nearly the whole disturbing 
 force was a force similar to permanent magnetism 
 making a definite angle with the ship's keel. Then, 
 deductively, using the means of these values, and com- 
 pounding them with terrestrial horizontal force in 
 different positions of the ship's head, on the principles 
 of Article 55, the resulting direction of the needle in 
 33 positions of the ship was found to agree with the 
 observed direction; only a small quadrantal difference 
 remained, which was evidently explained by the quad- 
 rantal term in Article 68, as produced by masses of 
 iron towards the ship's head. 
 
 Similar treatment of observations on three other 
 compasses in different parts of the ship gave nearly 
 the same value for the permanent magnetism trans- 
 versal to the keel, but smaller values for the longi- 
 
CORRECTION OF THE COMPASS IN IRON SHIPS. 177 
 
 tudinal values. It was certain therefore that there was 
 permanent or subpermanent magnetism transversal 
 to the keel (because no induction could account 
 for it) ; but it was difficult to say how much of the 
 longitudinal part was subpermanent, how much due 
 to the first term in Article 69, and how much to the 
 iron stern-post acting as a vertical magnet (Articles 
 60 and 53). 
 
 The laws thus ascertained were verified by placing 
 below the compass a magnet, in the position opposite 
 to the ship's magnetism, and at a distance which (as 
 had been ascertained by experiment) would enable 
 the magnet to produce an effect equal to the ship's 
 effect; and also applying a mass of iron at one side 
 of the compass to correct the quadrantal deviation 
 (Article 62). Then, in swinging the ship round, the 
 compass was found to be correct in every position of 
 the ship. 
 
 In the next experiments, a change of great value 
 was made in the practical operations; founded on 
 the following theoretical considerations. Conceive the 
 ship's magnetism to be resolved into two parts, one 
 transversal to the ship, one longitudinal. When the 
 ship's head is placed north or south, the transversal force 
 alone disturbs the compass, and the quadrantal disturb- 
 ance vanishes (Articles 65 and 66) ; and the transversal 
 magnetic part can be corrected by an opposite transver- 
 sal magnet broadside-on to the compass, whose distance 
 is determined without any calculation, simply by trying 
 its effect at different distances till the needle points 
 
 12 
 
178 ON MAGNETISM. 
 
 correctly. Then, in like manner, if the ship's head 
 is placed east or west, the longitudinal magnetism 
 only disturbs the compass, as the quadrantal deviation 
 vanishes there, and it is to be corrected by a longitudi- 
 nal magnet broadside-on to the compass, tentatively 
 applied. The effects of permanent or subpermanent 
 magnetism are now entirely corrected. In order to 
 correct for the induction-effect which produces quad- 
 rantal deviation, the ship's head must be placed in 
 azimuth 45 (nearly), or 135, or 225, or 315; there 
 will be no difficulty in ascertaining whether the quad- 
 rantal disturbance is such as corresponds to the effect 
 of iron in the direction of the ship's head : and, if so, it 
 must be corrected by iron on one or both sides, shifted 
 by trial till the correction is complete. 
 
 These processes were introduced by the author 
 in 1838, and they are still retained in use without 
 alteration. 
 
 80. Reference to the causes of partial failure in the 
 correction of the compass. 
 
 So long as the ship's magnetism remains unaltered, 
 and so long as she remains in the same region of the 
 earth, her compass will now be perfectly correct. But 
 it is necessary to examine the changes which take 
 place in lapse of time and in change of geographical 
 position. 
 
 The determining circumstances of a ship's subper- 
 manent magnetism, and the effect of time upon it, 
 including the effect of changing the position of the 
 
PARTIAL FAILURE OF COMPASS-CORRECTIONS. 179 
 
 ship's head while in the same locality, and subjecting 
 her to some degree of violence, were first investigated a 
 few years after the establishment of the principles 
 above described, by a 'Liverpool Committee' appointed 
 by merchants of the City of Liverpool. From their 
 researches it clearly appeared that the magnitude and 
 direction of a ship's subpermanent magnetism are con- 
 nected with the direction in which she is built, exactly 
 according to the law explained in Article 75 ; and thus 
 no doubt remained that the subpermanent magnetism 
 was the effect of terrestrial magnetism acting on the 
 iron during the heavy hammering to which an iron ship 
 is subjected in the operations of building. And it 
 was found that if, after launching, the ship is kept for a 
 time in a different position, and more especially if she 
 is subject to the tremor of steam-navigation, a large 
 proportion of this subpermanent magnetism vanishes: 
 a part however remaining invariable. Now the effect 
 of either making no attempt to correct the subper- 
 manent magnetism by magnets, or of correcting it 
 completely and then finding that, in consequence of the 
 decay of the subpermanent magnetism, it is now over- 
 corrected, may be estimated by reference to the con- 
 struction of Article 55, Figure 45. It is there seen 
 that, with a given change of ship's magnetism, the 
 error produced in the needle's direction will depend on 
 the magnitude of the local terrestrial horizontal force : 
 if the ship is in a part of the earth where horizontal 
 force is large, or where B E is large, the error of the 
 compass will be small : but if the horizontal force is 
 
 122 
 
180 ON MAGNETISM. 
 
 small, the error of the compass will be large. To meet 
 the effects of this time-change it is very desirable that 
 the magnets by which the compass is corrected should 
 be so mounted that they can at any time be easily 
 adjusted to the distance at which they make the com- 
 pass correct. Gray's Binnacle is arranged expressly for 
 this purpose. 
 
 The effect of change of geographical position is 
 sometimes very troublesome. If the subpermanent 
 magnetism is not completely neutralized by the mag- 
 nets, the remaining error, as above explained, produces 
 different deviations in different localities where the 
 horizontal force is different, and no table of errors 
 which applies to one place will apply to another place. 
 In the formulas of Article 68, and their explanation 
 in Article 69, it will be seen that induction produces 
 a term VN representing magnetism directed towards 
 the ship's head: and it is impossible to say, in the 
 operations for correcting the compass, whether this 
 term has a real value: and the compass must be 
 corrected as if it did not exist. But when the ship's 
 magnetic latitude is changed the value of V is changed ; 
 and when she goes into the south magnetic hemisphere 
 where the dip is reversed (Figures 21 and 37), the sign 
 of Fis changed; and these changes in the value of VN 
 produce a great change in the value of longitudinal 
 magnetism which was corrected as if it were constant ; 
 the correction therefore is no longer valid, and a con- 
 siderable error is produced in the compass. But upon 
 looking at the expression for N in Article 68, in which 
 
PARTIAL FAILURE OF COMPASS-CORRECTIONS. 181 
 
 the factor cos a is negative for iron which is astern 
 of the compass, it will be seen that ; though N may be 
 large in a merchant-ship, W"here the compass is .placed 
 very near the stern ; yet it will be small in a ship of war 
 where the compass is much nearer the center of the 
 ship. 
 
 There is another serious cause of error. The ver- 
 tical stern-post and rudder-post become in fact, by 
 induction, vertical magnets : but the upper end which 
 acts on the compass is charged with blue magnetism in 
 the northern hemisphere, and red magnetism in the 
 southern hemisphere. This change, however, may be 
 neutralized by adopting a suggestion of Mr. Rundell, 
 that another vertical bar be fixed in the ship on the 
 opposite side of the compass. In ships of the Royal 
 Navy, the compass is too distant from the stern-post to 
 be sensibly affected. 
 
 When due attention is given to these considerations, 
 and the ship is not very new, it has been shewn by the 
 writer, in the Phil. Trans., 1855, from examination of 
 the disturbances in several ships, that, with insignificant 
 alteration in the position of the magnets, a ship's 
 compass will be perfectly correct in all parts of the 
 world. 
 
 81. Continuation of the history. Investigation of 
 the effects of the ship's heeling. 
 
 In the Admiralty Manual of the Compass, and 
 in other papers, Mr. Archibald Smith has elaborately 
 discussed the forces acting on the compass. Those 
 
182 ON MAGNETISM. 
 
 which depend on subpermanent magnetism require no 
 special notice: but those depending on induced magnet- 
 ism are founded exclusively on Poisson's general equa- 
 tions given at the end of Article 70. These are the 
 last attempts made on formation of a theory: though 
 discussions of the magnetic phenomena presented by 
 special ships, leading to conclusions of great interest, 
 have been published by the same writer and by Capt. 
 Evans, in the Philosophical Transactions. 
 
 A very important field for the application of theory 
 is afforded by the consideration of a ship's 'heeling' or 
 inclination to one side. Usually, in an iron ship, when 
 her head is placed N. or S,, the ship's inclination 
 through an angle of n degrees disturbs the compass 
 through an angle of about n degrees ; but in some 
 particular instances, it has been known to disturb 
 the compass as much as 2n degrees. This effect is very 
 serious in those parts of the earth where the wind is 
 steady and the ship is inclined in the same direction 
 for many days or weeks in succession. All the preced- 
 ing investigations have gone on the supposition that 
 the ship was 'on even beam,' or that her deck was 
 horizontal. The investigations for the inclined position 
 are necessarily very complicated. We shall give an out- 
 line of them, as they proceed in sequence from those 
 obtained for the horizontal position of the deck in 
 Article 68. 
 
 Let h be the angle of the ship's heel towards the 
 starboard side : we shall suppose that, in the induced 
 forces,, the square of h may. be neglected. Also let 
 
EFFECT OF THE SHIP'S HEELING. 183 
 
 F be the subpermanent force acting on the red end of 
 the needle towards the fore part of the ship : 8 that 
 force parallel to the ship's deck which acts towards 
 the starboard side : K that towards the ship's keel. 
 When the ship heels through the angle Ji, the forward 
 force is not thereby disturbed, but the horizontal force 
 towards the starboard side becomes S cos h K sin h. 
 But the transversal magnet which has been correcting 
 the compass when the ship was on even beam, and 
 whose power therefore was S in the plane parallel 
 to the deck, now exerts the horizontal power Scosh. 
 Therefore the new force of subpermanent magnetism in 
 the horizontal plane, towards the starboard side, pro- 
 duced by the ship's heeling is 
 
 K sin h, 
 which produces 
 
 + sin h. K . sin A towards the north, 
 and sin h.K.cos A towards the east. 
 For the forces produced by induction, let x, y, 2, be 
 measured horizontally towards the north, horizontally 
 towards the east, and vertically, as in Article 68. 
 Consider cos h as = 1. Then, using the notation of 
 Article 68, 
 
 o; = r.cos 6.cosa.cos-4 r cos 5. sin a. sin A 
 
 + sin h . r . sin b . sin A ; 
 y = r.cos 6. cos a. sin A +r.cos b. sin a. cos A 
 
 sin h.r.sin b.cosA ; 
 z = r.sin 5 + sin /&.?'. cos Z>. sin a. 
 
184 ON MAGNETISM. 
 
 The new term introduced into x* is 
 sin L 2r 2 {sin6. cos&. cosa. sin A. cos^l sin&. cos5. si 
 and therefore the new term introduced into H. 2 
 will be 
 
 , _ T , ^ 6?m . sin b. cos b . cos a 
 
 81Iia.il. 31X1 24. 2 - * - 
 
 The new term introduced into xz is 
 s 2 .sin a. cos a. 
 
 and therefore the new term introduced into F. 2 - i 
 
 r 
 
 will be 
 
 . _ T _ . A _ cos 2 6 . sin 2 a) 
 
 sm ^ . F. sin ^4 . z, - - 5 - '- 
 
 Call the summed fraction R: (it will be seen that 
 R = M + PQ): then the term in question becomes 
 
 Combining this with the preceding, the whole additional 
 term towards the magnetic north 
 
 = sin h {HN . sin 2 A + VR . sin A}. 
 The new term introduced into xy is 
 
 sin h . r" { sin b . cos 5 . cos a (cos 2 ^! sinM) 
 + 2 sin b . cos b . sin a . sin -4 . cos A], 
 
EFFECT OF THE SHIP'S HEELING, 185 
 
 and therefore the new term introduced into JET.2 - f-^ 
 
 will be 
 
 6lm . sin b . cos b . cos a 
 sin h. H. cos 2A .2, - 3 - 
 
 = sin h . HN. cos 2A. 
 The new term introduced into yz is 
 
 and therefore the new term introduced into V. 
 
 
 will be 
 
 -L -rr A v l m (cos 2 . sirfa sin 2 6) 
 sin h . V. cos A . 2, - - a - ' 
 
 r 3 
 
 = sin h . VR . cos A. 
 
 Combining this with the preceding, the whole additional 
 force towards the magnetic east is 
 
 . - sin h {HN. cos 2 A + VR . cos A}. 
 
 Uniting the terms derived from subpermanent magnet- 
 ism and from induction, we have the following forces 
 introduced by the ship's heeling : 
 Towards the north, 
 
 sin h x {K. sin A + EN. sin ZA+VR. sin A}. 
 Towards the east, 
 
 - sin h x {K. cos A + EN. cos ZA+VR. cos A}. 
 
 The latter is the only force which disturbs the 
 direction of the compass-needle. 
 
186 ON MAGNETISM. 
 
 82. Examination of the heeling -disturbance, and 
 remarks on the possibility of correcting it. 
 
 The quotient of the deviating force by the terrestrial 
 directive force, on which the needle's deviation will 
 
 Y 
 depend, will be (remarking that ^y= tan dip) 
 
 sin h x \N. cos 2 A -f ( R tan dip + TT ) cos A [ . 
 
 No simple rule can be given for the position of the 
 ship's head which will make the bracket vanish : cos A 
 will be determined by a quadratic equation. 
 
 The first term has for factor N. Now in examining 
 Articles 68 and 69, it will be seen that N is that effect of 
 induction which puts on the appearance of a constant 
 magnetic force parallel to the ship's keel. The correction 
 of N by a magnet is of no avail in reference to the 
 formation of the first term in the last article. But 
 correction of JVby a mass of iron subject to the same 
 induction as the rest would destroy the term in the last 
 Article. In the ordinary place of the steering-compass 
 in a merchant-ship, it may happen that this term is 
 negative and large, principally as affected by the magnet- 
 ism of the sternpost : and the treatment of the heeling 
 error is very unmanageable. There appears to be no way 
 of determining the value of the bracket in different 
 azimuths, except by inclining the ship in different 
 azimuths. Here we see a great advantage in the use of 
 Mr. Rundell's vertical bar in front of the compass. This, 
 
CORRECTION OF THE EFFECT OF HEELING. 187 
 
 which is subject to induction, if so adjusted as to correct 
 W when the deck is level, will also correct Nin the heel- 
 ing term : and the part depending on cos 2A will dis- 
 appear. In ships where the steering compass is much 
 nearer the middle of the ship, N will usually be small. 
 
 Supposing then that N is put out of consideration, 
 the term that remains is 
 
 sin h x ( R tan dip + -=? j cos A. 
 
 Both terms of the bracket become large in high mag- 
 netic latitudes, where the dip is large and H is small. 
 If K be positive, or tending to draw the red end down- 
 wards (as will hold in the subpermanent magnetism 
 produced by the operations in the process of building 
 iron ships in north latitudes), the second term, which is 
 the larger, will be negative ; and, remarking the sign of 
 cos A t when the ship's head is north of the east-and-west- 
 line, and the ship heels to starboard, the red end of the 
 needle will be drawn to the west : when the ship's 
 head is southerly of that line, the red end will be drawn 
 to the east. Both cases are included in the seaman,' s 
 rule " the red end of needle deviates to the windward 
 side." In southern magnetic latitudes, it is the blue 
 end which so deviates. 
 
 When the ship's head is east or west, that is, when 
 A = 90 or 270, the heeling force vanishes: it is maxi- 
 mum, with different signs, when the ship's head is north 
 or south, that is, when A = or = 180. 
 
 The circumstance that the deviating force is expressed 
 
188 ON MAGNETISM. 
 
 by a multiple of sin h x cos A enables us to correct it 
 by application of a magnet. In Article 51, Figure 41, 
 putting h for <, that is to say placing a magnet in the 
 ship which shall be vertical when the ship is on even 
 beam and which will have the inclination h when the 
 ship has the heel h, we found that its horizontal force 
 on the red end of the needle is 
 
 When the ship's head is north, or A = 0, this force acts 
 transversely to the needle, and is wholly available (and 
 so, with changed sign, when the ship's head is south). 
 But in any other position of the ship's head, the force 
 acts obliquely on the needle, and must have the factor 
 cos A. The efficient force is therefore 
 
 ca . , A 
 
 sin A . cos A. 
 
 a 2x 
 (c a ) 
 
 This follows the same law as the force which we wish to 
 neutralize : and therefore, by proper choice of the poles 
 of the magnet, and by sliding it up and down parallel 
 to the ship's masts, a position may be found in which it 
 will entirely correct the heeling-error. 
 
 Unfortunately, the terms included in the bracket 
 both depend on geographical position, and the correction 
 which is valid in one part of the earth will not be valid 
 in other parts. The correction of the heeling-error 
 deserves, more than any other point, the attention of 
 practical magnetists. 
 
RECORD OF CHANGES IN TERRESTRIAL MAGNETISM. 189 
 
 SECTION XL 
 
 ON THE CONTINUOUS REGISTRATION OF SMALL 
 CHANGES IN TERRESTRIAL MAGNETISM. 
 
 83. General principle of photographic self-registra- 
 tion now usually adopted. Distinction of the magnetic 
 elements which are to be registered, and appropriate 
 positions of the recording apparatus. 
 
 The object to be attained is, to make an impression 
 depending on the position of some part of the apparatus, 
 without contact, or friction, or mechanical resistance of 
 any kind. Nothing is so suitable for this purpose as 
 photography. If from a minute source of light (as a 
 lamp shining through a very small aperture) light falls 
 upon a concave mirror, or upon a plane mirror assisted 
 by a convex lens, which is firmly attached to a moving 
 part of the apparatus ; then a spot of light (the optical 
 image of the small source of light) may be formed at a 
 proper distance, and the motions of the moving part of 
 the apparatus will produce corresponding motions of the 
 
190 ON MAGNETISM. 
 
 spot of light ; which, if received on photographic paper, 
 may be made to impress a permanent register of the 
 position of the spot, from which the positions of the 
 moving apparatus may be inferred. 
 
 It is now necessary to explain how the time is re- 
 gistered in combination with the register of the spot- 
 movement. For this purpose, the photographic paper 
 must be attached, either to a plane board which is 
 moved by clock-work uniformly in its plane in the 
 direction at right angles to that in which the motions of 
 the spot occur, or to a barrel which is made to rotate 
 uniformly and whose axis is parallel to the motions of 
 the spot. With either of these, the motions of the spot 
 leave on the paper a photographic curve, whose abscissa 
 represents time at a given length for an hour, and whose 
 ordinate represents a quantity proportional to the in- 
 strumental movement which causes the motion of the 
 spot. If we interrupt for a short time the beam of light 
 (which will cause an interruption in the photographic 
 curve), noting also the clock -time, we can mark off 
 accurately the hours, &c., on the time-scale. And if 
 we possess any independent methods of observing the 
 position of the moving apparatus at definite times, we 
 can, by adjusting the scale of ordinates to the spot- 
 position at those times, make it available for every 
 other time. 
 
 The elements which most conveniently represent the 
 state of terrestrial magnetism as acting at anyone geogra- 
 phical point, and whose changes it is desirable to record, 
 ar6j the position of the free magnet, the small changes 
 
RECORD OF CHANGES OF MAGNETIC DECLINATION. 191 
 
 of which may be conceived as the effect of a westerly 
 magnetic force acting on the red end the magnitude 
 of the horizontal directive force H and the magnitude 
 of the vertical force V. For numerical expression of 
 all the small changes of force, it is convenient to use H 
 as the unit. The changes of declination of the free 
 magnet are in the horizontal plane, and therefore the 
 axis of the barrel on which they are registered ought to 
 be horizontal: a horizontal position also, it will be 
 shewn, can be made available for register of the changes 
 of H : but for the changes of V it will usually be found 
 convenient, though not absolutely necessary, that the 
 axis of the barrel be vertical. 
 
 84. Record of the small changes of magnetic de- 
 clination, and evaluation of their scale. 
 
 The apparatus, as will be gathered from the last 
 article, is exceedingly simple : a fixed source of light ; a 
 concave mirror, firmly connected with the frame that 
 carries the magnet, and causing the pencil of light to 
 converge to a spot ; and the revolving barrel with hori- 
 zontal axis which receives that spot. Suppose now that 
 the distance of the concave mirror from the surface of 
 the barrel where the spot is formed is m inches. To 
 give to the spot a motion of 1 inch, the beam of reflected 
 
 light must have been turned through the angle : and 
 
 YH/ 
 
 therefore, as the direction of the incident light is in- 
 variable, the mirror (and the magnet which accompanies 
 
192 ON MAGNETISM. 
 
 it in its motions) must have turned through the angle 
 
 =7 . The direction of horizontal magnetic force has 
 2/7i 
 
 therefore changed through an angle represented by ~- . 
 
 This change of direction would be produced by com- 
 bining, with the northerly directive force, a westerly 
 force equal to 
 
 northerly directive force 
 2m 
 
 Consequently a motion of the photographic spot through 
 1 inch in the direction of the ordinate of the curve will 
 
 represent a westerly magnetic force equal to ^ of the 
 whole northerly horizontal force. 
 
 85. Bifilar magnetometer for record of the small 
 changes of magnetic horizontal force, and evaluation of 
 their scale. 
 
 A torsion-apparatus of any kind, which permits an 
 accurate measure to be made of the force that produces 
 any angle of torsion, would answer perfectly for this 
 object. But the kind of torsion which has been adopted 
 as most convenient is that produced by suspension by 
 means of two cords or wires, separated, both at the 
 upper place of attachment to a fixed beam or other 
 support, and at the lower place of attachment to the 
 magnet. 
 
BIFILAR MAGNETOMETEK. 193 
 
 Suppose an unmagnetic bar (as of brass) to be 
 suspended thus by two cords separated at the top and 
 at the bottom. The bar will take such a position that 
 the two cords will hang in one vertical plane. Let 
 the apparatus be so adjusted that the unmagnetic bar 
 takes a position in the magnetic meridian. Substitute 
 for it a magnetized steel bar; the steel magnet is in 
 the position which it would assume if perfectly free, 
 and therefore it exerts no mechanical effort to escape 
 from that position. Now turn, through a limited 
 angle in the horizontal plane, the substance to which 
 the upper ends of the two cords are attached. The 
 two cords are now no longer in one plane : and they 
 exert a force of torsion or wringing on the suspended 
 magnet. The magnet will yield to this, but not en- 
 tirely ; for, as soon as its position makes an angle with 
 the magnetic meridian, the earth's directive force tends 
 to pull it back towards the magnetic meridian, or to 
 resist the torsion-power produced by the bifilar sus- 
 pension. The magnet therefore will take a position 
 in which the torsion-power, produced by the circum- 
 stance that the two wires are not in one plane, exactly 
 balances the torsion-power produced by the action of 
 terrestrial directive force upon the magnet, now in- 
 clined to the magnetic meridian. 
 
 Now suppose the terrestrial directive force suddenly 
 to increase. It will more than balance the torsion- 
 power of the suspension, and will draw the magnet 
 nearer to the meridian. Suppose the terrestrial 
 directive force to diminish: the torsion-power of the 
 
 13 
 
194 
 
 ON MAGNETISM. 
 
 suspension will overcome it, and will turn the magnet 
 further from the magnetic meridian, till the balance 
 is restored. It is plain here that, by noting the 
 position of the magnet, we have the means of ascer- 
 taining the direction and magnitude of the changes 
 which terrestrial directive force undergoes. 
 
 It is indifferent whether the rotating apparatus 
 (or ' torsion circle ') be connected with the fixed beam, 
 so as to act on the two upper points of attachment, 
 or with the magnet, so as to act on the two lower 
 points of attachment. It is also indifferent whether, 
 in the former case, the two lower points are in the 
 longitudinal direction of the magnet: in the three 
 diagrams to which we shall now refer, we shall sup- 
 pose that they are in an inclined position. 
 
 Figure 56 is a side view of the magnet in an 
 
 Pig. 56. Fig. 57. 
 
 L-L 
 
BIFILAR MAGNETOMETER. 195 
 
 assumed position of the points of attachment : Figure 
 57 is an end view. These views shew that the two 
 wires are not in one plane: (the angle of crossing is 
 very much exaggerated in the diagrams.) Figure 58 
 
 Fig. 58. 
 
 represents the view from above, or the projection 
 of the whole upon a horizontal plane: this will 
 give the means of computing the torsion-strain pro- 
 duced by the weight of the magnet. 
 
 Let the distance EF of the upper points of attach- 
 ment be 2a, and the distance GH of the lower points 
 be 26 : and let them make the angle </> : also, let the 
 length of each suspension- wire be l\ and the weight 
 of the magnet W. The torsion of each cord will be 
 
 W 
 
 sensibly ; and the resolved part of this in the direc- 
 & 
 
 tion EGf, Figure 58, will be ^ ; and the momen- 
 tum of this to turn the magnet will be 
 
 
 
 KL being the perpendicular from K upon EG. But 
 
 132 
 
196 ON MAGNETISM. 
 
 EG x KL - 2 area of triangle EKG ab . sin < : there- 
 fore the momentum of the strain in the direction EG 
 to produce rotation of the magnet is 
 
 W 
 
 , ab . sin <f>. 
 
 A similar momentum is produced by the strain in the 
 direction FH : therefore the whole momentum of rota- 
 tion is 
 
 W.ab . 
 sm (f>. 
 
 Now let the upper suspension-bar be turned round 
 till the magnet is turned to a position at right angles 
 to the magnetic meridian. The momentum of terres- 
 trial horizontal magnetism upon it, by Article 21, 
 supposing it inclined to the magnetic meridian by 
 the angle 0, will be E . B. sin 6: and sin 6 will sensibly 
 = 1, not only when 6 = 90, but also when 6 = 90 4- #, 
 where a; is a small angle (such as we have to consider) 
 
 a: 2 
 which makes sin 6 1 + &c. We shall therefore 
 
 consider the momentum of terrestrial horizontal mag- 
 netism &s=E. J3. And, as this balances the momentum 
 of torsion, we have the equation 
 
CHANGES OF EARTH'S HORIZONTAL FORCE. 197 
 
 Now conceive E to be variable, and < to vary in con- 
 sequence. The equation of variation is 
 
 cos 
 
 Dividing this equation by the last, 
 
 $ -rt 
 
 cotan <f>.$(f> = -^. 
 
 Thus the ratio in which the Earth's horizontal mag- 
 netic force (E or H) varies is inferred at once from 
 B<p', that is, from the angular change, in the horizontal 
 plane, of the position of the magnet. To make this 
 measurable, let a concave mirror, or a plane mirror 
 assisted by a lens, be attached to the magnet so as to 
 partake of its angular vibration in the horizontal plane : 
 let light from a fixed lamp fall on it; and let it form 
 an image of the light upon a rotating barrel covered 
 with photographic paper at distance n inches. A 
 motion of the spot through 1 inch corresponds to an 
 angle in the position of the reflected ray represented 
 
 by -, and therefore to an angle in the position of the 
 
 mirror represented by ; therefore, for a motion 
 2tfi 
 
 1 inch in the spot, &<j> = , and 
 SE BH I 
 
198 ON MAGNETISM. 
 
 This is the change of terrestrial horizontal force cor- 
 responding to a motion of 1 inch in the photographic 
 spot: by means of this value, a general scale for in- 
 terpreting the values of the spot-motion on the ordi- 
 nates of the photographic curve can be formed. 
 
 86. Balance-magnetometer for record of the small 
 changes of magnetic vertical force, and evaluation of 
 their scale. 
 
 Let Figure 59 represent the balance-magnetometer : 
 
 Fig. 59. 
 
 a magnet to which is attached a steel knife-edge C, by 
 means of which the magnet vibrates in the vertical 
 plane; its knife-edge being supported by horizontal 
 planes of hard stone. The vertical plane being trans- 
 versal to the magnetic meridian, the horizontal di- 
 rective force has no effect on the motion of the 
 magnet : it is affected only by magnetic vertical force 
 and by gravity. 
 
 The red end of the needle is pulled downwards, 
 and the blue end is pushed upwards, by terrestrial 
 vertical magnetism. To maintain the magnet in a 
 horizontal position, its center of gravity cannot be 
 below the knife-edge C, but must really be somewhere 
 towards the blue end, as at G : the point G being 
 
BALANCE MAGNETOMETEK. 199 
 
 supposed to be connected with the magnet, and to 
 vibrate with it. Let V be the magnet-power of earth's 
 vertical force, B the magnet-power of the magnet; 
 then as in Article 21, the angle 6 being very approxi- 
 mately 90, 
 
 V.B=WxKGi 
 
 K being in the vertical below C, not vibrating with the 
 magnet. It will be remembered that, in that Article, 
 the units of statical and dynamical forces are connected 
 by a formula which does not contain g. 
 
 Now suppose the Earth's vertical force V to vary. 
 The only other element in the last equation which can 
 vary is KG. Hence we find 
 
 and, dividing this by the preceding equation, 
 
 = 
 v KG 
 
 But, as we cannot immediately measure S . KG, we 
 must resort to an indirect process in order to extract 
 a meaning from this equation. If the magnet is in- 
 clined through the small angle ty, S. KG will = CKx \jr, 
 and 
 
 SF CK 
 
 Now a value of CK may be obtained by causing the 
 magnet to vibrate on its knife edges, thus. Let / be 
 
200 ON MAGNETISM. 
 
 the moment of inertia of the magnet : and consider the 
 whole weight of the magnet, as assisted by the vertical 
 magnet-force of the earth as above mentioned (the petty 
 alteration of which will have no sensible effect on this 
 element) to be collected at a point which, when the 
 magnet is horizontal, coincides with K. Incline the 
 magnet through a small angle % : the angular moment 
 produced by its weight will be 
 
 W. 0/r.sinv W. CK 
 _____ *or--^ % , 
 
 (omitting the usual factor g, for the reason given in 
 Article 18) : hence 
 
 df ~ 1 
 
 The solution of this equation is, 
 
 W.GK 
 
 Let T be the time of a complete double vibration, in 
 which the variable term in the bracket increases by 
 
 27r ; then T ' = 2?r, or CK= ~r* Hence 
 
 V~ W.T 
 
 To obtain a value for /, take the magnet off from its 
 bearings, and suspend it by a single cord, as a free 
 declination-magnet ; the side which, when mounted on 
 its bearings, is vertical, being now horizontal ; so that 
 
CHANGES OF EARTH'S VERTICAL FORCE. 201 
 
 the same value I will now apply to its moment of 
 inertia in horizontal vibration which formerly applied 
 in vertical vibration. The magnet being now inclined 
 to the meridian by the angle o>, and the force which 
 acts on it being the horizontal magnetic force H, we 
 
 HB . sin &) H. B. co r , 
 
 shall have - j or = for the angular mo- 
 ment: therefore 
 
 H.B 
 
 ft) 
 
 and, if T be the time occupied by a complete double 
 vibration, 
 
 .B T'\H.B 
 
 T """" J A 2 * 
 
 Substituting this in the last expression 
 SV_T* H.B 
 
 . T' 2 H. B 
 
 ~ifiz cotan dip x ^r. 
 
 This supposes that the unit by which 8 V is measured is 
 the entire vertical force. If we prefer to adopt for unit 
 the entire horizontal force, we have simply 
 
 gy 2 1 ' 2 
 
202 ON MAGNETISM. 
 
 If (as with the other instruments) a concave mirror 
 be attached to the magnet and throw the image of a 
 fixed light to a photographic barrel (whose axis is 
 vertical) at the distance p inches : then the direction of 
 the reflected beam will be changed through the angle 
 2i/r, which will cause the light-spot to move through 
 
 2p . i|r inches. For 1 inch of motion, ty will = ^- , and 
 
 BF = JT^_ 
 H ~ 2p . T 2 ' 
 
 87. Results obtained from the continuous registers 
 of small changes in terrestrial magnetism. 
 
 When the sheets of photographic paper are detached 
 from their barrels, and a large number of these sheets 
 (extending for instance through a year or through 
 several years) are examined, they present the most 
 capriciously discordant appearance that can be imagined. 
 Thus, Figure 60 represents the curve given by the Hori- 
 zontal-Force-Magnetometer on a quiet day (1869, Octo- 
 ber 17) : Figure 61 represents that given by the same 
 instrument on a day of disturbed magnetism (1869, 
 March 10). It appears from such records that the 
 terrestrial forces are at every moment in a state 
 of change, though in very different degrees on 
 different days. The laws of change extend without 
 sensible alteration over considerable geographical dis- 
 tances: the writer of this treatise has compared many 
 photographic records made at the Royal Observatory 
 of Greenwich with those made at the same time 
 
PHOTOGRAPHIC REGISTERS OF EARTH'S FORCE. 203 
 
 Fijf. 60. Fig. 61. 
 
 ||o 25S2S2?-??*? 
 
 cm 
 
 -7 
 
 ^ 
 J- ( 
 
 -** 
 
 -** 
 
 -IP 
 
 KlHKN 
 
 I I I I I I 
 
 -15 
 
204 ON MAGNETISM. 
 
 at the Kew Observatory, and has not remarked 
 any sensible difference. In most cases, but not in 
 all, the disturbances in the east or west direction are 
 comparable with those in the north or south direction, 
 and greater than those in the vertical direction. The 
 periods of great disturbance sometimes occupy a portion 
 of a single day, sometimes several days in succession: 
 they are familiarly known by the name of 'magnetic 
 storms.' They are not connected with thunder-storms 
 or any other known disturbance of the atmosphere ; but 
 they are invariably connected with exhibitions of 
 Aurora Borealis, and with spontaneous galvanic cur- 
 rents in the ordinary telegraph-wires : and this connec- 
 tion is found to be so certain that, upon remarking the 
 display of one of the three classes of phenomena, we can 
 at once assert that the other two are observable (the 
 Aurora Borealis sometimes not visible here, but certainly 
 visible in a more northern latitude). 
 
 But when the ordinates are picked out from the 
 different sheets for the same hour of the day on every 
 day through a year or through several years, the 
 irregularities neutralize each other in a great degree : 
 and the mean laws of inequality of the magnetic 
 elements for different hours of the day have a very 
 close resemblance, as deduced from different years. 
 They are not however precisely the same : the change 
 in their type is gradual, but it does not recur in any 
 cycle of years or according to any other law yet estab- 
 lished. 
 
 Having ascertained, from the mean of all the photo- 
 
MEAN DIURNAL INEQUALITY OF EARTH'S FORCE. 205 
 
 graphic records during a year, the mean value of the 
 ordinate at each individual hour, and having compared 
 that number for each hour with the mean of all the 
 similar numbers for the 24 hours, we obtain the dis- 
 turbance at each hour; in westerly force, or in northerly 
 force, or in vertical force, as the case may be. Now if 
 we lay down in a left-hand ordinate the westerly dis- 
 turbance at each hour, and in an upright ordinate the 
 northerly disturbance at each hour, we produce a curve 
 of the singular form represented in Figure 62. The 
 
 Fig. 62 
 
 small figures on the curve give the solar hours. And 
 
206 ON MAGNETISM. 
 
 though there is a sensible difference (as has been stated) 
 in the forms of the curves for different years, yet those 
 characteristics of the curves upon which the eye rests 
 as marking its most striking peculiarities are repro- 
 duced with accurate resemblance in all. 
 
 The curves for the different months have a marked 
 difference. In the summer months, the curves are 
 larger and more nearly round, and the small appendage 
 about the early morning-hours (15 h , 16 h , &c.) is less 
 strongly marked: in the winter, the curves generally 
 are smaller, and the morning-appendage is more im- 
 portant. 
 
 If, instead of using solar hours to define the times 
 of measure of our ordinates, lunar hours (reckoned from 
 the time of moon's transit over the meridian) are 
 employed, we obtain a remarkable result. The solar 
 diurnal inequalities disappear entirely from Jbhe mean; 
 and we find that there is a true lunar tide of magnet- 
 ism, occurring twice in the lunar day, and shewing 
 magnetic attraction backward and forward in the line 
 from the Red Sea to Hudson's Bay. These forces are 
 however considerably less than those which follow the 
 law of solar hours. The mean diurnal solar inequality 
 
 may be stated as about + -^ of horizontal force: the 
 
 600 
 
 lunar is about + 
 
 - 12000 ' 
 
FORMATION OF A GALVANIC CURRENT. 207 
 
 SECTION XII. 
 
 ON THE RELATION BETWEEN GALVANIC CURRENTS AND 
 MAGNETIC FORCES : AND ON THE REGISTER OF TER- 
 RESTRIAL GALVANIC CURRENTS: WITH SPECIAL RE- 
 FERENCE TO DISTURBANCES OF TERRESTRIAL MAG- 
 NETISM. 
 
 88. Fundamental principles of the creation of a 
 galvanic current, and of its magnetic action : application 
 to the galvanometer and to the speaking -telegraph. 
 
 The subject of galvanic action in general belongs so 
 completely to another science that we shall enter upon 
 it here no farther than is absolutely necessary for ex- 
 plaining the relations of which this Section treats. 
 
 The simplest form of a galvanic battery is repre- 
 sented in Figure 63. A small vessel, of glass, or earth- 
 enware, or guttapercha, is nearly filled with dilute 
 sulphuric acid. In the acid are plunged two plates of 
 metal, selected principally for their difference of sus- 
 
208 
 
 ON MAGNETISM. 
 
 Fig. 68. 
 
 ceptibility to the action of the acid : great care is taken 
 to prevent the plates from touching. For one metal, 
 zinc (its surface usually being rubbed with quicksilver) 
 has. from the beginning of the science, been adopted : 
 for the other metal, silver, or copper, or (now more 
 commonly) graphite, a form of carbon extracted from 
 the iron retorts in which coal is distilled in the manu- 
 facture of gas for illumination. These two plates are 
 connected by a metallic wire: or, a separate wire is 
 soldered to each plate and the wires are brought into 
 mechanical contact (which, if the touching surfaces are 
 clean, is sufficient). Then a galvanic current or gal- 
 vanic currents pass through the wire. We are justified 
 in asserting this by observing that heat is produced in 
 the wire, sometimes sufficient to fuse iron and platinum : 
 that magnetic effects (to be mentioned shortly) are pro- 
 duced at every part of the wire: that time is required 
 for the transmission of the effect through great length 
 of wires : and that the disruption of the wire at any 
 point produces a spark. The phenomena seem to 
 justify us in asserting that a current proceeds from 
 
MAGNETIC ACTION PKODUCED BY GALVANISM. 209 
 
 each plate, the qualities of the two currents being differ- 
 ent: and we shall sometimes, for convenience of language, 
 speak of a "zinc current" and a "graphite current." 
 Mechanically, the effects of such currents passing in 
 the same direction may be considered as + and : but 
 when (as when originating from opposite ends of one 
 wire) their directions are opposed, their effects are 
 added together. 
 
 The simplest magnetic action produced by a gal- 
 vanic current is the following. The current as in 
 Figure 64 will deflect the red end of the needle from 
 
 Fig. 64. 
 
 the reader's eye. The current as in Figure 65 will 
 
 Fig. 65. 
 
 deflect the red end towards the reader's eye. The 
 
 14 
 
210 ON MAGNETISM. 
 
 current as in Figure 66 will deflect the red end towards 
 
 Fig. 66. 
 
 the reader's eye. That in Figure 67 will deflect it from 
 
 Fig. 67, 
 
 the reader's eye. The magnetic attraction is always 
 normal to the direction of the current; a singular circum- 
 stance, we believe, in physical action. The direction 
 may be remembered from the following fanciful rule. 
 Conceive an insect to travel along the wire, in the 
 direction of the graphite current, with his face 
 always turned (upwards or downwards, or horizontally, 
 as the case may be) to view the red end of the 
 needle. Then the galvanic power deflects that red 
 end towards his left hand. 
 
MAGNETIC INDUCTION PKODUCED BY GALVANISM. 211 
 
 In Figure 68, the properties of Figures 64 and 67 
 
 Fig. 68. 
 
 are combined, and the red end is thrown from the 
 reader with doubled energy. In Figure 69, the action 
 
 Fig. 69. 
 
 is multiplied to any extent. This is the construction 
 of the ordinary galvanometer, and also that of the 
 acting part in the common English speaking-telegraph. 
 
 89. Inductive magnetic power of the galvanic cur- 
 rent: its action on steel and on iron; formation of trans- 
 ient magnets ; registering-telegraphs. 
 
 In treating of pure magnetism we have seen that a 
 magnet-pole, which attracts the red end of a magnetized 
 needle, possesses the power also, in the operation of 
 double-touch, of drawing all the red magnetism of an 
 
 1 !_'> 
 
212 
 
 ON MAGNETISM. 
 
 unmagnetized steel needle to the end nearest to itself, 
 and thereby magnetizing that needle. And in like 
 manner, if a soft iron bar be presented to it, it converts 
 it for the moment into a magnet in a similar state. It 
 is therefore easy to conceive that the galvanic current 
 may be able to produce analogous effects. 
 
 The best form of wire for this purpose is a long 
 spiral. In Figure 70 is exhibited a simple spiral : but 
 
 Fig. 70. 
 
 the wire may be carried round and round so as to form 
 numerous layers, care being taken that the wire is 
 turned round always in the same direction. Such a 
 spiral constitutes a kind of magnet, which, though 
 acting feebly on external objects, is sometimes useful. 
 But its magnetic effect in the interior of the coil is 
 powerful. Conceiving a mass of red magnetism in the 
 interior, the imaginary insect which we have cited, in 
 crawling along the wire from the graphite end, with his 
 face towards the nearest part of the red mass, would in 
 every part of the spiral have his left hand towards the 
 graphite : and therefore the attraction of every part of 
 the coil tends to draw the red magnetism towards 
 
MAGNETIC INDUCTION PRODUCED BY GALVANISM. 213 
 
 the graphite end,\ and the blue towards the zinc 
 end. (If the direction of the spiral turns had been 
 opposite, the result wpuld have been opposite.) 
 
 Now if we insert ^n this coil a bar of unmagnetized 
 steel, as in Figure Tip the bar is instantly magnetized, 
 
 Fig. 71. 
 
 and becomes a magnetic needle with its red pole to- 
 wards the graphite (the direction of the spiral being as 
 shewn in the figure) . This process is extensively used 
 for magnetizing compass-needles. 
 
 If instead of the bar of steel we insert a bar of soft 
 iron (usually called the 'core'), the bar is magnetized 
 in the same manner as under other inductive force, 
 having its poles in the same position as those of the 
 steel bar just mentioned. But the magnetism is 
 transient, lasting only as long as the galvanic current 
 lasts. If the current be destroyed by interrupting the 
 circuit in any way, as by cutting the wire at any point, 
 or by separating two portions of the wire which are in 
 contact, or by separating the wire either from the zinc 
 plate or from the graphite plate, or by lifting either of 
 the plates out of the acid, in any of these cases, the 
 iron instantly loses its magnetism. And this property 
 
214 
 
 ON MAGNETISM. 
 
 is exceedingly valuable, because by it we can make and 
 unmake a magnet at a great distance, even several 
 hundred miles, and in any locality, and even in a 
 moving frame. 
 
 A convenient and powerful form is that of the 
 horse-shoe magnet, the wires being arranged as in 
 Figure 72. A piece of iron must be provided, to be 
 
 Fig. 72. 
 
 pulled by the two poles of the magnet. It is in this 
 form that galvanism is commonly employed for the 
 telegraphs in which permanent impressions are made 
 on paper at the distant station. 
 
 90. Spontaneous terrestrial galvanic currents : in- 
 vestigation of the magnetic effects due to them, and 
 comparison of these magnetic effects with the magnetic 
 disturbances recorded by the self-registering magneto- 
 meters. 
 
 In the ordinary system of wire-telegraphs, each 
 wire, when not used in the actual work of transmitting 
 galvanic currents, is detached from all galvanic bat- 
 teries, and is connected at both ends with the earth. 
 
TERRESTRIAL GALVANIC CURRENTS. 215 
 
 It was soon found that, when the wires are in this 
 state, galvanic currents sometimes pass through them 
 which are sufficiently strong to cause movement of the 
 galvanometer-needle: and (when a battery is placed 
 in the circuit for giving signals) sufficiently strong to 
 pervert the telegraph-signals. And it was at length 
 discovered that those currents, produced by the earth 
 only, occurred at the same times as magnetic storms. 
 In order to investigate the relation between the earth- 
 currents and the magnetic storms, two wires were 
 established in connexion with the Royal Observatory 
 of Greenwich ; one about 10 miles long, terminating 
 at Croydon, the other about 8 miles long, to Dartford : 
 each wire was carefully insulated in every part except 
 at both its extremities, which were plunged in earth, 
 and the two wires passed through two galvanometers, 
 one appropriated to each wire, in the Observatory. 
 Each of these wires might be expected to bring from 
 the earth at one end to the earth at the other end a 
 portion of the galvanism which is flowing through the 
 superficial strata of the earth. 
 
 As it was soon found that currents, weak or strong, 
 were almost always perceptible in the movements of 
 the galvanometer, a self-registering apparatus was 
 prepared. To the needle of each galvanometer a small 
 plane mirror was attached, and the light of a lamp 
 shining upon the mirror was by lenses made to con- 
 verge, to form a spot upon a revolving barrel covered 
 with photographic paper. Thus two registers were 
 obtained similar in general character to those of the 
 
216 ON MAGNETISM. 
 
 changes of Magnetic Elements, described in the last 
 Section. 
 
 The ordinates of these curves (considered as mea- 
 sures of the terrestrial galvanic currents passing through 
 their respective wires) were measured for correspond- 
 ing times. In order to determine experimentally the 
 sign to be given to a current, considered as positive 
 when its nature was that of a graphite current coming 
 from the distant station, a small battery was placed so 
 as to send graphite currents through the galvanometer, 
 and the nature of the movements produced by it was 
 noticed. Then it was conceived that each current 
 might be represented as the effect of one current from 
 the north and one from the west, the effect of each 
 upon one experimental wire being proportional to the 
 cosine of the angle made by that experimental wire 
 with the north and with the west respectively. Putting 
 a for the azimuth of Croydon from magnetic north 
 towards east, a for that of Dartford, G, D, N 9 W, the 
 currents from Croydon, from Dartford, to the North, 
 and to the West, respectively : 
 
 C= Nx cos a + W x sin a, 
 D = JV x cos a' + If x sin a'; 
 from which 
 
 _ _ r 
 
 sm (a a ) sin (a - a ) 
 sina 
 
 ^y _ c . , D 
 
 sin (a a') sin (a a') ' 
 
TERRESTRIAL GALVANIC CURRENTS. 217 
 
 and by means of these expressions, the intensities of 
 the northerly and westerly currents could be computed 
 for every instant. Then, assuming these to be the 
 representations of veritable currents flowing through 
 the upper strata but below the Magnetic Observatory, 
 and applying the rule of Article 88, the disturbances 
 of northerly magnetic force due to W were found, and 
 the westerly magnetic disturbances due to N were 
 found. These were compared with the actual disturb- 
 ances of northerly magnetic force (or variations of H) 
 and the actual westerly magnetic disturbances (propor- 
 tional to disturbances of declination) registered by the 
 Horizontal-Force-Magnetometer and the Declination- 
 Magnetometer respectively. And the results were as 
 follows : 
 
 (1) In the magnetic storms, the disturbances of 
 magnetism in the horizontal plane are almost perfectly 
 explained as the effect of the terrestrial galvanic 
 currents. 
 
 (2) On days of quiet magnetism, the magnetic 
 forces, produced by the earth-currents, follow a well- 
 marked diurnal law, which differs greatly from that 
 of the magnetic diurnal irregularities. 
 
 (3) The galvanic currents discoverable at the 
 earth's surface do not explain ordinary terrestrial 
 magnetism. If that magnetism is to be explained 
 by such currents, they must be very deep. 
 
218 ON MAGNETISM. 
 
 91. Note on thermo-electric or thermo-galvanic 
 currents, and on their possible connexion with terrestrial 
 magnetism. 
 
 In Figure 73, let A and B be two dissimilar metals, 
 
 Fig. 73. 
 
 soldered together at C ; and let their ends D and E be 
 connected by a wire. (The metals found to be most 
 favourable are antimony and bismuth.) Then if heat 
 be applied at their point of union C, galvanic currents 
 will be created through the wire DE, which can be 
 measured by the deflexion of the needle of a galvano- 
 meter. The current which issues from the antimony- 
 end of the combination is of the same quality as that 
 which issues from the graphite-end of a galvanic 
 battery. 
 
 If there be a number of pairs of bars of the same 
 metals, as in Figure 74, and if heat be applied simul- 
 taneously to all the points of junction C, C, C, where 
 the metals follow in the same order, the intensity of 
 the galvanic current through DE is proportionally 
 
THERMOGALVANIC CURRENTS. 219 
 
 increased. The several points C, C, C, are usually 
 
 Fig. 74. 
 
 brought very near together, in order to receive the 
 same degree of heat. 
 
 This is the most delicate method known for mea- 
 suring the intensity of radiant heat. It is thus that 
 the diathermancy of different kinds of glass and of 
 different crystals, &c., have been compared with great 
 accuracy, and that the radiation of heat from the 
 principal fixed stars has been made sensible. 
 
 Regarding the earth as a heterogeneous compound 
 of different substances which may possess in some 
 degree the properties of different metals, and conceiv- 
 ing (as is the opinion of many physicists) that there 
 is in the interior a great store of caloric, which may 
 heat the points of contact, some of them steadily and 
 some by occasional bursts of flame, it seems within the 
 
220 ON MAGNETISM. 
 
 range of possibility that such a combination^ of heat 
 with dissimilar substances, may be the cause of terres- 
 trial magnetism. But there is no evidence for this, 
 beyond mere conjecture. 
 
 It is worthy of remark that the isothermal lines 
 on the earth's surface bear a striking resemblance to 
 the lines of equal magnetic intensity shown in Figures 
 35 and 36. 
 
 On the whole, we must express our opinion, that 
 the general cause of the earth's magnetism still remains 
 one of the mysteries of cosmical physics. 
 
 CAMBRIDGE: PRINTED BY C. J. CLAY, M.A. AT THE UMVERSITY PRESS. 
 
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