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About Google Book Search Google's mission is to organize Ihe world's informalion and lo make it universally accessible and useful. Google Book Search helps readers discover Ihe world's books while helping authors and publishers reach new audiences. You can search tlirough the full text of Ihis book on the web at http: //books . google .com/ PhyncaUb. A TREATISE OH ELECTRICITY AND MAGNETISM MAXWELL VOL. II, 'Sontron HENRY FROWDE OXTOBD TTNIVIiBBITT FBBBS 'VPABZIHOUSS 1 PATB&NOBTSa ROW '^^ eiarenlion ^vts& ^stit& A TREATISE ON ELECTRICITY AND MAGNETISM BT JAMES CLERK MAXWELL. M.A. LL.D. BDIN., D.C.L.. F.R.SS. LONDON AND EDINBUKGH HONORARY FELLOW OF THIHITV COLLEGE, AND PR0FC5«0K OF EXPESIUENTAI. PHYSICS IN THE USIVEUSm OP CAMBSIDOE VOL. II SECOND EDITION ©xfotlr AT THE CLARENDON PRESS 1881 [ All rigiU rMtrvei J >}«.4.'-^ UZi- MBS.C W, ClFT «».&«. PATTESSOM COXTENTS. PART III. UACNBTISU. CHAPTER I. ■LKMSitTXBT THEORY OF MAOKSTIItM. Properlies of » iiinf[>i«t wbcu ttcled on liy ttic earth .. I D(£nitii>n of Uio axU of the niiignot nnd uf (h« directioil of magDBtic force I Action of luagneU on ODD ftQotber. Lew of magnetic force .. 2 DttGnition of nugnetio anita ntid tliotr iliuiciutiojia S Naturw of the ovidcocc for the law of magnetic force .. .. 4 Hagnetiem aa a mathematical quautity 4 TIm qnttntitieo of the opposite kiiiU» uf miigiictum in n Du^ct Mw alweyi exactly ciiiiil , i Effects of breaking a DiAgnct 5 A niagiiet it built up of parliclM each of irhicli ia n maf^iet .. R Tltoory of mu^ctic 'nmttcr' fi U^ietiiBtioD i« of the nature of a vector 7 Ueaning oftbe t«nn 'Mo^uelic Poluriution'.. ..' .. 8 Propcrtiaii of A magnetic piirticic 6 Definittona of Magnetic Moment, Intensity of IfagnettiatioD, and CompouGDta of MaijuetJiatiou S PoicDtial of a mugnetiswl dement of Tolumc 9 Potontial of a rongnct of finito stu. Tvo cxpre-dHona fur tliia potential, eoiT«spODding respectively to llie theory of polar!- xation, and to that of mnguetic 'mutter' tO Invet^tigatioD of the action of one magiiclic jxtrliule on luwtber 10 Particular casea 13 Poti-nlial euN]g}- of a inug;n«tinsny Held of force 14 On the miignetic moment and axis of a mrgnet 15 vi COKTESTS. 391. Ivs[mn»iun of the pntcntiul of h mognel m *]>)i<-rical barmonic* 392. Th« ccntro of » nutgoot oad tli* primary and BeotHulary axM through the centre .. .. .. ., I" 393. The aorth ciul of u niugnet iu tluN IrettiKe in that which poiDba north, aod thv loutli encl that which points aonth. Boreal magDDtiam i» that which is mpposcil to exist near th« north pola i>f the earth and the south eud of a niUjpMt. Auflrul nutgDClijiiH ia that whiuli bbloDga to the Noiith pole of tiic earth and the north end of a magnot. Austral nuLguetifia i% considered poailive 19 394. The <lir«<lion of ma^uotic force is tliat iii which aiutnil mi^ nctifn) tl^^d• to move, that ia, from aouth to north, and ihia u the positive direction of magnetic lines of force. A magnet l» mud to be magncLited from ita south eud tomrds ita north end 19i CHAPTER II. MABXBTIC rOBCK AXB aAOXKnO IKDUCTIOBC. 395. Magnetic forco defined with reference to the magnetie potential 21 396. Magnetic force in a cjlindrie cavity in a magnet untlonDly magnetized parallel to the axis of the cylinder 22 397. ApplicatioD to any magnet 22 398. An elongated cylinder. — Miignctia force 23 393. A thin disit, — Ma^etic iududiun 23 100. Relation between magnetic force, mu^ctic indnction, and mag- netizatioD 21 401. Ltne-int€^ral of magnetic force, ornagnetic |>ot«ntiaI ., 24 402. Surfnc*'iot<STnl of DwgDClic induction 25 403. Solenoidal difctriWtion of maguetic indnolion 2t; 404. 8ur&cea aud tubca of magnetic iuduGtion £7 405. Vector-pntential of magnetic indnclion 27 106. Itelations between tiio tcalar and the Tector-poteiitlal 29 1 CHAPTER m. UAavwnc souetoitw axd shuls. 407. Definition of a magnolio aoleiioid .. .. .i » ... i. 3] 408. Drfinittou of a complex ftolcnoHl and cxpnwton for tti ]»tential at any point 33] VII 400. Tlie potonlial uf a BUgnetlc sliell at any poict is tli« product of its FtrcDgth raulliplwd hy tlie iiolid augle iiti butimlary Hib- tendfl at (be point 32 410. Anullier mi'thud of proof 33 411. Tbc potcntiiil at a point on the positire (d<l« of a sbcll uf titmi;^b <t> exceeds that on tbo nonrcMt point on t]i« nr-gativn W(I«by4lF* 34 412. lAincllur iJiKtribiition of nia^uititim 34 413. 0>inplex lamollar distribution 34 1414. Potential of a Buleiioidal magnet 35 415. Potcotiul of u lumrllnr muj^ii-'t .. 3S 416. Vcctwr-potculial of a laniclUr tnfignct 36 '417. On tbe solid angle subtended at a given point by a dosed curve 36 418. The aotid Mxglc cxpmm^l bj- tbr kti;-lb of a curv« ou tlie s]ilier« 37 419. Solid angle found hy tn-u line- integral ions .. 38 420. n expreaied aa a determiuant 39 421. The Mlid ntigte in u cyclic rniictiaii 40 432. Theory of tlie Toctor-polential of a closed curv« 41 1 423. Potential energy of a niaguetie slidl placed in a ua^ietic field 42 CHAPTER IV. ixbt'CKD lunxrnzATiox. ' 424. When a body under the action of magnetic force becomes itself mafpietixed llie pbenomMioD is called maj^iietio induction .. 4 1 425. Uu{{iM>tiu induction in different (ulMtaiicu 4A ■ 42fi. Delinition of the cocHicicnt of induced ma^etiaiticin .. .. 47 427. Ustbonmtical tlieor)' of majjucUc inductiou. Poisson's method 47 1 428. Faraday'a Doetbod 49 420. CitM of a ImmIjt Nurruiindcd by a ini^pttic mfdium Al 430. Pouson'a pfaygic«l tbeoiy of the cauM of induced mflgnotiBm .. £3 CHAPTER V. PABTICOLAB PIIOBI.BM1I IN MAOKETIC tSDUCnOK. 131. Tbeoryof a hollow Bphcricol abell S6 ^432. Caae when K ui large 58 433. When i" = I 58 134. Corresponding case in two dimeBuous. Fig. XV 09 |<t3S. Case of a solid spbere, Ibe ooediuients uf magnet iutioi) being diflcreiit in difl'crcnt dircction> 00 Tiii COITTBSTS. ^^^^^^ Art. no* 436. The nine ooefBcieuts rcdaced to six. Fig. XYl 6| 4S7. Theory of »n rllipAoid BClod on ]>y « uoiform mu(pi«tic foKC .. 438. CiteMOf TifylUt Mid of vcT)- long c11ii>ii(ii'1it iS'X HlatenKut of problcois solved by Neumann, KirclthofT. Mid Groan 4'I(X Uclhod oFKpfwvximAtion to n nolntion of the gcnenl |>robIi when K is vcary atuM. Mng^ctic twditot tt-nd towards of most iulense magoHio force, aiul diamfi^-ndic bodies teud to places of maltott foroe 4il. Ou ship's magnotism CHAPTER VI. WEBKR*8 TBEOBT OF INDtTCKD MAUKKTISM. 442. Expei-iineutA iudicatiuiT a maxiinain of moffiwtisaliOD .. 41,'!, Wrbcr'* muthcmiktiad theory nf temitorary uioiiuiTtiznltoii 4'I4. MotHftcation of the tlioory to account for residunl ini^fnetieatton 446. £x|ilanation of phenomeua by the modified theorj 4 16. UiiRDL'tijuitiou, denw^ctization, and ivmsfpieliiation .. .. 6^ 447. EfTocts of nngnetlEatioii on Uio diiii«n*ios> of the lungnet 448. £xi>erim«nt8 of Joule 449. 4. to. 451. 452. 453. 454. 455. 456. 457. 4. -(ft. 459. 4«'>, 4ill. CHATTER vn. MAOKETIC XKASl'RKlHaTe. SoBpeniioD of Ibe ma^et McUuhIk of otworvntion hy mirror and ecalo. Photographic m«lbod IViocipIc of oollimatioo mnployed In the Kew muj^ietomeleT .. DetemiinatioD of Uie axti of a nu>gnut uid iif Uic diroction of the borintDtal oooipouent of the ma^etio foreo * Meaaorament of the morociit of a nngont and of tlM intoimty of the liorinonlal oomponeiit of mofpictic force Obwrvntioiu of tlollnxtun ,-, ' iqq Uetbod of tang«iUi Mid method of ainen .. „ „ ,. 102 OhaemtMO of ribwtiona I03 miminattOQ of the effects of mafriiettc indni-tion tU^H Statical mMhod of tneaBuriug the bon»>iii*l force I0^| Biftlar su«pen»ioti loJB Syiitifin of oWmliona in an ob*erv«toTj .. „ .. .. 113 OtxonrnliuD of the dip-eirclc .. ., „ it; C08TENT5. ^^^m iX AH. f«* ^_ 402. J. A. Broun'* in«fl)btl of ourr«ct)oa .. 11t> ^M63. Joatc's nupeoBJon 116 464. Balance v«rtii.'iU force magDetometer .. .. 118 460. 466. »467. 4G8. 469. 471. 472. 473. 474. CHAPTER VIII. ox TKRBESTBIAl. MAOXensU. ElemcntsofUio magndic force 121 ComliiDatioD of tUe r«sulU of tie tiiajtnetic survey of & country 1 22 Deduction of f1i<! rxpuni^ioii of tlic mngiivtic potcutinl of tlic earth in f^herical liaimonics 124 D^fiiiitiim of tlie eHitii* m«giietiii poles. They are uot at the extri-tniticit of the ningnctic nxia. t'alte potcK. I^iey do not oxUc on tbe earth's surface 124 Gaun' calculation of tlic 24 oocfRcieDta of Ifac firftt four lior- mouica 12S Sepojution of external horn internal cauaea of mafpietic force .. 125 The solar mid luuur viu-iutioiia 126 The pir iodic varint long ,. 120 The dislurtioDcea and Ibcir period of 11 yean » 127 Ktdlexioiia on magnetic iuvestig«tiona 127 PAKT IV. RI.ECTROMAOKETISM. CHAPTER I. BLCCTROMAONETIC FORCS. Int Ontef* tlisco^'cry of the action of an rlcctric cuircnt on & BUgnet .. 129 i7n. The apace near an electric current ia a uugnctic field .. .. 129 ^477. Action of averiicsl cnrrent i>n a onagDct 130 ire. Proof that tlw force due to a straight cnrrenl of luileHuitely ETMt lengtb vurica ini-er>ely «• tlie dinlunoe 1 30 l79. KlMtronugnctic mcseure of the current 131 CONTESTS. Ah. 480l Potential lunetiuu tlu« to n atmiglit canviit. It u a fuactioo of iDnn; vnlues II 481. The MtioD of this ewreiit compared with that of « magnetic Bh«ll baling ail infinite draiglit edgti and extending on one Mi(l(- or ihiK cdgo to infinity 482. A nmall circuit acts at a gtvat diiilanco lilcc a magnet .. 48$. Deduction from lluN of the action of a cloiul circuit of any form aitd Hixe on any point nnt in tlic current iUnlf 484. Couiparifoii botweeD the circuit nud a muKnelic efaell .. 485. Uagnetic poti'utltil uf a clniMid circuit 134 486. Conditions of continuoim rotation of a mognel nl>nnt a currout 487. Foriu of ibc magnetic etjuipoleatial iiurfaoee due to a cloiiid drcBiL Kg. X\TII .. .. , 488. Untual action I»ct«eeii uij syafem of magneta and a dosed current 489. BeHctioi) on the circuit 490. Force acting on a wira carrying a current and placed in th» m^ietic lield 401. Th«orj of clectroinnguclic rotation* 192. Action of one electric circuit OD the whole or any i>ortioa of another 493. Our method of invcftigatioo is that of Fnraduy 494. Illustration of the method ap|ilied to parallel currvnta .. 496. IMnienHone of the Quit of current 496. Tlie wire ia urKvd fh>m the sido on wldcb its magnetic action Kircngllienii the magnetic force and tovard* the side on which itoppOMxit 497. ActioiU of an iufiulle straight current on any coitcnt in ita plane .. 498. Rlateineut of the hwa of electromagnetic force. Magnetic force dne to n current .. 499. Gnterality nf thne laws 500. Forc« aaing on a dnmit |>Uccd in tlie magnetic field .. .. 801 . Qectromagnolie force is a mochanical force acting on the con- doctor, not on the rloclrie earrsnt itaelf I4Q CIIAPTKR n. amide's avxsnoxjKUt op the MtnuAt AcnOH op BE-Bcrtl CDBKENia. 503. ARipfcrv'iiiBT«Eligatiouof tbe law of force twtireea tlw elemrnU of electric cunvula Art. 503. SOI. 805. toe. '607. mkw. ... 612. SI 3. S14. fil5. 16. 17. .618. tl9. 520. 521. 522. 823. 521. 523. OOKTEXTS. ^^^p XI Pu* His method of «x)ierim«titiDg 147 Ani|>irc'M Iinluico 118 Amptrc'fl dm exporimoiit. Equal aod opposilc ciitreiits nen- tnlise each oUmt 148 Second ex|wriincnt. A cronkfitt conductor is tiquivulmt ta a rtrnigbt onr earn,-in» the mme current H9 Third GXpcrimenl. The oction of a closed current a* an d«- nient of luaoLher current in [lerpeudiculor to that element .. 149 Four^i cspcrinK'iil. GijuhI cnrreDts in sfEt«ius gcomctricall}' simitar produce etjunl forccH 150 In all of tti«tie ex|)crimcnlii the ncling cnnvnt iit a cloned one.. 192 Botb circuitti may, however, for mathcmntical purpoRRK W con- ceived as cooaistiuK of i-lciucntury porliODB, xnd the action of tbe oireailit u.-> tbn rcjiultnnt of the action of tlicM elemento 1 S2 NeceKfliry form of the relations betwoen two «lemcntniT por- ttoiM of lines 162 The {j^mctricnl quantities which determine their rclatire posi- tion 153 Fonn of the components of their mutuul ucttou .. .. .. 154 Ke§o)uttou uf iheiN! in three directiontc, pni'nllcl, n«[icctivcl}', lo the line joining them and to the elements themselves .. .. 153 a«i>eral expression for the ucttun of a finite current on the ele- ment of nnothcr 155 Condition funuahed by Amp^'s third case of equilibrium .. 166 Theory of the directrix and tbe <]ctenniniui(j< of elcutrody mimic iKtion 157 KxprcMion of the delermtDant^ iu terms of tbe CMiiponcnt* of the vector-potential of the cnmMit.. ^ •• •• •• "^^ Tlio port of th« forc«i which U indntc-rminntc can he expresse<l as the space* variation of a potential 158 Complete exprcKMon for tbu octjuii between two finite ciurront* 1 59 MntunI potential of two closoil current* 169 Appropriateness of quaternions in this inrcdtigation .. .. 153 DeterminalioD of tlie form of the functions hy Amptrc'a fourth cue of equilibrium IdO Tbe eloctrodynamic and electromagnetic units of currcnta .. 160 Final expremon* for cteclromugncliu force between two elc- mentJi Ifll Four different admisaible forms of the theory Itil Oftbow Anp^'sb tobepreJerrod 162 ■liili ^^^^xii ^^^^f CSOKTBIITS^^^^^^^^^B M ^^^B. CHAPTER ni. ^^^^^^ OK TRB mnvcnoji op elcotrio ctmssKn. ^^^V i^ ifi) ^^^H 629. Tbe luuthwl of thin trcntii« fDunilLil on thnt of Fnruilay .. 164 165 167 167 ^^^^1 533. Induction hj llic motinn <>f the onrtk 168 ^^^B 634. Tbv i-lrctromotiva force due lo iuducUou do6B not depend on ICd les 1 1«» ^^^^ &37. Un of tli« galniDometer U determine the time-integral of the 171 172 ^^^H S39. Matbctnatical eipressioD for the total enirent of induction 173 174 ^^^H Sll. Hi« mrtlKwl of fttnting the lnw> of induction vritii reference to 175 ^^^H 542. Tlie luw of Lenx, and Neumauu't tlieaiy of indoetion 177 ^^^H 513. Hc^mlioltz'tt dninetion of induction fiYim th» Bi«^ntcal action ^^^^P of curratta by the prii>ci]>le of oonseiratioa of energy ,. 177 179 179 ^^^H CHAPTER IV. 1 ^^^^^^ OK niK mnicnoK or a ccokkxt ox mscF. 1 1« isfl ^^^^ 518. Oiflervui.-« 1i«t««en tlu« caae and that of a tube eoDtaimng a fl isfl ^^^H 549. If there is motnentum it is not tbat of tbe maving et«ctricitf .. i^B ^^^^ 350. NerenlldnH llie pbentmienK ani exactly tukalogout to Omm oI fl i^M ^^^H Ul. An eleelric rurrent luta rner^, which nw; \m calW diwtro- 1 IBS ' ^^^^1 fM?> Thu Trait* lu In fnrm a djnaaueal tbenrv ufntortrtc carrebta.. J C05TESTS. xB "Alt. $53. Hl56. 557. 058. 559. 560. 561. 662. 563. 664. 565. 5«6. r 668. 569. 570. 71. ri 72. 673. 671. I 76. 76. 77. CHAPTKR V. OK IBB BQUATIOKS OF HOTION OF A COXSROnO STWKH. LsfiTaiiKe'a metbod furaUbe§ ap]>ropriate idcu lor tho rtady of Uie bigbcr ilvniuDicnl adeatts 183 Tbcw ideas tonet be tmubted from tnaUKiiiatka) into Ay- Bunical luiigoaga •• •. •> 166 DegnMarfrMdom ofBeoniiMtod Nptom I66 0(n«raUz«d meaning of rclocity 187 Oenenliited meontDg of force 187 Qracndixed mcanuig of momentum snd impulM I87 Work done by « *m«II impvlM 188 Kiu«tio«i>ergy in tonoB of momenta, (7*,) 189 Hamillou't oqiutioiis of mutiou 190 Kinetic energy in t«rm* of lb« velocities and momenta, {T^j) •■ 191 Kinetic energy in terms of velocities, (T^) 192 Itdntiont lietweeu T^ and T^, p und q 192 Momenta ami product* of inertia nnd moliility 193 NecesMry conditions irliich these ooellicientfi must satisfy -. 191 Relation betTr««u ro«tli«mttt4cal, dyuuiuico], luid cltictricid idcus 195 C IT AFTER VI. DTKAUICAL TIIKOHX Or KLKCTKOMAOKCTIBII. The cle«tnc cnrrcnt (iMiBMaM energy 196 The current is a kinetic phonomnDon 196 WoHc done by electromotive force 197 The mart general esprewjoo fur tlie kiuetic energy of a i^tem iiidading electric eumtnta 198 The devUicil variables do not ag^iear in this expremioa .. 199 Mevhiuiicnl force acting on a uvitduutor 199 Tbt part depending on. products of ordiuniy relocttica «ad ttreugtbd of currents does not exitt 201 Another cxprriinentnl teit 203 Piscufsion of tho cleclronioilim forOe 20S If terms iuvolviug products of Telocitics and cnrrtnts existed they would iutrodace electromotive forces, which are not ob- •erred ,. S06 CHAPTER VII. TH80HT or ELBCnuO CiBCCITS. The eWlrokinetic eoetgy of a sj'stvm of linear circuita .. SbctramotiTv force in cnch circuit .. 207 .- 208 Xiv ^^^V OOlTTETrTS. Art. 880. Eloctromngnetic force 581. OiMof two drcuito 209 582. TLoory of induced currents 210 583. U<«bHiiicii) action between the circuits 211 684. All tho pliuiinmcnn of tlm mtitiinl ticttnn of two cjrcuitit dopcnd on a single quantit}', th» iratentinl of tbc tvo circuits .. .. 211 cuAPTEft vm. eXPLORATlOX or TIl» nBLD lir HEAKS OF THE SECOKUAKT CIICVIT. 683. Tbfl electrokin«tic Riom«nt<im of tlic KOCODdnz}' circuit .. .. 213 686. Esprctaed AS a liue-iuH^ral 212 fiST. Any ^jfitcni of i-ontiguuuK vtrcuita ia etjuivaleut Ut tbc ctrcait formed by Uicir extorior bounding .. .. ,, „ 213 688. Electrokiiietic monivntTUii expressed na a surfiH:e-iiiteg;ral .. 213 689. A cronkod portion of a circuit (-(luivalifiit to n ntnugbt [lortivn 214 S90. £teotn>kiuetic momentani at a point expressed as a vector, 'SI .. 2 ] S 691. Itjt relation to the magnetic inductioD, tB. £<juiiUi(iu (A) .. 215 592. JostifiMtion of thcac naiDcM 31S 593. Cowveulions vrith respect to the signs of tranalations and rota- tions 217 59-1. Theory of a eliding piece 218 695. R1ec!truuiotive force dac to tbc motion of a conductor ,. .. 219 596. FJcctraniignetic force on tbe sliding piece 219 597, Four d^rfinition* of a line of magnt'lic inilaction 220 698. (leueml e<|uations of electromotive force, (II) .. .. ,. ., 220 699. Aualj'iiis of (lie electromotive force 233 600. Tbo g«ncral cquutions referred to moving axes 224 601. Tbe ntotion of tbe axes cbanges notbiug but tbe apparent valiM of tbe dcotric puleotitd .. .. 225 603. ElcctiMiMgnetic forc« oo n condnclor ,. 325 603. Electromagnetic force on an dooMDt of a ooDduding body. £<|uatioitt (C) 227 CHAPTER IX. OKXS&AL BQCATIOKB OP THE KLXCTROHAOICSIIC FtEU), R04. Recapitulation ..> » . G05. E<)uaUonii of magaetixalioD, (D) .. .. 93 60$. llotatioD l>el«TDn inngaetir force and electric current* .. .. 231 607. Equations of electrk curreoto, (B) , .. .- 232 608. Eqnationa of electric di>iilacciDeut, (F) S3j Xti ^^^^^ COSTEXTSI. JUt. p^ 689. Tlie torn tding on a ptrticlo of « rahdmoce diw to its laagqct- inliiM) 353 040. ElcctmowgDctic force dao to an el«ctric current {Mstog Uiroagb U „ 254 641. EipluMtion of tbete tonen bj Iho liypotluus of itran in • mediun 253 642. QoMral character of the itreis required to proJnee iIm: [^coo- raeua 2S7 S43. WliuD there ti do magnotJuttion the stress is » tension in the direction of the lines of magnetic force, oomlMued with a [ireauire ia all directiuua at right angles to these lines, the magnitude of tlie teiwtou and pruMuro betBg -—• &', where 4 is the iDAgnetic force 2fl8 6H. Korce acting on a conductor oarrjing a ourreut 2S9 645. TlK-ory of stKwt in a tuediuin as xtated bif Forada/ .. .. 2S9 646. Kumcrical viduc of magnetic tcimion 260 A|<|H-nd)i I 2CI A))i]Ciidix II 2G2 CH.\PTEK XII. cvaMBKr-eatsa. 647. Daflbntioti of aotureiit-slieet 263 618. Curreat-funclion 263 649. Electrto potential 264 &IH). Theofy of stead)- ciim-nia 264 est. 0ms of UDiform CDDductirity 2G4 652. UsgnatM aetton of a curreDt-altect with ctoeod ctimnta .. .. 261 663. MagnMlc potential duo to a citrreiit-tib<.'>:t 664. Inductioti of currents in a sheet of infinite eoodnctivity .. 626. Such a sheet i« i(ui>erTiou« to mafpietic actwu .. .. „ Ml 666. Thcorjr of a iilnne camnt-sliMt • 667. The atagnetie fnnclions vxpressod as derivattros uf a uugle fonetioa 2G 658. Action of a TariaUe tnsgnette ^uleni on tiie nliMrt 27^ 669. \^'hcn tiMre b no external action the carrenta d«ci9, and their nagtMtie actioo dimiiushc« as if the sboel had awved off with MBrtaut vslocdty A 271 'B60. The oumnts^ excited bjr the iastantaneoiu intrrwIuetiaQ of a DBgnHic qrstent, prvdace an eflect eqninlriit to an image of that systera 37i ^^^^^^^^^^^ COSTBNTa. ^^^^^^^ xvii ^Rk ^^^^ Vmtn ^HC61. This imago moves avny from its original positioo with v«lo- ~ dly H 273 »$62. Tnil of imugea fonnccl hf a ma^olic Bjnit«m in coDtiuuooa motion 272 663. UatfaematioAl ei]>resaion for the tffcot of ibc iiiduiNxl ctirrenlM 273 664. CaM of tlw niiifDrm motion uf ft magnetic pnio „ „ .. 273 665. Value of the force acting on the magnetic pole •• >. .. 274 666. Cbm of curvilinear mutiuu 275 667. Com of motion ncnr tii<^ i^gn of the nhnot 278 663. TheoT}- of Arago's rotating disk 275 669- Ttuil of imajtei tu Iho form of a helix 278 670. Splierioil cum-ntiilici'tii 279 671. Tli« vector-potential 280 t672. To produce a field of eonat«Dt Diaguetic force within a spherical %Un 281 673. To produce a constant force on n suspended coil 282 674. Cun«iit» iiarulld to a plane 282 678. A {done electric circuit. A xpherical Kbell. An i:llipiiuidal ieheil 283 76. A (olenoid 284 77. A long •olcnoid 28S 678. Force near the ends 286 679. .\ iinir of induction ooil* 286 680. Proper thick new of wire 287 tendlcu solenoid 288 CHAPTER XIIL PAKAU.K1. CCBHKlrtS. 6S2. Cylindrical conductors 290 683. The cstenial imkgnetic action of a cyliudric wire depends ouly on the whole ciirrrnt through it 391 684. The vector-potential 292 6S6. Kinetic energy of the current 292 686. Ttepulftion between tW ilirevt mid the n^tum current .. .. 293 6B7. TbiiMou of the wiree. Amp^rv's «sperim«ut 293 6B8. Self-induclion of a win doubled on itaelf 294 689. OuTsnta of vM)-ing inteoMtj' in acylindric wii^e 295 6fW. BeUtion between the electromotive force and the total current 296 691. Geometrical mean di»tuace of two figures in a plane .. .. 298 ^692. Particular <WM 2»9 ^B93. Applicatwo of the method to a coil of insuhkl«d wires .. .. 301 H TDL. II. b covnsn. CHAPTER XIV. Art. P»» Il|t4. I'aUititial Ana to « xpWicnl bowl 303 llDfi. M>liil uiiKl«mbt«Dilcil 1i,v a circloat any point 30a ttt>i). I'oiuLiliul energy of iwo droukrcunvaU 306 U97. MoiTimit of tho CQuplo ndiDg botwom ttvo coila 307 008. VftUiw of />; SO? tW\. Allrodioo liHumcu two]MnJI«l oireulftr vtureuta SOS T(lt), CnlvMlalion of iW cM>IBctrobi for n coil of finite ti-ction .. .. 308 Till. I'liU-ntinl of two |Mmllol circles cxprvned b; elliptic intt^nis 3d9 703. IJuonfrom' rtMiiwI n ciifuUr coi-nMit. Fig. XVIII .. .. 311 rOS. [lilTerauliiJ «>)iitttion of tbo |>oleiitittl of two droll* 312 TOI. Api'mxiriMtiou wlim tli»cir<^araT«iT Dcftf OMMMtber .. 313 Pt\ fcVi-lticr u|>]>FvxinAlion 3H riMl. (\iil uf luiuiiuum pvlf-iwluclion 31G Aptw-ioUx I 317 ApiMUtHxIl ... .. 320 A|>|pw<lii. UI ' •• •■ S21 CIIAITER XV. KuvntoMAoxmc ncsimriatxis. TpT. StMid*rd gklnMOMton dad «nihiv« gthtnaaal TVA. i.\MlntatiMoffti«MMkideua n% lUMMMlftMl thpoi; 4f Uw panwoMtn- riix. rHMJ^ib «t U* iMfMrt giinMaitar aa4 the Tit. 0*K«MiM«w«Mkal^««il .. „ „ n». i a wiKAS »B w i fc few Mr<M i i«- .. „ .. nx otMiBiiii wfci^»twa... .» „ ^ )9««^ rh<r»tkMMi»«< Iks «*••<> I COKTBNTS. ^^^^ 724. Weber's elect rod jTiaiuotneter 337 72fi. JouIc'h cuireiit-wi-iirbcr 341 727. Suction of nolwioiit? 342 T28. TTniforni tone uonnal to Ku^peiid«<d coil 342 729. Etectrodj-munumiTtcr with ttmiun-iirm 343 CHAPTER XVI. ■LBOTBOMAOSETIC OBSEBVATtOXS, OWrvation of vibnitioiDi 344 Motiou iu a lognriUimic npirul 34S RMtiUncnroeciltfttioas ID a rceUtingmediuia 346 ^'aluefi of Buccesaive cEoiigatious 347 PatA aud quieiita 317 PonilKin of equilibrium determined from tliroe sitcccssi^-c eloD* ^liom 347 Dctermiuution of tliu lognritbDiJi: decrement . , . , , . . . S4S Wlien to ttn[> tJir cxporimont , . ,. .. .. ... .. .. 348 Dntcrmi nation of the time of vibratioD from three transits , . 346 Two series of observationa 349 Com^ioti for nm}ilitud« and for dumping SSO Dead brat galranometcr 330 To measare a coostaul current wilb tbc Kalvatiometer ,, .. 331 Best an^c of ilFflcxioii of a, inn^itnt gnlvnnometa- 353 Beat metJiod of introducing the current 352 McaMirem«Rt of a currcut by tlie firiit elun^aliou 333 Tu malce a »i.-H<.-!i of iilwiTvutiiiiiH uu u constant current . . . . 354 Uctbod of multiplicutinn for fdcbtc currentH 3M Ut^mirrmcnt of a transient current by tint elongation , . . , 3SS Oari-oction for dauipiiijf 366 Series of obw.-rTatii>nK. Ztiriiekvitrfwvii method^ 357 Ul-cImkI of nioltipli cation 359 t i CHAPTER WW. COUPABISOX OF COIU. 752. Elocbtiml mi'Aiuremcnt fooictiraea moro accurate tbon direct measurement 301 753. Deteniiination of 0, 363 764. Delvmiination of 9, 363 755. Determination of ll>c mutual induction of two c<h1s .. .. 3R3 756. XX-termi nation of tlie self- induction of a coil 365 57. CoinporiaoQ o( the self-induction of two coila 367 ba XX C0STKNT3. CHA1>TER XVm. ELECTBOUAflXimC UKIT OY KKSISTAXCK. Iri 758. Definition of resistance 759. Kirchhoff'ii mHhod 368~ 760. Weber's inctiioil by tniiMiicot cuiTcnta 37( 701. His method of ohscrvatioD , ,. ., 37J| 7B2. liVeWs niBlliod by (lumpinf; STI 763. TliiimiKiii'H mi^tboil hy « rcrolviti^ coil 37| 7G4. Mnthcninlical theory of tlie rcrolring c»il 765. C'ttleululioD of the ruristonce 7flii. Correi'tioOii 871 7K7. Joule's calorimctric method , . . . 87| CHAPTER XIX. COUPAMBOK or TRX KLSCTBOSTATIC WITH TIIE KLBCntOMAOKBIIC CHITS. 768. Nature nad imporUiic« of the invMti^tion _37d 769. Tlic nitig of the iinito i« » velocity 379 770. Current by convpction 380 771. Weber aiid KoliIniusc!i'*melbod 380 772. Tboniaou'a metliMl liy tepanit« eleolroroeM- and elcdrodyn»- inomrler 38^ 773. MaxwcII's method by eombiued elcclromdrr and cledrodyoa- niometer 381 774. Etectromi>fp«tic mea.iarcni«iit of the eupucity of a coudcDMr. Jcnkin* method .. ., 775. Uclhod by an intermittent current . . 776. Condcmer and AVippc ae aii una of ^VIieatstoae^B bridf|« . . 38fl 777. Correction wtien tlie action in too mpid 3S 778. CnpAeity of a eondeosor comporwl with th« sclf-iuductioD of a •oil 387 779. Coil and Gondcnaer coubioed . , 38^ 780. Electrootatie nioamre ot reaiiriance aMnjinml wilh itt electro* magnetic mtuunt S9l CIIAPTEH XX. ELECTXOlUnKBTIC THEOHy OP UflHT. 781. Compariiwn at tim imygxTli** of the «ti!ctr«iBBgn*tio mMium witi tbcM of the medium ia the uudulatory tbeofy of light Art. 782. 78a. ■ 78*. 786. 786. 787. 788. 789. 790. ;791. 793. 793. 794. 795. 796. 797. 798. 799. 800. 801. 802. S03. 804. 805. I k ft' 806. 807. 80S. 809. 810. 811. 812. C0XTRKT3. ^^^^^H XXI Eoei'gX or light during its propBgation .. .. 394 Eqniilton (if propagutiuti uf nn vltxiromOiftKtic <]iHturt>auoe . . 391 Solution wlicn tliu mrdiiim IK H non-conductor .. ., .. 396 CharacleriBtica of trave-propagfitioD 396 Velocity of propngution of ploi'lroiiiMfttiftiL- ilialurliances . , 397 CompiinRon of thie velocity witli tliiit uf lt)j;1it , . . . , . 397 The q>ecific iii(liictir<> ca|>ncity of n <licl«<ctric is tho sqasn of its index of rufmclJou 398 Compnrisoti of tlu^xc qunntilicn in tlio cane of jMiiulSa . . . . 398 Tlicorjr of plane waves 399 Tbe electric diHpliiociueul and the magnetic dislurtiitncfl are iu thv plutii! of lliti wii\-i--froni, nud piTpuudiuuIur to eacb otli«r 400 Encrgjr and 6tre»8 during nulistiim 401 PreMurtexertt^l liy li;{)it 402 Kquationit of niutina in a cryNlulltiscd meditiiu 402 Propagation of plnnc waves 403 Only two waves are prup»^t«d 403 The iJieory agree* wilU that of Freuiel 404 B«lfttion bctwctn electric conductivity nnd opncity. . . , . . 401 Oompurison with fact« 405 Tranapaineiil melab 406 Svluliuu of thi- ciiuntionii when llin tnnHuni is a conductor . . 406 Cnsv of an infinite medium, ttie initial slate tn-iug given . . 406 diaracteristics uf diiTatiiou 407 Dixlnrliaucc oftlic eUvtromngnctic field when a current be^ios to flow 407 Jtapid approxiuialion to an ultimate stati! 408 CHAPTER XXI. MAOSK-nC ACTIOS OX UOBT. PoMitile forma of tlie ridiition between ma^etiBm and light . . 410 The rotation of tlic! pUnc tif iioUrixaliou b)' magnetic uutioii. . 41 1 The lawx of the pbciiomeua 411 Venlct's discovery of ncghtive rututiou Ui ferromngnctic media 411 Rotation produced by quartx, turpentine, &c., iudepeudeully of mn^netiiin 413 Kinenial)C«l .innljfiB of the phenomeiia 413 'Die velocity of « circularly -polurixed ray is diflcireut accordin|[ tif its direction of Mtotioa 413 Ri^hl and lefl-ltHndcd rays , 414 In iiK^ia which of IhcniMlveB have the rotatory property the velocity ia dificrcnt for right and left-handed eonfij^iatious 414 coirrEKTS. lit. FH«f 815. In media Acted od by magnetisiu (be velocity is diflermt for oppiiBtt« <lireoliom of robitUia 41SS 616. Tliti lumitiii'crouii dinlvirbaiicc, mnthomnticall; conMdercd, u • vector 415 817. Kiiiemutic ci]uatiuiui of ctrcuUrlj-polnrixed liglkt 41BH 81B. Kindic piiid p[)ti:ntinl pncrgy of the modinm 417fl 819. Couditioii of w a V6- propagation 417H 820. The action of im^Tietiaui munt depend on tt reul rotation nlieut H die dirtvliiui of Uic tnAgnctic fuive nn an nxix 418^ 821. 81itt«rin«nt of tlic results of the annlysis of tbc plieaomenoa . . 418 832. Hypothc§ia of iuo)«ciihir rortiees 419 823. Variation of the vortices aeconJinR to HeluilioU/a low . . . . 420 824. Vnriiition of the kinetic energy iti the disturbed medium . . 420 Expresaiou in terms of the current and the velocity , . . , 421 The kinetic onerg}- in the GUM of phme w&vcB ^^^ b The equations of motion 4231 Vidoi-ity of n circulaj-ly-polurixod my 422 The magnetic rotation 423 lt«3carches of Verdet 424 Note on a mechanical th«ory of mulcvular vorttMu . . . . 42fi CfUPTEU XXIL rBRBOMAUXKTISM AXD DIAMAOXUTISU KXPLAIXBC SY KOLBCtTLiS CURUKNT*. 832. Mngnetiim i* a pWnomcnon of nioleculci it 833. The pbenomcna of magnetic molecules nuiy be imitolecE hy eh:ctric currents 43 834. DiJftT«uce Ijctween tho olcmentary theory of continuous tnagncte and the theory nf motccuUr cuneota 43( 836. I^iniplieity of the electric theory 431 836. Tbeoiy of a current in a perfectly conducting circuit ., .. 431 837. Cu« in which the current is entirely due to indoctioa . . . . 433 838. U'vImt'm theory of diatiwgnetiam H B39. Mngnccrystalli^: induction ii 840. Theory of a perfiBct ooDdndor 433 841. A medium containing perfectly conducting *|>licncBl molrcides 434 812. SlechaBtiad NC4iou of DLaftuelic force on th« cunmt irhich it cxcrtM 43^ 843. Theory of a molccuk with a priu)iliv« ourreut 433 811. Modificaliooa of Weber's tfavoiy 43G 8tlj. ConBeqacncoH of the tbeoi; .. 4SftH CONTENTS. XXlll CHAPTER XXIII. :oKiK8 or ACTioir at a putasoe. |lit Fkae 846. QnnnlitJeit wliicli enter into AinjiiTd's furmutft 437 847. B«btive motion of two electric pnrticlcii 437 848. B«lntivi! moltou of four electric iwrtivlcvs. Fethoer's llieory . . 438 849. Two new foroK <if Amjibrn'ii formuln 439 850. Two difl'erciit cxprcsftioQs for the forec tictireon two electric particles in mulioQ 439 831. Tlic9c Arc ilue to Guum und to \Ve))Cr resjKctivel)' . , . . 440 8S2. All forcrs muet be coaKixtciit with ttic pnnciplo of the oou- »en-a(ioD uf energy 440 |'863. Wclicr'* fomiuta ia vouMatent with tlus principle liut tlint of Onniis it not 440 854. Hdufaoltx'a dt'iluctiona from Wobor'B formula 411 85fl. Poteuti*] of two currents. 442 866. Wcbcr'n tlwory of the induction of electric currents . . . , 44 J 867. Segrogxling force iu a conuluctor 443 658. Oue of moviog condudora 444 869. Tlie formula (if flnniu IciuIh to on erroiicouM rcmtit 445 860. That of Wchcr Bgrrrs with the phenomena 445 861. LetKr of CauMio Weber 446 662. Tboiffy of RicnuuiD 416 863. Tlwwry of C, Npumatiit 446 864. Theory of Iletti 447 863. Repugnance to Ike idcu of a medium 448 , 866. TIic idea of a medium cannot he got riil of . , 448 PART in. MxVQNETISM, CHAPTER r. ELEMBNTART THEORT OP HAGNBTISH. 871.] Certatk bmlios, as, for instance, the iron ore railed load- stone, the rarth itM>If, and pitices nf Bl«d witicli have been «uh< jcctcd to <!crt<un treattnont, are found to posdeas the fullowiug properti«0, and arc called Mo^uetd. If, near any part of the earth's surface except the Magnetic Polra, 8 magnet be suspended so as to turn &eely ahout a vertical BKis, it will in general tend to set itself in a certain aeimuth, and if disturbed from tliis position it will oBcillate al>ont it. An nn- nugnetized body haa no iuch tendency, but is in equilibrium in all aKimulhs alike. 372.] It is fuund that the force which acts on the body tends to cause a certain line in the body, called the Axis of tlie Magnet, to become psialle) to a certain line iu space, called the Direction of the MagDctie Force. Let OS suppose the magnet suspended so as to be free to turn in all directions about a fixed point. 1^ eliminate the actiun of its weight we may suppose this point to be its centre of gnvitf. Let it come to a position of equilibrium. Mark two points on the magnet, and note their positions in space. Then let the magnet be placed in a new position of njiiilihrium, and note the positions in space of the tvo marked points on the magnet. Since the axis of the magnet coiiicidt-s with the direction of Iiagnetic force in both poiiitionit, we have to find that line in he magnet nhich oc'Ctt|iics the nme position in space before and VOL. II. B after the motion. It apiM^arr, Troni tlic tlioory of the motion c| bodies of iDvariabli! Tom), that such a Uiw iiWayn vx'kIs, aud tbafl a motion Mjtiivnlviit to the kctuul motion Hii{;ht< linvv takcQ placq by nmple rotiitioii round thU line. I To linJ the tine, join the first «n<i last portions of «i«;h of the marked point*, niid draw pbn^ hisecting ibcuf linoK at rifiht anglcn. 1^0 intersection of these planes will he the line roquirvd, which iudiciitew the direction of the axis of the magnet and th« din-ctioD of the magnetic force io apace. The method just doBoribed is not oonvenient for the pmctic determination of those directions. We shal) return to tJiis aubje when we treat of Magnetic Meaaiiromente, The direction of the mag:nctic force is foand to be different different parts of the earth's surface. If the end of the axtti of the ma^et which points in a northerly direction be marked, it has been found that the dinoctioo in n-hich it sets itself in ^ueial deviates from the true meridian to a considerable extent, and that the marked end points on the whole downwards id the nortJMrn^ hemisphere and upwards in the southern. ^M The azimuth of the directioa of the ma^etic force, measured frym tlii? irue north in a westerly direction, is called the Variation, or tlie Thliij^iietic Ueclination. The angle between the direction of_ the magnetic force and the horizontal plane is called the Ma^c Dip. These two angles determine the direction of the mag^e foive, aud, when the magnetic inteusity is also known, Itio magnetl force is complete.'!)- determined. The determination of tlw vale of these three elements at different parts of the earth's surface, the discuwion of the manner in which they vary aceonling to the place and time of observation, and (be investigation of the caus of the magnetic force and its variatioos, constitute the science Tevre«trial AUgnetism. ^d 3~3.] Let us now suppose that the axes of several magnets Iiav^J been determined, and the end of each which points north marked. Then, if one of these be freely suspended and anoUier brought near it, it is found that two marked ends repel each other, thi a marked and an onmarked end attract each oUier, and that ti unmarked ends repel each other. If the magnets are in tJie form of long rods or wires, uniformlj and longiludiaally magnetized, (see below, Art. 36-1,) it is fo tfiat the greatest a^uifentatiun of force occurs when the end o| one magtkci ia held near the end of the other, and tlial UV OF UAOMTIC FOBCB. ■. pbenoiiK-ia can be ncconnted for by supposing that like dulft of the magnets ivpol rach othor, that unlike eD<U attract each otiicr, ftoi] that tlia iuUrmcdifttc pnrt« of the nin^cte Iiave no wnsible nuttial aetion. Tbo onds of a loDg thin ningnt^t arc commoolj called its Polea. the can of an indelinitely lliin magnet, uniformly maifnetiEed throughout its IvngUi, Uie c:xtreinitio» act as centr«e oF force, and the niA 6i' the niugn^t apjK'-iirs devoid of inii(>notic action. Id mil actual nui^eUi the magnctiitation dcviut^if from uuiforrnity. so that no single poi»t« can be lakea an the pole*. Coulomb, how- ever, by i»ing long- thin rods ningnntiEcd will) aire, succeeded in ottobluibing tiw \aw of force between Ivro magnetic pokv*. Tie re/,u/iion heltcrfn Urn n<ii/nelic i>qU» h in He slral^St Cmejoiatttg ^_ (Mem, and i» nummtaily c^iia/ fc (it product ^ ike atreitgtks <if ^H tiie poie* d'mdvd dy the i^uare ^the duUtnte betwecH (hem. ^B 874.] This law, of course, assumes that the strength of each ^^lole is mefljiured in terms of a certain unit, tie magnitude of which ^^may be deduced from the terms of the law. ^t The unit-pole is a pole which points north, and is such that, Tr^irhcn placed at unit distance from another unit-pole, it rcjiels it with unit of forc«, Uic unit of force being dcGued as in ^VrU <j. A pole which points couth is reckoned negative. If OB, and Mj arc tlic strengths of two magnetic pol(«, I Uio distance Iwtwcvu them, and/ the force of repulnion, lU expressed numcrieally, then _ w,Wg /- -fi-- But if [«»], fi] and [f] be the concrete units of magnetic pol^ I and fo[G«, then it follows tliat or [».] = [£Br-'J/*]. The dimensions of the unit pole arc tlwrcfore J as rejpirds length, (—1) as rc^rds time, and } as rcgnnls moss. These dimensions are tlie same as those of the eleetrostatic unit of electricity, which is specified in exactly the sanM way in Arts. 41, 43. * H» KtpuniiiiinU CM BuipuUHn aitb t]lc ToraUm Bgkluicc are oonUtnvil la the Xtvtoirt »/ tkt AeuAmg oj Pkrrtf, 1760 9, an>l la Unt'* Traiu dt eiftlfm, IoulUL ft B 3 tt Er.EMIWT.VIlY THEORT OF ilAONETISM. [375- S7S.] Tlie ncctimoy of this law may be considered to lure been eatatiliiihed hy tJie expcrimcntB of Conlomb with the Toraion BaUnce, and cotifirim'd by Uio experiments of Gauss and Weber, aod of all ob.tervera in mi^nRtic obnerviiforics, who ani every day making n]ea8Ui«raents of magnetic i]tmiitities, and who obtain nsulls which wculd be incontiuit«nt with each other if the Uw of force had been erroneously assumed. It derives additional Miip|K>rt fro m „ it« conBist«ncy with the laws of electromagnetic phenomena. ^H 876.] The quautity which we have hitherto called the stlrenijtl^^ of n polo may also be called a quantity of ' MagiKtiam,' iirwvjdcd h we attribut« no properties to 'Ma^etism' except those obaerved^| in the poles of ma^ets. ^^ Since the expresgion of the law of force between given quantities of 'Afannetism' has exactly the same mathematical form as the law of force between quantities of ' Electricity ' of equal numerical value, much of the muthiimatieal treatment of magnetism mustt b« similar to that of electricity. There are, however, other properties of iiiii<^cts which mtisl he bonic in mind, and whicb may thruw sonic light on the electrical properties of bodies. 4 377.] The quantity of magnetism at one pole of a magnet is always equal and opposite to that at llie other, or more generally thus : — In ee^ry Magnet Ikt Mai quantity of Magneiitat (reckoned al, braically) U ten, Heooe in a field offeree which is tmiform and parallel Uirotigbout tliC space occupied by the magnet, the force acting on tlie marki end ofihe magnet is exactly cqmd, opposite aud parallel to tluit the nnmarked end, so that the revultant of the forces is a 8lati< couple, tending to place the axbi of the magnet in a determinw direction, but not to move the magnet aa a whole in any direetion. This may he easily proved by putting the magnet into a small TPMcI and floating it in water. The vessel will turn in a eer direction, »» as to bring the axis of the magnet as near ae poesibl to the direvtion of the curtli's magnetic forc<>, but there will be motion of tlMj vesael as a whole in any direction ; «> that tberc cai he no exMcs of the force towards the north over that towanlti i ■outh, or the reverse. It may also be shewn from the fact Ihai magnetizing n piece of steel docs not ultcr iln weight. It doe« al the apparent position of its centre of gravity, causing it in 4 IIAOSETIC * MATTER.' Iatibi<ic3 lo sUifl along lUe ucis towHnls llic nortb. T\>e centre of iaertia, as detennined hy the ])li«n»in«tiii of rotation, remainn Dnaltered. 378.] If the middle of * long tliin miignH be vxamined, it is found to possess no magnetic [rroperticit, but if tfa« magnet be broken at that point, each of the pieces is found to Uiivc u nut^oetic pole at the place of fracture, and this new pole ik cxiictly equal nod opposite to the other pole belonging to Unit piece. It is impotfiiblc, either by magnetizntioti. or liy breakitig inBgni-ts, or by any other means, to procure a magnet whose jmjU-s arc un- Initial. If vrc break the long tbin mngnet into a number of nbort pict^'8 ^^vc shall obtain a series of sboi-t maj^ete, each of which hiui [xiU-s ^n" nearly the same strength as those of the original long magnet, ' I'his mnlliplicatioD of poles is not nwessarrly a creation of energy, ^^or we miut remember that after brcukiu^ the magnet we have to ^^o work to foparate the parte, in conecriucnce of their attraction for One another. 379.] liCt OB now put all tbc pieces of the mngnet together I** "t first- At each point of junction there will bo two poles ^^zactly equal and of opposite kinds, placed in eontac;t, so that (heir ^Binilttl action on any other pole will be null. Tlic magneto thna ^ffeebaill, has tbcrefon; the same projicrtics a» at first, namely two polca, ODc at each end, »)ual and opposite to each other, and the part between these pole* exhibit* no lungnelic action. Since, in tbiK mw, we know the long mngnet to be mode up of little short niogneta, and ainoe the phenomena are the same a* in th<' ca»e of the unbroken magnet, we may regard the magnet, ^^ven before lieing lirokcu, as made up of small particles, each of ^Kihich haa two equal and opposite poles. If we suppose all maguela ^Ro be made up of such parlides, it is evident that since the Vmlgebraieal quantity of magnetism in each particle is zero, the qoantity in tbe whole magnet will also bu zero, or in other words, its poles will be of equal stren^h but of opposite kind. ^H Tieory ^ ifaynetU • Mailer' ^B 380.] Since the form of the law of magnetic action is identical ^Hrith that of electric action, the nnie rc-a^ons which can be given ^Bbt attributing clrctric phcnomeaa to the action of one '0uid' ' ■ or two ' fluids' c»n alw Ix; used in favour of the cxi'^tcnoc of a BMgnclic nuitt«r, or of two kinds of mngnetic matt<T, lluid or 6 T,I,EMI!START TIIEOBT OP UAGVETISM. vm 1 Otherwise. In fact, a theory of mi^etio matter, if need in a purely niath«mjitic«l senw, cnnnot fail to explain the phenomena, provided ncir laws arc freely introduced to account for thv nctiul fact*. Onu of theeo new lavrs must be that the ma^ette fluids cunn'it jiaiw from one molecuh) or particle of the mairnet to another, I>ul that the proccMt of magnetization oonsixtii in eeparatia^ to a certain «xt«nt th« two fluids within each particle, and caurin^ the one fiaid to bo more concentrated at one end, and the other flnid to bo more conoentrated at the other end of the particle. This is the theory of Poisson. A particle of a mnpnctizahle hody jb, on this theory, annlo-joiis to a email insulated conductor without char^, which on the two- fluid theory contjiins indefinildy Xar^ but exactly cjiiaJ qiumtitieK of (he two electric-itiest. When an t^lrctromotiifo fwrce acU on the conductor, it Mparntei' (he elect riciticn. canning them U> beooi manifest at opposite sides of the conductor. Iti u similar munnfti according to this theorjr, the magnetizing foreo OHuses llie two kind* of magnetism, which were originally in a neutrah'zed state, to W separated, and to appear at opposite sides of the magnetized particle. In certain mbetancee, sttch as soft iron and those magnetic substances nhich cannot bo permanently magnetized, this magnetic condition, like the eloctrif ligation of (he conductor, disappears wfa the inducing force is removed. In other euh<^aDcea, such as Btcwl. tlic magnetic condition is produced with difficulty, and, w produced, remains aOer the removal of the indncing force. I'his is expressed by saying tliat in (he latter case there is Coercive Force, tending to prevent alteration in the miignetizati< which must bo overcome before the power of a nugnct enn either increased or diminished. In the case of the electrified li thii> would corrcjiiKUKl to a kind of electric resistance, which, imli the renistunce observed in metala, would bo eqoiTalent (o complete iaeulation fur electromotive foreca below a certain value. Tilts theory of mognetUm, like the corresponding theory electricity, is ericleiitly too large for the favta, ami requires to restricted by artificial conditions. For it not only givos no why one body may not dilfer fmni another on aecoiint nf \m more of both Huids, but it enabUw as lo mty wluib would be pn)p-riieB of a body containing an excess of one tnagnelin fl ii in true that a mson is gircn why such a body cannot tioi^ >lik«9 MAONRTTC POLARIZATION. ^ but this TMmn i* Only iotrodiici^d m an ancr-tlioa^ht to explain UiiB partjciiliir fiict. It docs nut (jrovr out of the thcon*. S81.J Vi'c must thervfoix' Bcck for n moilu of i-xpression vrhich lull not bo atpnble of csprvMiin^ too much, and which sluill leave room for the introduction of now idvas ua Wietc art- duvolopcd from iiRW fnctK. Tbi?, I think, iva mIiuII nhtnin if vrc In-i^in by suying M tbul Ihc particles of a nia^not arw PoUrizi-d. ^^^^^B Meaning of fie Urm 'Polarization.^ ^K Wlion » particle of a body poHsesMO!* properties related to a " OHrtain line or direction in the body, and when tin' body, retaining theae properties, i§ turned Bo that this direction is revened, then if as regarda other bodies these prapprtiea of the i)artiole are reversed, the particle, in reference to these properties, is said to be jK-larized. and the properties are said to couetitute a particular kind of polarization. Thus we may eay that the rotation of a body about an a\i9 constitutes a kind of polarization, because if, while the rotatidn continues, the direction of tlie axis le turned end for end, the body will be rotatinfj in the opposite direction as regards spiico. A oonducting particle through which there is a cnrrcnt of cloc- tricity may be said to be polarized, becyiusc if it were ttirmil round, and if the current continued to flow in the »imc din^vtion as re<;ar<.ls I the piirticlc, its dir«(ion in space would be rcvcrHcil, In short, if any muthcmaticsl or physical tjiiantity is of tJic Datur« of a vector, as defined in Art. 1 1, then any body or purtiole lo which thiit direcU'd (jiiantity or vector belongsi may I* said to Iw Poliiriziil *, bi'ciiiiKi' it huK opposite properties in the two opposite directions or )>oU>k hC the direct<-d <{imiitily. The poles of Mu- ciirth, for exaniph', have reference to its rotatum, and liave aecordinyly dill'crcnt namex. ■ Tliii worl PolnHttlixii hM txifn iibhI in % wnit* ni>C cimdiil^^t with thi* In Optica. >ili«ra » T*y of ti(rli> i* nfrl to Iw p<>laiiied vhuii il luui |>r<ijii-jiJa' nlatiii); to it* •i'its. whith lur iileiiticnl (m nppontc tiHtn of the t*y. Thii kin'l <>( jn'taritiiii'Mi tVttn W aofttktT kind -f Direct-d Qiinallty, wbld) nuv bo «a11«I ■ IHpolar l)mnili[y, ta opficidllnii Ui tlw )urmi-i kind, vbluli iuut (hi calltul tlnlpolar. wiMa ■ illi'iUr i|iut>ai(y 1> liini*<I aii^ (in i>n<! it ruiiinliu llif umc u bcroni. TeoaiooB «n-l FmanrH in wljd l>odM*, Eitmvi.iiui, <.Viii[im»i'iiii^ >uit Diitortinn* Mill 11104 uf (hf D|>ti(]kl, clactriol, (md Diagnttic propenioi of ur^ratalliUHl bodlu an illpAltr <|iM»tiliH. kllji I , {fodvcod hv ninRn'tiinn in lmi>f«K»i IioJIm of t»i«liiii Ilia pliuw ifif )• f lliv Inti.Wil U|f!"- '' ''It* iiint:ii' lixti Itulf. It UDipulBr [-rvpoty. IW r..— •;.'.• ,'->.(Mrt7 nifHrml t« in .Irt. 303 it sbu uiiipolMr. \ 8 ELKHEITTART TnEORT OT MAONETlSir. Meaning fjfthe term ' MaynetU Poiaritalion.' S82.] In spcakinf? of the Rtut« of tho pnrtkW of » inftf^et as ms^i'tic polarization, wc imply thftt «K^h of the BniiiHcst parts into which ii mugnct may bo iliviilod has certuin properti<« related to a d«6nite dircctJon through tlio parlicl«, eallod it* Axis of Uagoctization, niul tliut. thn prnpertit-t) n>Iat«d to one ond of tlii* axis are opposittt to tlie propcrlit-n reUt^-tl to the other en^l. Th« pTO]x>rtic« which wo attribute to the particle are of the «amo kind tu> tho^e which we observe in the complete magnet, and in BMstiniing that the particles posses-i tliese properties, we only a«9«rt whul we can prove by breaking the magnet up into Binall pieces, for eacli of these is found to be a magnet. M^ Propertiet of a Magxelized ParlicU. 383.] Let the clement if-riiyth be a particle of a magnet, tot us assume that its magnetic properties are those of a magnet the strength of tvhosc ponitivo potc is m, and whose lengrth is ■/«. Then if P is any point in space distant r from the poflitire pole and / from the ncgativo pole, the ma^etic potential at P will - due to the poritiTe pole, and , due to the negative pole, r''^[''-r). (I) If di, the distance between the polcM, i« \-er7 small, wo oiay put /— r = (/»eoti€, (2\ where * is the an^le between the vector drawn from the magn to P and the axis of the magnet, or ( — mdi r=-^-c08«. MagHttle Sloment. b.] The product of the length of a unifonnly and longitud- inally inagtK-tized Itar magnet into the strength of its pu&itiTO pole is called W* Magnetic Moment. Initiuitg of Ma^lhathn. The intensity of magnetimtion of a magnetic particle is tlie 1 uf its magnetic moment to itw volunw. ^Ve •liull denote it by ■The maifnvtization at any point of a magnet may be define hy its intensity and ita direction. Ita direoUoD may be dennwl itfl direcUoo-oosinea A, fi, r. 385-1 PTTie COMIftNKSTS OF MAGNETIZATION. 9 I. I C«mjionrHl4 (}f Jifa^iiefhaf4on. the magnvtixiitioii nt a point of n inar;n<!t (boinj; a vector or Oirc«1<Ml quantity) muy hv cxpTc»c(I in t^rms of He throe cum- poa«nto rercTTiKl to the axc« of coordtoAtos. Calling tbcse A, £, C, A = l\. B = ItL, C= !», and the numerical value of / i« given by the equation (4) I 7/« = ^ + J!» + C». (5) 885.] If the portion of the. magnet ivhicli we consider \» the difTereDital element of volume (/xi/^</;, and if /denotes the intensity of magnetization of this element, ita magnetic moment is Iiinl^dx. Suhetituting this for meh in equation (3), and rememberia^ that where ^, 1, C are the coordinates of the extremity of the vector r drawn from the point (x, y. :). we find for the potential at the point [it li due to the magnetized element at {x,y, t), {^(e*)+-B(l-y) + C(C— -)} ' 'i^'f^'l'. (7) To obtain the potential at the point {(, ij, () due to « mi^et of 6uite dimensions, we must find the integral of this expres.-ion for every element of volume included within the space occupied by the magnet, or L ''=///{'' (f-') + 5 {l-.y) + C(C-^)} ~ da,d,dz. (8) Int«grated by ports, this becomes where the double integration in the first tliree terms refers to the Buriace of the magnet, and the triple integriition in the fourth to the space witJiin it. If /, n, R denote the direction-coBincs of tlie normal drawn outwards from the element of surface dS, we may write, as in Art. 21, the sum of the first three t«rn», Ijk ff(lA-i-mB+ ttC) ' dS, where the inl^ration is to be extended over tlie whole surface of li« augnet. 10 KLBHESTART THEORY OF MAGSETISSr. If we now introdace two new symbols a and p, ilelinet! Iiy the equations v = U ^mB+nC, ,dA fIB dC\ — — r the expreseion for the potential may be written 88(i ] Thin expnwssion is identical with that for the electrii poti'niiiil due to a boily on the surfiici: of which there is an elc trifientjon whose nuHiioe-density is /t, while throughout it« substance tliorc IB a hwlily elect rificut ion whose volume-density is p. Hence, if vfv iixKumo a and p to lie Ihe mirfiice- and vol unie-densi ties of the i diittribtition of nn imaginary kuWUiiiix-. which we have caltedfl ' inagnftic niatt^T,' the [lotentinl due to this imaginary distribution™ will lip idcRtieal with that due to the actual mn^notization of every element of the ma^et. The surfaoe-density a is the resolved part of the intensity magnetization / in thcdireetion of the normal to the Kurfaec drawn outwards, and tlie volume-density p ia the 'oonvcrgcnoe* (bm Art. 25) of the magnetisation at a ^iven point in the nia^et. lliiM method of reprebenting the action of a magnet as Am to a distribution of ' magnetic matt«r * is veiy oonvciiieol, but most always remember that it is only an artificial method of representing the action of a system of polarized particli's. ;ry I on 'J Oh fie Action <^oite Magnelic MolecuU oh atio4ier. 387.] If, as in the chapter on Spherical Harmonies Art. tS9, d d d we make 1-/- dA~ dx'' "dg ■""^+"37' (0 where /, m, « arc the direction-cosines of the axis i, then tli^ potential due to a magnetic uiolecule at the origin, whose axis parallel to i^, and whose magnetic moment \» Wi, is (= wlioiw A, is Ihc en«ine of thr angle between iJ, and r. Again, if a M<eond magnetic molecule whoxe moment b m^, i whose axis Is {larallel to A,, ia placed at the extremity of the radiua vector r, thi> jiolential energy dne to the action of tl»c one magnet! on the iithiT i« 587.1 FORCE BETTTEEN TWO MAGN"EnZED PARTICXES. 11 r= Hit dK dk. 1 1B,M.^ rf" lii.dk. & tit*jn. = 4i-'{*'u-3A,X,), (3) (*) mako with ench 7«rc fij. ia the co»ine of tho angle wliu'li the nxt^a other, And Aj, A, are th« i^OHmca of tho auglcti which tliey make with r. »Let us next dotermme tlie moment. cT tin- cfiuiile with which the fint magnet tends to turn the second mund its cenlre. I Iiet OS 6Dppo§e t]ie eecond magnet turned throu{;h an an^le d^ in a plane perpendicular to a third axis ^3, then the work done RgaiDSt iiw magnetic forces will be ^ ^ ilip, and the moment of the force* 00 t)i« mngnct in this plane will be TTie acdtal moment aclinff on the SL-cond maffnet may therefore »c considered m the resultant of two coaples, of which the fintt. leta in a {dane iiaratld to the axes of both mnffncts, and ttMide to mereaAe the angle between Ihcm with a force whose moment is ^«m(^.^.). (6) while the second coupU; acts in the plane passing through r and the axis of the second mn^^net, and tt-nds tn dminish the ang-Ie between these directions with u force ■ _^-!!!sco9(r^,)8in(rAJ, (7) whore {riy\ (r-i,), (*,i*j) denote the angles between the lines r. To determine the force acting on the second magnet in a, direction frallr'l to a line h^, we hare to calculate I si's = — MiBi, —J—, by Art. 128ci = 3X,5^(^«-5M,)+3^,,'!^^'A,+ 3^"'^'A.. (10) If we Hup|w«o tho Bcluid force compounded of three fefw«, H, If, and //j, in the ilircctiona of r, i, and i^ respectively, then the >roe in the direction of i. i» AjA+fr-^//, *^//j, (") 12 ELEireSTAItY THEORY OP MAGNETISM. Since the direction of ^, ia arbitrary, wo mufit faave /I, — />J, y/^=^i!^A. The force It '\b a repulsion, tending to increase r; /T, and act on the eecond mngnet in the directions of Uie axes of tb« and second magnet rpspcctivoly. Tliis analyGig of Hie foru-s iictinf^ between two email tnaji^ei was first given in t^mas of the Quaternion Anslysis by Profeesor Tiiit in the Qiiarterly Mali. Jouri. for Jan. tSGO. See obo bis vork OD Quat^miont, Art. 414. Particular Po»iti<mt. S88.] (1) If A, and \ are each equal to 1, that is, if tlie of the nm^^net'S arc in one straight Hnc and in the samo direction, Pi^ = 1, and the force In'tween the magnets is a repulsion ie+A+^=-^- (^^i The negative b!^ indicates that the feroQ il bd itfoution. (2) If A, and A, are zero, and }iy^ unity, flu am of the ma^ot* are parallel to each other and perpendicular to r, and the force is a repulsion 5«,^ In neither of these cases is there any couple. (3) If A, = 1 and A, = 0, then ttj^ = 0. (1 TT»o force on tlic second mafpt-t will be — 1-i in the dircctioi of its axis, and the couple will be — ^^ , tending to t*irn it parall to the iirst magnet. This is equivalent to a single force — \ -' acting jiantllcl to the direction of the axis of th« second and cutting r at u point two-thirds of its length from m^. t| r«. 1. Tims in the ligtirc ( I ) two msgnets are made to ilokt on water, i rOSCE BOTWEEK TWO 8KALL MAGXETS. 18 ^, bein^ in the direction of the axU of nr,, hnt hitwing its nwn axu »t right nagiea to that of m^ , If two points. A, h, rigidly connected with m, and m^ respectively, are connected by moans of a string T, .he Bj-stpm will be in equilibrium, provided Tcuts the line m, m^ at right angles at a point one- third of the distance from m^ to m^. (<) If we allow the second magnet to tura freely about its centre till it comes to a position of stablo i^nilibrium, /^ will then be a minimnm as regards 4,, and thcri-rorc the resolved part of the force due to m.j, taken in the direction of ^,, will ho a muximum. Hence, if we wish to produce the greatest possible magnetic force at a iTtn point in a given direction by means of magnets, the positions of whose centres are given, then, in order to determine the proper diTct-tiuns of the axes of these magneto to produce this c9oct> we TQ only to place a magnet in tlic given direction at the given point, and to observe the direction of stable ctiuilibrium of tlic Axis of a second magnet when its centre \\» placed at each of the other given poiolie. The magnets must then Iw placed with thoirax(« in the dircclioiii indicated by that of the second magnet. Of couree, in performing tbia experi- ment we muMt lake nocount of terrestrial magnetism, if it exists. Let the second magnet lie in n posl- 'tion of stable a|uilibrium nH regardit ita direction, tlien since the couple acting on it vanishes, the axis of the second magnet most be in the same plane with that of tlie first. Hence d the couple being -^ (sin (^1 ,*^— 3 cos (i, r) sin (r Aj)), we find when this ie zero tiUl(^,r) = 2itkU{rk^, tan //, M, fl = 2 tan Rm^ H.. a ^ ' D K Fig. 3. (16) n (17) (18) (19) When thta portion has been taken up by the aecond magnet the value of Jtrbeoomee rfr or AVq rfi. ^vb«r Ii«re ig ia in the direction of the line of force due to m^ at i^. 14 ELEJIENTAKY THEOBY OF MAOKETISU. HODOC (2oy Hence the ^vc^nd magDct will tend to more towards [Joccs «^| greater rM(iIt«nt forre. ^ Tlic furcc on thu si-cund ma^ot may be dccompoecd into a force R, n'liirli in this caw is nlivikys attractire towards tbe first mu^ct aud 11 forov 11^ jmrulli:! to ttiu iixts of the first magnet, where CT| lllj A, lOIDBj (2l" '^ -/SAif+l In Fig. XIV, at the end of this volume, the liuee of force and vquipotentiiil surfaces in two dimendons are drawn. The ma^ct* which produoe them arc suj'iiosi'd to be two lonf> cyliiidncnl rods^ the Kctioiia of which iire rcpivwi-nlcd by the circular blank sp. anil these rods are magnetized transversely in the direction of t uri'ows. If we remember that there is a t«n3ion alonir the lines of force, i' is eney to see that each maeiiet will t4-nd to turn in tlte direvtion' of tbe motion of the hands of a watch. That OD the nght hand will also, as a whole, tend t« more towards tlie top, and tbat on the left band towards the hoi of the page. On He Potential Ktifrpjf <^a Magnei plaevi t» a Magwtxe Titld. 8B0.] Let V be the magnetic potential dne to any sjrstem < magnets acting on the magnet under considemtioD. Wu aliall call ^the potential of the external magnetic furou. ^M If a small magnet whooe strength is iw, and whose leuf^th is d*^^ be placed so that its poisitive pole is at a point where the potential 18 V, and its negative pole at ii jwint whore the polentJul is ¥\ tbi potential energy of this magnet will be w^T— f"), or, if d$ SMjosuml from the nc^tive pole to the positive, If / is the intensity of the magnctitation, and A, ^ v its dir tion-coeincs, we may write, fiiiJ* = Idxdtfdt, , dv dr dv dv and, finally, if A, /I, C are the components of magnetization. 590-] POTESTUL KNEROy OP A MAOSET. 15 (.1*1 ». " ' ^"'\ > < • (») CO tliat th« exprcKsion (1) for the [lotc-iitial vni-Tgy t^ the clement ||»f the nacgiwt Ijecomv* : ITo obtain tlie pot«ntial encTfiy of a raagnet of finite siiw, wo FDUst inte^rrate tliis cxpreeaion for vxety cleroeot of the inagnitt. e tbue obtain RS th« v»Ihc of the potentiftl cnt^rgy of the magnet witli reiipect ^^U> tbe in»gn«tic fit-ld in whicli it is pWcd. ^B The poleiiliai energy is hero expressed in terms of the eoniponentu ^^of millet tzutiou and of those of the magnetic force arising from extcrnid vituse*. By integration hy pnrts we may express it in terms of the dtstiibntion of mugcietic mattvr and of magnetic potentiaJ ^=f}\Ai^B^^Cn) FdS-jjJF {'^ + ^^ + If) ,/W^A (4) where tt'h'' are the direction-cosines of the normal at tho olumciit of suT&ce i/^. If we substitute in this equation tlie exprossionti for the snrEncc- and volumc-den«ity of magnctie nuiltor iw ^vvn in Art. 38G, the cxprcofion becomes H #-= fJrffdS+ fjjypdxdydz. (5) H We may write equation (3) in the fonn H fr=~JJj(.4<t+B0-i-Cy)dxdydi, (8) ^pvhere a^ ^ and y sae the components of the external magnetic force. ^^^V 0» lie MagMlie Moment and Axi* of a MoffMft, 390.] If throughout tbe whole space occupied by the magnet the external mngnetic force is uniform in dirt'clion and magnitude, the compODciitK n. j3, / will be conslaat quantities, and if we writ» ffjAdxdydi = iK, JJjsdtdydi = mK. jjj CJjedyd: = « A', (7) the int^rations being extended over the whole substance of the UMignet, the value of Wmay be written r=-A'(/a + «.^ + ay). (8) 16 ELEJIESTAHt THBOBY OP MAGNETISM. [391- In this oxprcsiion A *<< ■> sre tlic jirfction-cciBinee of tho axis o\ the maf^i-t, HD<1 A' is tho magnetic moment of the ma^ct. If < is tlitt Bn^lti which the axis of the magnet makes with tlic diivctioD of the mitgiiutic force ^, the value of ff' may be written Jr=-J.^oo9f. (9) If the magTiet ia siiepcnded so as to be free to turn abont a vertical asie, as in the cnsc of an onlinar]' compass needle, let the azimuth of the nxis of the magnet be <^, and let it be inelinrd $ to the horizontal {ilune. Let the force of terrestrial magrnctism bo in a direction wlioi^ azimuth is 2 and dip C ^hen a = ^ cos f COS 6, (9 = •& cos f sin 8, y = ^ sin f ; {10) f = COS ^ cos ^, ffl = cos sin ^, n = sin tf ; ( 1 1) whence W= — Jf^ (cos f oos tf cos (♦— 6) + sin Csin 0). () 2) The moment of the force tending' to increase ^ hj turning the mngnet round a vertical axis is I — J— =— A'^coafeo«fl8in(^— 8). (IJ) On He Expantion ^ tie Pofmtuilofa IfaffUffl in Soliil llamonic: 391.] Let rbo the potential due to a unit pole placed at the point (i, jf, C), The value of V at the point x,y, e is r= {(f-«)» + (^-,)« + (f-r)'}-*. (1 Thii expression may be expanded in term« of splierical lukrmooica, with their centre at the origin. We have then r=ro+r,+r,4&e., (»; V^ts-,T being the dialanee of (£ *f, () from the origin, (3 '^i pi • n whcTA in (»1 To det«nniiie the value of the potential enei^py when the magnet is placed in the Beld of force expressed by this poteatial, wo have to intes™'* *•"? expression for ff-'in equation (3) of Art. 389 with reapect to ^, y »n«l -". considering f. ^. fand r as crtustants. If wc consi«ier only ibe terms introduced by f\, f\ aui] jr result will depend on the following volaino.iti(egnilB, 392-] KXPANSIOtf OF THE POTENTIAL tvZ TO A MAGNET. ■ 17 lK=jjJA<ltdyiz, mK^fjfsdxdsdi, nK=jjfcdxdydt; (6) ^L=JJfJxdxdfd3, M=ffJBsdxdydi, N=fjj(kdxdyds\{7) P=JjJ(Si+ Cf)d^d9d^, Q =JJJ{Gf+Az)d^dydt, K B=fff{Jj, + 3x)dxd,dz. (8) We tUuR find tor the tbIoo of the poUintiul energy of the tnagoet j<la«ed in [)rc«(!nce of the unit polo ftt tbv [Kiiiit ((, >}, (). I l-lf~y)+y,i(23t~?r-L ) +C{iy-l'-M)-i-3iP,iC+Qa-t-Rin) +&«■ This €XprcenoD may also 1» rognrded as (he potcntinl energy of thv unit pole in presence of the nia{>iiot:, or moio Bimply u tlie potential iit the point f, >;, ( due to the magnet. I On Ai« Centre of a Magnet and U« Primary and Secondary Axe». 392,] This expression maybe simplified by altering the directions of thw ooordinatea and the position of the origin. In the first place, ve Hhall nuke the direction of the axis of x paraU«l to the I, a xifl of the nm^et. This is ecjaivalent to making H '=1. m = 0, n = 0. (10) If we change the origin of cwnlinatos to the point (*',y', s"), the dircvtionia of th« axes rpmaining uncliang«d, the Tolurae-integrala IK, mK and mK will remain unehangit), but the others will be altered as follows : V=L-tK^, ir=M-mK/. N'=N-ttKA (U) P'=P-K(m^-i-n/). <^=Q-K{njr+U'). ir=R~K{{y-i miT). (l2) If we now make the direction of the axis of x parallel to thu ftxia of the magnet, and put ('■>) V=- (13) 2^^' '-JC' ' K' then for the new axra M and iV have their valtics imclianged, and lie valne of J/ becomes J (i/"+ jV). P leninins unchanged, and Q Jt vanish. We may tltenfore write the potcntia] thus. FT.KMKSTAnT THBOBT OF SrA0SBTrSH. "We liavc thus fowD<l a point, fixed with respect to the magoot, such ihitt thi; second bvrm of the potential auumes the aioKt nimpHH form whi-n tlitu point ia tnken as ori^o of coordinates. Tim poi^^ we then*ror<> <Iclin« as the centre o( the niag;net, and tlie *xi« drann throufrh it in the direction fonnerly defined as the directi< or tlie Ris^ctic oxiii may be deliued as the principal axis of magnet. We may aimplify the result still more by turning the axes of jr and s ronnd that of X through half tlie angle whose tangent P •^ — i,, lliis will oaaee P to 1>econie xero, and the Gnnl ft M—JV of the potential may bo written H This is the simplest form of the first two terms of the potential of a magiiet. When Iho axes of y and x are thus placed they he culled the Secondary axes of the magnet. We may u\m duti-nnine the centre of a msf^et by finding the positiun of the origin of coordinates, for which the surface-integral of the SEiuarc of the second term of the potential, extended oye r , a Bpherc of unit radius, is a minimum. ^M The quantity which is to be made a miuimtim is, by Art. 141, ^^ 4{L' + .ir'-i-,V^~M.V-NL-r.M) + 3(F^ + g*+S'). (16) The changes in the values of this quantity due to a change of position of the origin may be deduced from equations (1 1^ and (li Ilenoe the conditions of a minimum are il (2L—M-A')+ 3uQ + 3mlt « 0, ^ a«i(2JA-A'-£)+3/fl+3)iPiaO, }■ (l| iH{2iV-L-5[) + 3mP-t-3 I q =: 0.) If we assume / = 1, n = 0, m=0, these eo&ditions b«ootD« 2 L-M—A' = 0, e = 0, A =a 0, (I which are the coodilioixa made vte of in the pr«TioiiB invc gation. Thia inTostigation may be compared with that by which potential of a syatcm of f>nvitating matter is expanded. In latter case, the most ooDvenieat point to issumo as the ori is the centre of gravity of the systt-m, and the most conven axes are the prineipal axes of inertia through that point In the case of the mai^net, the point oorreM[>ondiiig to the cent of gravity is at an iutinitc distance in the direction of the CONTESTION RRSPECTISO SIOSS. I 19 Bnd the point wliicb we coll the centre of the magD«t is n point lutvin^ (lifTerent properties from those of the centre of gravity. The quantities L, J/, jV correspond to the moineutji of inertia, and P, Q, A to the products of inertia of a material body, except that L, Jtf and .Vare not necessarily positive quantitiea. When the centre of the ma^^net is token as the origin, the •phcricat harmonic of the second order is of the sectorial form, having its axis coinciding with that of the magnet, and this is irue of no other point, ^^'hen the magnet is eymmctrical on all sides of thia axis, a« the case of a figure of revolution, the term involving the harmonic of the sGoond order disnpp<;srs entirely. S93.] At all part« of the earth's sarface, except some parts of i« Polar rc^onu, one end of a magnet points towards the north, ^ftt Iciut in a northerly direction, and the other in a southerly on. In speaking of the ends of a magnet we shall adopt the popuhir mi'thod ol' calling the end which points to the north the north eiKl of the magnet. When, however, we l^pcak in the langtuige of the theory of magnetic (luids wo shall uho the words real and Auntnkt. Boreal magnetism is an imaginary kind of matter uipposcd to he most ahunduut iu the northern purls of the earth, and Austral magnetism is the imaginary magnetic tter which prevaiU in the southern regions of the earth. The nelism of the north end of a magnet ist Austral, and that of the south end is Uoreal. When therefore wo speak of the north aod sooth ends of a magnet we do not compare the magnet nith the earth as the great magnet, but merely express the pusition which the magnet endeavours to fake up when free to move. When, on the other hand, we wish to compare the distribution of ima- ginary magnetic Suid in the magnet with that in the earth wc shall nw the more grandiloquent words Boreal and Austral magnetism. 894.] In Kpi-»king of u field of magnetic force we shull um the phraae Mognetie North to indicate the direction iu which the north end of a comjiaKs neetlle would point if placed in the Held of force. In speaking of a lin« of magnetic force vm Rhall always KUpjiose it to be traced from magnetic oouth to magnetic north, and ithall otll this direction positive. In the same way the direction of magnetization of a ma^et ia indicated by a line tirawn from the south end of the magnet tonarda the north end, and the end of tbe magnet which pointA north is reckoned the posibive end. Hbo ma the Bnatt Kuagi ■ the I 20 ELEUEKTABT THEOBT OF UAQNETIBM. [394. We Bhall consider Austral magnetism, that is, the magnetism of that end of a magnet whiob points north, as positive. IF we denote its numerical valne by m, then the magnetic potential and the positive direction of a line of force is that in which V diminishes. CHAPTER II. MAGNETIC yOBCB AND MAGNETIC INDUCTIOS. I 896.] Wb h«vc already (Art. 385) determined the magnetic tcntial at a given point <lti« to n maf^not, the tnagnetization of which 18 given at every point of its substance, end we hare Hhenn that the maUicmatical rcKult may W L-xpre«eed either in terms of the actual oiitguetization of every elvmctit of the magnet, or in terma «f an imaginary distribution of ' mngnvtic matter,' partly condensed on the surface of the magnet and partly dillb.tcd through- out its suhstanee. The magnetic potcati»1, as thus defined, i* found by the same znatlieniatical proceee, whether the given point is outside the magnet or within it The force exerted oo a unit magnetic pole placed ttt any point out«ide the magnet is deduced from the potential by the Ewne process of difTcrciiti^ilion as in the corresponding electrical problem. If the com|>oneuts of this force arc a, fi, y, dF . dF 4F H' ^=-. m •& ■ " ~ rfy ' ' ~ ./r *'* H To detcrmino tiy experiment the magnctie force at a point witJiin Htbe magnet we must begin by removing part of the magnetized ^uobatance, ao a» to form a cavity within which we are to place th« awgnetic pole. The force acting on the pole will depend, in gei>enil, on the form of this cavity, and on the inclination of the widls of the cavity to the direction of magnetization. Uencc it is ncccMsary, in Older to avoid ambiguity in speaking of the magnetic force ^^irithin a magnet, to Bi^-cify the form and position of the cavity ^pritliin which the foroo i» to be measured. It ia monifeat that when the form and pn«ition of the cavity is specified, the point ffithin it nt which the magnetic pole is placed must be regarJi'd as nt 11 BUGKETIC FORCE AKD HAONCTIC INDUCTION. no longer witbin tlie substanco of the nui^et, nnd thcitifoK Dm ordinaT}- methods of determtDtng; the force bccom« nt once nppltotble. 396.] Let 119 now consider » portion of H miif^i-t in n-Iik'li the direction and intensify of the niii<piftiitat.ion nrv anifonn. Witliiu this portion let a cavity be hollowed out in the form of a cylinder, the axis of nhieh is parallel io the direction of magntfttzatioo, unit let a msgnEtic pole of unit stroiif^th be placed ftt the middle point of the axis. Sinue the gcncmtiiig line* of this cylinder arc in the dJrecUoi of magnvti/iitiou, there will be no snpcrfieial distribution of mstg- nctinn on the curved surface, and since the circular ends of the oylitider are perpendicular to the direction of magnetization, thertfll will be a unitbrra Ruperfieial distribution, of which the surfaoe-m density is 7 for the negati^'e end, and — / for the positive end. Let the length of the axis of the oylinder he ■>&. and its radios a. Tlien the force arising from this BUpt-rlicial distributtoQ on a magnetic pole placed at the middle point of the axis is that dae to the attraction of the disk on the positive dde, and the repulsion of the disk on the ne^^tive side. These two forces are equal and in the same direction, and tlicir sum is From this expression it appears tiiat the force dcprnds, not on the abwlule dimensions of the cavity, but on the ratio of the len^^tfa to the diameter of the cylinder. Hence, however small we make the cavity, the force arising from tJio snriiicc diBtributiun on ile wullf will remain, in general, finite. 307.] We have hitJierto supposed the magnetization to be nuifonD and in the same direction throughout the nhole of the portion of the magnet from which the cylinder is hollowed out. \Vhen the mugnetiution is not thus rcHtTict«d, there will in general be a distribution of imaginary magnetic matter through tho Bubstance of the magnet. The catting ont of the cylinder will remove part of this distribution, but since in simihir solid figurrw the foreoa at corresponding i>oint« arc proportional to the linear dimensions of the figumt, the ultemtion of the force on the nagnetJo jmle due to thr volume-density of magnetic matter will diminish indeiinitoly as the size of the cavity is diminished, white the effect due to the surface-dcnnly on the walls of the cavity remains, in genemt, finite. If, tbeiefore, we Mxume the dimensions of tins cylinder so smal' MACirrnc poiica re a cavitt. 23 E pthi I that the mn^etirjitioD of the {lurt rvmovi-d may bo rcfrarded at* evtirywliere parallel to the axi* of the cvliinlcr, and of constant mafc^itdde /, the force on a mag:Dctic polo iilucwl at the middle point of the axis of the oylindrical hollow will he comjioundoJ f t«-o forces. Tlie 6rst of tbeae is that due to the diKtrihiition .of magnetic matter on the outer surface of the iniig-m-t, and roughout itfi interior, exclusive of the ijortion hollowed oiit^ llie oompooeots of this force are a, fi and y, derived from the {loteutinl by equations (l). The second is the force R, acting along- the axis f the cylinder in the direction of ma^etizatiou. The value of ibis force depends on the ratio of the len^h to the diameter of the oylindric cavity. 398.] Cate J. Let this ratio be very j-reat, or let the diameter of the cylinder be small compared with its length. Expanding the t>n for /! in t«rms of ^, wo find (3) ity which vanishes when the rati" of i to a is made infinite. Hence, when the cavily is a very narrow cylinder with its axis parallel to the direction of mnf^netisnatiun, the magnetic force within ihe cavity in not afTi-ctod by Ihe snrfucu djutrihution on the ends of llie indcr, and the components of this force are simply a, A y, where ^ovhiii a = — dr y=-di- 0) shall define the force within a cavity of this form as the magnetic force within the magnet- Sir William Thomson has When we have shall denote it called this the Polar definition of mngm'tie force. to consider this force sus a vector we occasion by^. S99.] Caae IT. Lot the Icn{»th of the cylinder be very small eompand with it« diamct«r, eo that the cylinder becomes a thin disk. Expanding the exproesion for S in terms of - , it 1)ocomcs « = -^|'-;nS-H (5) E the nltimate value of which, when the ratio of o to ^ ia made iDttnite, is 4x1. H«;nt«, when the cavity is in the form of a thin disk, who«e plan« Dormal to the direction of magitetizaliou, a unit taagtuetie poU U MAOKETIC PORCH AKD UAOSETIC IITDUCTIOK. placed at the middle of the axis experienoee s force 4*7 in tho tlircclioD of magnctiiatioD arising from iWe su))er(icifll tQagnetism on the circalar Burfaoes of the disk *. Siooe tho componeata of / are ^, i? and C, the oomponcDts of this force are ivA, i-aB, and 4ii(7. This murt \» ooin|>Qui)d«d with the force whose oomponcnte are a, ff. y. 400.] Let the actual force on the unit pole be dcnol«d l>y Uie vector ^9, and its compODontc by a, b and c, th«D a = a + i-nA, \ b = fi + iiiB, I (6) AVe shall doGno the force witliin it hollow disk, wlioso plane sides are nonanl to tho direction of mn^netization, as the Magnetio Induction within the magnet. Sir William Thomson has called this tho Electromagnetic deJinition of magnetic force. Tlic three vwtorti, the magn«tixntion 3, the magnntic force ^, and the mugDCtic induction t8 are counncUtd by the vector oquuttoQ '« tine-Imkgral ^ Maffnttle Fone. 401.] Since the magnetic foro^, as defined in Art. 39d, is that due to tho dif^tribution of free magnctiMm on the surface and through the interior of the magnet, and is not affi-ictcd by the suHkofr- mogDetism of the canty, it may be derived dinx-tly from the general expression f<Hr the potential of the magnet, and (he line- integral of the magnetic force token along any carve from the point A to the ]>oint B is 'litiere Fj and /^ denote the potentials at A am) B respectively. * Om tht /vret ftUki* carkla o/ tiktr form*. I. Anx owTov cTWTMss. The Uavt atvBiiK ftoca Um MrfMc-nuq^Mlkni k l«r MH * ia Uiv (bsctUHi of tile aotiiial to Uu pkaa of Iha cm*mb«, wImt* « U Uw ■ogle b«tw««a tikii uoriDkl oiul Uia dlractkn «f HtgnaUiMiga. Whan tha tnvMma li pMalUl to the ditoelifn of luis^ni'iinitDa Uic ruro* » ^h* anfiiotu foitv ft ; vham dio e r u T M ia la pnrpendicDUi u i^ directlui of nu^MtUMion Uin fonc b Uia miKiutio Indncci^ !&. H. In *a clooeiud cjliiidw, tha alia nt wikit makaa an v^U t nltti ilia dinolluu of uiagBHixatloo. iIm &«m< ariiias rrvoi tho tmUtn-vaipteti^ it itf ud *, pitpaaiicuUr l« tbt axil ia iW |iUaa ooMlaliiiqg Du Ktk and tha dkn-Usa uT nu^MtindoN. S. In • iiltn tL» faro* afUog from ■ntboo-magnctlan la | a / la Uu dlioction uf piUBallwlkM 402-] BDBFACE-INTEORAL. 25 Surface- littoral (^ Magnetic Iiuluctioit. 403.] Tko magnetic iiiductioa throtigh the suriaco S U deEned the value of (h« inlnnal «=// S co« |</5, W J where ® denotes tlie tnsgnitude of the magnetic indaction st th« ^•element of surface dS^ and < the angle betwoeD the direction of , the indoction and the normal to the element of eurfaoe. and tho iut^grition is to be extended over the whole surface, which may bo either closed or bounded by a closed curve. ■ If a, 6, e denote the components of the magnetic indnction, and If M, a the direction-cosines of the normal, the eurface- integral may be written ^^ Q =jf{ia + mt>-i-uc)dS. (10) I If we subKtitute for the components of the magnetic induction their values in U-rn» of those of the magnetic force, and the magnetization as given in Art. 400, we fmd q=jj{ia + mfi^nY)^Si-'iirff(lA + mB+7iC)dS. (II) ^■extends is a closed one, and we shall investi^to the value of tho ^■two Urma on the right-hand side of this equation. ^1 SinoQ the mathematical form of the ri^'lation between magnetic ^■/orce and free magncli:nn is the wime as that Iwlween electric force and fVee oloctricity, we may apply the n-sult given in Art.. 77 to tho Hntt tcnn in the value of Q hy substituting a, /J, y, the componeiitK of magnetic foroe, for X, Y, Z, the com|)onent» of electric force in Art. 7", and M, the algebraic sam of the free ma^etism within the dosed surface, for e, the algebraic sum of the free electricity. We thus obtain the equation % jf{ta+iii^+nY)dS=^ inM. (12) Since evtry magnetic particle has two poles, which arc equal in numerical magnitude but of opposite signs, tho nigvbntic sum of tho mn>^ctism of the jiwiiciv is u^ro, Uence, those particle* which are entirely within the clond aurtiiee S can contribute nothing to the algcbiaic sum of the magnetism within S. The 26 MAGSETIC FOKCE ASD MAGNETIC INDUCTlOS. [4O,;. value of 31 must therefore depeod only on those magnetic particles^ which are cut by the surface S. Hj Consider a Bmall element of the magnet of lMi;rth t and trans^* verse section *', magnetized in the direction of it* len^h, so Hut the strength of its polee is m. The moment of this snoAll magne^H will be M«, and the intensity of ita ma(*iietization. being- the ratit^l of the magnetic moment to the volume, will he (13) Lot this small magnet be cut by the surface S, so that the direction of magnetization mates an angle / with the normal drawn outwards from the surftce, then if i/<S denotes the area of thescction. l^ = dScost'. (H) The negative pole — m of tliis magnet lies within the snrfnce S. Hence, if we denote by dM the part of tlie free mag;neti*ra within S which is contributed by tbia little magnet^ dif = -m = -W, = — 7co9e'rf5. (ll To lind M, the algehnie snm of the free magnetism witliin til closed Hiirfiicc St wc must integrate this expression over the elc surface, ao that H^-Jflco^t'dS, or writing J, li, C for the com{>onent>! of magiirtintion, mad I, m,k for the direct ion -cosines of the normal drown outwnrdtt, 3twi-ff{lJ + MB^HC)d8. (1 This ^ves ua the value of the integral in tho second t«nn equation (11). The value of Q in that equattOQ may therefore 1>0 found in teTons of equations (12) and (16), Q = 4v3I-4v3l=0, or, lie turface-ittlegral tjf tit nayntlie imluction tirougk any titrfiin M sen. 403.] If wo sEsiimc as the closed surfuov that of tbv ditTcrcntiai cli'inent of volume 4x4ydt, vre ohtiUD the equation da M d-e dg *%-'■ {" TI1U ia the solenoidnl condition which i« always saliaiied by tl ooinponeDts of the magnetic induction. 405] LIKES OF MAONRTIC mnPCTION. 27 ^^ Sinrc tlic di»trit>Htion of magnotic indutitton U Golotioiilikl, the ^H^^actTon tliruugli any «urrace lioutuletl by a closed curve de];)endft 6n1y on tbe Torm and {Kmition of tJio dosed carve, and not on that of the »urriuM> itttftf. I40(.] Surlacea at every point of nhieh la + mb + ne = (lit) an called Surfuccs of no iDdiiction, and the intersection of two euch BnriaraM ix catted a Line of induc-tion. The conditiouK thiit a curve, 9, may be a line of induction arc H A eyetem of lines of induction drawn through every point of a Bclosed curre forms a tubular surface called a Tube of induction. The induction acroea any section of ench a tube is the eame. If the induction is unity tlie tube is called a Unit tube of in- duction. ■ All that Faraday * eays about lines of Diagnetic force and niiig- netic spbondyloids is mathematically true, if understood of the Unee and tubes of magnetic induction. The mn^etio force and the magnetio induction are identical outside the magnet, but witliin the substance of the magnet they most be carefully distinguished. In a straight uniformly mag- ■ netizcd bar the mngaetic force due to the magnet itself is From the end which points north, which we call the positive pole, towards the south end or aegatife pole, both within the magnet and in the spAoe without. Th« magnetic induction, on the other hand, is from the positive pol« to the negative outside the magnet, and from the negative pole to the iiositive within the magnet, so that the lines and tubes of induction are r&^entering or cyclic figures. The importance of the magnetic induction as a physiutl <[uantity will Iw more clearly seen when we study electromagnetic phe- nomena. When the magnetic field is explored by a moving wire, as in Fataday s Erp. Ret. 3076, it is the magnetic induction and not the magnetic force which is directly measured. I Tie Fechr-Potdiflal qf Shffnef.ic Induetion. 405.] Since, as we have shewn in Art. 403, the magnetic in* doclion throngh a surface bounded by a closed curvo deiM-nds on • £q^ Bet., —n«» xxtiii. 28 MAGNETIC FORCE AKU MAONBTIC INDUCTION. [4O5. thfi closed carve, and not on the form of the snrfivce wliicb is bounded by it, it most be possible to determine tito induoUoD tluroug^ a closed curve by a process depcaditi;; only on Uie nature of tliat curve, and not involving the conslrucUon of a sarlaoei forming: a diaphragm of the curve. This may be donv by finding a reetor $f related to 39, the maprneE induction, in such a way that the line-inUgtal of BI, extended round the clo«cd curve, is equal to the surfiieo-intcg-tal of 9, extended over a surface lionnded by the closed curve. If, iu Art. 24, we write F, G, U for tlie componenta of 9(, and a, b, c for the components of id, ne find for the relation bctwc«n these compouenta dff dF i=-U~ -^.- (21) djf ^' ' ^ m* das dy Tlje vector 91, whose components are F, 0, II, is called the vector- potential of magnetic induction. If a uixgiictic molecule wlioee moment is n and the direction of J vrho«e axi» of mngnetization is (X, ft, r) bo at the origin of 00-j urdinateii, the potential at a point (ir, y, t) distant r from the origin is, by Art. 387, V rfu djf <lz' r d* d' = iw(x didi-*-''d^. + '' d^>7' m He which, by Laplace's (equation, may be thrown into the form d ^^d d^l d , d ds.1 The <juaQtJtica a, b may be dealt with in a similar manner. From this expreesion G and // may be found by aymmeiry. We! thus seo Uiat the vtctor-polentlal ai a given point, due to a mi^r""*'*"^ particle placed at the origin, is nnroiirically equal to the magnetic moment of the particle divided by tlie square of tho radius vector and multiplied Ly the sine of the angle Ix-lweeii the; aria of magnetiziition mid tho radius vector, and the dinxtion of the voctor-iiotcntial i« perpendicular to the plane of the aiia of I TWrOB-rOTEKTTAL. 29 nugnetization and the radius vector, and in such that to ao eye looking in the positive direction alon^ the axis of magiictiziition the vector-potential is drawn in the direction of rotation of ttic handfi of a watch. I Hence, for a ma^ct of auv fonii in which A, B, C are the compoDonta of magnetization at the point xy:, the com|)onent» of the vector-potential at the point f r| C, are F= ^=m^t~^%)^^s^.\ (22) Fwherc p in put, for coneisenesB, for the reciprocal of the distance between Uie points (^, i), C) <"><! [x, y, 2), and the intogratious arc cxt<^ndcd over the sjwce oceu)>ied hy the magnet. 406.] The ECahir, or ordinary, potential of magnetic force, BAiI. 3SS, Iwcomes when cxprcsstid in the same notation, I Itemcmbering thai ~ = — ^, and that the integral /// ^(S+|?-S)^** ' has the value —iir(A) when the point {$, ij, f ) ia included within tbe limits of integration, and is zero when it i^i not so included^ (J) being the value of A at the point {S, v, C)i ^'^ 8^*1 ''"' the value I of the x-component of the magnetic induction, (21) The Bret term of this «3ipre<don is evidently —"Jr, or a, the ^iRiponent of the magnetic force. 7/f 30 MAOJTETIC POECE AKD MAGKBTIC IKUfCTlO^r. The qiuwtitv' under the integral sign in the second term is zero for every fiU'mL'nt of volume except that in which the point (f, tt, () is included. If th« v«ltw of ^ at the point (f, »j, f ) ia (A), the value of Ui<; fvcond term ia easily proved to be 4t(^), when- (J) ia evidently zoro at all ])ointK outside the maffnet. We may now write the value of the ir-componest of the mn^etii! induction „ = a + i-!i{A), (25) an cqtiaUon which ie identicul with the first of thoee given in Art. 400. "Hie equations for 6 and e will also a^ree with tlioee of Art. -100. We have already sceu thut the magnetie force {*) ia derived firom the Bcatar magnetic {lotential I' by the application of Hamittoo'l operator V, so that no may write, aa in Art<. 17, *=-Vr, (26) and that this equation is true both without and wtttiin the maj^net. It nppoars fnjm the present investigation tliat the mo^pietie induction © is derived from the vector-potential 91 by the appli- eution of the itame o|>erator, and that the result is true within the Biagiiet as well as without it. The application of this o]wrator to a vector-function prodaceO) in general, a M^alar quantity as well aa a vector. The scalar part^ however, which we have called the eonveigenoe of the vector- function, vanishes when the vector-function satisliea the saleooidi M condition dF dG . dU ^ loidnU (27™ By differentiating the expressions for F, 6, Jf in equationa (33), we find that this equation is satisfied by these quantities We may therefore write the relation between the magnetic ioductioD and ita vector-potential which may be expressed in worda by mytng that the magnetic induction is the curl of its vector-potential. Sec Art. 85. « CHAPTEE in. MAOyETtC 80I.BKO1D3 AND SnZUA*. On PaHieutar Forma of Ma^tiel*. 407.] Ir a longf narrow filament of TQagnetic matter like a nire is ma^etized everywhere io a longitudinal directiou, then the product of any transverse section of the filament into tlie menu mtcnsity of the ma^etization across it is called (he strength of the magnet at that section. If the filament were cut in two at the section without altering the uiagnetizatiou, the two surTaecs, when ei-parated, would be found to have eqiiid anil opposit« qtiun- IbitioH of Euporficial roa^vttzation, each of which is numerically equal to the etrength of the magnet at the section. A filament of magnetic matter, so magnetized that its strength to the mrnfi at every section, at whatever part of its length the section l>c made, is called a Magnetic Solenoid. Ifnr i« the strength of the solenoid, da an element of its length, r tlie distance of that eleioent from a given point, and c the angle which r makes with the axis of miignetizatiou of tlte element, the ^noteotial at iJw giveu i>oint due to the element is ^^^^tegrating tliie expression with respect to », so ss to take into H account all the elements of the solenoid, the potential is found r, being the difcianco of t'he positive end of the solenoid, and r, titat of the oegutive end from the point where V exists. Benoe the jiotentiul due to a solenoid, and coneequeuUy all ite maglMtio eflcctn, dei)end ouly on ita strength and the position of flitir COB c mdr , N «r*, ■ Sw Sir W. Hiuiuoii's ' UstbgniAtiol Tbcory of Hasnctiaa,' PkU. fraM-, 1S50, Btprint. MAOITETIC SOr.TiSOIDS LVD eilSLTA iU ends, and not at all on it« form, w)>i'tJi«r straight or curved, between these pointa. Hence the ends of a solenoid lUAjr be oallcd in a strict eeu^^ its polea. ^M If a solenoid forms a closed curve the potential due to it is zen^^ at ereiy point, bo that such a solenoid can exert do ma^ottc action, nor can tte magnetization be discovered n'ithoal breaking it at some point and separating the ends. ^M If a mai;nct can be divided into solenoids, all of which ciihc^^ form clomd curves or have their extremities in the outer surGioc of tbe niHgnct, the magnetization is said to be solenoidal, and, sioce tbc action of the magnet depends entirely upon that of 11 ends of tbo solenoids, the distribution of imaginaiy magnetic matt will be cnlirel}' sniHrrScial. Uonce the condition of the magnetizatioa being solenoidal is ^ + dg ds where A,S,C xsq the components of the magnetization at anjr point of the magnet. 40B.] A longitudinally magnetized 6lament, of which thestrcngtJi varies at different ports of its length, may be conceived to be made lip of a bundle of solenoids of different lengths, the sum of litrengtlis of all tlie solenoids which pass through a given aecti< being the magnetic strength of the filament at that section. Ueai any longitudinally magiietizod filament may be called a Complex Solenoid. If the strength of a complex solenoid at any section is m, then the potential dae to it« action is lade '=/: ■■- da where m in variable, r^ Tf J r d» ■.'!i lliiii ahens that berades the action of the two ends, which in this case be of different strengths, there is an action duo to tb distribution of imaginary magnetic matter along the filament with a linear density d^ ifa^uetie Shell t. 400.] If a thin sliell of magnetic matt«r is magneti»-d id SITRt.LS. S3 direction ovt^rywliere iionns) to its EiirTnc*. the intensity of tlu! Bin^tiL'tiKiilioTi at nny pWn multiplied l>j- the thiebncsB of tJic ,th4«l at that [i1a<:o i» callotl thu Strvn|;tl> vf tlie ms^etio bIicII Uiat plsoo. If thfl »twi»frtli rif a iih<'ll i^ cvcrywUere cqiml, it is cnlled a Simple CDAgiictic »boll ; if it varies from ]x>int (o puiitt it may be ctmccived to l>f mude up of n niimWr of simple Hlmllg saporpoeed and overlapping c«c-b other. It i» tb«refori; calW a Complex Hpia^ctic Mhcll. ^R Let dS lie an clcmont of the xurftce of th« tthdl at. Q, nod ^ ^Hhe strength of the Khttll, tlicn tlio potcntinl at any poiiit^ P, due ^Bo the clement of the shell, iit dF= * ^ rf5 ooa t. ^ wlieie e is the anple between the vector QP, or r and tlio normal drawn from the positive eide of the ehi-ll. »But if d» is the «itid »nf>le subtended by dS at the paint P I'dti =dS coat, whenee iiF= <t> dai, and tlicrcfore in Uic caw of a simple mngnetJo hIicII ir, fie poienlial due to a magnetic xhell at any point is tie prodtiel it* ttrengtk inla lie loiid angle tubt-tnded liy ita edge at (he given point*. 410."] TTie same result may be obtained in n different way by ■uppoRing the magnetic alicU plaoed in any ficit] of mn^netic fnrce, and determining the potential energy due to the position of the •belt. If F ia the potential at the clement dS, then the energy due to this element is rf/- ^y jr. , „ or, li« preiltiei of tie tirenglA of tie thcU into tie pari of lie turfacc-inltgral of V due to tie element dS qf tie tkell. Henoo, integrating ivilb reM]>eet to all micli elcmentN, the energy due to the position of the ohell tn the field ia equal to tiic product of the Btrengtb of the shell and the Murface-integrol of the magnetic iadaetioQ taken over the surface of the shell. Since this surface-integral is tltc name for any two surfaces which • Tbia UiiKtfCBi ii iliM loUkiM. GrNfmf Theory ^ Tnrttlriat M<ign<iUki.\iZ- .. VOU n. D 3« UAOKEnC SOLEJTOIDS AND SUEU^ have the same bounding edge and do not include between them , any centre of force, the action of the msg&etic shi^l depends on|^ri on the form of its edge. ^^ Xow rap|>oae the 6old of force to be that due to • msgnetic pole of gtrengtJi m. We have seen (Art. 76, Cor.) tliat the surfaoe- int«fn^ over a eurface bounded by a giren edgv is tho product of tho strength of the pole and the solid angle subtended by tb* edge at the polo. Honce the energy due to tho mutool actio of the pole and the shell ts <t>M(i), and this (by Greeo's theorem. Art. lOOj is equal to the prod' of the atrvngtb of the pole into the i>otential due to the shell the p»lt^ The |K>tfntial due to the sliell is tberoforc 4>a>. 411.] If a magnetic pole » titarttt fn>m n point on Oxe nt^ti aurface of a magnetic shell, aud travels along any path in spiic« so to come round the edge to u point olose to wIktd it started but on the poiiitivc aide of the shell, the «olid angle will vary continuoas), and will increase by 4ir during the process. Tho work done the pole will l>e i-a<t>m, sod the potent.iul nt any point on po«itive wide of the shell will except tlial at tlic neighbouring poi on ihc negative aide by 4ir<l>. Jf a magnetic shell forms a closed Kurfu<.'o, the pot«atial outside tho shell is everywhere zero, aud ttiat in tbc space witbin ia everywhere 4«4>, being positive when Uic positive side of the shell is inward. Honce such a shell exerts no action on any magnet placed either outside or inside the shell. 412.] If a magnet can be divided into simple magnetic shells, either closed or having their edges on tlie suriacc of the magnet, the distribntioD of magnetism is called Lami'llar. If ^ is the sum of the strengtlu of all the shells traversed by a point pasving from a given point to a point xjfe by a line drawn witi the magnet^ then the conditions of lamelUr magnetization ar« ^'rfp' -^-Jf ^-dT- The quantity, ^, which thas completely determines the inagnct- ixation at any poiut may be called the Potential of ^faguetizatioo. It must be carefully distingui^ed from the Magnetic Potential. 413.] A magnet which can be divided into complex mag shells is sid to have a complex lamellar distribution of netism. The condition of such a distribution is that the lines 415.] roTSNTIAt. DUE TO A LAMELLAR MAQ!fET. 35 I magndimtioii niuat be such that » systotn of siirruoos citu be drana cutting tliem at right ungL-s. Thk condition is 03tprcs>e<l by tlie rell-kiiowti m{ nation Pormt of the Potffaiiafa of Solfno'idai atitl Lamellar Manned. 414.] Tbo g«oeral expression for tho scalar potential of a Dtagnrt fwbere p denotes the potential at {3),y, i) due to a unit magnetic pol« plnvcd at ((, it, 0, or in othi-r worde, the rcciprooti! of the distaum betwran {(, ij, (), the point at which the pot«ntid is ■neasuFcd, and (x, j, s), the jKisition of the element of the niiignvt .to which it is due. This quantity mny be iutt-gr»t«Ml by parts, as in Arte. 96, 386< r=fjp {Al^ Bu + Cn) dS -ffjp (g + ^^ + ^) ./.<fy^. * »: wbere /, m, » are tlie direction-cosines of the normal drawn out- wards from dS, an element of the surface of the m^net. When the nui^ct iif solcnoidal tho expression under the integral aign ia the second tvrin is aero for every point within the magnet, M that the triple integral is zero, and the scalar potential at any p<Aiit, whether outside or inside the magnet, is given by the soriacc* integral in the first term. I The Ecuitu- potential of a solcnoJdal magnet ts therefore com- pletely determined when the normal component of the miignet- ization at every point of the surface is known, and it is iudeiienilenl of the form of the xolciKiids within the ma^'net. 415.] Id the case of a lamellar magnet the mugnetizatJon is determined by ^, tlie potential of magncLizatioii, so that dg He expression for V may therefore he written Integrating this expression by parts, wc fiutl dp A = 5—1 09 L = -3-- k //*('|-|- 1)«-///* (0-^ -S)^*^'- DS ^A »e MAGNETIC SOI.EITOlDa IKD SHILW. I4I6. Th« «fCoii<l term is stiro iitiiois Uic point (f, ij, () is included in the mxffno^- '" whicli cnse it licwimeji 4 57(0) where {4.) ia the «d»e of 4 at tho point (f, rj, (). Th« smrface-iiitcgral may be expressed in terms off, the line dniwn from {j.j, r) to ((, ij, f), and $ the angle whi<;)i tluK line makea witli the normal drawn outwards from JS, so that the potential may be written r=y//?j*cosflrf5+4Ti(^), itAuiK the xefond term is of course zero when the point {(, 7, f)l ntft inclixlcd in tltc vubntanee of the magnet. The pot«-ntial, f', expressed by this equation, is continuous ei at the Ktirfuce of the magnet, witere ^ becomes suddenly zero, m^ if wo write a s: jj ^^ eoi lis, and if ilf is the value of i2 at a point just within the surface, flj that at a )>oint close to the first but outside the sor&oe; or Fg = r,. Tlic quantity fi is not continuous at the surfiiee of the magnet. The component* of magnetic induetiou are related to £1 by is]ttntions « =~ J=- rfQ </a da dx ' ' ~ rfjf ' * ~ dt 416.] In the case of a lamellar distribution of magnetism we may aI>o simplify the vector-potential of magnetie induetton. Its x-cooipoiieiit may be written -m/i-ri)''^'^' By integnttion by ports we may put this in the fonn of surface • in tegral or ^"//'("S-ffV ITie other components of tlii> Tectnr-i)oieiitial may be down from three exprasaioiut by making the proper suLstatutiona. On Bond jHfftet. 417.1 We have already proved Uiat at any point P ifao [•oli^nti 'Htl^^J SOLIO ANOLBS, 37 dae to s msfpietic slicll iis «qiiiil to tliu eolid »ngh siilttcnduil by the e<\ge of the elipll tnultipllvd by tlic >>ln-D^th of tfav slii;Il. As we eball have occasioa to rt-fcr to Bolid an^^leti in thv theory of electric currents, n'c Khali now explain how they nrny Iju mui^iieJ. KDefiiution. Tha mlid anisic eiibtondvd at a jg\\c(t point by n osed curv« is imtUiurGd by the area of a splutTiciil surfaco whow (Dtre is thu ipvcn |ioiiit and whose raditis is unity, the outline of which is tnu-'LYl by the intereuctloR of the radius viTctor with the sphere as it traces the closed curve. This area is to be rcckoaed positive or negative (ioeordiii>f as it lies on the luft or the right- hand of th«! path of the radius vector a« sc«q from the ^vvn point. J^ (f> 1) he the (^iven point, and let {ji, y, s) \w w point on the closed curve. The eoonlinates .r, _^, z uro functions of t, the len{*th of tli« curve rockonvd from a given point. They are periodic fuDction« of ', rtwurrin;; wheiwver » is increased by the wholo U-n^lt of the elOKol curv4;. We tnny caleulatc the solid aug'Ie m directly from the dcliiiitioa iJius. Usin^ iqihcriciil coordioati's with centre at (f, t), ()• ^^^ patting x—i = f siuflcoa^, j>— ij = rsialJsin^, i—^= rco«fl, we find the area of any carve on the sphere by infegrattng I w = f(l-f:os0)di>, I or, OBUi^ the rectangular coordiuateit, ^vtt intention being extended rouni) the curve «. II If the axis of ; j>assc8 oncu thnrugh the el<ised curve the first Bterm ifl 2t. If Uic axis of e does not jugs through it this term T^iszero. 418.] This method of calculating & solid angle involves a cboi<.« of axes which ta to sotne extent arbitrary, and it does not (U-pviid solely on the oloeed curve. Hence the following method, in which no snrlace is mpposed to be constructed, may be statod for the suke of geometrical propriety. Aa the radios vector from the given point traces out the closiil curve, let a plane passin>; through the given point roll on the closed curve so as to be a tan';cnt plane at each point of the curve ^Bn sueoosston. Let a line of unit-length be drawn frotn the given ^Boint [wrpendicular to this plane. As the plane rolU round the 38 IIAQSETIC SOLESOIDS AKD SHBLLB. [4'9- closed curve the ntTCmit}' of the pcrpcndicnUr will tnce s second dosed curve. IjcI the loDgDi of the second cloved curve be v, Uten tlie oolid angle Mubtcnded l>j the lint closed cun'o is This follows fron> tliu well-known theorem tli&t the am (^ s dosed curve on a ^hcre of unit nwliuK, togt'thiT with the circnm- ferenoe of the polar curve, is numerically equal to the circuDifermce *f ■ grent drclc of the si>here, This constnictton is somt-timos convenient for calculatinff the »>lid angle sul>teinlnl hyx n-ctilin«^iir figure. For our own purjKise, which is to form clear ideas of ithymciii phcoomena, the rollowiop mcfliod is to be pn-A-rred, as it employs no conitructions which do not flow from the physical datdi of tJic problem. 419.] A dose<l curve » is given in spam, ani) we have to find the solid angle vulitvniled hy < at a given poiot P. If wo consider t)ie solid aiiglc us the potential of a magnetic shell of imit strength whose edge ooincidec with the dosed eurv«, we iDHst define it as the work done 1>y n unit magnetic pole against the magnetic force while it moves from an infinite dislancd to tJie point P. Hence, if « is the path of the )>ole as it approaches the point P, the potential must bo the result of a line-integnition alon^ this path. It must also be the result of a line-int^-gnlton along the dosed curve i. The pro])er form of the cjtprcssion for the solid angle must therefore I>e thai of a double integration with rus|iect to the two cur^'cs » and <r. Wlieii P ii at an inllnite dixtnnce, the solid angle is evidently zero. As the point P approaches, the clowd curve, as seen from the moving point, appears to open out, awl the whol« solid angle mny Iw conceired to be generated by the op|<«nm( nK>lton qS the^j diHereiit dements of the closed curve as the moving point op>^| proadm. ^^ Ah Uio point P moves from P Ui P' over the dement da, the element QQ' of the closed carve, which wo dciiolc by d; wiU ^^ change its position relntivdy to /', and the line on llie unit .^phera oonvsponding to QQ' will sweep over an area on the Kplivriuul surfara, whiuh wk may write r/H> ri'/x/.T. (I)j Til (litd II lul IIS <iti|i]i«iiu> P flxi<d while the olomd curve ia movt-dl parallel (o itwilf ttiroiigli a dlalunou dv etpwl to PP" tMit in titc uppiMilr dimi-linn. lint h-laUve nii>ti«iu of tlio {loint P will be thi^H sunie H» ii> I li« roul uaw, ^^ I 420.] GESERATIOK OF A SOLID AITOLB. BQ I During th» motion the olcmcnt (^Q' will generate itn urea in the form of a paralliflogram whose sides are paralU-l Rail equal to Q(^ MMJ Pi**. If we construct a pj-ramid on thin pHralldogram u base with its vertex at P, the solid angle of this jiyramid will te the increment (/» whioli we are in search of. To detennine the value of this solid angle, let and ^ be the angles wliich ih and d<r make with PQ reupcct- ively, and let /}> be the ang^le between the planes of these two anglvM, then the ares of the projection of the psrallelo^ni <fj .da on a plane per- T to PQ or r will he dtda sin 9 Bin ^ sin <^, eioce this is equal to f^rfw, we find du = ndsda = -^ sin eia ^ sin <)itU da. n = ^ Biu$ sin ^sin tp. (8) (3) 420.] We may express the angles C, 6', and ^ in termH i.f r, and its diflVreatial coefiieientB with roxpect to » aud a, for Dcnoc OOS0 = dr „ dr coa0'= T-. d^r (^) - , . ™-v = -i-. and Binflsinfl'ooa* = r -r-T tat dn ^ did We thus find the fullowing value for 11', "■4['-(|)']['-&']-iO'- (^> A third expresBioQ for ri in terms of rcctangulur eoordinute* may be deduced from the consideration that the volume of Hie pyramid whose solid angle is du> and whose axis is r iit Bat the volume of this pyramid may also be expressed in terms of the projectiotta of r, d*, and da on the axis of a, y and f, as a determinant formed by these nine projections, of which vre must, take the third part. We thus find as the value of 11, f-», n-», (-', n = A ^. ^. i^, da da da da dif dz dt dt d« («> *0 MAOKETIC SOLENOIDS ATfD SIIELIS. ityoT Iose« in lio, indfl 'Flits «xpre«Kion giv«jt th« yaloe of n frefi from th« ambiguity of ei)^i iiitniiliK'ol by ct|iiAtii>n (5). 421.] TIk' viilui;! of w, tlic solid angle subtended by the «Io arvp ut Uic point /', may now be vrriiu-n wkcro the intrarfi^ion tnlli rus{M.'ct to « is to be extended ooinpl«lely round tlie cloMfd cur^'o, ood that with rv^pcct to a from A a lix«l |M)iul (in tbr ctirro to the point P. The constant Uq is tbe value of tht' colid Kiiglit at thv point ^. It is zoro if J is at an iwti uite distance from iiw clu«ed ourvv. The vnhio of u at any jtoint P is inilependvnt of (In* form' tlie curve between J and P prov(di.-d tlint it dws not [<asi> Ihrnii^ the niiiKnctie shell it««lf. If the shell be sup|>osed iiifinit^'Iy Ihio, and if P «i>d ^aro two ]>oiuts close tofrrther, but Poo the ixMcilivai and /** on th« nefjative surface of the shell, then the curves JP and] JP" must lie on opposite sides of the edge of the shell, so that PJi in a lint- which with the infinitely short line P'P forms a closed circuit vmbrseiuff the edj^. The nine of a* at i' exceeds tlutt nlP' , by it, that is, by the surfaee of a sphere of radios unity. llenoi-, if a closed curve be drawn so as to pass once througli the shell, or in other words, if it be linked ooce with the of tl»e shell, the value of the integral fffUtitr exteiid«d both euni-cs will U' Ir, lilts intr}^) theivftnv, eoosidered as depeadin^ only on the «lowd curve t and the aibitniy curve AP, k as instance ot ftutetton iif multipU* x'uluos, nnce, if we pom &om A \jt> P\ diSkrvnt |<ath« the intc^n^l will have dlflereiit values to th« number of tiuura which the corre XP » tviaed rouad the c«rt« a. U eM ftm of th* ranrv betw««a J a»d ? on he ttaasCamed nito,aw4be« fey «Mmnt>m notkat without iBtns«vtiiiy the carve ^ tht iatagnl will h«n lh« wm vahn for both coma, hot if dvuir the tBHMfcnBklmt it ia l ewaeto lb dMnl cnrv a txans tlx^ valuMof iheibhxnlwiUJi^by 4t«. fl ir « aiKl • ai« aay tv)\^ ekot^ <«rm> n ^acn tka, tf Ihif air^ ««* KeVmI h><wlhM. lh<> i«le«nJ Mtmilei eaaa immI Wtft If thcT an tatuvtwiOHU • \.p\-^ la t)-.v ssntr ttTr^-tk^a. th« of the uMig«l i» *•> h » pMwhK huwewr. fcv two I VECTOB-POTEimAL OP A CLOSED COBTE. 41 to bo intcrtwineid alt^Tnately in op[)oiiit« tlirectionB, mo tlmt th«y mro inMpuruMy linked together though the value of tliv itiU'gruI La zero. Sw Fig. 4. It was tlie divcovery by Gauss of this very integral, oxiiresflitig the work don« on a magitetio pole while dc- acribii^ a iilosed curve in prt>sence of a closed deetric current, and indicating the geometrical ooniMsion between the two closed curves, that led liini to l»mt-nt the small progress made iu the (icomt'try <ii' PnHition since the Lime ui Li-iLuitz, Euler and Vandermondc. We have now, bow- ^' * ever, some progress to repori, chiefly due to Eieuiaun, Helmholtz and Listing. 423.] Let Ds now investigate the result of integrating with nspect to a round the closed curve. One of the terms of n iu equation (7) is If wc now write for bre«ty "'Hi-'- "'Z;!^. "=51%^ w tile iot^TiilB being taken once round the closed curve t, this term j of n may be written rf^ (/sy/ the correspondtti^ term of f TlJs will be dnilff Collecting all the tvnoE of Xi, we may now write ■ tit} 'IGyn rdf diUo \j, rff di >fO <IF. d^ Thin quantity is e\-idently the rate of decrement of w, the miigtu-lj<! jwtcntial. in passing along the curve a-, or in other words, it \» the ma^etic force in the direction of da. By a»«umiii^ da stKceesively in the direction of tho bx« of , y and ;, we obtnin for the values of tlie compouents of (ho lAiagnetic force 42 UAOXETIC B01.K?fO]1>3 AiTO SHELI^ rf« dU tiG^ _ ^/w _ ^ _ J/f ^ ~ f/i, ~ rff rff ' _ rf<.> _ rfO _ ///* '' ^ tic di dn Tho quantities F, 0, //»re the components of the vector-iiotffntiil of ttio mafrnetic shell whose etrength i» xtnity, aad whoae ^ge it the curve a. Tbey are not, like the scalar potential •, function* having a series of raluee, but sre perfectlj detcnDioatc for every point in space. The vector-potential at a point Piae to a ma^etic shell bounded hy H closed curvo may be foand by the following geoatetrkalj oonstniclion ; Let a point Q tiavcl round the closed curve vrith a velocity nHmericnlly equal to its distance from P, and let a second point R start from a fixed point A and travel with a velocity the direction of which is alvrays parallel to that of Q, but whose magnitude is unity, ^Micn Q has travelled once round the closed curve join JH. then the tine JJt represents in direction and in numaiieal mngnitnde the veelor-potenlial due to the closed curve at P. PftrmtUit Eatrgg tfa ihfnttk SietlpUetil ta « AfofMiit Fte!4. 438.] We hare almdy shewn, in Art. 410, that the potential energy of a shell of strength ^ pbced in ft magnetic Seld whose potential is T, is where f, », ■ arv the directtoa-coeuMS of the nonul to the drawn ftviu the ]M<»itivo side, aad the sorftoe-iuteyiiil u OTtir the shell. N'ow this swfitee^talwgnl nwy bs tzwlomed into a line-in by nMau of lite Teetoc>pet«aiti«l ef tlw m^petio Geid, and m xuKf write i J'^-^/Cs+elfffJjA (»; w1h«* (be inteptntiM is exIanM «m* nMoi the chwed enrre » which f^>ni>« the tOite ef IIm H ii ^ iw tiu sMl the dirsetiQa of di being oppoeit* Xa that «t the hawls of n wntch whc« vwwid tram the p.iMlit>->i«Wf>f tbs>WU. If we iMW Mpv«e thni the B Myw He S«U is tfel Am bi « 423] POTESTUL OF TWO CLOSED CURVES. 43 ^^wtii second roogUDlio slioU wIioni> Btrong^-h is ifi', we rnny det«nniDe the value of /"iliroctly from the rccalto of Art. 416 or from Art. 405. If f, m', «' W the Oirectioii-«OKineii of tli<! nomiKl to tlie element dS' of tiie •ecoix) (hell, we have ^=*-Jh-fA-''f,> \vTe r ig the distance between the clement ilS' and a point on th« boondary of the first shell. Now this Burface-iDt«i;ra1 msy he converted into a line-integral ronnd tlie boundary of the second shell ; viz. it is -// iW. (14) Id like manner =*-/i A' d»'. Snbetituting those raluea in Uie expression for Jf we find 'whtrft the Intef^tion is cx(<-nili.-d once round * and once round /, Thia exprcKKion g'lvcf the ]<ot^ntiul energy due to the mutual actit>a of tite two shellK, and is, ajt it ought to be, the same when t and / are interchanged. 'Diiit expression with ita sign reversed, when the strength of each shell in unity, is called the potential of the two closed curves t and s'. It is a quantity of great importance in the (heory of elcyitric currents. If we write « for the angle between the direclionH of the elements dt and ds\ the potential of » and t nay be written (16) JJ^dsJ^. It b evidently a qnantity of the dimension of a line. CHAPTER IV. IKDCCED lUCKETIZATIOy. 424.] Wk have hitlierto considered Hw actual (Jwtribution of BUkgnetiEatioH in u m&^et w given explicitly amon^ th« dita or the invtwtigxtion. We b»ve not made soy usttimptioD w to wtiPttitT lliiti iiugtKtinition is pcnnanout or t«Rii>orary, except in thoeo \axia of ottP Teuonin^ in which we havo £U)>iM3cfd the magnet broken np into Gmall purtions, or small portions remoTMl frona ttie inti;^itrt in such a way as not to alter the ma^uetixatioQ c^ •oy part. NW havo now to coonder the magnetization of bodies with mpact (o tltK node in which it tuj Im prodaoinl and changed. A bar n( irou It^ parallel to the diicetion of the earth's magnetio fovet is found to l)««onM n^netac. with its poles tamed lb« np- poaita way IVmu thov^ of the rartb, or the aame war a> those of ft compan Modh" in ^tallp c^uilitiriura. Any )«)cr<p nf snfl ircoi placml in a aufiMtae SeU is found to rshilni nayneli.- pTi^ivNiMa. IT it be phwvd in a part of the fidd wber« the MMKnaic fi<n« it gnai,, as bekw«ea the pnles of a honse-shoc — gw rt . the numwftisnii of the iron b ecppw inteiwp. If the iran if T<rmi^\-<^ frxMn th* t w n o t t io fwM. its mapMlM properties an Iffwally WYfckwM".! n* iMw uy ar eatiiv^r. If tha nagnetic properliM (/ the i^m A^^wi Mtlurty oft the mugMtic tmt of tk ti.-ld nWh tt ' and \'anuh wk<« il » noKMml &<aa thr it M l,N»1l•.^■ >- - -■'•^. livMS «hi<h i* aoA in th* BM^nriif) M ahu miA i« thr hiM«l MMMVL II w n^ to bm^ n jq^ il a pr n wawe*-: .«|l. Vu the wo*S^- SWih • I 425-] SOFT AKD HABD STBKI,. I » up the tnagndic state eo rcaility ne soft iron. TIjp operation of Uamiiicnitif, or nny oihi.T kind of vibriition, allows liari! iron uudvr the intiiinK'i* of magnetic force to assumo Ihe magnetic s1at« wore readily, and to part witli it more readily wben the taagoetixxng for(H4 is rcmovcf!. Iron which is nin<^ftica11y liard is also more •tiS'to bi-iul and more apt to break. The prooMMw of hammering, rolling, wire-dramn^, and snddea cooling tend to hardvn iron, and that of annealing' tends to ioRcn it. The miLf^etic as well as tlio mechanical dideroncee between steel of hard and »oft tomptr arc much greater than those between hard and mfb iron. Soft sU-el is almost as easily ma;;nctizi?d and de- niAgiietiz4.-d as iron, while the hardest Et«el is the beet material for magnets which we wish to bo permanent. Cuct iron, though it contains more carbon than steel, is not CO retentive of ma^etization. K a magnet conlil be constntctvd so that the distribution of its magnetization is not altered by any mugnctic force brought to act upon it, it might be called u rigidly mn^etized body. The only known body which fullils this condition is a conducting oircuit round which a constant electric current is mjide to flow, Such a circuit exhibits magnetic properties, and mu}' therefore be called an elect romiignct, bat tbeso magnetic properties are oot nffeetcd by the other magnetic force* in the 6cld. We shall return to this subject in Part IV. All actual magnets, whethor mode of hardened steel or of load- stone, are found to be affected by any nugnetic force which is brought to bear upon them. It is convenient, for scientific purposes, to make a distinetJOB iMtwccn the permanent and the temporary magnetization, dclining the permanent magnetization as that which cxist« independently of the rangoetic force, and the temj>orury iniigneli/.attou as that wliich depends on this force. We mu«t obterve, however, that this diittinction ia not founded on n knowledge of the intimate nature of magnetizable subetances : it is only tJie expresiiion of an hypothe«i8 introduced for the sake of bringing caloalation to bear on the plienometui. Wo shall return to tlie physical theory of mngnetizution in Clmpter VI. 42iS.] At present we shall inn'stigate the temporary magnet- ization on thr nwiimption that the magnetization of any particle of the mibstitnee deiwnds wicly on the magnetic force acting on INDUCED HAONBTIZATlOIf. 4 « are fouB^^ ttiat particle. This magnotio force may arise partly from esteroal cnum'ii, and partly from the temporary niugnetizatioa of neigh- bourinj^ partklcs. A liody tlius roagnctized in virtue of the aotioa of magneli' foroc, is wiid to be ma^otlzed by induction, and the magnetizatioiii ii anid to bo induced by the niagnetiziug foroe. 'Hie inajtiiiHization induced by a given magnetizing force diflvrs in difli>rt^nt suhstaDoea, It is greatest in the purvet and sofleet iron, in which the ratio of the nuignetlzatioD to the magneUc forottj may reach the v&luc 32, or even 45*^. Other subaUuices, anoh as the metals nickd and cobalt, are cu)>able of an inferior degree of magnetization, and all sabstancea whfu subjected to a suffieieDtly tAnog magnetic force, are found to give indicatione of polarity. When the magnetization ts Id th« same direction astfae fi>n-e, m in iron, nickel, cobslt, &c., the eubetauce k tailed mitgnciio, t'erromagnctic, or more simply Magnetie. When the indnced magnetizatioo is in the direction opposite to the magoetie foioe, as in bismuth, ke^ the Eub«t»nc« is said to b« Diamagnetic. ^u In all these iRib«taoces the ntio of the magnetization to the^| magitetio force which produce* it id exceedingly entail, being only ^' about —tWvn >» ^^^ '^*^ ^^ bismnth, which it the most highly ^y diatnagnetio aubetanco known. ^M In ery«t«llized, »tmincd, and organiEed substances tlie directloD of the magnrtixalion dc<i-< not always coincide with that of the nwguatie forw which produces it The nJation between the 000- ponenU of magnettaation, referred to axes fixed in the body, and tht««> of the ntagnetto force, may be expnseed by a system of thm tin<«r equations. Of the nine coefficietito involved in these equa* tions «« dull shew that only six are independent The phenomena afbcdiaaof th» kind are classed uadcr the name of Slagnecrystallio lltKOOSttHI^ Whea placed ia a field of u^inetao fbttn, nyatals ind to set ihewMtlvn* so that the axis «t s;ntAt^ parami^niitiii. or of kwt 4iMas«netM. iA^KtM i« parslM la th* l»e« tf See Alt 4SS- In spft irMt the directkta of tW mnailiialiiM - u that of the aucavlw foR« at tha poiu. and ttt ^mB Iha BUgnstia foiw U« niafiw^ualiea ia aawly - • naUa, .Vw« Jrt« «qk .s,« jK C|«L. 11 I I PBOBLEM OF DtDUCED MAOSETIZATIOK. 47 IAs ihe magnetic force increases, however, the □ugitL'tizatioa in- etoucs nore slowly, aod it woald appear rrom experimeiitti dctcribcd ia Cbap. VI, that there is ft limitiiij> valao of the mii^nrtuuitioa, bejond which it cannot p«s, whalvrcr b« the raltie of Mm mognt'tic force. In the following outline of the tbeory of inducnl magi>^i^i''>> we shall be^n by EUpposing the magnetization profKirtion*! to the magnetic force, and in tbo same line with it, IJkfinitum of the Cbefieteni ^ Inductd MagntiiMiion, 426.] Let ^ be the ma^^netic force, defined as in Art 398, at KOj point of the body, and let 3 be the mafrnetization at that point, then the ratio of 3 to ^ is called the Coefficient of Induced Ibgnetization. Deooting this coefficient by «, the fundamoatnl tsjuation of indQCcd magnetism is 3 = .«. (1) ■ The coeRinent k \» positive for iron and {)arainagnetiu subKtancvri, and negative for bivniuth and diamagnt'tie substances. It reacbe* the value 32 in iron, and it \» »aid to be large in the case of nickel and cobalt, bnt in all other casefl it ia a very email quantitv, not greater than O.OOOOl. tThe force ^ arises partly from the action of magnets external to the )>ody magnetized by induction, and partly from the iiiduoed magnetization of the body itself. Both parts satisfy the condition of having a potential. ^7.] Let y be the potential dae to magnetism external to the body, let H be that due to the induced magnetization, then if V is the actual potential due to both causes »u=r+a. (2) Let Uie compooenls of the magnetic force ■^, resoU-cd in the directions of x,g,t,\>o a, fi, y, and let those of the magnctizatioa 3 be J,.d, C, then by equation (1), Multiplying these equations by dx, dy, dz teepectively, and adding, we find ^ Adx-i-Sdjr+Ctl: = *{ad4S+fidjr+yJt). 48 INDCCED MAOKETIZATIOK. [427- Bnt sinoR a, and y are derived &oni the poteotial U, w« maty writv tlie second memljer —iciiU, Hciive, if M is constant tbrou^hant the substanee, the Rni meml tniiDt ftlao be a eompl«tc difTerontial of a functioD of jt, y ukl which we shall call ^, and thu oqiintion bi^comcs vb«n ate A = '^, B = 34, (5) The magnet ization is therefore lamellar, as defined in Art. 412. 1 It «'«» shewn in Art. 386 that if p is the volume-density of &ce mn'metisni. which becomes in virtue of oquation* {3), "^-ydi But, by Art. 77, ''=-'W fda d0 4iy dy ^ is -f-£-- Henee (l + 4«)p = 0, vrbeooe ^ = f6) ihrougliout tJie substance, and the mognotiiation is therefore sc4e- noidal as well as Unx-llar, Sve Art. 407. There is therefore no free ■Qaguetism except on the boandi stirfnce of tlie body. If r be the normal drawn inwards from tl surface, the magiKtic BnrfRoo-densi^ ia "-% (') The po(<-ntial U due to this nagnptization at any point may therelorv be found (Wm (he mriaee-inte^nl The value of Q will be finite and continuoas everrwbere, and will Mti»fy lAplan-'s pqaatioit at emry point both within and without (h«< ■nrfiii'e. If wv di9tin{;ni:»b by «a aetvat the value of n iwt»i*le the MiHace. and if »' be Um nonoal drawn ootwnnU, ire have at llir suHikv (8P tt'^u: (!W 428.] POiasOlrS METROD. 49 ^ + 3^ = -4ir<r. byArt.78, We may therefore write the surGk«c-coudttioa IciMic tlie determination of tliu magnetism induced !u a homo- geneous iMUttpic body, bounded by a surface S, and acted upon by external nui^nutiu forces whow potential is ?', may be reduced to tlie following umtliomutiuil problem. We mu»t find two funotioni; il and il' mitisfyingf the following^ conditjonn : Within the surface S, Cl musi be finite and continuous, and must Kstisfy XiHplact^'s eipiation. Ontxido the surfa^'e 8, Cf must be iinite and continuous, it must vnuiHli nt an inlinite distance^ and must satisfy Laplace's equation. At <f\tiy point of the surface it«i-If, H = il', and the dcrivalives of A, Q.' and F with respect to the normal must satisfy equation ThiM method of treating the problem of induced magnetism is due to Poit«OD. The quantity i which he uses in his mcmoiTs is nut tlie nme m «, hut is related to it as follows : *-BK{i-l) + 3i = 0. (11) The coefficient r which we have here used was introduced by J. Neumann. 428.] The jiroblem of induced magnetism may be lrc44t<^ in a different manner by introducing the quantity which wo have L'slled, with Faraday, the ilagnetio Induction. tThe relation between ?*, the maguetic induction, ^, the magnetic Ibzc^ and 3, the magnetization, ie exprceeed by tiic cijuattoa « = ^ + 4ir3. (12) The equation which expresses the induced magnetization in terms of tlie magnetic force i» I I I '*. (18) Tqt,n. ]Ieace, etiminatiDg 3, we Bod © = (1 + 4.,)^ (U) as the relation between the mftgnetio indactioD and the magnotic force IB Bubatanees whose magaetization is indaced by mognvtic force. In the rooiit general caw « may be a function, not only of the position of tbe point in the substance, but of the direction of tbc vector ^, bat in the case which we are now consideriag jc is a numerical (|uantity, If we next write fi = l + 4ir«, (18) we may define n as tJie ratio of the magoetio induction to the magnetic force, and we may call this ratio the magnetic indoctiTo cajnoity of tbe snbetanee, thus distinguishing it from k, tbe co- efficient of induced magnetization. If we write U for the total magnetic potential coropouoded of T. the potential due to exti'mnl csuecs, and Q for that due to the induced msignetization, wo may express a, 6, e, the components of magnotic induction, and a, fi, y, the components of magnetic force, as follows : jf/ JU ' = ">' = -** 35" * J The components a,&,e satisfy the Bolonoidal conditioa 4a d6 de (»•) (17) Hence, the potential U must satisfy Laplace's equation d*U iPU d^V , . at every point where ft la oonstant, tlint is, at every point within the homogenoouH substance, or in empty q»w. At the surfiice itself, if i- is a normal tlmwn towards tbe magnetic euWance, and v" on« drawn outwards, and if the symbols of qnan* titiiMi outside the sub«tance are distjngaished by accents, the con> ilition of continaity of the magnetic induction ia dv dp dv df Ju dv 0; (19) 439 ^Kor, by eqaatioDB (16), FARADAY'S THKOBY OP HAOSETrC ISDCCTION. 0. 6t (20) do "^ dv' II, the coefficient of indtiotion outside the maici^et, will be uuit^ unless the enrrounding' medium be mttgoetic or disma^etio. |h If we stibslitate for U its vnlue in terma of Tand ii, and for H fi its value in terms.of k, we obtain the same eijuatioD (10) as we arrived at by Poiason's method. ITbe problem of induced ma^ctism, when consideri'd wi(h respect to the relation bctwcrcn miignctic induction and ma^ettc force, corresponds exactly with the problem of the conduction of electric enrrente thToa;fh hctero^neoue media, ae g-iv<-n in Art. 310. The magnetic force h derived from the magnetic potential, pre- cisely as the electric force is derived from the electric potential. The mafpietic induction is a quantity of the nature of » fli:s, and satieGes the same conditions of continuity as the electric current does. In isotropic media the matfnetic induction depends on the mag- netic force in a manner which exactly corretiponds with that in vrliicb the electric cnrrent depends on the electromotive force. ^B The specific magnetic inductive capacity in the one problem corre* ^^ xponds to the specific conductivity in the otlior, Hcnoc Thomson, in his Ti</>ry o/ Indnetd Miiffnntign (Itepmif, 1872, p. 48<), has called this quantity the permea&i/ify of the medium. I We are now prepared to consider the theory of induced ma^etiam fnmi what I conceive to be Faraday's point of view. When magnetic force acts on any medium, whether magnetic or ■liaiiingnetic, or n u(ml, it produces within it a phenomenon cjilltnl Magnetic Induction. Magnetic induction ia a directed quantity of the natnni of a Bus, and it satisfies the same conditions of continuity as electric curreuta and other lltuces do. In isotropic media the magnetic force and the magnetic induction arc in the same direction, and the magnetic induction is the product of tie magnetic force into a quaiitity called the coefficient of imluctjon, which wo have expressed by i*. In empty space the coefficient of induction is unity. Tn bo<lii'ft tapable of indnced magnetization the coefficient of induction is l + 47R = /t, where k is the i]nantity already defined as the co* efficient of induced magnetization. Im 4S9.] Let f^ ^4' be the values of fi on opposite udes of a surface 52 INDDCBD MA0NET12AT10N. ccpAriiling two media, then if F, f" are the potentiiUs in tin tvio media, tlie iiiflgnciio forces towards the surface Id the two media dv . <ir are -r- »nd -r-r av an Tb« qimntitie« of magnetic induction through the element of; rarface dS utb n-j-dS and fi' -jy dS in the two media rupcct- ivcly reolcoiii'd towards dS. Since the total flux towards dS is zero, dF ,Ar Bill by the theory of Uie potential near a surface of density tr, I '»^+M't-t = 0. Hence dV dV , If X, is the ratio of the Bujicrficinl magnet ization to tite normal force in tlic first medium whose oocfGcient \t ft, we have 4 Hence V, will be positive or negative BCondingf «$ it i» gTcat«r or le«$ than ft'. If we put n t= ivx+l and fi'= 4iric'+ >. In thin exprowion k and k' arc the coefficients of induoed Baag-J ni-tiKatioii of the fimt and second media deduced from expennentoj made in air, and k, is the coefficient of induced mitgnetixatioo of tlie tint medium when surrounded by the second moilium. If k' itt greater than «, theu k, is m^^tive, or the apjiareufcl RULgni-tiTntion of the 5ni medium is tn the oppositv direction from the nisgiii'ttzing foi-ce. lliun, if a veKHv! oontainiug a weak aqueous solution of a para- magnetic saii of iron is su^ndcd iu a strouger solution of the same salt, and a«ted on by a magnet, the vetiwl moves bx if it vnn magnetixed in tlto opposite direction from that in which »^^ magnet would set it«clf if suKpcnded in the same [i1m«. ^M This may be explained by tiw by^thesiM lluit the solution in '" tlic vo&scl is really maguetixed in tiie lainc dirwtioR aa the mag- nelao force, but that the solution which surrnunds the resatil is mi^nctizcd more slrongly in the same direction. Uenoe the vessel ta like a weak mngnet placed between two strong ones all m.*"! 1 430.] POISSOS'S TUEORY OP MAGNETIC INDPCTrOS. Kd » netizcti in the same direction, so that op])OKi(>^ poles aro in contact. The north pole of tlie weak taigvet p»tiit« in the same direction as thow of the strong- ones, but since it is in contact with (he south pole of u ittronj^r tna^ct, there is an excess of soutli tnii|^iieti8in in the neighbourhood of its north pole, whicli cau»>ii tlic small magnet to appear oppositely magmetii-eti. In Komc suhslanc-es, however, the ajiparent magnetization ia ne^tiv« even when they are sufi])ended in what is called a vnctium. If we atcume « = for a vacuum, if will lie negative Tor thcte sut»tances. No substance, however, has been discovered for whieli « ha» a negative valne numerically gi^eat«r that — , and therefore for nil known Habst-ances fi is positive. Subit«nO(« for whit-b k is negative, and therefor.) ju le«s than unity, arc culk-d Diama^et-ie eubgtanoes. Those for which k \* poaitivc, and ft greater than unity, are culled Paramagnetic, Ferro- mngnotic, or «'mpiy magnetic, substances. Wc shall conndcr the physical theory of the diamagnetic and jnramagnctic properties when we come to electromagnctism. Arts. 831-815. 430.] The mathematical theory of magnetic induction was flret givvn by Poitison *. The physical hypothesis on which he foundtNl his theory was that of two magnetic lluid>«, an hypothesis which lias the same mathematical advantages and physical difBculties as the theory of two electric fluids. In order, however, to explain the fact that, though a piece of soft iron can be magnetized by todudion, it cannot be charged with unctpial quantities of tlic two kinds of magnetism, he suppoees that the BuhstAUc<- in general is a non-conductor of these fluids, and that only certain Kinall portions of the substance contain the fluids under circumHtanccft in which tbcy arc fiv* to obey the forces which act on them. Thwc nnall magnetic elements of the substance contain each pre- cisely equal quantities of the two fluid", and within each element the fluids move with perfect fr<'edwm, hut the fluids can never pa»» from one magnetic clement to another. Tlie [iroblem then-fore in of the Winn- kind as that relating to a number of small coniluctom of eli'(;tricity disaemiuated through a dielectric in>uhiting medium. The conductors may be of any fomi provided they are timall and do not touch each other. If ibvy arc elongated bodies all turned in the same general • aiORMm tU rituttiut, IBM. 04 IFDUCED UAOKETIZATIOK. L430- ** = f»i dirwUon, or if lliey arc erowded more in od« dtrectioc than another, th« mixUuia, as Poisson himself eli«we, will not be isotropic. Poiseon thcrefure, to avoid usoteas inlri«acy, examinee the cose in which L>Hi.'h mugiHitic element U spherical, and the elements are dibsem- ioati-d without regard to axes. Ue supposes that the whole volome of at) the magnetic elemente in anit of volume of the eubstanoe ia t. We have already considered in Art. 314 the eleotrio oondoctivity nf a medium iu which small spheres of another medium are dis- tributed. If the conduotivit; of the moditim is /i,, and that of the spheres p,, wo have found tliat the conductivity of the composite syetetn le Futting p, = I and Hj = x, this becomes l + 2it This <]mintity fi ii the electric condoctivity of n median eon- Hitrtin^ of porrectiy conducting spheres disseminated throug'h a m«iUum of ootidiKiivity unity, tiie aggrt-gute volume of the spheres in unit of volume U-ing It. The *yiubol fi uhni rvpmvnt* the oocfficieot of magnetic iodnctien of a milium, counititing of it{)herM for which the pcrmeabtltty it infinite, disst'miiiaU-d ihrou^ffa a mMliam fur which it is nnity. Hi« itjubol i. which W0 shall call INiiMon's Ma|>ncttc Coefficient, rtfumaata thv ratio of th« volume of the ma^etic elements to the whola Tolunt of tlw substance. Th« symbol K i» kai>wn as Neumann's Cuefficicot of Magnet- tsattun by IndiHtion. It ir morv coovmii-nt than I\>moo'b. The «yub><l >i we shall call th« OwtScicnt of Ma^faetie IndactiotL Ifa advanti^ is that it fiKilitatca the tnuuformati.iu of magBctJe probkua int» problem* retattaf to eWtridty sad h«au The rclaticawuf thewthrtvaymbolsarvaafBDows: i=: U If M ptll k - M. Ow mIm ffivM hy tUUa a* KpsuHots M 4 4 4 d 43a] poisson's thboet of magnetic isopctiok. 55 soft iron, we find i = \^. This, aecordiBg to Poisson's theory, IB the ratio of the volume of the magnetic molecules to the whole volume of the iron. It is impossible to pack a space with equal epheres so that the ratio of their volume to the whole space shall be ao nearly unity, and it is exceedingly improbable that so large a proportion of the volame of iron is occupied by solid molecules, whatever be their form. This is one reason why we most abandon Foieaon's hypothesis. Others will be stated in Chapter VI. Of coarse the value of Poisson's mathematical investigations remains imimpaired, as they do not rest on his hypothesis, but on the experimental &ct of induced magnetization. CHAPTER V. PABTICULAR PROBLEMS IK MAOSETIC HTDDCTKHT. A Jfolhte SpHerieat Siett. 431.] TuK firet cxwnple of tli* complete Kolution of a problem in mnpiftic iii^ductioD wan tliat given hy Poisson for the com of « hollow ephi.'ri<-al «hell act«d on by any magnetic force* whatever. For Mitnplii'ily w« Bball suppose tlw origin of the nuif^netic forcee tn bo in tho epftM oat6)d« the ehdi. If f <leuot«8 the potential <luc to the external magnetie ayat«m, we may expand T in a series of solid harmonics of the form where r is the distance from the centre of the sbdl, S( in a sarface b«TmAnio of order i, and C, is a coi-fficicnt. ^1 This Bcne« will be convcrgrnl pTx>\-idL'd r is lees than the dtntauee^^ nf (he nvnrMt magnet of the sj-nt^m which produce* tbi« pot^ntiaJ. Hence. lor the hollow spherical shell and tl»e tpoee within it, &i» expansioii is eonTFTgmit. Let the external mdiue of the ehell be a. and the inner rwdins «j , and let the {>otential due to it« induced niagoetian bell. The form of llie f\inet)oa Q will in general !•« diflrrrat in the hollow vpnce, tn tfam substance of th* shell, and in tito space beyond. If we expand these ninctions in harmooie eeriea, then, confining oar ittmtioB to those tcnns which inralre the ntbre hannonie St,^^ thiill find that if il, t* that whieb comspottds to the hollti*^^ s)iA«> within the shell, the expnnnon tt Q, mmt be in pc«itjVB har- nuwic* nf ' '' ..> .f, .^ >*. becnuse the potvMal moA not beeone infinite n .; >)ihvTv whive ntdius iE«|. lo the mbaUnoo of the shell, when: r^ hm l<etw«ea «, nnd «,, | the wrtN BUtjr cwatnin K^th [v«iti>e wd MgMiTe po«*» oT r, t'f t)M> form J,^f*-f i£j5,r-e'i\ Outside the oWll, «Imi* • is gnmWt than ■,, auH* U» aavs 43 1 J BOUOW SrnERICAL SI1BI.L. 6i I must b« ooDVCTgcnt however great r may be, wc mmt have only nogntivo i<owew of r. of the fonn The condition* wlitcli umxt be satisfiod by the function Q arc; rt niuttt he (I) linit«, nod (2) continuous, and (3) must vaoiwh at it« dixtAnre, and it must {*) cTuywliere nitisfy Lnplace's On norouDt of (l) ^i = 0. On iKxount of (Z) when r = u, , and when f = it,, On aooount nf (3) A, = n, nad the condition (I) \» KatixGcd everywhere, wnce the fun<-tions are hannonie. But, bcsidcH the«e. there are other condilioim to be iiatisficd at the inner and outer curfaee in virtue of eqiiatioD (10), Art. 127. At Die inner aurface where r t=. <7,, da, da, dF ilr dr dr atxl at the outer vurfnce where r = a^, dV (,+4„)'^_— . + .,,,- =0, -(>^-)'^-'^- B K T- = 0. dr (3) From these conditions we obtain the equations and if we pot ^. = ^ : 7-irrr. («) (i + 4,.)(2'+i)*4(4«<fi('> >)('-© ) we find . M-H. A, = ~{A^^f ,(i+ I)(l - (?1) ).V, C. (9) ^«=-4,«.[2, + l+4ir«(.-+l)(l-(^')"*')]A'<C.. (10) B^ m <«i(2i+ 1) V'*'A; C|. (I I) S,=-ii,Ki{2i+i+i^M(i+l)){a»'*^-a,"^^)N,C,. (12) TheNe qnanlilie* being euWituLed in the liarmonic cxpunstons j>ive thi> part of the potential due to tho magnetization of the shell. Tlie quantity Iff '" alwayit positive, since l-f 4irK can never be B^ativc. Hence A, \» always nogative) or in other wordts th« UAaKETIC PBOBLBHS. I kction of the magnetized »l>el) ou a point within it is alwayi poHcd to that of the cxtvnial magnetic force, whether the she |HiramagTietic or diumiigiK-tic. Tlie actual value of tJie resultant putootiul u'ithin the »hull in or (l+4w«)(2»+»)*jV|C!S.''- (H)' 433.] When k is a large number, as it is in the cut of soft: iron, then, unlvsH tlie shell is very thin, the magnetic foree within it^i is l>ul a omall fraction of the external force. ^^ In thiK way Sir W, Tliomsoa has reudercd his marine galvano-^^ met«r independent of external magnetic foroe hy enchwing it in^ a t'Uhe of oofl iron. 4S8.] Tlie cAne of grefltest pra<!tica1 importanee is that in wlik (Ml. Is this cofe (") 9(l+4,r«) + 2(-U«)'(l-{^f)' ^,— 80«)'(i-Q)').v,r^. J»,= I2wita,'>',C,. 5j = -4»«(3 + 8ir.)(V-V)'V,Ci. The magnetic foioo within the hollow shell is in this nase and equal to C.J 3('-H") « 9(l + ««) + 2(<w.)»(l-{a)) If we msh (o determine « by measnring the magnetic within a hollow shdl and nuuparin^ it with the external fortx, the limt value of tite tJiicknMs of tke ibeH nay bo Iband (Vom the mjuattaon Ihlin^Tlttfir fcnrt ia^de the shell is tbes half of its nke ontadt. Siui'c, in ths nan of mw, ■ i» a number between SO aad 30, ihii-kuMM of lb» ■btU OHfbt to l>o sUmt 1^ boiMlradth part of r ladius. Tlii> awtlvMl i» a)>ptiakt>le naly whui the value of « bry*. When it is v««> omall the value U A^ Umnn UKnnbJc' Moe H di^icDila on tiM *>{uar» t^ «. idt. 1 d 4M0 srnERrcAt, snBtL. SS For a nearljr solid epkere with a very small spherical hollow, llTK 3 + 4ir« 4gic 3+4 vx (18) I fron I I The whole of this inreBtigntion mig4tt have 1»en jf^aoed directly from that of oondnctioii through u Hjilierioal stiell, as given ia .312, by putting ;fj = {^\-^Avi)k^\a the exprassiona there given, remembering that //, and A. in thi' [iToblem of conduction are eqni- ralent to f^ -f- ^, and C^ -f A., in the problem of magnetic induction. 434.] The corrcHpomling solution iu two diinensions is graphically represented in I'ig. XV, at the end of this volume. The lines of tiuluction, which at a distauoc from the centre of the figure are nearly horizontal, are ri^jirescnti^d as disturlx^ by a cyllndric rod magnetized tran)iven>ely and placed in its position of stable equi- Ubrium. The lines nbich cut this system at right nngliM represent tbe eqnipotential tiiirfacm, one of which is a cylinder. The targe dotted circle represent* tl)i> Kcctiou of a cylinder of a paramagnetic eubstance, and the dotted horizontal stmight lines within it, which are continuous with the external lintr-s of induction, represent the lines of induction within the subsbiuce. The dotted vertical tines represent the internal ei|uipoli>nIinI snrOiccs, iind are c<^ntinuous with the external syst^'m. It wilt be obeervt-d tbul tlic lines of iodoction are drawu nearer t^tgethi^r within the substance, and the equipoteatial surlaees are vepurutcd farther apart by the paramag- netic cyliuder, which, in Ihc language of Faraday, conducts thtt 'Uoee of indnctioD better than t)ie xurrouiiding medium. Tf we consider the system of verlifjil liaes as lines of induction, and the horizontal syitem as ei)ui[iot<:ntiul surfaces, we have, in the first place, the caac of a cylinder magnetized transversely and placed in the poKition of unstable ei^uilibrium among the lines of force, which it causes t^ diverge. In the second place, considering tbe large dotted circle as the section of a diamagnetic cylinder, t}i« dotted straight lines within it, together with the lines external to it> represent the eflect of a diamaguetic substance in separate iog the lines of induction and drawing together the e(|ui potential sar&ccs, such a substance being a worse conductor of ma^etic iulaction than the surrounding medium. 60 MAOSETIC PROBLEMS. [43- Catt of a 8pher« in teiUi (Ae G*e^!euft of MagntUMtiim are Dijfereni in Different Birevtitm*. 435.] Let a, fi, y be the compDnentd oi magnetic force, and A, B, C those of the nia^iu'tiKatioii at a.ay point, then tite moet geoerai lincitr relation between theac ({uantitics \s given by the equations A = '■,a+;:^/3+ j,y,j where the eoeffieicnt* r, /», q are the nine coefficient of magnet- ization. Irft lis now Bi])i»iee that theec arc the conditions of magnet- ization within a iiphere of ladiHti a, and that tlic magnetization at every point of the Niilwtanoo w uniform and in the aame directiim, having the components A, H, C. L«t ti8 aUo vu)>poK4> that the external magnetizing force ie also tinifonn and puralk-l to one direction, and has for its componcote I. y, z. The ralne of T is therefore r = -{.Vi-+f> + ^r), (2) and tliat of 0.', the potential of the magnetisation outside the ^herc, i«, by Art. 391, At a^ ^ ^ cr=^^{^*+5^+c4 (3) The value of XI, the potential of the ma^etixution withia the Thu actua] potential within the qihore W F+ti, to that we ohall hnvc for tbs compoDents of the mag^otic force within the sphere y=Z-ivC. ) Hence (l+J»rOJ+ iwp,B+ jTftC.r,X4ftr+y,2; Stilving then eqiuUiona, w» find i I CRYSTALtIXE SPHEBBi I I (8) ^ where Ifr{^ ''1 + Ji'('-»'i-ft?j + f,r3-ft?,) + (8'')'-0. ^P( = ft - i " (£« ?.-ft n ). to-., wlwMe i> is the detvrmiuant of ihe ccx^fficients on the right nd« of c(|aittions (6), and 1/ that of the coefficients on the left^ Tho new system of copfBci«nU p. if, / will be symmetrical only wh«a the system p, q, r a ttymmetrical, that is, wlien the co- rfficienta of the fonu p aro vquui to the corresponding ones of tb« form q. 436.] ^Tbemotnentof tbecoaplet«DdiDgtottin) the sphere about tlie axis of j from 1/ towards z is found by eonsidcrin" the foroea 011 ton elementary volume und tidiing their sum for the whole sphere. The result is = %-.anP,'ii'-9:y^Hft'-r:)TZ+J{q,'Z~n;Y)). (9) If we make X = 0, r=Fcos0, Z=Fmn0, thi« corresponds to a magnetic force F in the plane of y:, and inclined to y at an angle 0. If we now tnrn the sphere white this force remains constant the work done in turning the sjihere will f" be / Ld0 in each compIet« revolution. But this is equal to j;r««'?-«(/V-?.')- (10) Hence, in order that the revolving vphorc may not become an ioexbaustible source of energy, /),'= j,', and similarly p/= y/ and These conditions hIicw that in the orig-iual iH|untioDs the coeffi- eieot of .B in the third equation in equal to that of C tu the second. > Ilk aanalitT of tbe cwBi':! hiIh p uul f limy be hIiowd oi fi'Uuwa : Lst the fonu Mto; oa tnn •lilioro turn it ationt » iliunrtvr irlim- (ilrwtfuu-vodinai arc K f, ' thfoogh Ml Nistc >f : Uicn. If W douotc tin mngj <;>f \ht iphorc, wo L>vu, by An. 430, 811I if tlw KIM of cwrdlribUi bn filed In tbe iplitn wa htv« in <gii««queiioc of tbo raUtiiMi Bjr-(rV-2/i)>«,«tc. B«M« v« nwv pnt Tlikt tlw nrolrinc «ph«n niaj' ni)t bocam* k »ource of enoiuy. tbo ciprtarion cm the light hmul o< llMlMt vqtutiun iiiuat b« » pnrflKt Jiiruniiiikl. Utioue. lioM A, B, C ■fB Wninr ItndiiHu of i. Y. Z, it follow* Uot W it t> ijcuKlivCiu fanctluii uf JT, 1'. Z, amA 1^ rcquirad mull i> iX once ijeducwi. Sw >bv Kir W, TliuiiiNo'i R«i|iiiiit of Paptn tm Ettdriettjf and Majnaism, pp. 4S0 461.] 92 SUQXETrC PBOBLKHS. and 80 on. Ueaco, the ayetem of eqnations is eymmclrioil, iind Uto equations become wbeo referred to the priDvipnl »xt» of m«{p- Detizatioo, A = S = i + J"'-. 1+5'"-* X.1 Y. C = z. (uy llic moment of tlio couple tending to turn t)i« sphere round th« axis of fit i= JM» f.— f. XZ. In most cases the diS^encos between the cocSicient« of magne ixatioa in different direction* are very snuU, so th»t ne may pat ( 1 1 0^1 lei I Tliio is the force tending to turn a crystalline sphere about asiii iif X from / toiriirdis /. It aliraj-* tends to plac« ttie axis greatest magnetic eoefiieient (or least diamagnetie coelRcientJ parallel to Ute line of magnetic forec. Tlie oorrosponding case in two dimensioiu Lt represented I-'ig. X\l. If we ynppose tlie upi>er aide of the figure to be towards the north, till' figure reprewut^ the lines of force and equipotential wirraeea as dinlnrbed by a trau9verM>iy magnetiied cylinder placed with the north side eastwards. Tlte re»uIUnt force tends to turn tlw cylinder from eaat to north. I'he large dotted circle repretieiits a aection of a cylinder of a crystalline enbstaiMe wliicli has a larger corfllcient of indoelioB aloof; an axis from north-«a»t to south-west tltau along an axw fnun north-watt to •onth-euL The dotted lines within lli« (Side n-pn-M^nt the lines oTindootiofi aad the e^uipotential cur&cea, which in this cbod are not at right angles to each other. The T«»uttaBt (bn» on the cylinder ia avideiiUy to turn it from east t*t nortli. 497. 1 Tlte eace of an e)tips«Mit placed in a field of mufnrm and l«t«Uel Bta^Betic forea hm booi nlvnl in a nnr ingcniooB manner U r is the piitcati^ at the pmt (»,/, s^ ^M to the gnritntioii «i a My of My form of luufotm <hnly a a,-^„ £:»the ELLIPSOID. > poUittinl of tlie magnetidm of the same body if aniformly m^^ tuAiv^ in the direotion of « with the intennty 1= p. dV For the value oi — -j-hx at any point ie the excess of the value of r, the potentinl of the luidy, ahow l", the toIhc of the potential when the body \» moved —his in tho direction of «. If we sup|>o«ed tht! Imdy »iuftc<i through the di»Unee — d;r, and ita density chang<>d from p to — p (that is to tay, made of repulsive instead of attractive matter,) then — j~hi would bi' tin- i>otentiul doe to the two Iiodies. Now consider nny rlomotidirj- portion of the hody fontainin^ a volume hr. \\» (jiHintity is plv, and corresponding to it there is an element of Ihn ithilVd Wly who^e quantity ia ~ptv at a distance —hx. Tlic cHect of theeo two flt-rat-nta ia equivalent to that of a magnvt of strength pBr and length hx. The intensity of magnetization is found by dividing the magnetic moment of an element by its volume. Tlie rcEnlt i* phx. Hence — t- 8* ia the magnetic potential of the body magnetized with the intctuity phx in the direction of z, and — XT '* *^' ^^ the body magnetized with intensity p. Tliis potential may he also considered in another light. The body was shifted thri>ngh the ditttance —hx and made of density — p. Throughout that [lart of space common to the body in its two po«itionB the density i» zoto, for, as far as attraction is oon- iU!m«Kl, tho two eqoal and oppodite dcusitteif annihilate each other. Th«Ta remains therefore a shell i>f positive matter on one side and of negative matter on the other, and wc may regard the rosultsnt potential as due to these. Tlie thioknoM of the bbell at a point where tho normal drawn outwards makes an angle < with the axis of f is 07 COS ( and ita density is p. ITic surface-density is therefore /)&« coa r, and, in the cose in which the pot4>ntial '< — >~i "i^ surface>deoMty is p eog t. Id this way we can find tlie magnetic potential of any body uniformly magnetised parallt't to a giv<-n dircution. Now if this uniform magnetization is due to mugnelio induction, the mag- netizing force at all points witiiin the body niiul also be uniform ami parulleL Tilts force conststs of two part*, one due to external causes, and 64 MAOHBTIC PROBLEMS. [43; ^ th« otker due to the magnetisation of the l)ody. If lln^refore th« exteniftl ma^etii^ force U iiaiform anil panillt^l, Iho magneliti force' due to the magneti]'3tion must also l>o iinifomi and panllel fi all {lointa within the body. Ueoce^ in order tliat this metliod may lead to • solution of the problem of magnetic induction, -fz must bo a linear fuDct)<»i of llie coordinate* x,y, s within the body, and therefon; F mxiai a quadntiv function of the coordinates. Now the only cases with which we are act^uainted in whiob is a quadratio function of the coordinates within the body are thoM iu which the body is bounded by a complete surface of the seed degree, and the only case in which such a body is of finite dimen KiouK is when it is an ellipsoid. We shall therefore apply tb' method to the case of au ellipsoid. Let be the equation of the ellipsoid, and let •t>^ denote the definite integral 7. *Afl»j.A»iaa-i.rf,s\f^.i.Aii\ • "/ Then if wc make inalc M=iv^ .**« d(i^r X=i '4 tlic value of the potential within the clliiwoid will be ^=_|(i,« + Jfy+3V) + oon9t. (4) If the ellipsoid is magnetized witli uniform intensity / in a direction making angles whose cosines are /, m, m with the axes of s, g, :, so that the components of nia^netizatioD arc ^^ A=Ii, B^tm, C^Iii, ^ the potential due to Uiis magnetization witlun the ellipeoid will be a^-I{Lh!+Mmfi-NK:). (a) If the ext«rDal magnotitiag force is ^. and if it« componenta are X, i', /, iU potential will be ■Hie oomponont* of the actual magoetiring force at any point within the body are therefore x+JA r+j?j/, z+cK (rj • Sot TVuMnn luul T»H"< SMniat Fhtonfig, { C23l I 438.] ELI,IP30n>. 65 I The nuwt geiiCTHl relatioiui betwifen tho mngncttzalion ani) the nugtielixin^ force arc g^ven by thiwo linear eciuationit, involving nine coitffii'k-nttt. It is neoessary, lioweror, in ordor tt> riillil tltc condition of tho coniM^nration of energy, ttiat in the «a«e of ning'ia-ttc joduction tlirM of theitc shoald be equal respectively to other three, •0 that we shoukl bave c ^ K\(x+AL)+K\(r+Bi/)+K,{2+c.y).) From thOM! <ffunti<in£ we may determine A, B and C in terms of X )', Z, imd tliis will give the most general tiolution of tho problem. The potential outiiide the ellipsoid will then l>e that due to the ougTieti/Ation of the ellipaoid together with that due ia the eictenial magnetic foroe. 43B.] The only case of practical importance is that in which «'. = «"» = "'s = 0. (9) We have then ^ = B = '-T^'- (10) -/l^^' If the ellipsoid has two axes oqnal, and is of the planetary or tKl form, ... Ifi in-'*), )• (") (12) If the ellipsoid is of tho ovaiy or clon^^tcd form i^j, = _a,( log ) JV 1 1' l+e = -..(! -,)(f...S-0 In the €■« of a sphere, when e = 0, i = Jf=iVa-Ji (13) (14) (13) roL. It. 66 MAOKETIC PBOBLEltS. [43* *ctf % Id th« cose of a very {lattened planetoid L becomes in Ibe limU cquul to —in, and JTand I^ bvoome — k'-* In the case of a veiy eloa^t«d ovoid i and Jf approxim&t to the value —2ic, while .V approximates to the form nnd vanieliOK wlu-n « = 1. It appours from these results that — (1) Wlii'ii K, the coefficient of magnetization, is very Bniall,^ whether positive or nc^tive, tlie induced ma^netizatioD is nearly equal to the magnetizing force multiplied by k, and is almost independent of the form of the body. (2) When ic is a targe poiiitive quantity, ibe magnetization depesda principally ou the form of the body, and is almost indepvodont c^ the precise value of k, except in the caae of a lon^fitodinal for acting ou an ovoid so elongated that A'x is a small quantity tboug R is large. (3) If the value of « could be negative and equal to — ahouM have iin infinite viduo of the maguctizaiion in the case of a miignetizing force actinjr normitlly to n flitt plnte or disk. TlHta^ abKurdity of (bin reitull confiniut what w« »aid in Art. 428. ^M Hence, ex peri men tit to determine the valac of k may be made on bodies of any form, provided ■ it very sniull, as it is in the CMC of all diamognetic bodieei^ and all nuLgoetic bodies except ii nicko), and cobalt. If, liowercr, as in the case of iron, k is a laign number, expen- ments mode on aphcres or flattened figuru are not suitable to^i determine k; for instance, in tiie case of a sphere the ratio of th^| magnetization to the magnetiung force is as t to 4.32 if ■ = 30^^ as it is in some kinds of iron, and if k weru inlinite the ratio would be as 1 to 4.19, so that a rery amall error in the dcterminatioa of the magnetization wonld introdtioe a very large one in the value of «. ^J But if we make nsc of a piece of iron in the form of a Tei^| dongat«d ovoid, then, as long as Ak is of modentte value com- ' paml with unity, we may deduce the valno of * from u determination of the magnetization, and the smaller tho value of ^V the more__ accuTutv will be the value of «. ^1 In fact, if Nfhe made unall enough, a snuill errw in the valo^fl 438] CTLisraa. 67 I » of A' itaelf will not introduce macb error, so that ne may use any etoDg&tcd body, hucIi as a wire or long rod, instead of an ovoid. Vfe mnst remember, howeii-er, Uiat it is only when the product Xk ia small compared with uoity iliat this Etibstitution is allonable. la laot the distributitm of ma^ctiem on a long cylinder with flat cads does not reeemblc that on a long ovoid, for the free mag- netism is very much concentrated towards the ends of the cylinder, vrb«reas it varies directly as the distaouc from the equator in tlie caueof the ovoid. The distribution of electricity on a cylinder, however, is really comparable with that on ao ovoid, as we have already seen, Art. 152. These results also enablo us to understand why the magnetic nuHnent of a permanent magnet mn be made so much greater when the oui^nct lias an elongated form. If wc wore to ma<;ncti7.e a disk with intensity / in a direction normal to its surfai-e, and then Imiv« it U> itself, the int«ri»r [larliclos would experience a constant demagnetizing force equal to 'In/, and this, if not snRicient of itself to d<!Klroy jmrt of the niognetixutioD, would soon do so if aidod by vibrations or changes of tvnqHTuturc, If we were l^i magnetize a cylinder trunsvereely the demagnet- izing force would be only 2 it I. If the magnet were a sphere Uie demagnetizing force would 1m. s«/. In a dixk magnctlM-d transversely the demagnetizing force is a' ' /, and in an vlongatod ovviii magnetized longitudinally it ia least of all, being 4 n -^ 7 log ^ • Hence an elongated magnet is less likely to lose its magnetism than a abort thick one. The moment of the force acting on an ellipsoid having dJITerent na^etic coeSicienta for the three axes which tends to turn it about the axis of x, is Hence, if k^ and xg are smalt, this force will depend principally on the crystalline quality of the body and not on ite shape, pro> vided its dimensions are not very unequal, but if Cj and c, are coasideiahle, as in the case of iron, the force will depend principally 00 the shape of the body, and it will turn so as to set it« longer axis parallel to the linee of force. 68 UAONETIC PBOBLBUS. [439- ] >t4 If ft gufni^icntly itrong-, yet aniforoi, field of nuigTieUo force oodM 1*c obteincnl, an Glongated isotropic diaina^netic body vould also set iUelf witk its longcitt dimecuion parallel to the lines of magiiietJo foroc. 439.] Tlic qtientioD of tlie diMtribution of the tnagnetization of OD ellipsoid of revoliiUon under the action of any magnetic forcee bas been investigated by J, Neumann*. KirchhoCTt has extended, the Dtethod to tbe «a*e of a cylinder of infinite leof^ acted on bj any force. (Ireen, in tlie 17tli wction of 1ii» Eisay, kas given an invest igation of tbe distribution of magnetitim in a cylin<ler of finit length acted on by n uaifonn external force pantllvl to its axis. Though ftome of the vtepa of this invc«tigntion are not very^_ rigorous, it is probnble that the result repreocnts roughly tlM^| ncttml mngnetization in this moot important c*»e. It certainly vxpress^if very fairly the transition from the case nf a cylinder for whii'h k is a large number to that in which it is very small, but it Ihils entirely in the cose in which k is negative, as in diamagitetM substancee. Oreen Bnds that tbe linear density of free magnetism at distance x from the middle of a cylinder whose radios is a and] whiMo length is 2A u X = xxX/a -p ► «"■ +b"^ where pis* nomflrical quantity to be found from the eqitatioa 0.a31«l>3-2log.i) + 2ii»— ^. The following are a few of the oorresponding- values of p and k. IC P > f ■0 U.803 0.07 SSft.4 0.0 1 9.137 0.08 GS.Ol a.ot T.5I7 0.03 4«.416 0.09 G.SI9 0.10 tMTf D.04 0.1437 1.00 90.18S 0.04 0.000] 10.W liJ9i 0.00 0.0000 Msxno * imapau • tWbkU >sV t iV*,M •J'*" v*^^*L 44aj FOBCE OK PA8A- ASD BIA-MACSETIC BODIES. 69 I Wbea the leo^li of tlie cyliiKlcr ia great compared with its nditiit, tlic whole (]iuintity of free mag^nt-tism on either side of the middle of Ike cylinder is, lis it ought to be, Of this i/jjf is on the flat end of the cyliinler, and the di^tauce of the centre of gravity of the whole quantity M from the end of the crltiider w - • P When K is vexj" small p is large, and nearly the whole free magnetism is on the ends of the cylinder. As k increases p diminishes, ami the free magnetism ia spread over a greater distance from the ends. When « is inflnite the free magnetism at any point of the cylinder is simply proportioDal to its distance fron the middle pointy the distribution being similar to thnt of free electricity on a conductor in • 6eld of uniform force. 440.] In all substances except iron, nickel, and cobalt, the oo- cfEcicnt of magtietizjition is so small that the induced magnetisation of the Iwdy produce* only a very slight tdtcrjition of the forces in the magnclio field. We may therefore w^^nme, iw a first approx' imation, that llie actual mikgnctie force within the body is tlie same u if the body had not been there. The siijH-rfii:ial magnetization dF dV ot the body is therefore, as a first approximation, « t-i where 2— ■« the tste of increase of the magnetic potential duo to the external magnet along a normal to tlie surface drawn inwards. If we now calculate the potential due to this superticial distribution, we may nse it in proceeding to a seeoncl approximation. To find the mechanical energy due to the distribution of mag- netism OD this first approximation we must find the surface-integral taken over the whole surface of the body. Now we have shewn in Artv 100 that this in equal to the volume-iotegral ^=-Kfff'(^\ 4r ) A^Jydt taken through the whole space occupied by the body, or, if R is the resultant magnetic force, E=-\jjf^JPdxdsdt. Xow since Uie work done by th« magncUc force on the body ro MAQVETIC PBOBLCMS. during a diBp1ac(>ment fi^ is Xim where .V ui the mcchiuiciil force in the direction of x, and «nc« f Xtx-hS = constoot, which shews that the force acting on tho body is as if every part of it t«nded to niovi> from places wlicr« ^ is less to places where it ia greater with a force which on every unit of votunto \» If K is negative, as in diatnanTtcttc bodies, this force is, as Farads first showed, from stronger to weaker parts of the magnetic Most of the actions observed to the case of diamugnetic bodies depend on thia properly. J 441.] Almost every part of magnetic science fmdtt tt« use in navigation. Tbc directive actton of the earth's magnetiKin on thej oompnes needle is the only mctiiod of lUM-erlAining the ship's cou when the sun and stars are bid. The declination of the needle I the true meridian seemed at lirst to be a hindrance to the appli- ration of tho eompnes to navigation, but after this drRiculty lud been overcome by tho constniction of magnctio chart* it appeared lively tbiit the desalination itMlf would assist Ui« nuiriner in de- termining his fhip'^t plucc. Ilie greatest diHiculty in navigation bad always been to ascertain tb« loQgitndo; but ainee (be declination is diflercnt iit dtlfcrent^H points on the same parallel of latitude, an observation of the de-V^ clination t<^ther with a knowledge of the latitude would enable the mariner to find his position on the magnetic chart. But in recent times iron is so largely used in the construction of ships that it bas become impossible to use the compass at all without taking into uceount tbe action of the ship, aa a magnetic body, on the needle. To determine tho distribution of magnetism in a mass of iron of any form under the influence of the earth's magnetic force^ even though not subjeetv<d to mechanical strain or other distur ances, ia, as we have aren, a very difficnlt problem. In this ease, bovever, Uie problem is simplifwd by tho followU ronsideratioui. 44'.] 8BIPS MAOSETTSM. 71 I I » The compass is supposed to be placed with its centre at n fixed point of the Bliip, and so far from any iron that the nm-jnt-lisin of tbr Dccdh^ docs not induce any perceptible magntdiiim in the ship. The rizc of the compass needle is Bup|)09ed so small that we may regard tli« mn^ctic force at any point of the needle as tbe same. The iron of llie ship is supposed to l>c of two kinds only. (1) Hard iron, mai»nrtiM'(l in n ctmstant manner. (2) Soft iron, the magiiotiKat ion of which i» induced by the earth or other magnets. In Htrictnees we must admit that the hardest iron is not only capable of induction but tiiat it may lose part of its eo-called pemanent magnetization in variolic way«. The sofleiit iron id capable of retaining what is called resUwal magnetization. The actual propertifs of iron cannot bo accnmtcly represented by fiiippoaing it compounded of the baid iron and the soft iron above deGned. Itut it has been found tlial when n ship ia acted on only by the earth's magnetic force, and not subjected to any extraordinary gtrees of weather, the supposition that the magnetism of tbe ship is due partly to permanent magnetization and partly to induction leads to sufficiently accurate results when applied to the correction of the compass. The equations on which tbe theoiy of the variation of the com- paas is founded were giren by Poisaou in the fifth volume of the MAaoirn dt VliitUuf, p. 533 (1824). The only assumption relative to induced magnetism which is involved in these equations is, that if a magnetic force A~ due to external magnetism produces in the iron of tbe ship an induced mag^netization, and if this induced mugnctlKiition excrLf on the compass needle a disturbing force whose components are A", I", Z", then, if the external magnetic force in altered iu a given ratio, the components of the disturbing force will be altered in the same ratio. It is Inic that when the magnetic force iw-ting on iron U vciy great the induced magnetisation is no lunger pmportional to th<! external magnetic force, but this want of proportionality is quite insensible for magnetic force* of the magnitude of those due to the earth's action. Hence, in practice wc may assume that if a magnetic forc« whose value is unity produee» through the intervention of the iron of the ship a diniurhing force at the compass needle whose com- ponents arc a in the direction of «, dia that of y, and y in that of «, 72 MAGKHTIC FROBLKHS. [443 the components of the disturbiti^ force due to « force X in the | dir^f^tjon of a witl he a A', (/,V, and,,ffX. If therefore we Assume axes fixed in the ship, m ihut x 'a loward* the ship's bead, g to the starboard eide, and s toiranls the Iced, and if X, )', X represent the componente of the earth's mag^ncttc force in these directions, and A'', Y', Z" th« components of the com- bined magnetic force of the earth and ehip on the compose needle, - r=r+j,T+«.i'+//+i2, [ (1)1 W z'= z+gX+ir+kz+R. ) In these equations a, 6, e, d, «,/,y. A, i are nine oonstant co- 1 efficients depending on the amount, the arrangement, and Ui«l capacity for induction of the soft iron of the ship. F, Q, and It are constant quantities depeoding on the permanent i moffuctization of the ship. It is evident that these equations arc sufficiently general if , ron|:;netic induction is a lint^ar function of magnetic force, for they i urv neither more nor less than the most general cxi>re«>ion of a vcvtor lU a linear function of another vedor. It mny also be shewn that they are nut tM> general, for, b; pro]K'r arrangement of iron, any one of the eocfficivnts may be mmlc lo vary indo[iendently of the othcnf. Thuii, a long thin md of iron uiMJer the action of a longitudinal roognetie force acquirer poles, the strength of each of which ia numerically equal to the cross aection of the rod multiplied by the magnetizing force and by the coefficient of induced ua^et- isation. A magnetic force transverse to the rod produces a roach feebler magnetization, the effect of which is almost insensible at a distaoce of a few diameters. If a long iron rod be placed fore and aft with one end at a dUlanee x from the compass needle, measured towards the sbipV b(^, then, if the section of the rod is J, and its coefficient of^j ntagn<!tiratii>u k, the ctreDgtli of the pole will be JkX, and, >^| Am — , tlip force cierted by this pole on the compass ncvdlu will be II .Y. Tlic rod may be supposed so long that tfae effect o^Hj the other pole oa the eompim may be neglected. ^H We have tliua oblaiuod the means of giving any inquired ralaa^| to the ci-effieioni it. ^M If we place another rnl of •eetion S with one extremity at th4^| ■ome point, dislauL « Ironi the eum^wH toward the h^ad of ^i^| I 441.] ship's maonettsm. 73 I Tttwelj and extending to starboarcl U> sucli a disUnce Ihnt llie distant pole produce no itonstblc i-fit-ct od the compass, the di»- tarbin^ force doe to tliix rod will he in the dircctiou or x, and equal to — ,- , or if ^ = — , the force will be jr. This rod therefore introtiucpa the cof flioient i. A third roil cxtiMnlin^ downwards from the some point will introduce the coelTieitint c. Th« coeEBwents d, e,/ may he prwlueed by th»e rodfl ertending to bead, to starboard, and downward from a point to starboard of Ui« compaea, and y, i, it hy three rods iu pAralk'l direotioDs from a point below the coropans. Henoe each of the niD« coefficients caa be ecparateiy varied by means of iron rods properly plwwd. The quantitie* P, Q, It ure eimply the compODeatl of the force on the oompaai arixing from thv permanent magnetisation of the ship logetlier with that part of the inducwi mag'ncti/ation which is dnc to the action of this permanent mii^nrtizution. A complete difcuiwion of the eqiiativn« (1), noil of the relation between the tme maf>nctic courst- of 1he ship and the course as indicated by the compass, is given by Mr. Archibald Smith in the Admindty Manual of tie Devialion of the C'lapant. A valuable graphic method of investiffating the problem is there giv«n. Taking a fixed point as origin, a line is drawn from this point repntDenting in direction and magnitude the horiKotital part of tbc actual magnetic force on the compass-ni.'edle. Ax the ship is swung round so as to bring her head into difTorent azimuths in saccosiott, the extremity of this line df«;ribcM a curve, each point of which corresponds to a particular azimuth. Soch a curve, by means of which the direction and magnitude of the force on the comjuifi^ is given in terms of the magnetic course of the »lii[>, is cstllfd a Dygogram. There are two varieties of the Dygogram. In tlie first, the curve is traced on a plane lixod in space as the ship turns roimd. In the second kind, the corvo is traced on a pluue fixed with respect to the ship. The dygogram of the Grrt kind i* the Ltma^n of Pascal, that ofUiK second kind is an ellipse. For the construction and use of tbese curves, and for many thvoremn n* interesting to the mathe- niatieian sw they are important to the navigator, the reader is referred to the Admiralty Mamiaf of (At /Mvialion t^the Compau. CHAPTER VI. WRBEIt's THEORY 0? IKDITCED MAOHSTISX. 44S.] Wb h»ve seen iliat Poieson mippoeeB tho magiielization ot iron to c^onsist in « sepantion of the mafrnctic fluid* withiD e»ch SMgDMic tnoleonle. If we wish to avoid Uiv usumptioit of tiui psislt'iicc nf niajjni-Uc fluids, we may rt«t« ihc sam* tbeory inl another fonn, by Mjing lluit racli molecule of the iroii, whiin tbe-l ma^nrtiiiiti; furcv nets oa it, becomo* a magnet. Weber's tluiiiy diffcw fntm Uii» in assnmin^ that tbc molMukal of Lhe iron an> always nuifrneU, vvea before the application of the magnetising forvo, but that in ordinaiy iron tiuf magnetic axes of thf moleoulfti are turned indilfercntly in every direction, ■o tliat tho iron as a «ho1« exhibits do mnf^nrtic proporliw. Whrn a luagnetit' force acta on the iron it tends to tarn the L«f tliu molecule* all in one direction, and so to cause the iron, I'whole, to become a majn>et. If the axes of all the motcculoe were eei panllal to tmeh other, the iron would «>xhibit the gmtan't intensity of magnetization of whii.'h it » r«]«b|e. Hi'neo Weber's tb«<ory impji*^ the existenee of a limiting iuleusity of maf^netintion, and the experimental rvidenoe that swh a Hmil exists is tbeivfism nere«saty to the tlworjr. K\peTtmieot» sh«Hria(; an appnrMch to a limiting valoe of BMgnelitatnm hav* tw»n made by Joule * and by J. Muller t. The miMcintttto nf Berta t m aleetnttjr^ iron deposited nnder tW action of majpMkio Rhm Awuuk Um Moot ewnpleto evidem of lbi» limit.— A filvet wiiv w«t tmmuliMl, and « x%rj mutmt lior on the • iMMbyn>Mk^M»,t»^m,ia»i «%&««. till. I 445-] TH8 MOLECtlLES OP IBOS AKB SIAONEre. 75 metal mu laid \mn by making n fiDo lonj^itiKliiuil Mnitch on the varnhih. The wire was tlicn immersed in n solution of a suit of troD, and placed in a iiiaf^nAtiv Bcid vritli the scratch in the direction of a line of ma^elic Force. By nutking the wire titc oatiitide of an electric current through the aolution, iron was <lepo»it«d on the narrow expojunl surface of the wire, mo1ecn1i> hy molecule. Th« filament of iron thns fbrRM-d wita then exHmiued maguetieally. Ila magnetjo moment was found to be very grout for so small a manf of iron, and when a powerful magnHizini^ force was made to act io the same direction tlio iiioroaac of temiKirary mngnettzatton was found to be very small, and the pcrmnncnt magnetization was not altered. A magnetizing force in the reverse diniction at once reduced the filatnent to the condition of iron magnetized in the oidinarj' way. ■Weber's theory, which supposes that in this case the mn;jnetizing force placed the axis of each molecnlc in the same direction during the instant of ita deposition, agrees very well with what ia obserrecl. Beetz foond that when the electrolysis is continued under the action of the niagnetixing force the intensity of ma^gnetiicatioQ of the sabeequently deposite<l iron diminislie^. The axes of the molecules are probably di'fle<^ted from the line of magnetizing force when they are being laid down side by side with the mole- ciilea already deposited, so that an approximation to parallelism can bo obtained only in the case of a very thin filament of iron. If, as Weber supposes, the molecules of iron are already magnet*, aoy magnetic force sufficient to render their axes parallel as they are electroly tically deposited will bo sufficient to produce the highest intensity of magtietizatton in the deposited filament. If, on the other fuind, the molecules of iron are not magnets, Imt are only capable of magnetization, the magnetization of tho deposited filament will dejwnd on t!ic magnetizing force in tho famv way tn which tliat of soft iron in general depends on it, Tlie expcn'meut* of Bcctz learc no room for the latter hy fothcsis. 448.] W« sliull now aanime, with Weber, that in every unit of Tolutne of the iron them are » magnetic molecules, and that the magnetic moment of coch is m. If the axes of all the mulecules were placed parallel to one another, the magnotic moment of tho unit of volume would l>e 3f= ttlM, 76 WEBER 8 THEOBT OF INDUCED MAONETISJI- and this would be tho great«)>t inU-nsity of magnetization of which ' the iron is capable. i Id Hie unmagnetizod atatc of ordinary iron Wchcr oiippmcs tbc axes of H& iiiolei'uli;)< to bo placed indilfi^rently in all direction*. To ezprcsR this, we may suppotte a itpbere to be described, and a radiuM dratvn i'roiii the centre {WTallel to iJie direction of tlie axis of eaeli of tlie n molecules. The distribution of the extiemitiee of these radii will express that of the axes of tlie nioleculee. lo i tlic case of ordinary iron tiiese n jioints are equally distributed over every part of" the wirfac^ of the sphere, eo that the number of moleculeis whose axes make an angle less than a with the axis ofris „ -(I -COB a). and the number of molecules whoso axes make angles with that ofa^ between a and a + daia therefore - cin a^a. This is the arrangement of the molecules in a piece of iron which ^ has never been magnetized. Let us now supjKxse that a magnetic force X is made to act on the iron in the direction of the axis of x, and let as consider ' a molecule whose axis wah ori^iially incliDed a to tlio axis of x. If thit; moltTulo is perfectly free to turn, it will place itself with its oxiK jwrullcl lo Uic axis of x, and if all the molecules did so, the very itlighte«t mngnetixing force would be found sufficient to develojie the Yciy highest degree of nuignctizutiou. This, how- ^ ever, is not tlte case. H The moItH^ules do not turn with their axcn pnrnllcl to x, and ~ this is either becauae each molecule is acted on by a force tending to pnwrve it in its original direction, or because an equivalent! effect is produced by tlie mutual action of the entire system of] moli-culn. Weber adopts the former of them suppositions as the simplest, and suppoBi-* that each n)ol<«nle, when doRccted, tends to retnrn to its original ixwition with a force which is the same as that which a majn^etio foivv D, arting in the original diivction of it« axis, trould produce. The {Hwition which the axis artiially assamcs is therefore in the] direction of the ntmltaat of X and I). Irt-t JI'll ivpn-wiit a ni-i'iliin ul'a siihere whose mdius ivprasents, on a certain wuih), Uw force H. I d 443- DBFLBXIOS Of iXE9 OP HOLECCLBS. 77 Ld the niiai, OP be pamllct to the uxiit of a particnUr molcculo in its on'giiuJ position. Ijet SO repreMnt on the mmv iwalo the miignctizing foroct X which i» Kuppcwed to Ml Trom S lovrurilit 0. Tb*D, if Ihc mokrule \a aut«d OD by the forw A' in the diredion SO, »n<I by a force I) ia » dirwtion jxirttli;! to OP, tho orifjimtl direction of it» axin, it« axis will fit itself in tlie direction SP, that of the re-iult^nt of X and O, Sinoc the axes of the moIeciiIeH are ori^nally in all directions, P may be at any )M>int of the sphere indifferently. In Fi^. 5, in whioh A' i* lt;M than 2), SP, the final poiiition of the axis, may be in any dire«tion whatever, but not indilTerently, for more of tJie I molecules will ha?e their axes turned towards A than towards Jf. In Fig. 15, in whioh X is {i;rpater than i>, the axes of the molecules FSg.S. Kg.». Hence there are two dtiTercnt canes scoordiiig as X is leas or greater than Jf. II 1/et a = AOP, the oHgioat inclination of the axis of a molecule ^B to the axis or«. ^^^K ff = JSP, the inclination of the axis when deflected by ^^^H the force ^^H ft = SPO, the angle of dt-flexion. ^^^H 80 = X, the magnetizing force. ^^^H OP = Jf, the force tending towards the original position, ^^f 8P = R, the resultant of A' and G. ^K m = magnetic moment of the molecule. ^^ Then the moment of the statical couple duo to .V, tending to diminish the angle 0, is ml. s MA'itind, tho moment of the couple due to D, tending to increaaa 0, ia mL=. mi7siaj9. 78 WEBERS THRoay OF ISDCCED HAONETISM. Equaling these values, and remetubering that J3 k a—0, we find . . If una -,, tan« = T> — r, iu to determine the direction of tlie axia after dedexioa. Vfe have next to find ihe intensity of ma^uetization produredl in the niiiw l>y the force A', and for this puqxiae we must resolve 1 the inaynolic nionient of every molecule in tlie directiou of s, add all tliL'se resnlved parts. Tlie reaolvod part of the moment of a molecule in the direcfaon of .7 is HI cos 0. The Dumber of molecules whose original inclinations lay botv a and a+ila is « - »in ada. We Uave therefore to integrate 2 = f - cos sin ai/o, (sl rcRXtmbering that $ isa function of a. We may express botJi $ and a In terms of R, and the cxproeion] to be intcgtated l>ccom«8 tlie geneml int^^ml of wlncli is Id the fimt ca§e, that in Tvhich .V iH ten Umn J), tlie limits of integration are H = J)+X and K = It -X. Id the second cosv in which X is greater than J), the limits anj JimX+J) and B = X-B. 1= ~mn. When X is Ices than S, Wbcn J is equal to 2>, Wbeo X is greater than D, and when X becomes infinite According to this form of the theory, which is that adopt by Weber*', as ttu maguetixtDg force increosos from to i7, • TImm {■ wvio nibtala In Ik* Ceramk g<nai bj- Wsbcr (TVwh. Atad. Sat i. B. in (1613), or Pen.. Jra- Usavii. p. !«; (1812) m (ba noOl d (Us iatrgriUft. ^•bfaorwhUsniutllTwIvUB. Bkfcnnwlaii ^ i-M> J*^^JC'U■^.^D' 444-] LIMIT OP MAOSETIZATIOir. 79 nmgiietization increases in tlio some proportton, Wb«n Uie mag' tuilizing forv« attains th« value J/, the magnetization is two-tbirds of ita limiting value. When tlie magnetizing force im further iit- creawd, the niagnetiiiation, instead of increasing indefinitely, tends tonarda a finite limit. Fig. 7. » Tb« Uw of magnetiiation is expressed in Fig. 7, where the mag- netuiBg force is reckoned from towards the tight, and the mitg- iwtitation is expressed by the vertical ordinatt^s. Weber's own experiments give results in aatiafact4>ry accordance with this law. It is probable, however, that the value of J) ix not the same for all the molecules of the same piece of iron, so that the transition from the straight tine from to £ to the curve beyond E may not be 00 abrupt as is here represented. 444.] TTkj theoiy in this form gives no account of the residual tnagn«ttzation which is found to exist atiter the magnetizing force is removed. I have therefore thought it desirable to examine the results of making a further assumption relating to the conditions under which the position of equilibrium of a molecule may b« permaneutly altered. Let UB Buppoeo that the axis of a magnetic molecule, if dcfleelcd throogfa any angle less than ff^, will return to its original position nhen the delleeting force is removed, but that if the deflexion $ exceeds ^g, then, when the deflecting force is removed, the axis will not retom to its original position, but will be per- tnaDently deflected through an angle /3— ^g, which may be called the permanent «<■/ of the molecule. This assumption with respect to the law of molecular deflexion is tM>t to bi^ regarded as founded on any oxa<^'t knowledge of the L intimate i<truclure of bodies, but is adopted, in our ignorance of tlie trtic state of tb« case, as an assistance to the imagioatiott in following out the «{>eculatiiia suggestvd by VVcbcr. Let l^JDtiafla. (9) 80 WKBEirS TnEORT OF INDITCED KAGSBTiair. [444- then, if the moment of tli« couple ftcUof; on a molocnlo is loses tb&n tub, thcro will be no ponnaiii'nt dt-flcxion, but if it excpwla "w/f ■ there will Im! tt p(;nri»nvHt change of thi> position of i!K|uiHbrium. To tnice the results of tbis eupposition, dctrcribc n Hpbore whose centre ifi and ratlins OL = L. As long as A' ia less than L everything will be the same ax in the case already considered, but at! soon as Xoxcet^K L it will bcf,'in (o proda«e a permani-nt detlfxion of some of the moleculefl. Let ua take the case of Fig. 8, in whieh A' is greater than L hut leiia than D. Through S as verti^x draw a double oone touching the sphere L. Let this cone meet the Kpherc D'mP and Q. Then if the axis of a molecule in it^ original [wsition lies belweca OA and OP, or between OJi and OQ, it will be dellected through an angle less than ^3^, and will not be permanently di-floeted. But the axis of the molecule lie* originally between OP cud OQ, then a couple whose moment is great«r than £ will act uiwn it and will deflect it into the position SP, and when tlie forw X ecwea to act it will not resume its original dir«'etion, hut will be per- manently set in the direction OP, Let xa put L = Xsin tfj where 0„ = PSA or QSB, then ail those molecules wbo«e axes, on the fonner hypoti would have values of $ between 0^ *^ '^■'o will he made to have the ralne $^ during the action of the force X. During the action of tJio forco X, thererore, tbom nwloeal whose axca when de6iected lie within either xheet of tlie doubU cone whose semivertical angle is d« will be arranged as in th« fonn«r ease, but idl thorn whose axes on tl>e fonn^tr thts^ry wotild^^ lie outside of the*e sheet* will be pennaneiiUy deflected, so tha4H| their axes will form a dense fringe round that sheet of Uie cone whieh lies towards d. aw >h\JM d 445] MOniPIED THEORY. 81 I Aa X increases, the mitulicr of molecules belonging; to tli« oone about S continually iliminUbex, and when X becomes eqiiul to J) all the molecales have been wrenched out of their former positions of equilibrium, and have been forced into the fringe of the cone Toiind A, BO that when X becomes greater than D all the molecules form pari of the eono round A or of its fringe. When thv force A' in rcmoTcd, then in the case in which X is IcM titan L cvcrj'thing returns to On primitive etiito. When X is between L and J) tlien there ts a cone round A whoKc au^e and another cone round S whose angle liOq = 0,-p^. Within theae conca the axes of the moleculce are distributed uniformly. But all the molecules, the original direction of wbo»e axes Hy ouUide of both th^j^e cones, have been wrenched from their primitive positions and form a frinjre round the cone about A. If X is greater tlian I), then the cone round B is completely dispersed, and all the molecules which formed it are converted into tiie fringe round A, and are inclined nt the angle ^g-f ^y. 446.] Treating this ca«e in tiie same way as before*, we find • (Tfcp nanlta gi*Bli in th« tott may ho ohXtAoci. with one lUglit raosption, by tile ptoeeon givon b«l»w, ihti i<l*li>iii»iit nt ihn niuUlioil thaary of Art. 414 bdng M fglloir* : Tha &iU of ■ nia^iMtic iiiu1k»1?, if ilvllix^liat Uiroiij^li au angle l*" (bui 8, will rMuni 1u ita uHifinkl noiritioii whea the dpfloitinij fiirvii <i niiii<iT*il ; but wlxn th* 6tB»*lMk nomli 0, tho TorHi tending to oppOH the dpfletinu givw my »n<l ponnila tka malcculD lo be ddlcirUd Into tbB wmc diioctiun u lliuH irhuns HcfleKJun u A., >ad whsn llw JitlfciJnit f<ir«a la r«mi>vc4 the ni'ilocula t.\ktm up > direciioa panllcl to tluti of the moliKuti) wImm dvflraioQ wu >>•■ Tlila dimclion ouy bo call*] tL> peniuuiunl nt of Om mukouloi. la (Iw our X :■ L --. D. lh« c^ptoHon / (or tbu m&gnelio momont conMnUnf tiro put*, tliv tint ef which !■ dun lo ths noEMulo* witkin tha oonsa A OF, HVQ and U Ui be found iiiwixly lu tu Act. 443, duo r^^aril h«iuit had to tho limiui of integraUon. BaAimag to ¥ig. 8 v* find for th* mfouiI pari, aoconKn); to ths above itatsiiuint of 1 .en Projootioa of QP on W fMmenASPn = -r^^ . nio t«o parta i^gotliM whoa taiuc«d give tbo malt in the teit. Whui X > /*, tha tatcttral a|[aiu oiiuiiittii nf two inu-li. mic of whitih (• to bo takca OTM-tliacuna A Of a* in Art. 448. The tusoii'l puiia, l>^>((C)i .ct, rroiofitfon of BP oa BA {tnneaaJSPx — * jjk ■ OP The raluv of J En Uua Ca*^ ati«n reddMd. dtffon from tho valne gimi la tbft (eit In thv thud tens, vb, : w« bar* then ^ , ». iniluad '■f ~ « v ''^'" ""^^ '^ *'^ S JC^ V A oba^go im tlis taU« nf miinrrfal Tallies glrtn tn tlio toxt will be that ohr'ii X - S. 7, If, iku tarrerpmlhijf iatu«« uf X will ba SS9, 917, DSO. TttMo ciiugoi do not TOU U. O 82 WEBEBS TnBORT OP ISDOCED MAGNETISM. [445- for the intensity of the tWDpomry magnetitttion during' the iiction of tbe forai X. which is euppoM'd to u«t on iron wbich hm Derfr beforv b^ f n magnetized, 2 X When A' ia lees than L, I~ -M-ft- ' 3 D 'A-'h When A' is etjnal to L, Wh«n X it between h and /), When X \i etiual to Z>, Wiien X i» greot«t than 2>, Wluin X is infmit«. / = Jf. 6J»Z* When A' is less thsn L the magnetization follons the former luw, and is proportional to tho magnetizing force. As »oou as X uxL-eeds L the magnetization assumes a more npid ntbe of increase on account of the molecules beginning to be troniferred from tbe one cone to the other. This rapid increaue, bon-ercr, soon cornea^ to all cntl ns the numl>er of molecules foiming the negative coBi^| diminishes, and at laut the magnetiiatioD reacbca tbo limiting Taliie M. If ne vera to aaaume that the valiieR of £ and t>t D are difTerenfl for diOiorent moleeulea, w« should obtain a rasult id vhicb tbe different stagn; of magnetization arc not m distinctly marked. Tlie nndaal inatniettxation, /', produced by the magnetiziog force X, and observed after the force has be«n removed, is as follows : When X \a leas than £, Mo residnal magnetizatioa. When X is between h and B, r-.J^(i-g)(i-^. •Iter Iha gaiMnl fbnwiOT of llw can* «f TtMBpana; Vif— finHim givta ^i* m)>H o( r la Uii MM «f rt(. S b •/;. n* nJiM »f r h tin cMB uf IV- > BUT h* laMi la tika I '1 llf AlTD RBSTOFAL MAOKETIZ.ITIOK. 83 'Wbeo X it) cqutti to 2>, "When X w greater than J>, "When X is infinite, li we make J/ =1000, i = 3, /I = 5, we Bud tlie following values of the temporary and tlic residual magnetization : — lUatfncitldng Tamponuy MHgiiiiiiutlon. X / 1' 1 J33 3 267 3 400 i 729 260 9 837 410 6 864 485 7 882 537 8 897 574 1000 810 These reeult* arc laid dun-u in Fig. 10. . J^^^^^^Si^taiBiaiAmrlMtmMaMtm Kg. 10. The eurve of tempoiary maj^aetizatioii is at 6nt a straiglit line 84 WEBRR's TnEORY OP INDCCED M.VOSRTISM. [446- t, and™ from X=0 to X=:£, It then rises mom rapidly till X = J), nud 1L8 X iocrcAGes it uiiprotictivH it« horizontal asymplotv. The L'urvc of rocidual iiiii);netirjit.ion bpgins when X^L, and approaches an asymptoti- »i 11 iliiilanoc = .St Jlf. It must bo rcm(;mlj<.Tcd that tlie n.'«idit»I miif^etutn tliuB foimtl correspondB to tbc case in whicii, when lh<t fxtcmnl Toroc is rmioved, there is no domagiictizirijr force arising from t^v diotnbiition (it inagnetifim in tlie hody it^lf. Tlie calculationH are therefore ap- pltciibli- only to vory eloD|^ati-d bodiat inagiivtizvd longiUidiiiHlly. lu thu cam of itiiort, thick bodies the residual mn^ctism will be dimini«licd by the reaction of the free magnetism in tlie m&« way as if an external reversed magnetizing foroc were made to act upon it. 446.] The acieuUfle value of a theory of \\\i* kind, in which w< make so nmny aKBiiinptionii, and introduce «o niuny o^uMtable constants, cannot be t-^tiniated merely by iIk numerical agreement with certain sets of expmmonts. If it liax any ralue it ix because it enables us to form a mental image of what talces place in a piece of iron during magnetization. To te«it the theory, we shall apply it to the c»»f in which a piece of iron, after being mibjectcd to B magnetizing force A'q, is again subjected to a magnetizing force A*,. If tJiv Dew force X, acts in the same direction as that in which Xg aoted, which we shall call the poeitire direction, then X,, if leaa than X„, will produce no permanent set. of the molecules, and when X^ is removed the rcsidtial ma^pictization will he the same as that prodnced by X„. If A', is greater than A'^, then it will produoe exactly the §amc efftvl as if .V„ had not acted. But let U£ suppose X, to act in the motive direction, and let us Xo = Zeosecilj, and X, = — ituwoetf,. suppose A9 A', increase numerically, i?, dimiuishi-v. The Rrst molecniefi OQ which A'l will produce a permanent deflexion aru tliose which form the fringv of the oo»o round A, and these have an iuclinatioB when undcflected of 6^ 4 ^o- As soon aa ff,— ^^ becomes hws than ffo + ft ^^^ process of de- magnetization will oommence. Since, at this instant, 0t m 9„-{- 2^, .T,, the force required to Wtjin the demagncl ization, is )et» tJum K^, the force which pri>duct-d thi- majjnoliitalion. If the value of J) and of L were lie same for all the inolecmie the Bliglieat incre.i«- of X, wouM wr(-neh the whole of the frin^ of muleculi-s whose axea have the inclination i^+^o inln a poai. ;4 446-] UAONRTISU ASD TORSION*. I * I ft tioD in wlLi<;fa fheir axes are inclined ^i + ^ to the native axis OB. Tlioujjh the doronfrnettzatioD does not late place in a uinnner so Kiul(]«ii ii« t)ii)i, \l fjikcj^ place m rapidly as to aiTord some conlirinu- tioQ oflbin modo ofcxplitinm^f the process, Let U8 naw ««[ii)oec tlmt t>v ^iviiijif ii proper value to the reverse tone Xi we havt; exiiotly dt-magnctizcd the piece of iron. Th« axoK of the molvculcs will not now he arranged indilFer- enllv in ull diTvctioD!<> a« in a piece of iron which has never been magnetiKed, hut will form three jfroiips. (1) Within a cone of scminnglu 9i—0o surronndins the positive pole, th« uses of the molcciito; remain in their primitive positions. (2) Tlio tuime is the ease within a cone of semian£;lc &^—Pq sorroundin}; the negatix'e pole. (3) The directions of the axes of ull the other molecules form ■ conical sheet Eurrounding the ncg^ative iKile, and are at an inclination O^ + fi^. When .1^ is {>rcat«'r than J) the secoud group is absent. ^Vben X| is f>roat«r tlian D the first group is also absent. Tlie stxlc of the iron, therefore, though apparently demognctizml, is in a different stut« from that of a pieoc of iron nliich lias never bevn Di»{rn(-ti£ed. To shew this, let lis consider the effect of a magnetising force X.^ acting in cither the ponitivc or the negative direction. The fir«t pcmiBiient i-fft^ct of such a force n'ill hu on the third gronp of molecules, whose axes make angles = O^-h^a with tlie negative axis. If tlie force X, nets in the negative direction it wilt begin to produce a piTniancnt effect a« soon as (J^ + A) l>econ)es Icsh than tf]-fA). Uxt 18, as Koon a* .V^ becomes greater than X,. But if X, acts in the positive direction it will begin to remagnetize llie iron as soon as 0^— ;3 bi^eoraes less than 6^ + ^,,, that i^ when 0, K flffZiS,, or while X.^ is still much less than X^. It iippcarn therefore from our bypothertis that— When a piece of iron is magneliiied by means of a force Xg, its ningnetiBm cannot be increased without the application of a force greater than X^. A rtverBe force, less than X^, is snfBcient to diminish ita magnetixation. If tbo iron is exactly dema^etizcd by the revereed force X, , tin n it cannot be mn}fncttiK-d in the reversed direction without the application oi' a force greater than .V| , but a positive force less than ^tt 86 WEBEBS THEORY OP INDUCED MAfiSETISU. X, is sufficient to hegin to rcmagneUze lite iron in ite ongioAl direction. Thccc rotiiiltK arc onneiiHtiHit with what hu been a«laa11y obMrve hy Riteliie*, Jncobif. Marianinit, and JooleJ. A very complete account of the relations of the magnet izatios of iron and stcrl to inagnctto forces and to mechanical strains ii g-ivcn by Wicilcinarin in liis Galvanitnus. lly a detailed com- pariKon of the effects of ma^etization with those of torsion, bfl shews that the ideas of elasticity and plasticity which we derivsl from experiments on the temporary and permanent torsion of wireaj can he applied nnth equal propriety' to the temporary and permanent magnetization of iron and steel. W?.] Matteuecill found that the extension of a hard iron hax during the action of the magnetizing force increases it« temporary magnetism. This has been confirmed by Wertheim. In the ease of soft hare the magnetism is diminished by extension. The permanent magnetism of a bar increases when it is extended,^ and diminishes when it is compressed. Hence, if a piece of iron is first magnetized in ono direction,' and then extended in another direction, the direction of magnet- , izatioD will tend to approach the direction of extension. If it h4^| compressed, the direction of magnutizatJoa will tcrnd to bccomc^^ normal to the direction of comprvsstun. This explains the tmiiU of aa experiment of Wiedemann's, enirent was i>a»f«d dowimani through a vertical wire. If, eithu during tlic |)ait»agc of tlio current or after it has ceased, the wir bo twistod in the direction of a right-handed screw, the lower eai becomes n north poU). sa I F^.II. Fig. 12. • pail. .Wo.?.. IS83. % Ami. Or VUmie H 4i Ptf*rf«, lUS. f p-ig,. akh., isai. CTTASOE OF POBM, 87 I ^ ^ Here the aownirard current majrnetizes every part of the wire in a tan^ntial direction, as indicated by the lettcra .V.S'. llic twisting of the wire in the directioo of a right-handed screiv the portion ASCH to be extended alonij the diagonal AC and coinpTcswd ulong tho diaj^onat BD. The direction of magnet- iuition thvrcforc tends to iippruach AC and to recede from BD, Rod thus tJic lower end becomes u north pole and the upper eail a vouUi pole. Effect ^ ilagnHicaiion on tie SHineniion* of the Magnvi. 448.] Joule*, in 1812, found that an iron Imr hccomes length* ened when it is rendei-ed magnetic by an electric current in a coil which stirrouuds it. He al'terwardsf shewed, by pladng tiie b*i in water within a glaan tube, that the volume of the iron is not augmented by th» magnetization, and concluded that it« tisasverse dimensions were contracted. Finally, he paseed an electric current through the axis of an iron tnbe, and back outside the tube, so as to make the tube into a closed magnetic solenoid, the mugnetization being nt right angloi U> the axis of the tubu. Tho length of the axis of the tube was found in this auc to be xhortoned. lie found that an iron rod under longitudinal pressure is also elongstrd when it is magnetized. Wlien, liowever, the rod is under considi-rable longitudinal tension, the effect of niagnetizntion in to rfiorlen it. TliiM was the case with a wire of a quarter of an inch diameter irben the tension exceeded 600 poundj* weight. In the case of a hard steel wire the effect of the magnetizing force was in every case to shorten the wire, whether the wire was tinder tension or pressure. The change of length lasted only as long as tho magnetizing force was in action, no Hlti^rution of length was obecrved due to the {)crmancnt magnetization of the steel. Jonle found the vlong:itioQ of iron wireu to be nearly prciportional to the (quare of the uotual mngnetixut ion, ko that the tiriit etfect of a demagnetizing current wbk to Hluirten the wire. On ihe otiicr hand, he found that the ahortening effect on wirea ondcT tension, and on stoet, varied as the product of the magoot- iiation an<l tlie magnetizing current. Wiedeinann found that if a vertical wire ia magnetized with its " Siiuwoi'i ^niHil/ o/ Bi«lH<Uv. vol, riii. n. S19. ^^_ t fAiC Jf(V.. \Ut. 88 webgb's teeobt of induced maonetism. [448. Boutb end uppermost, and if a carrent is then passed downwards through the wire, the lower end of the wire, if free, twists in the direction of the hands of a watch as seen from above, or, in other words, the wire becomeB twisted like a right-handed screw if the relation between the longitudinal current and the magnetizing current is right-handed. In this case the magnetization due to the action of the current on the previously existing magnetization is in the direction of a right-handed screw round the wire. Hence the twisting would indicate that when the iron is magnetized it expands in the direction of magnetization and contracts in directions at right angles to the magnetization. This agrees with Joule's results. For further developments of the theory of magnetization, see Arte. S32-845. CHAPTER VII. UAQKETIC UEASURGUeSTS. » I I 44D.] TuK principal magnetic mcaeunemoDts arc the deterniina- tioD or Uie nutgitctic axi« mid maf^netic moment or a inagnet, and that of tJie direction ftnd inti-nsity of tbo magovtic furco at a given pbce. Since theae meaaurcmente are made ncnr the Eiirfacu of Iho earth, flie magneti are alwaj-s acted on by gravity as well us by terrestrial mapnetisni, and iiiiiue the mugtiets are made of iAimiX tlieir mng- netisni ia partly pGrmanont and partly induced. Tlie permanent maf^etism is altenxl t>y cbaiiifes of t'em[iL>niturc, hy at.rong in- duction, and by violent blows ; tlic tniliin^d nia^iietisin varies with every variation of the external magnetic force. The mottt convciiient way of (>bscr\'ing the force acting on a magnet is l>y making the magnet free to turn about a vertical axis. In ordinary compaMcs this is done hy balancing the magnet on a vertical pivot. Tlie finer the point of the pivot the smaller is the moment of the friction which interferes with the action of tlie magnetic force. For more refined observations the magnet is soapended by a thread composed of a silk fibre without twist, either single, or doubled on itself ii xullicient number of times, and eo formed into a thread of parallel lilires, each of which supports as nearly as possible an equal part of the weight. The force of torsion of mich a thread is much leas than that of a metal wire of equal strength, and it may be calculated in terms of the ob- served azimuth of the magnet, which is not the case with the force arising from the friction of a pivot. The suspension 6hre can be raised or lowered by turning a horizontal screw which works in a fixed nut. The (ibro is wound round the thread of the screw, so that when the screw is turned the suspcosion fibre always hangs in the same vortical line. 90 MAGNETIC MBASUREMENTa. [45< r thfl on of^ fl ;bt est «s- ireo- The snspensioii fibre cnrrit-e n small horizontal dirided iiircle called tbe Toreion-circlf, «nd » slirimp with an index, whtoh eaa he plaoi'd so that tlic iiidox cuincuk-e with any ^ven division of Uio toreion circle. The stirrup is so shaped that the nia^^t Ijsr can bo tittiM) into it with it* uxie horizontal, and with anj one of its four sides uppermost. To osL-ertuin tlic zero of torsion » non-mai^etic body of snme w^iffht as tie magnet ia pL in the stirrup, and the poi.ition the tnrvion circle when in e<}uilihriui asciTtaiucd. Till" miig^net itself is a piece har(l.t4;miicrtil steel. According (lauss and Weber its length ought to be at Iwist eight times its greatest transverse dimension. This is neees- mry whi-n permanence of the direo- tion of the magnetic axis within magnet is the most important eoa sideriition. Where promptness of i tion is required the magnet ahonld be shorter, and it may even be ad- viauble in o1:)E«rviiig sndden altera- tiona in magiMttic force to ase a bar magnet ize<l transversely and sus- pended H-tth its longest dimeUfcioQ Tertical *. ^M 450.] The magnet i« provided wit^^ an arrangement for ascertaining ita angular position. For ordinary pur- pn^i^s its ends are pointed, and a ^' *' divided circle is placed below the ende, by which tlieir positions arc rend oiT by an eye placed in « plane through the nutipension thread ami the point of the needle. For more oceiimtc observations o plane mirror is fixed to the magnet, so tltal the normal to the mirror coincides as nearly as possible with the axis of magnctitation. This is the method adopted by Gsnsa and Weber. Another method is to attach to one end of the miignet a l«ns and to the otlier end a scale engraved on glaM, the diHtunce of tbe lens 5m, JrimriMttr, Vor.SS. ISU. 450-] TUB MIRROR JrETOOD. 91 » from tie scale being- equal to the principal focal length of the lens. The straight line joinings the zero of lh<? eealo with th« optical centre of the leas ought to coincide as nearly ti* possible with the msffiHitic axis. As these 0]>tical methods of asoeriainin^ the angular position of mspended apparatus are of great importance in many physical researches, wc shall here consider once for all their mathematical ticoiy. Theory of lie Mirror Metkoi, VTe shall euppoee that the apparatus whose aagul^ position is to bo detormincd is capable of revolving about a vertical axis. Tilts Kxis is in general n tibre or wire by which it is saspendi-d. Tltc mirror should be truly plane, so that a scale of milliractirs may Ix- «e<tn distinctly by reflexion at a distance of several metres from the mirror. Tlic normal through the middle of the mirror should pass through tbt axis of sofipension, and should be accurately horiKontal. Wc shall refer to this normal as the line of collimation of the ap- paratus, Having rotighly ascertained the mean direction of th« line of collimation during t)ie experiments which are to be made, a tele- scope is erected at a convenient distance in front of the mirror, and a little above the level of the mirror. The t«letM»j>c is capable of motion in a vertical plane, it is direete<l towards tlic suitjiensitm fibre just above the mirror, and a fixed mark is erected in the line of vitiiDn, at a horizontal distance from Uio object gli>s9 equal to twice the dintance of the mirror from the object glass. The apjiaratus iihould, if possible, be so arranged lliat this mark is on a wall or other fixed object. In order to ^eu the mark and the t>uspeni<ion fibre at the same time tlirongh the telewope, a cap may be placed over the object glass having a slit along a vertical diameter. This should be removed for the olh^r oboervatjons. The telescope is tlien adjusted so that the mark iw seen distinctly to coincide witJi the vertical wire at the focuH of the telcHcope. A plumb-line is then adjusted so as to pass close in front of the optical centre of the object glass and to hang below the teleiwopc. Below the teIe»cope and just behind the plumb-line a scale of equal parts is placed bo as to be bisvct«d st right angles by the plane through the mark, the suspension- fibre, and the plumb-line. The sum of the heights of tlic scale and the 93 MAGNETIC MEASURKMESTa [450- objed; glass should bo equal to twice the height of the mirror from Uio Q<ior. The telescope being now directed towurds tho mirroFj will see in it the reflexion of the ecale. If the pmrt of the «cal where the plumb-line crossee it appears to coin<;ide with the verliaii vire< nf the teleecope, thea th« line of collimation of the tnin coincides with tiie plane throiipfh tho mark nnd tho optical centre of the object ^lass. If the verticul wire coioeides wilil anv other division of the xenle, the ungtiUr po>ition of the line of oolliination it! to be found an followa :— Iji't the plittie of the paper be horizontal, and tet the various points be projected on this plane. Let be tJie centre of th< object glaa^ of the telescope, P the fixed mark : P and the vertical wire of the telescope are conjugate foci with re»j>eot to tlie object glass. Let M be the point where 01" cuts the plane of the mirror. Let M.\' be the normal to the mirror ; then OMS'=. is the ang\ which the line of collimation makes with the fixed plane. Let J/< be a line in the plane of OM and UN, euch that MfS = OJCVf then S will be the part of the scale which will be seen by reflexion to coincide with tho vertical wire of the telescope. JS'ow, since r. I — ¥^ ^ jl/.V IS horizontal, tho projeefed angles 03fy and .A'JI/i$ in figure are equal, and OJUS=20. Hence 0S= OMiaQ20. We hare therefore to measure OH in terms of the divieionfi of the scale ; then, if i^ is the division of the scale whidi coincides with the phmib-line, and » the obMrred division, «-»,,= OMtm29, whence 9 may be fonod. In measuring 03f we must remember that if tho mirror is of gliiBS, silvered at the bdclc, the viKgal in: of tho reflecting eur£Ke is at a distance behind Uic front sarfni 4 URTH0D9 OP OBSERVATION. * of the g^ass = - , vhere t ia the tJticksess of the glass, and n is tb« index of n>rmction. We must also rcmcmlwr tlint if the Hue of suepension does not pa^ tltrougli the point of Mlt^iioii, the position of 31 will alter with &. Kciice, when it is pufutihle, it i» adviwildc to make the centre of the mirror votncide with the line of itiui|M>n!iii>n. It it alao itilvisahle, espeoiall}' wlitm larf>;fi angular motions haT<! to be obiterved, to make the scale in the form of n concave oj-liodrio Rtufair, whoee axis is the line of suspension. The angles are then ob«erved at once is circular measure without reference to a table of tang«nl». The scale should he carefully adjusted, an that the axis of tiie cylinder coincides with the suspension fibre. The Dumbeni on the scale should always run from the one end to the other in the same direction so as to avoid negative readings. Fig. 1 G Fig. 16. represents the middle portion of a scale to he luied with a mirror and an inverting teleeooije. This method of observation is the best when the motions are slow. The observer sits at tlie telescope and sees the image of the scale moving to right or to left past the vertical wire of the telescope. With a clock beside him he can note the instant at which a given .division of the scale passes the wire, or the division of the scale which is passing at a given tick of the clock, and he oan also record the extreme limits of each oscillation. When the motion is more rapid it becomes impossible to read the divisions of the scale except at the instants of rest at the extremitjea of an oscillation. A conspicuous mark may be placed at a known division of the scale, and the instant of transit of this ! mark maj be noted. When the apparatus is very light, and the forces variable, the .motion is so prompt and swift that observation through a telescope MAGSOTIC MBASmiMIEirfS. [45 q would be useless. Id this cum tlio observer looks at tlie scald (linclly, And obwervva the motiuus of thu iinag« of tlie vortical win throwD on the scale by n Ump. Tt is munifest thut iiincc thi> im&g^e of the ecale reflected by th< mirror anil refrat-U'il by the object glass coiocidee with the ver wire, the imaifc of thu vertionl wire, if suf&oientfy illuminated, will'' coincide with the «cnlo. To observe this the room ts darkened, and the ooDCGQtrah-d rays of a lamp are thrown on the vertieal wire towards the object glass. A brii^ht patch of li^ht crossed by the shadow of the wire is seen on the scale. Its mottoos can bo followed by the eye, and the division of the scale at which it come to rest can be Bxcd on by the eye and read off at leisure. If it he' desired to note the instant of the passage of the bright spot past a given point on the scute, a pin or ■ bright metal wire tnay placed there so ns to Ihwh out at the time of posaage. By substituting a small hole in a dia]>hragiii for the cross wire the image becomes a small illuniiuutcd dot moving to right or loft on the scale, and by btuhxtitiiting for the scale a cylinder revolving by clock work about a horizontal axis and covered with photo-^ graphic pajwr, the spot of light tracK« out n curve which can afterwards rendered vinble. Each abscissa of this curve corre«poi to a particular time, and the ordinate indicates the an^ jKisiiion of the mirror at that time. lu this way an automatio system of contiimouH registration of all the elements of terrestrial magnetism hax been established at Kew and other observatories. In tome «we* the teIe»cope in dixpt^nsed with, a vortical wire 18 illuminated by a lamp placed behind it, and th« mirror is a concave one, which forms the imago of tlio wire on the scale a4M| a dark lino across a patch of light, ^H 461.] In the Kcw portable ap|i«nitiis, the magnet is made in the form of a tube, having at one end a leas, and at the other a glass ecale, so adjusted as to be at the principal focus of the leu Light is admitted from behind the scale, and after passing through the lens it is Tiewed by means of » telesctipc. Since the scale is at the principal focus of the lens, rays Ironi any division of the scale emeigo from the lens parallel, and if the t«!e«e4)i>e is adjusted for celestial objects, it will shew the scale in optical roincidenco with the crow wire* of tlie telescope. If ; given division of the ecale coincides with the intersectjon of tl orosa wires, then the line joininer that division with the optic centre of the lens must be parallel to the line of oollimation oC. be I he^ it a^ rire loft photo-^y !«nb«fl >ponda^| nguhu^B '41 ical^ ■ 45271 DIBBCTIOS OP MAOKBTrc TOBOB. »» the tetascope. By fixing tht> magnet and moving the t«1oRCoptf, we may ascertaiti tbe angular value of tlie tlivisioiw of tin- kchIc, und (hen, when (he ni3<;net is suspended and the poaition «!' the tt'lu- Kope known, wv may determine the poeitioa of the mugnot at ony inetant by reading off the division of the scale which coiuuidee with the CTOfS wires. Tbe telescope ia aupporbed on nn ana which ia centred in th« line of tho suspension fibre, and the position of the tele!ico[>c is read off by vornicrs on tho azimuth circle of the iuatrament. This arrangement is snttabk- for a small portable magMctomelcr in which tho whole apparatus is supported on one tripod, and in which the oedlUtioos due to accidental disturbaQcen rapidly subside. IDeUrmiHotion of tie Direction of the Axis of the Magnet, and of tie Direction of Terretlriat Magitelitm. 4fi2.] Let a syntein of axes be drawn in a magnet^ of which tiie axiK of : is in the direction of the lengtli of the bar, and x and y perpendicubir ti> the itideH of the liar supposed a paratli'topipod. Let /, in, << and A, fj, v be the angles which the magnetic axis and the line of eollimution make with these axes respectively. Lot J/ he the magnetic moment of the magnet, let IT be the horizontal component of terrestrial magnetism, let Z be the vertical oompouent, and let fi he the azimuth in which // actM, rockoncd from the north townnJs the west. u Let f be the obscrvni axinmlh of the line of collimation, let ^k be tho azimuth of the stirnip, and ^ the reading of the index "of the torsion circle, then a—(i is the azimuth of the lower end »of the suepenaon fibre. Let y he the value of «— ^ when there is no torsion, then the moment of the force of torsion tending to diminisli a will be where r ig a cocfiidcnt of torsion depending on the nature of the fibre. To determine \, fix the stimip m that y is vertical and up- wards, a to tho north und r to the weat, and observe the aKimuth f of the lino of collimation. Then n-move the magnet, turn it through an angle v about the axis of t and Teplac« it in this inverted portion, and observe (lie azimuth f of the line of coU ^^imatiou when g la downwards and x to the east, MAOSETIO MEASCHEMEST9. Hence Next, bftiig the stirrup to the suspension fibre, and place magnet in it, adjusting it carefully so that jt may be vertical an' upwards, tlieu the moment of the force tending to increase a is MHsia m sin (5— a— | + ;)— r(«-j8— y). (4JJ But if C is the observed azimuth of the line of coUimatioa f=a+|-A. C^J BO that the force may be written J///«ni«sin{«-f+i-A)-r(C+X-^-j9-y). (i When the apparatus ia in equilibrium this quantity is xero for a particular value of f. ^H MTicn the apparatus never eomes to rert, but murt be observeJ^^ in a i>t:itc of vibration, the viiluo of ( corresponding to the positios of equilibrium may be calculated by a method which will described in Art. 735. When the force of torsion is small compared with tJie moment of the magnetic force, we may put i—(+l—K for the cine of that angle. m If we give to ^, the reading of the torsion circle, two diflcren^^ valuee, ^ and ^,, and if d and Ci oro the corresponding values of C ioa^^ en^B or, if we put J/// sin « (C-C) = r (C,-C.-ft +^). (i-U — — = /, then T = Mllma mV, and equation (6) becomec, dividing by J/ZTsin m. If we now reTcme the magnet so tlint y is downwarfa, Okdjust the appatKUiK till f is exactly vertieal, and if C is the ne^, valoeof Uic aEiraulh, and ff the corresponding declination, ^M y-f '_/+A-t'(r-X + |-/8-y) = 0. whence i*g = i(f+n + *''tC+r-308+y)). (10) ^ OBSKBVATrOS Of DEFLEXION. i»7 I I Th« reading of tlie torsJon circle should now be iidjuHtoil, ao that Uie coefficient of r' may l>e an nearly aa poa»ible aero. For ihie parposG we must clet.i.'rmiDe y, the value of a—fl when there ia no torsion. This may be (lone by placing a uoq- magnetic bar of the Eame wetfjht ae the magnet in the stirrup, and deteriuining: a— ^ when thcr« is ctiuilibHum. Since t is small, great accuracy is not requirnl. Another method is to use a torsion bar of the same wei);ht us the mmgaai, containing within it a very small magnet whcee n)af>nctic moment is - of that of tho principal ma^et. Since r remains tlu: itame, r' will become nr, and if ^j and (,' arc tJie value* of <"«« found by the torsion bur, »= i(C. + Q+i»T'(C, + C,'-2(5+y)). (12) Subtracting this equation from (II), 2{— l)(/3 + y)=(« + ^)(f. + fn-(l+}0(^+O. (13) Having' found the value of j3-|-y in this way, ^, the reading of the torsion circle, should be altered till C+r-2O+y) = 0, (14) as nearly as possible in the ordinary position of the apparatus. Then, sinoc t is a very small numerical quantity, and since its coeffleient is very small, the value of the second terra in the ex- pnasioa for 2 will not vary much for small errors in the values of f and y, which arc the iiuautitiea whose vaJues are lea^t ac- curately known. The value of h, the magnetic declination, may be found in this way trith conaiderablo accuracy, provided it remains constant during the cx|>crimcntB, ko that n-e may a^ume E'= E. When gnat accuracy is required it is necessary to take account of the variations of S during the experiment. For this purjHiBC ebaervations of another suspended magnet should be ma<le at th« ■una instants that the diflerent values of ( are observed, and it If, ij' ate the observed azimuths of the second magnet corresponding to {and (", and if 2 and 1/ are the corresponding values of &, tlieu Ueooc. to find the value of ft tve must add to (1 1) a correction The declination at the time of the first obBvrvation ii therefore « = i(c+c+i-t')+^'(c+r-2^-M- (16) VOL. U. 98 MAOSETIC MEASmtBSrESTS, [453. To fiud the direction of tiie magnetic uxis n-itliiD the miignct fubtraet (10) from (9) and add (15), i / = A+HC-n-i{7-l')^4''(C-r+2X-»). (17)1 By repeating thu tfxpcnraents nitb tUe bar on iU two «dgea, bo that tlie uxix of ■s in Tcrticallj' upwards and dnwntrards, we can find tlie value of m. If Ike axis of collicnation is capable of ad- jnstmvnt it ought t<i be made to coincide with the niB^etiti axis IIS nearly as po^naiblc, so that the error arising from the magnet not being exactly inverted may be as small aa poiwible*. On the Measurement of Ma^etic Foreei. 453,] The most important mcasurementn of magnetic force are^ thoae which dt-termine M, the magnetic moment of a magnet,! and //, the intensity of the horizontal component of tcRCStnalj magnetism. This is generally done by combining the resnlts of j tno ex))eriments, one of which determines the ratio and the oth«r{ the product of these two quantiticB. Tha intensity of the magnetic force due to an infinitely smull magnet whose magiictio moment ie ^f, at a point di^tAut r from the centre of tlie magnet in the positive direction of tlie uxis of Uie magnet, ia -=? (Ill and is in the direction of r. If the magnet is of finite siz« bub | spherical, and mngnetized aniformly in the direelion of itN axiiiij this value of the force will still bo exact. If the magnet ia m BolcDoidal bar magnet of length 2L, If the magnet be of any kind, provided its dirocniuonfi are alll small, ctnnpared with r. It = 2~(l+J,l+J^-^ + &c.), (3) where A^, A^, &c. are eoeHiHcnts depending on the di»lributioa of the magnetization of the iMr. Let If l>o the intenftity of the horizontal part of tArmtrial mtignettim at any place, il it directed towards magnetic north. Let r l>e meastired towards m^nclic west, then the magni-lic force at. the extremity of r will be // towards the north and R t')warda I • Sm ■ rapor oD ' ImjicrfsM IttTCndnii,' b* W. S»»». rA. Mi (1«6[^), p. }li>. TrttM*. a. S. Bilm I 453-1 DEFLEXION 0B6KRVATI0NS. m tbe west. The resultant force will make an angle with tbe magnetic meritliaD, mcmured towards the weet, and such that B = lltao0. (i) n Hence, to cIet«rmino -jj we proceed as follows : — The dir««tion of the mugnetic north having hven asecrUtned, a magnet, whoKC dimensionit nhould not be too givatj \a, anspendud aa in the former experimentit, and the dellecting magnet M ia placed 80 tliut \is centre is ut a disttuico r from that of the gus- peuded magnet, in the same horizontal plane, and due magnetic The axis of M is earcfiillv R<)juste<l so as to be horizontal and in the direction off, Tbe 3u»[>i-nili'd magnet \* observed before M is brought near and also sAer it is placed in position. If is tlie observed deilcxioo, we have, if we use the approximate forniuin (1), M T- H=Y^°^-' or, if we use the fornnU (3), l+Ji~+J^^+&c. (6) («) I Here we muHt bciir in mind that though the dcHexifio can be ohwrved with great acciirncy. the distance r between the centres of the magnetH is a quantity which cannot be precisely deter- mined, unlv«a both magnets are fixed and their centres dellned by murks. Thi* difficulty in overcome thus : Tlie magnet .1/ Im plucird on a divided BMie which extends eOBt and west on both sides of the Eut>[>eDded magnet. Tlie middle point between the ends of M is re<'koned trlio centTo of the magnet. Thin point may be marked on the magnet and its jioifition observed on the scale, or the ponitiunM of the ends may lie obaierved and Ihe aritlimetic mean taken. Call thiit <>,, and let the line of the suapemiioD fibre of tJic suspended tuugnet when produced cut the scale at fg, then ri=*j— *„, where «, is known acouiately and a^ ap- pr^xiniately. Let 0, be tlie deflexion observed in this position of M. Now reverse M, tliat U, place it on the scale with its ends rrversed, then f) will be the same, but M and vf,, A^, &o. will have their signs changed, so tliat if dj u tJte dcllexion. I ff 1 ! - - T. r,Manl»„= i_J, _+4 — _& 2 J/ ('> B 1 100 HAGKCTIC UEASUBBUfiXTS. [45^ 'liking the aritlimi.'tica! mean of (6) and (T), 1 U 1 (0)1 Now lemove M to the vest sido of tb« Euepoodwl ma^Dot, place it with its centre at tlie point marked 2«o— «| on the scaler Let the deflexion when the axis is in tho first poeition be (?,, and when it is in the eecond 0^, then, as before, Let us suppose that the true position of the centre of the ei peoded magnet is not «„ but Ja+<''> t-'"^ fi = r— ff, rjsr+o-, (10) and since -^ may he neglected if the measnremeats are car made, we are sure Hint we may take the arittimetical me*n of f)' and r.j" for r'. Hence, taking the arithmetical mean of (8) and (9), 1 /A I - j^f*(tantfi— tan^j + tan^,— tantf^) = l + ^-i-+&c-, (12) ot, making j (tan(Ji~tan^+tAn^j-taD0J = D, (U\ 464.] We may now regard J) and r as capable of exact detei minatioD. The quantity ^ can in no case exceed 2X*, where L is half th« length of the ma^fuet, so that when r is considerable compared with Z we may ncglo«t the term in J^ and determine the nUio of // to M nt once. Wa cannot^ however, aasame that A^ is oqaftl to 2L*, for it may be \tm, and may even b« negative for a mngn«t whose largest dimensions arc trnnsTerae to the axis. Hie in vJ,, and all higher terms, may t«fely be ncglectod. To eliminate A.^, repeat tlw exprriment, using distances r^, fjT^ ftc, and let the v^ues of Ji m 2M , 1 'ii "»• ^|i A &c. &o. DEFLEXION OBSERVATIONS. 101 If we suppose tli»t. the probable errors of these equatiofui are e4]UAl, iiM th«y will be if thcj dopen<l on the determination of J) only, nod if tlierc is no unccrtuinty tiboiit r, then, by muHipIyiog each eqnatioD by r~^ aod ndtlJit^ the results, we obtain one e<]uation, and by multiplying^ eiu.-h cquntion by r''' and adding we obtain another, accordinj; to the general rule in the theorj- of the com- bination of fallible m(.'H«urc8 when the probable error of each equatioD is supposf^ the same. Let UN write 2(01-') for /),r,-» + i>,r.-» + Z)jr.-» + 8tc., and uac similar oxpresxions for the sums of othi'r groups of symbols, Ifaea the two resultant e<iuations may be written 2 M I » and J,{S(7>r-»)£(r~'<')-£(2>/-'')S(r-')} = 2 (ft-") 2 (r-«)-S (Dr-^) 2 (r-*). The value of A, derived from these equations ought to be less than half the square of tlie length of the magnet M. If it is not we may suspect some error in the observations. This method of obaer\'ation and reduction was given by Gauss in the ' First RcjKirt of the Magnetic Association.' When tbo observer can make only two series of experiments at 2M distances r, and r,, the values of - . and A^ derived from these experiments are r,"— r. If iDy and hD^ are the actual errors of the observed deflexions Z>i and b^, the actiuil error of the calculatetl result Q will be r * r * If we snppoeo the errors ?7>, and ft/)j to be independent, and that the probable raluo of either it tl), then the probable value of the error in tho calculated value of Q will be S Q, wher« - 1" A. r "• MIOSETTC MEASntRMBSra. If wo sapposc tbat one of thceo dUlnnccw, tay tli« smalW, ii <;ivi-n, the vnlue of the greater <Ii<itiiiioV may be determined to M t(> make bQ a minimum. TbJs couiiitJoa leads to an crjuutioii of the fiHh do^rcM) in r,", n-hich hu only one nsal root greaiat than fj*. From this the Lwgt value of'/-, Ik Tound t« be /■, = 1.318flr,' If ODe obsen'ation only is taken Ibe Ix-Mt distance ift wlien where i/> is the probable error of a measurement of defiexion, and br IK the probable error of a measurement of distance. MelimI of Sina. 455.] Tlie method whieh we have juet deeoribed may be called the Method of Taoffentt!, bt'cutiec the tangent of the deflexion is a measure of the magnetic force. If the line r, . iustend of liein^^ measured cant or west, ih adjujrted till it is at right niiglefl with ihu uxih of the deflected ma^et, then S is the Kume as bi^fore, but in order that the suspended mag-net may remain pcqiendteular to r, the revolved (lart of th« force If in the direction of r miiHt be niual nod opposite to S, Ilenee, if is the deflexion, H = II sin 6. This method in ealle<l the Method of Sine*. It can be applied, only when /f is lf» than //. In the Ken- jiortable a]>paratit8 thiit method ia employed. The eutipeuded magnet hang§ from a part of the apparatus which revolves along with the telescope and the arm for the defle^-ting magnet, and the rotation of the whole is measured on tlio azimuth circle. The apparatus is first adjusted so that tlie axis of the telescope coincides with the mvaii position of the Hov of coUimation of the magnet in iU undiirturbed nlul«. If the magnet Is vibrating, the true azimutJi of magnetic nortlt \» fotmd by obecrring the ex- tremities of the oscillation of tlie transparent acole and nuiking tb« proper correction of the reading of the azimuth cirelo. The deflecting magnet is then placed u|>on a Mtraigbt rod which posMS through the axis of tlio revolving apparatus at right angle«: to the oxia of the telescope, and is ndjuFled 6o that the axis of the deflt-i'ting magnet is in a line poHHing tlirougb the eent(» of the snspi'iidtd magnet. The whole of tlie revolving apparatus is then moTod till the Uu^ • See Mty-t Mayn^Unt. I d 456.] TIME OP VIBRATIOS. 103 I I I of ooUimAtion of the saspondwl mag^net afrain coincides with the axis of the telescope, and th« new azimnth rtadinf- is contctwl, if neeesaary, l>y the mean of the ecalo readings at the extromitic« c^an osnllation. The difTereRce of the corrected azimuths gives the deflexion, alU-r which we proceed as in the method of tangents, «xocpt tluit in the expression for 1) n-e put sin Q instead of tan 9. Id thia method there is no correctioo for the torsion of the eue- pending fibre, since the reUtive position of tho fibre, telescope, a&d majcnet is the same at every observation. Tlie axea of Uie two ma^ets remain alnays at right nn{;1ca in this method, so that the correction for length can he mora acoaratcly made. 456,] Having thus measured the ratio of the moment of the deflecting m:^rnet to the horizontal component of terrestrial ma^- netixm, we have nest to find the product of these qiiantilies, by determining the moment of the couple with which terrestrial mng- netism tends to turn Uie same magnet when its axis is deflected from the magnetic meridian. There are two methods of making this measurement, the dy- samicalj in which tho time of vibration of the mngBct under the action of t^^-ricstrial mnguetism is observed, and the statical, in which the magnet is kept in equilibrium between a measurable statical couple and the magnetic force. Tlic dyniimical method requires simpler apparatus and is more ■CGnrat« for absolute menaurement^, but takes up a considciablo time, the statical method admits of almost instantaneous mcasure- mml, and is therefore useful in triicing the changes of the inten^itj of tho magnetic force, but it requires more delicate apparatus, and is not m accurate fur absolute measurement. jWrAiorf «/■ Vlhratiom. The magnet is suspendetl with iU( magnetic axis horizontal, and is eel in vibration in ainall :irc». Tiie vibrations are obwrved by mans of any of the methmU already dtwcribed. A point on the scale ia chosen corresponding to the middle of the arc of vibration. The instant of paa.inge tlirough this point of tho scale in the positive direction is observed. If there is safii- cient time berorc the return of the magnet to the same point, tlie instant of juusage through the point in the negative direction is iihu> observed, and the process ia continued till a + 1 positive and 104 MAOSETIC HBASCREMBKTS. [456. » no^tive pasmf^es have been obserri'd. If th« vibrations an too rapid to allow of every consecutivo paamge boin^ obcerved, everr third or every 6Ftb passage in obwrved, oare bein; taken tliati the observed paessges are alternately po.iittv« and iwgiUve. Let the observed tiroes of passage be we put 1 r,. r,, r»^„ then then TV,.! is the mean time of the potitive pusa^a, and oagh to agree with 7",,, , the mean time of the negative passagee, if tha point baa I>een properly chosen. The mean of theao resulta ia to be taken an ti\a mean time of the middle pasmgc After a large number of vibrations have taken jilaoe, bnt befora the vibrations have ceased to be distinct and regular, tho observer makeK another series of observations, from tvliich lie dcdnces th mean time of the middip pas^afr*- of the secom) series. By calculating the period of vibration either from the fi HCries of observations or from the E«cond, he ought, to be able be certain of the number of whole vibrations which have tak |>hice in the interval between the time of middle pastxage in (Jii> two Mtrie*. Dividing the interval between the mean timen of middle [WK-^age in the two series by this number of vibrations, the meua'' time of vibration in obtainivl. The observed time of vibration is then to be rcdnc«d to lJi< time of vibration in infinitely small arcs by a formula of the sama kind as that used in pendulum obttervations, and if the vibrations are found to diminiali mpidly in amplit«idc, there is aoolher cor- rection for resistance, ttee Art 740. These corrections, however, ara very nnall when tlie magnet bangs by a Hbre, and when the arc of vibration vs only a few desreois. Tito equation of motion of the magnet ts Jiff where 6 is the angle brtwi>en the magnetic axis and the direction of the foTOO //, A is the moment of ini-rtiu of the magnet an iHi*i>ended apparatus, .1/ is the magnetic moment of the magnvi // the intensity of Uie hori«ont«I magnetio force, mnd J/Z/r' tJi ooeffioient of torsion : t' ts dvlenniniHl as in Art. 452. and i« vciv smaH quantity. The value of tf for equilibrium is % = {-^j' • n ^^ «iia!l angie, 1 re 01 i 457] BLIMISATIOS OF IITDOCTIOK. 105 MU = Mtd the eolutioB of th« equation for small values of Uie amplitude, is where T in tlie periodic time, and C tbe amplitude, and 3/7/(1-, t')' whetioe we lind tJie value of MU, Here T is the time of a poinjiletc vibration determined from obdervatioD. J, the ntoment ol' int-rtio, in founil once for all fur tbe ma^ety eiUier by wt^ghing ttnii meuKunnfr it if it iit nf » regular figure, or by a dynaaiieal procuss of coin[iiirisoD with a body wbwe moment of inertia is kuowu. ^ Combining tlita value of M/f with that of ^ formerly obUinud, weget J/« = (J///){|j = y,^/*^. 457.] We have suppoiied that // and ,1/ continue constant during tbe two series of experiments. Tlio flugtuatiuus of // may be aecertained by simiiltuiKmis observatiouH of the biRlar magnet- ometer to be preacntly d<.WTib^-(l, and if the magnet has been in use for some time, and i* nut exposed during tiie experiments to changes of temperature oi' to conoussion, the part of M which de- pends on permanent ma^ietism may be assumed to be conrtant. All steel magnets, however, arc capable of induced magnetism depending on tlie action of exti^rnal mugnetic force. Now the magnet when employed in the deflexion cxpcrinaent* i» placed with ita axU cant and we^t, to that the action of ter- restrial nmgnetiam is transverse to the magnet, and doc^ not tend to increase or diminish 31. When tbe magnet is made to vibntlv, its axis is not-tb and south, so that tbe action of terrestrial mag- netism tends to magnetize it in the direction of the nxin, and therefore to iDcrcaee its magnetic moment by a quantity i fl, whore it is a coefficient to be found by experiments on the m»}^ii!t- There are two ways in which this souroe of error may be avoided without calculating i. the experiments being arranged so that the magnet shall bo in the came condition when employed in deflecting another magnet and when iteelf swinging. I I 106 MAOSBTIC MEASntEMKKTS. [451 W(! may plitcc tho d<;lloctin^ ma^ct with its »xi* potntuif; north, aL a. (lisUitcu r from tho centre ot tlifl etip|i#nd«d inagn«t, Ihv line r miiking iin iingk- wliow cosido in V j willi the magnetic meridian. Tlie iietion of the tlf(i«clin{r miM>Dct on tlie sii^ix^ailed one i» then At right unifies to its pnrn direction, tttid ia equal to Hera Jf is the magnetic moment when the axis points tiorth, RK in thir experiment of vibration, so tliKt no corfts;tion has to li^^l muilc for induction. ^^ Thi* method, however, is extremely difficult, owing to the largo errors wbieli would be introduced by a Blifflit dioplaeement of the deflecting magnet, and as the correction by levorsiiig the deflecting mugnct iff not uppliL-ablc hero, this method i» not to be followed except when the object is to determine tlie eocffieicnt of induction, 'llie following method, in which the magnet while vibrating fn-od from the inductive action of terrestrial magnetism, is due Dr. J. P. Joule*. Two magnets are prepared whose imigiietie moments are nearly equal as possible. In the deflexion experiments these mag^J net* are used separately, or they may be placed simultanoousl] on opposite sides of the suspendnl magnet to produce a greater deflexion. In themt ex[H-riment« the inductive force of terrestrial^' magnetinm is trantiverM to the axis. ^M Let one of these msgiietw be suxpended, and let the other be placed )iaral)el to it with itit centre exactly below that of the Bits- ponded magnet, and witli iU »x\s in the same direction. The force which the fixed magnet exerts on the sospended one is in the opjiosite direction from that of terreatrial magnetism. If the 6xcd magnet 1>e gradually brought nearer to the snapended one the time of vibration will ineT«i.->e, till at a certain point the equilibrium will cease to be stable, and hi^yond this point the suspended mugne^^ will make oscillations in the reverse pontion. By experinicntin|j^| in this way a position of the fixed magnet, is found at which it ' exactly neutralizes Uiv effect of terrestrial magnetism on tlie iras- peiMled one. Tho two magnets are fastened together so as to ^__i, parallel, with their axcH turned Uie snme way, and at the dintanO^H jiiet fotind by experiment. Tliey are then suspended in tlio ueua^V way and made to vibrate together through small aiva. • Prec. PItit. S; KauAnUi, Mkrdi li>. ISOi. I IKTEKSlTr OP HAOKXTIC FOBOB. t07 The lower maj^Det exactly neutralizes tUe effect of terrestrial niagnetiiira on the upper one, and since the maijTiets are of equal niuinent, the npper one oeutralizes the inductive action of tlie earth on tliG lower one. H The value of 3/ is therefore the same in the cxperinii'Dt of ^^vihration a^ in the experiment of detlesion, and no corri^ction for induction ii* required. 468.] The most accumtc method of asoertaininj^ the intensity of the horizontal magnetic force is that wliieh we have just described. The whole series of experiments, however, cannot lie performed with sufficient accnraey in much tesH than an hour, so that any changes in Uie intensity which take pJaoc in periods of a few minutes would escape oIiMTvation. lleiiw a didVrent method is required for ob- M.-rving' the inten«i(y of the majs^ietic force at any instant. The statical method eon^ists in defieeting- the magnet hy means (if a statical couple acting in a horizontal plane. If £ he the moment of this couple, M tlie magnetic moment of the m^not, Jf the horizontal component of terrestrial magnetism, and 9 the deflexion, MIIsin$z= L. llenoe, if L is known in terms of 9, Mil can he found. The oouplp L muy be generated in two waj-», hy the torsional ity of It wire, iix in the ordinary torsion halancv, or by the ight of tlie sui^pi-ndcd appnnitui;, us in the bifilar euspenition. In the torsion bnlancu the magnet is fastened to the enil of a vertical wire, the upper end of which can be turned round, and ita rotation mcosurvd by means of a torsion circle. We have then L = T{a—a„—0) = MllsmB, Here a^ is the vnhic of the reading of the torsion circle when the oxia of the magnet coincides with the magnetic meridian, and a is the actual reading. If the torsion circle is turned so as to bring the magnet nearly perpendicular to the magnetic meridian, no thut then I .=1-^. or r^,a~<,,-^"-^.(r)^Mn{\-\ff% MH=T{\+\ff*)(a-a^~~ + 6'). By observing!}', the deflexion of the magnet when in equilibrium, wo can calculate Mil provided we know r. If we only wish to know the relative viiiue of // at diflerent times it is not neoessary to know either M or r. Wo may easily determine r in absolute meo-iure by suspending 108 MAOKKTIC lIBASCREKCrrS. a non-magiietic body trom tlie mme wire and observing its time ortMciUiition, thc-n if A in Ibe momont of ioertia of tbts body, am T th« time of a complete vibration, 4 It' .4 Tbe tiut! objection 1o tbe luo of the torsion balance i» that tb« zero-reading a„ in liable to change. Under the constant twiEtinp fbrcv, iirii^![i(; iwm the tendency of the magnet to torn to the north, tl>e wire graduMly aoquiFeti a permanent twist, eo that it becomes noccsMry lu ili^U'rinine the zero-reading of the torsiou circle afrc»h at ithort iiitervatn of time. BiJUar Siupention. 459.] The method of sn^tpending the magnet by two wires Dbrw was introduced by Gausa and Weber. As the bifilar s pennon is naml in many electrical iostruroents, we shall investigattt it mora in detail. The general appearance of the suspension ia shewn in Fig. 16, and 1%. 17 represents the projection of the wires OD a horizontal plane. AB and A'B' are the projections of the two wires. JA' and BB" are the lines joining the upper and the lower of the wiree^ a and 6 are the lengths of the lines AJ" and SR. a and ^ their aziniiiths. ff' and H'' the verti^nl comjicnenUi of the tennons of the wires. Q and Q" their horicontal component*. i llifl rerticml distance between A.f and BB'. Tl)e forres which act on the magnet ar^— its weight, tbe conpl arising l'Tt<m tcrreetrial magnetism, the UmJon of the wires (if anyj and their tensions. Of these the etTects of magnetism and of ti»«ion are of the nature of couples, lleoee the resultant of the tcUN<>ns nniit eonaist uf a vertical force, equal to the weight of the nagnet> together with m coaple. The reenlunt of the vertical components of the leosions ts thervfore alone the line wboM pn^ tioo is 0, the intersection of Jj" and BB", and either of lines is dividej in in the ratio of tt' to M'. Thi> boritoiital euinpoaents of the ttnaiou fonn a coaplo, are tberefont e>)iial in magnitodv and pMiQal in duMtioo. Colli: eitber nf them Q, the moment of the couple which th«y form ia when PP" is the diateaea betvaan the pwalU Uaca JM ud JTJF, in^^ 459-] BiriLAB SUSI'BNSIOS. 109 Tu find Lhe value ot Z vrts bave Uic cqautions of momenla QA=Jf'.AS = tF'.A'B', and the gcomctriml cc^uation {AB + J'li') py= ah sin (o-fl). wbeoc« wc obtaiD, (2) Rj-l*. Fig. ir. ir n ii tfao mass of the ButtpondiHl appnmtiiH, and ^ tho intciifity of gravity, ;r+ »"= mg. (5) If we also writ* r- W'=»mg, (6) I wo find i= i(l-ii»)«i^ ^am(a-^). (7) llw vsliw of X is therefore a maximum with req>oct to s wfa«n s no MAOSETIC UEASUREMESTS. [459- is zero, that is, when the weight of the 8usi)«D(l«d muss is equally borne by thv two wim. ^M Wo may mljnst the U'Dsionx of Ihc wires lo i>quaUty l»y observingf ^^ the time of vibnition, mid milking it u minimum, or we may obtain ^i a Kclf-ucting uiljustmc^nt by uttaching the ends of Ihv wires, u^f in Fig. IG, to » pulley, which turns on its axis till the t«iisioiu^^ arc equal. The distance of the upper ends of lUu nuiipcniiion wires is re-j gulated by mCAiiti of two other pullies. Tlic dUtanee bctwoon the>j lower endK of Uie wires is also capable of adjuatnient. By this adjustment of the tension, the couple artslD^ from the! tensions of the wires becomes The moment of the coaple arising from the toraion of the wirei i is of the form j (y—^), where T is the sum of the coefficients of torsion of tlw wires. The wires ought to be without torsion when a = ^, we may I then make y = a. The mumi^ut of the couple arising from the horizontal mo^etio | force iH of the form where i is the magnetic declinafion, and $ is the azimuth of the tis of the magnet, Wc shall avoid the introduction of unncceeeary eymbols without siicriGciDg generality if we sssume that the axil of the magnet ia parallel lo SB", or that ji = $. The equation of motion then W-comes A ^J^ = J/// sin (8-ff) + 1 ^ «y sin (a-(Jj+r(o-»). (SjJ There are three principal positions of this apparatns. (1) When a is nearly equal to A. If 7*, ts tbe time of a complete j ovcillation in t^is position, then iv'A I ai „„ , , mg^T+itl/. (9) r.» 4 i (2) Wlien a ia nearly equal to 8+ir. If 7, is the time of complete oscillation in tliis position, the nortb end of the magar' being now turned towards the south, iii*J lab ^ - i T "'■*■'•— *'"• *'*•) The quantity on the cigfat-haod of (his equation may -be m: I 459] BIFILAH SFSTENSTOX. Itl ^f M small as we plense by diminisliing a or i, bat it mwit not ht made ac^stive, or the equilibrium of the magnet will bevome un- Btxhle. The magnet in thia ]>oaition forms an inatrument by which small vanatiouB ia the dirixtion of the maj^netii; force Piay be rendered sensible. I For wbeo i~$ is nearly equal to ir, na (8—^) is nearly equal to 0—h, and we tind J. 1 l** .r»r By diroinishin^ the denominator of the fraction in the InH t«rin we may make the variation of S very large eomiiared with thitt of f>. We should notice that the coetHi'ieut of 8 in this expri^itsiriii is negative, so that when the direction of the magnetic force turns in one direction the magnet turnn in the oppo&itii direction. (3) In the third jxHjit.ion the \x\>\k:t part of the suspension- BplKirattn is turned round till the axis of the magnet is nearly pcrpendicuhir to the ma<<;ne)tic meridiao. If wc make $-h = -+(f, anil o-tf = /3-ff', tlie equation of motion may be written (12) If tb«re is equUibriuin when II — If^ and f = 0, (13) (H) and if ff is the value of the horizontal force corretponding to a •mall angle S", Vand H «na V V ^■^nysin/S+r^ Id order that the magnet may be in stable equihbrium it ia nece«MUy that the numerator of tlie fraction in the s^-ond member ■hoiUd W poaitirc, but the more nearly it approaches sero, the tnon: «en«itive will be tlic inittrument in indicating changes in the value of the intensity of tbe horizontal component of t«'rre«trial magDetiiim. The statical method of estimating the intensity of the force depend* upon the action of an instrument which of it«elf assume* (-^myc08j3 + r \ (15) iia MAOKETIC MEASUREMENTS. [46a diflbrent positions ni* equilibrium for dilTeTent Tallies of the Toret. Heuoe, bj nieaaii of a mirror attacbed to tbe magnet and throning * 3{>ot of light upon a photographic surface moved by clockwork, • cun'e may be traced, from which the inteaeitj* of tbe force at uijr inirtimt m»y bt- dctormin(Hl according to a scale, which we may for tlie pr^'sent consider uu arbitrary one. 460.] In an ub8ervat'>ry, where a continuous system of reffie* tratioa of decliaatioa and iutenttity is kept up either by eye ofa torvatioD or by tlm automatic photographic method, tJte abttolut valuex of the ilfcltnutioii nud of the intensity, as well as tbe positioa^ and moment of the magnetic axis of a magnet, may be detemuned to a greater degree of accuracy. ^M For the di>cli no meter g'wcs tbe declination at eveiy instant afTecte^H by a constant error, and tlieblRlar magnetometer gives tbe intensity at every instant multiplied liy a constant coefficient. In the ei- pcriments we sab«titute for $, S' + Sg where 2' is the reading of tb^. declinometer at the given instant, and 8^ is tbe unknown bat eon^| Btant error, so that i'*f ^ is tbe true declination at that instant. lu like manner for //, we substitute 67/' where //' is tbe reading of the magnetometer on it^ arbitrary scale, and C is an unknown but cooHtant multiplier which converts these readings into absolut« measure, so that CJI' ia the horizontal force at a given instant. Tlie experiments bo determine tbe absolute values of the quan- tities must be conducted at a sufficient distance from the declino- meter and magnetometer, so that the dilferent magnet* may not sensibly disturb each other. The time of every observation most be noted and the corresponding values of h' and //' inaettcd. Tbe equations are tlien to be treated so as to 6nd 6^, the constant error of the declinometer, and C the cot-fficient to l>e applied to the reading of the magnetometer. Wlien these are found the readings of both iustriunents may be expressed in absolute measure. The absolute measuremeots, however, must be freij^uently rep«Ated in order to take aocotutt of changes which may occur in the magoctic axis nod nuiiniettc momeut of the magnets. 461.] The moihods of determining the vertical component of the terrestrial magnetic force have not been brongbt to the anm^ degree of precision. The vertical force murt act on a magnet which tnms about a horizontal axis. Now a body which turns about a hori- Kontal axis cannot be made so seasitive to the action of small forcn as a body which is snspeaded by a fibre and turns abont a vertical axis. Uesides this, the weight of a magnet is so largo compara^^ 461.] DIP. » » » I with tbe magneiic force exerted upon it that a emnU displocv- ment of the centre of iDdrtia bv unequal dilutittioa, &c. produces a gKAtjet cflect oa tho poeitioa of the ma^et than a confii<Ier»lilc ebao^ of the magnetic force. Uenoe the measurement of the vertical force, or the comparison of the vertical and the horizontal forces, ia the least periVct purt. of the system of ma^etio measurements. TTie Tertical part of the magnetic force is ^nerally deduced from the horizontal force by determining the direction of the total force. If i be the angle which the total force makes with its horizontal component, i is called tbe magnetic Dip or Inclination, and if ff is the horizontal force already found, then the vertical force is 77 tan i, and the total force is //sec /. Tbe magnetic dip is found by means of the Dip Needle. The theoretical dip-needle is a magnet with an axis which passes through ita centre of inertia perjjendicular to the Tiiagnetic axis of the needle. Tbe ends of this axis are mode in the form of cylinders of small radius, the axes of which are coincident with the line passing through the centre of inertia. Ihese cylindrical enda rest on two horizontal planes and are free to roll on them. When the axis is placed magnetic cast and west, the needle is free to rotate in tbe plane of tbe magnetic meridian, and if the instnimeat is in perfect adjustment, tbe magnetic axis will set itself in the direction of the total magnetic force. It is, however, practically impossible to adjust a dip-needle ¥0 tliat ita weight does not influence ita position of eijuilibrium, because its centre of inertia, even if originally in tbe line joining the centres of the rolling sections of the cjlindrical ends, will ccaec to be in this line when tbe needle is imperceptibly bent or un- eqaally expanded. Besides, the determination of the troe centre of inertia of a ma<>net is a very difBcutt operation, owing to tbe interference of the magnetic force with tliat of gravity. Let us suppose one end of the needle and one end of the pivot to be marked. Let a line, real or imaginary, be drawn on the needle, which we shall call tbe Line of CoUimation. Tbe position of this tine is read off on a vertical circle. Let $ be the angle which this line makes with the radius to zero, which we shall suppose to he boriiontal. Let A be the angle which tbe magnetic axis make« witlt the line of coUimation, so tliat when the needle is in this position the line of colliinatton is inclined $+K to the horizontal. VOL. XI, 114 MAGNETIC MEASUREME5T8. Lctp 1)0 the perpendicular from the cmtn; of inertia on the plaoe on whioh tlie axis roilii. then p will bo n function of d, n-hul«vi bo the Khajie of the rolling »urf<iccs. If Iwth tho rolling siAtio of the endH of th« axis urc circular, j> = e—a ein (fl + o) (: where a ie the distance of the centre of iaertiii from the line joininff the centres of the rolling eeotions, and a is the an^le whidi Uoe makes with the Uoe of ooUimation. If M is the inapietic mometit, m the maae of the ma^ot, y the force of gravity, / the total magnetic force, and i tho dip, th< by tlic conservation of onergy, when there is stable equilibrium, itlooB[0-i-K—i)—mgp (! must be a maximum with reapect to $, or oe ] .'IP (I JW/sin (ff + X— i) s— Bij^, = IW^flC0«(^ + o), if th« ends of the axia are cy)in<lnc«l. Also, if 7* be the time of vibration about the positioD of eqn librium, ■ /* V ■<»''^ /J MI+Mpamn{9+a)= -yj- {4 where A i» the moment of inertia of the needle about its uda of rotation, and is determined by (3). Id det«TmiDing the dip a readiDg- u taken with the dip circle in tbe DUgnetic meridian and with the graduation towards the weet.^ Let 0, be this reading, then we have jl//sin(^j-f X— i)= ni^<ico8(d, + a). {i The instrumeDl U now turned about a vertical axis through 180*, so that the gnuluatton is to the east, aod if tf, is the new reading', J//»in (flj+A— x+i) = wyaeoa (ff(+a). (6) Taking (0) from (5), and remomboring that dj ia nearly equal to I, and 9t nearly eqtud to %—i, and that X ia a Rmall angle, such that rngaX may be neglected in compariaon with 311, ^M 3fl(0i~0. + v~2i)= 2mffacmict»a. {^^ Now take tlic nuignot from its bearings and place it in deflcaioo appantua, Art. 453, so as to indicate iu own magnet moment by the dellexion of a suq>cndcd magnet, then 3f=kf*ir2) where J) is the tangent of the deflexion. IB 1 DIP CIBCI.!!. Nexi, reverse tbe msgnetisai of tbe needle And dctcnnine its Bew magnetic moment J/', by observing a new deHcxion, the tan- gent of which is 1/ j/'= J^V/i/, (9) wUenoe MI/= M'D. (lOj Then place it on iU bearings nnd take two readings, 0, and $^, in which d^ is nearly ir + t, and S^ nvarly — t, J/'7sin(flj + A'— IT— i) = mgaaM{e^-^-a), (ll) i/'i8in(dj + X'+i) = m^flco9(a« + a), (12) whence, as before, if'/(tfj— d,— IT— 2i) = 2in^(»co8icoBa, (13) and OB adding (7), Jf/(fl,-tf^+if-2i)+jl/7(tfj-fl«-ff-2i) = 0, (14) or J>(fl,-fl, + »-20+ jy(^,-fl4-ir-2i) = 0, (15) wheDco w« find the dip 1 ■ i?(gi-<?34^) + -P'(tf,-<i«-ir) ^ (ICJ lere i) and i/ are the tangents of tbe dodcxioDB produced by the needle in it* first and second magnet ixations respectively. In taking obiwrvatione with the dip circle tbe vertical axis is carefully silju.tti<d no that the plane V'aringw upon which the axis of Uie magnet re^ are borixontsl in oveiy azimuth. The magnet being magnetized so that tlie end A dipG, is placed with ite axis on the plane bearings, and obdvrvati'iOB are taken with tbe plane of the circle in the magnetic meridian, and with the graduat«;d side of the circle east. Each end of the magnet is obeervi-d by means of rending inicroeoopes carried on an arm which moves concentric with the dip circle. ITie cross wires nf the microscope are made to coincide with the image of a mark on Mw niiignct, and the position of tbe arm » then read olT on tbe dip circle by means of a vernier. AVe thus ohlaio an observation of Uie end A and another of the end B when the gradnations ai-e east. U is necessary to ohsorv-o botli ends in order to eliminate any error arising from the axle of tbe magnet not being concentric with the dip circle, The graduated side is then turned west, and two more obscrva* ttona aie made. The magnet is then turned round so that the ends of the axle are reversed, and fonr more observations arc made looking at the other side of the magnet. 1 3 110 MAQKBTIC WRASCBEUBST8. i The mafrnetizsHoD of Ibe ma^et is then rereraed bo that the end B dips, the magDetio tnoniont is ascertained, and eij^ht observa- tions ar« takoQ \q this state, and tlio sixteea obserratioos combined., to determine the true dip. 462.] It is found that in spiU) of the utmost care the dip, ns tbiisi dcductd from obsiTvutions muile with one dip circle, differs pur- eeptibly from thut di-duocd from oWrrutioiis with another dip eirele at tlic Mmc pliicr. Mr. Brotiii hits pointed oat the eflV diie to ellipticity of the benrini^ of the nxlc, and how to w>rrec it I>y taking observations witli the magnet mngnetiM-d to differen strengths. The principle of this method may he stated tku. We »h*ll suppose that the error of any one observation is a amiiU quantity not exceeding a degree. We shall uUo Hiipposc that nome unknown but re^^ular force aets upon the ina^et, disturbing it fhim ita true position. If X ts the moment of tliia toroe, 0^ the true dip, and 6 th« observed dip, then L = Mlsia{d~$a)> ('^ = MI{e-e^), (18 sinee 0—6^ is small. It is evident tEiat the greater M hMomea the nearer does the needle approach its proper position. Now let the operation oE taking the dip be perforuiod twice, first with the nu^netixationl fs^xnA to Ml, the (ficate&t that the needle it capable of, and next* with the magnetization eiual to ,1/j, a much smaller value but Kufliricnt to make the readiu^ distinct and the error still moderateu^M Let 9, and 0^ be the dips deduced from these two sets of obattrva-^^ tions, and let L ho the mean vsloe of the unknown disturbinff force for the eight positions of each determination, which we shall auppoae the same for both determination*. Then If we find that several experiment* give nearly njnal values for L. then we may consider that B^ muitt be vety nearly the true v«lae of the dip. 46S.] Dr. JohIc has recently construeted a new dip-circle, which the axis of the neodle, instead nf rolling on horixontnl agat pUnes, is slung on two (iUments of silk or spiilKr'it lUti-jid, the Henoe tf„s3-' (20)" 463-] JOULES 8t;8FBM.SIOM. 117 I I I of tlie flUmcnt* bcin^ nttnch^d to the arms of a delicate balance. Tb« niit of Uic ncvdle thu« rolls on two loops of silk fibre, and Dr. Joule ftt\t\» tliat its rrccdom of motion is much greater than whim it roll* on agtile plance. In Kip, 18, NS is the n4MdIe, CC is its axis, conn&Ung of a ■tnight cylindrical wiru, and PCQ, P'C'Q' arc the filaments on whicli the uts rolls. POQ i» the balanoo, consistinfp of a doubl« bent luvor siipjiortcd by a wire, 00, stn'tcl>i'd horizont- alljr between the prongii of a forked piece, and liaviiij; a counterpoise Ji which can be screwed up or down, so that the iMtlance is in neutral equilibnuD) about 0. Id order that the needle may be in neutrsl equilibrium as the needle rolls on the filamcDtfl the centre of gra- vity must neither rise nor fall. Ht^nce the distance OC must remain ccniKtunt an the iiectllo rolls. This condition will be fulfilled if the arms of the balance OP and OQ are equal, snd if the filaments are at rif^ht angles to the arms. Dr. Joule finds that the needle should not bo more than five inchirK long. When it is ci|»ht inch«i hmg, the bcmling of th<! nt-nlle tind^ tu dimiiii«h thi; apjiurcnt dip by a fraction of a minute. The axis of the nocdle was origiiinlly of Bt«ol wire, stniighteuetl by Iwing liMught to a re<I lii'at while ^itn-tcht'd by a weight, but l)r. Joule found that with the new suspL-ntiion it is not nocwssary to use steel wire, for platinum and even standard gold are hard enough. The batanoe is attached to a wire 00 about a foot long stretched horiz^Hitally between the prongs of a fork. This fork is turaod round in azimuth by means of a circle at the top of a trii>od whteh supports the u hole. Six complete observations of the dip can be PtK-l8. 118 U&OITBTIC UEASTBEMCSTS. obtained in one hour, and the aven^ vrror of n nogle obwrvattun is a fractioB of ft minut« of arc. It is proposed that the dip ncedl« in the Cambridge Pti;si«^^ Labontory shall be obscTvod by meanit of a double image iMlrq^l meat, eonsistinfr of two totally rullMting prisms pbocd an in Fig. 19 nod mounted on a vertical graduated circle, so that the pluDc of reflexion may be turned round a horizootal axis nearlj coinciding witJi the prolongation of the axis of the suspended di{ needle. The needle is viewed by means of a t«Icscope pla behind the prisms, and the two ends of the needle are seen togethi as in Fig. 20. By turning the prisms about the axis of tlie vertit circle, the images of two lines drawn on the nuv<lle may be ma to coincide. Tlie inclination of the needle is thus detennined the reading of the vortical circle. ""~~-... / = The tots) intensity /of the toagnvtic force in the Une of dip n be deduced as follows from the times of vibration 7„- 7,, Tg, ia the four positions alrmdy deKcribed, The values of J/* and J/' must be found by the method of dcflcxi and vibmtion formerly described, and A u the moment of inertia the mu(^ut about its axle. The observations with a magnet anspended by a fibre an much more acearate that it is usual to deduce the total force from' the horizontftl force from the equation where / h tlio total force, Jl the horisontal force, and $ the dtp. 404.] The procc« of detonniniug thi^ dip being a tedious one. w not suitable fur dct«rminiiig the continuwut variation of the mogtMtti I TERnCAL FORCE. 1 r force, .tlie most convenient instrument for continuous observa- tions is the vertioil force maf^Dctomi-tcr, wbich is simply k mag^«t balanced on knife ed^^s so as to be in stable equilibrium with it« magD<>tic axis nearly liorizoiital. K Z is the vertii'fll component of the magnetic force, M the mof^etic moment, antl the small an^le which the magnetic axis makes with the horizon JHZ = wya cos (a—$), where » is tlic man of thi; mag-net, ^ the force of ^mrityi a the distance of the centre of gmvily from the axis of suspension, and a the angle which the pinoe through the axis aud tlie centre of gravity makes with the ma^uetic axis. Hence, for the small variation of vertical force t2, there will be a variation of the angular position of the mngnot 16 such that MlZ= n>yaBin(a~-0)80. Tn praetiee this instrument is not used to determine the abtiolute value of the vertical force, but only to register its email variatirns. For this purpose it is sufiieient to know the absoluM valuL- ol Z ilZ when = 0, and the value of d9 B The valne of Z, when the horisontal fojcc and the dip are known, is found from the equation Z = //tan 9„, where 6„ is the dip and (// the horizontal force. To find the deflexion due to a given vnrintton of Z, take a magnet and place it with its axis east mid west, und with its centre at a known distance r, east or west from tlie declinometer, as in ex- periments oil deflexion, and let the tangent of deflexion be Dy. H Then place it with il« axis vertical and with it« centre at a ^ distance r^ above or below the ociiLre of the vertical force mag- netometer, and lot the tungcnt of the deflexion pro<Iuccd in the magnetometer be />;. Then, if the moment of the deflecting , magnet is M, Henoe 2.v = irvi>, = 2r,»/>s- i^^Jl^^^. do A The actual value of the vertical force at any instant is JZ Z==Z^ + 49 where Z„ is the value of Z when 6= Q, For continuous observations of the variations of magnetic force 120 HAQITETIC HEASUBEHENTS. [464- at a fixed observatory the Unifilar Declinometer, the Bifilar Hori- zontal Force M^oetometer, and the Balance Vertical Force Mag- netometer are the most convenient inBtruments, At several observatories photographic traces are now produced on prepared paper moved by clock work, so that a continnous record of the indications of the three instruments at every instant is formed. These traces indicate the variation of the three rectangular com- ponents of the force from their standard values. The declinometer gives the force towards mean magnetic west, the bifilar magnet- ometer gives the variation of the force towards magnetic north, and the balance magnetometer gives the variation of the vertical force. The standard values of these forces, or their values when these instruments indicate their several zeros, are deduced by frequent observations of the absolute declination, horizontal force, and dip. CUAPTER VIII. ON TEBRESTBUL MAGNETISM. 465.] Ora knnwk-dcc of Terrestrial MagTietiBni is derived from U»e fUidy of llic dilstrilnition of miifpictk' force on the earlb's sur- fac« »t nnj- one time, nod of the changes in that dietributioa at different tinu<». The rDi4pii>tic force nt any one place and time is known when itK three courdinalog urc known. These coordia»ti.-8 may he given in Uie form of the dL-olinatioo or azimuth of t)ie foree, the dip 01- inclination to the horizon, and the total iuteoeity. ^e mcwt convenient method, however, for iDveatiguting the g«nend diittribtition of magnetic furce on the earth's surface i» to con»Jd«r the maj;niliidea of the tliree compoueuta of the force, A' = IIvo» h, dir.-ct«d due north, I y= Ntin 4, directed due west, ( (1) Z = Htan 0, directed rerticsllf npwards, I where JI denotes the horizontal force, A the declination, and the dip. If r is the maj^nctie potential nt the earth's surface, and if we consider the citrth a sphere of radius a, then a eoB I dk ir where I is the latitude, and X the lon^^ittidc, and r the distance bom the centre of the earth. A knowledge of Tovvr the surface of the iitrth may be obtained from Uie observations of horizunlul force alone as followi*. Let Vg be the value of F ut the true north pole, then, taking the line-integral alonf; any meridian, we find, -I for the value of the pot«DlJal on that meridian at latitude /. KAOSmSV. [466. ThiiE the potential mair be founil for any point on ti>0 Mrtb's Eiirfacv provided we know thu valuo of A', the northerly oomponctit at cxf.ry point, nnd ^, the value of Tat thv polu. Sinoo tlie (onxa depend not on the absolate value of F on its derivatives, it ie not necesssry to 6x any particular val' for f„. Tlic value of V at any point nay be ascertained if wo know the value of X along any given meridian, and also that of Y tlie whole snrfiice. ilnS Let ■'1 where the integration is performed along the given meridian from the pole to tlie pAiallel /, then r = r,-ajYcoBUk, (sj where the integration is performed alon^ the parallel t from tb (riven meridian to the required point. Thcso methods imply that a complete magnetic survey of th« earth's surface has been made, so that the ralue^c of .V or of F or of both are knonni for every point of tbo earth'« tur&ec at n given epoch. liVhal wo actually know are the magnetic com* poDcnts at n certain number of vtationx. In the civilized parts of the earth these ttatiooa arc comparatively numerous ; in other places there are laT;ge tracts of the earth'* siirfacc about which we have no dats. Sfaffjulic Surrrjf. 4C6.] Let us sup{>oce that in a country of moderate size, who: greatest dimensions are a few hundred nJIes, obeervatioiu of tl: declination and the horizontal force have been token at a ood- siderable number of stations distributed &irly over ibv country. AVithin this district we may suppose the value of /' ta be pMsented with sufficient aocutscy by the formula whence X=><i + *,/+fl,A, (T) Let there be m statioas whose latitodes are t^, t,, ... &c and longitudes A,, X,, &e., and let T and F he found for each sUtioa. Let i,mi^{ty and \, = jX{X), di MAGNETIC SCETET. 123 /g knd Ag maj be called th« latitude and longitude of the central Btatiott. Lot Xa«-S(.T), and ToCOb/.^ - 2{rcoB0. (lO) thea Xg and fg are the raliiea of X and K nt tlif tmagiDary central HtatioD, then Xx=X^ + Ml~l,) + B,(\-\^), (11) rcoB;= l'„co»/„ + 5,{/-/„)+5,(A-Ao). {17} W« tiavc » MjoatiDns of the form of (11) and n of the form (12). If we denote llie j>roI>able error in the det*rmin«l ion of A' by f, »iul that of y coft I by t], then we may ntlcuiate ( and t] on the suppontion that they arise from errors of obserTation of // ^ and >. (Let the probable error of // be ^, and that of h, d, then since iX = COB d .4 /I-H<iin bM, P (» = .*« cos" 6 + rf« //* wn" fl. Similarly t)> = ^' »in* 8 + </*/?' co«« 8. If the ruriatiouj* of .V and K from their vataes as given hy equa- tions of the form (H) and (12) oonsiderably exceed the probable errors of obeerration, we may conclude that they are due to local attraction!!, and then we have no rea^D to give the ratio of f to r| any other value than unity. Aoeording to the method of least squares we multiply the equa- tions of the form (II) by ij, and those of the form (12) by f to make their probable error the enme. We then multiply each equation by the coefficient of one of the unknown qiinDtities £,, IB^, or Sj and a<Id the results, thus obtaining; three e(iuution)) IVom whiohtofind.fl„£j, 5,. in which we write for conciseness, di=S (/»)-«/„', &, = S(/A)-«/«X„. *, = 2(X*)-«V. P, = 2 llX)-nl^X„ Qt = S(;rco« l)^nl,y,i^l„ i»,»2(AJr)-i.AoXo, Q,= S(Arco«/)-»\,j;co»/„. By calculating B,, S^, and B^, and substituting in equations ' (II) and (12), we can obtain the values of X nod i'at tiny i>oint within the limits of the survey free &om the local dt»turbani;e9 ^ will 124 VBB&liTBlAL UAGKKTISU. [467. ;« which nro fonod to exist where the rock near the etation is msf;netic, as most igneous rockti are. Siirvvys of this kiud can bo nuule only ia ooontriee where mag- iictic in8trumeDt« can be carried ohout and set up id s gT^nt: maoy slations. For other parte of the world wo must be content to find j the dintribiition of the magnetic elements by interpolation betwoeo I their vahiee Ht a few stations ut (frcat distances from cacb otbor. 467*] Li*t us now suppose that by processes of this kind, or] by thr cqiiividcnt gnphical process of constructin^f charts of the lines orc({<ial values of the ma<;oetic elements, the values of A' and y, and tliciice of the potential F, are known over tho whole surGic«j of the globe. The next stt'p is tu expand F in ihe fonn of a siTies of spherical surface harmonics. ^j If the earth were m»|rnctir.od uii i form ly anil in the tame directiou^l tliroujifhout ltd interior, I' woiiM lu' it harmonic "f the 6r»t degree, ' the mag'netic mcridinuK woulil be great circleit pa:«ting through two ^^ magnetic polew diametrically opposite, the magnetic equator would ^| he ft great circle, the hori'/ontul forco would be equal at all point* ^^ of the magnetic e<|uator, and if Jl^ is this constant value, the value ut any other point would he If = Iff, cos C, where f ia the magnetie latitude. The vertical force at any point would be ^= 2J7,BiD/',J and if 6 is the dip, tan 6 would ho = 3 Ian T. In the case of the earth, the magnetic equator is defined to bej the line of no dip. It U not a great circle of the sphere. The magnetic polos are defined to be the points where there is^ DO borizoiitat force or where the dip is 00*. There are two such p'jints, one in the northern and one in the southern regions, but they are not diametneiilly op{ioBitc, aiul the line Joining th«in not parallel to the maguetic axis of the earth. 468.] The magnetic poles are the jmintc where the value of on the surfitoo of the earth is a maximum or minimum, ot etationarj. At any point where the potential is » minimum the north end of the dip-needle point* vertically downwards, and if a compass- needle be placed anywhere umt such a jioint, the north end will point towards that point. At points where the potential ts a maximum the soutJi end of tlie dip-nwdle points downwards, and the souUi end of the ""-"p—- »i Di«dle point« towards the )>oint. ^M If there are p minima of J' on tho earth's surface there must he^^ ^—1 other points, where the north end of tfao diji-uradle pointy, 4 ] I I I 470.] MAQinrnc tfattius of trr EAiiTn. 12s downwards, l>ut whirn tho oonipiMit-nAodle, when carried in a circle raund the point, instead of revolviiif; i>o that iN north end point* cnnstantly to Ute centre, rerolves in the oppottite dirt'cttoii, so as to turn Bometimes ita north end and sometimes it« south end towards the point. If we call the points where the potential is a minimum true north poles, then these other points may be trailed false north polea, because the compass-needle is not true to them. If there are p true north poIe», there must he^— 1 fal«e north polea, and in like manner, if there are q true south poles, there must he f— i false south poled. The number of poles of the same name roust he odd, so that the opinion at one tJme prevjilent, that there are two north poles and two ."iouth poles, is erroneous. According to Gauss there is in fact only one true north pole and one true south pole on the earlli's surface, and therefore there are no false poles. The line joiaini; these poles is not a diameter of the earth, and it is not parallel to the earth's magnetic axis. 460.] Jlost of the early inveatig-itors info the nature of the nrth's ma{rQ(^tiaiu endeavoured to express it as the result of the action of one or more har ma^ets, the position of the poles of which were to he determined. Gauss was the firet to express the distribution of the earth's magnetism in a perfectly Reneral way by expanding its potential in a series of solid harmonics, the coefficient* of which he determined for the first four degrees. These coeffi- cients are 2't in number, 3 for the first degree, 5 for the second, 7 for the third, and 9 for the fourth. All these terms are found nraessary in order to give a tolerably accurate representation of the actual state of the earth's magnetism. Tojind villi Part oftJit Obtervfd Magnetic Force u due to Szfemal and what to Internal Cautei. 470.] Let us now suppose that we have obtained an expansion of the magnetic potential of the earth in spherical harmonicti, ooniiatt^t with tho actual direction and mngnituilc i>f the hori- zontal force at every point on tJie earth's xurfiK'c. then GausK ha* shi-wn how to determine, from the ohsorvcd vertical force, whrthcr the magnetic forces arc duo to caused, ^iich us magnetization or etectric ciirn^nli, within the earth's surface, or whether any |»rt is directly due to cause* exterior to the earth's eurfaco. Let y hit the actflal potential espanded in a double series of I ii|)heri>ca] harmouics, • ;* 126 iESTRTATi MAG17ET19X. [471- -('+» The first eeries repreeente the part of the potentift! doe to eauttS* exterior to the euih, and the second serie* repMsftnta the |Nu:t to causea within the earth. The observntions of horizontal force give wa the mim of the» series when r = a, the mdius of the earth. The term of the order t is The obBcr\-a(.ionB of Tertical force give us and the term of the order i tn aZ is Hence the part due to external causes is ^- — iTTi — ' and the part due to causes within the earth ia il't-aZ, B,= ii+l The expannon of T lia« hitherto been calculated only for m«in vala« of F at or near certain epochs. No appreciable port] of thin m«ui value appears to he due to causes external to tbej earth. 471.] We do not yet know cnoug^b of the form of tha 1 of the solar and lunar parts of the variations of T to l^ Hit metAod whether any part of these variations arises from maffnetic force acting from without. It is oortain, however, aa thfl caleulatioDa of MM. Stonoy and Clianbers have shewn, that 1 the priiK))«] part of these variations cannot ariw from any direct^! niH^oetic action of the sun or moon, supposing- these bodies to be ^^ nuignetjo *. 473.] The principal changes in the magnetic force to which attention has been directed are as follows. • Pnibaor IlanulelD of FruiM kM dbronrol ■ pwiaje ehMgs tn tfw ch la S9JIS iby*, BlwaM ■lacUjr W ' •jnodi« rarolotioa cf llio kid, u deduoad (ruia tki o1i«mitian of >bd lyoli •lenrntK, ttia p<vlad of which which ik<rihs^| S«Jia day*, >I<iu>M oMUjr mmI (o Oak 3t Ita; loduoad (ruiB tki olwvTation nt > bd ifoli mw U* MOlor. TUi nwUiodnf diieowriaK «>»* "»■« of reOtfsw of th» iiaiiii mM bdy «f tb* van 17 lu aflecti » IW nwffnMlo maSbt k tbe Bnt tmHaliiwnl nf Ilia r«pkjiiiMt bt AUrnMiHii of tU lUbt M AalrooMnr- Ahtd., Wiok, Jniw IB, 1971. Hau Prvt. ^ s7Ko». 1», 1871. U74-] eOBTEBBASBAH OR CBLBSTIALT I I T. Tia more Bfgular Varlationt. (1) The Solar variittioos, dcpcndio^ on tLo boor of th« day and the time of the ycsr. (2) Tlic Lunitr vnrinliont!, tlppendtng on the moon's boor angle and on hi?r other demouts <>f portion. (3) Thcec variutions do not repent Ihom^^elves in difTorcnt years, but Ewem to be subject to a variation of longer period of aWut eleven years. (4) BeftideB tbia, tbere is a secular alteration in tbe state of tlie earth's ma^etism, wbicb has been ^ing on ever since ma^nutic obmrvationn have been made, and is producing changes of the magnetic elements of &r greater magnitude than any of the varia- tioQH of dntall x>vriod. II. The Disluriaacet. 473.] Besides the more regular cbangcif, tbe magiietic eleinenU are subject to sudden disturbanees of greater or less amount. It i« found that these disturbatici-s are more powerful and frequent at one tjme than at another, and that at times of great disturbance the laws of the regular variations are masked, though they are very distinct at times of amull disturbance. Hence great attention has hetn paid to these disturbances, and it has been foun<l that dis- turbaaees of a j»iiicu1ar kind are more likely to occur at eertaia timea of the day, and at certain Ecasnns and intervals of time, tJtoDgh each individual diHturban<'e appears quite irregular. BcGides these more ordinary diflturhances, tbere are oecanionally times of excessive disturWuce, in which the magnetism is strongly disturbed for a day or tvro. These are called Magnetic Storms. Individual disturbances have been sometimes observed at the same instant in stations widely distant. Mr. Airj' ha^ found that a largo proportion of tbe disturbances at Greenwich correspond with the electric currents cullovted by electrodes placed in the earth in the neighbourhood, and are such a» would be directly produced in the magnet if the earth-current, retaining its actual direction, were conducted through u wire placed tiwderatath the magnet. It has been found that there is an e[Mcb of maximum diHturbanoe every eleven years, and that this appears to coincide with the epoch of maKimum number of spot^ in tbe »un. 474.] The Beld of investigation into which we are introduced 128 MAQXETISJf. 4r-i by tlie study of terrestrial inagnetitm is «s profound u it u CX' U'lmivc. We know that the aun and moon act on the enrth'N magnetimi.l It has been proved that this notion cannot be exjilaiiied by «U[>-1 poniii)* the^e Ijodies magnets. The action is therefore indirect. Is the cusie of the sun part of it may be tliermal action, but in the caite of the moon we cannot attribute it to this cause. Is it |k»- sible that the attraction of tJiese 1>odies, by causing: strains in tfa« interior of the earth, produces (Art. 447) changes in the magnetism already existing in the earib, and so by a kind of tidal action causes the semidiurnal variations ? Hut the amount of all these chani^ is Yory smalt compared witJ the great secular elianges of the earth's msgoertiani. IVltat cause, whether exterior to the earth or in its inner depth;* produces such enoroions chanties in the earth's magnetism, that its' magnetic poles move slowly from one part of the globe to another ?^i When we consider thst the intensity of the magnotizatioa of th4i^| great globe of the earth ts quite componble with that tvhich we produce with much difficulty In our atoel ma^ets, these immense^: changes io so large s body force us to conclude tluit wc are not yd^l acquainted with one of the most powerful agents in nature, tho bceue' of whoee activity lies in those inner depths of the earth, to tlie knowledge of which we have so few means of access. PART IV. ELECTHOMAGNETISM. CHAPTER I. »LBCTR0MA0NBT1C FOBCB. 47S.] It lutd been noticed by many differeDt obaervers tliitt in n-rtain cnsce m^netisTa is produced or destroyed in needlra by electric discharjr^ through them or near them, aod coiijectures of various kinds had been made as to the relation between roag- n«tisra and electricity, but the btwe of these phenomena, and the form of these rvUtions, remained entirely unknown till liana Chri»Uui Untted *, st » private lecture to n fctv advaueod students it Co]><!tihBgeD, obtcrved that a wire connt'ctinpf the cuds of a voltaic Ixtttcry afleeted u magnet in its vieinity. This discovery be [MihliKhed in a trmct entitled Experimfitta cirea effectum CouJIicliit HUttrici in AfHm Majfutiieant, dated July 21, 1820. ExjX'riinento on the relation of the ma^et to bodies charged with elei:trieitv h^ been tried without anv result till Oriited endeavoured to uM-ertain the etl'eot of a wire ieated by an electric current. He diHcovered, however, that the current itself, and not tbt* heat of the wire, was the cause of the action, and that ihe 'eleotrio conflict acts in a revolving manner,' that is, that a magnet placed near a wire transmitting an electric current tenda to set itaelf perpendicular to the wire, and with the same end always pointing forwards as the magnet is moved round the wire. 476.] It appears tlierefore that in the space surrounding a wire * Sm nAoibaT aeoount of Ontvd'a dfacovvrr in k lottRr trtaa Pmrouar lUnaUoi la ths Ltfi «f F»ni4ag b; Dr. Bcmoe Jcoico, vol. ii, ji. 30fi. TOL. U. K ISO ■LECTROMAGSETIO TOROl. 1 oi transmittinf; nn electric current a [nuj>Dft i» acted on by forces tlcpciulciit on tlic iiosition of i\w win? und on the streo^li of ihv curn-nt. Tlic hi>im;i; in wliitili tliOMB foreea act may thcn-fwre W coneitlvred ui a magootic flelil, au<I we amy study it in tlic came way ft> we hnve already studit d the f\M in the neighbourhood of ordinary magnets, by tracing the courae of the lines of magnetic forco, and measuring the intensity of the force nt every point, 477-1 ^'' ^^ begin with the case of an indefinitely long etraigb wire carrying an electric current. If a man were to place bimselj in imagination in the position of the wire, so that the cmrrent should flow from his head to his fcot, then a magnet suspended freely before him wotd<I set iteelf so that the end which gmints ttOr(J^_ would, under the action of the current, point to his right hand. ^| The lines of magnetic force are everywhere at right angleis to planes drawn through the v/'in, and are there- fore cireW each in a plane perjiendicular to the wire, which passes through it« centrt^l The polo of a magn«t which point* north, ii^^ carried round on« of those cireles from left to right, would eipericncc a force acting always in the direction of its motion. Ilie other pole of the t>amo magnet would esjierieng^^ a force in the opi>oeito direction. ^^^H 478.] To comjwm thc«e forces let the wire be apposed vertical, ood tJic current a de^^ scending one, and let a magnet bo placed M^| an ap))aratus which is fr«e to rotate about a vortical axis coinciding with the wire. It ic found that under these cireunistancee the cumnt ha)i no I'lTcct in causing the rotaliuo of the apparatus as a whole about itoclf as oa axia. Ht^ncc tlia action of the vertical current oa the two pobw of the magnet is such that the etaticid moments of the two forces about the current as an axis are equal and oppont«. Let M] and ai, he the strengths of the two poles, r, and r^ their distances from the axis of Uie wir«, 7*, and 7*, the inteniiiticM of the magnetic force doe to the current at the two ih.Ioi rfspectively, then the force on «, is m, 7",. and once it is at right angles to the axis its moment is m, 7*, r^. Similarly that of the force on lh« other pole is m^T^r^, and since thccu is no motion obMTVed, IV- SI. 48o.] BLECTROMAONBTtC POTESTTAt. X = ~2i^, I But vu know that in nil magnets M, + Mf = 0. Ileoce 7, r, = T^ f,, or the e1«ct<on)ag:n«H« force due tn a iitmig^ht ciirrent of inliRite len^i is perpendiouUr to the current, and varies inversi^Iy oa the distance from it. 479.] Since the product Tr dejiciid* on tlio strength of the current it may be employwl ait a measure of Ihe current, Thi» method of measurement is difiercDl from that foonded upon dec- tro«t»IJo phirorimena, and as it depends on the magnetic phenomena produced l>y olpctric currunts it is called the Ki«;tromagm'tio«y>iteni of measurement. In the elwtriiniiignetic system if i is the current, Tr ='2». 480.] If the tvire be t;iken for the axis vf :, then the rectangular OompODCDt« of T OTO Ilere Xi£r-f Tdjr+ZtU U a complete dlfferentiftl. Wing that of 2tt»n-'^+C. BeDM: the mognetto force in the field can be deduced from a potential function, ati in aevcral former instancee, but the potential is in this case a fnnction baviuff an infinite seriea of values whoso common dificronce is 4iri. Tl)e differential coefficients of the potential with rettpect to the coordinates have, however, definite and single values at every point. The existence of a potential function in the field near an electric currant is not a self-evident result of the principle of the con- servation of energy, for in all actual currents there is a continual expenditure of the electric encr;,'y of the battery in overcoming the reeietanee of the wire, so that unless the amount of this expenditure were ncc«cnt«iy known, it might be suspected that part of the taergy of Ihe battery may bo employed in causing work to be done on a ma^et moving in a cycle. In fact, if a magnetic pule, St, moves rotmd a closed curve which embraces the wire, work is actually done to the amount of 4irmi. It is only for closed paths which do not embrace the wire that the line'intcgraJ of the force vani«but. We must therefore for the precent consider the law of force fliid the extKtciiev of a potential as resting on the evidvDOO of Uifi experiment already described. K 3 132 ELSCrnOMAOKRTtC FOBCR. '1 ea 1 it« it« cait I 4S1.] If we consider the Kpatw RiirToiinding: an infinite ittraigfat liue we sball en tbiit it is n cvclic span^, bocansc it returns into itself. If we now vonoeive a plau«, or any otber surface, com- mencing at the Ktraiglit lino and extending on one side of to inlinitf, tliie etirfiico may lie regarded as a diaphra^i wbichl reduces the cyclic iqnce to an acyclic one. If ^m any fixed point lines be drawo to any other point witbAUt cutting the diaphragm, and tbe potcntiul bi; defined as tlie line-inte^al of the force token along one of Ihesu lines, the jiotential at any point will thon have K single dvfiuitv value. Tlic Diabetic field i» now identical in all respects with thai d to a magnetic ibcll coinciding with this snriace, tbe ntr^gth of the shell being t. Tliin ^ihell is bounded on one edge by the infinite struight line. The other part« of its boundary are at an infinite dii^ljince from the piirt of the field under consideration. 482.] In ull ucttial exjiertmenta the current fonns a closed circuit of finite diiiifrnsions. We shall therefore compare the magnet aetion of a Unite circuit with that of a magnetic shell of vhiob circuit is the bounding edge. It has been shewn by namerona experiments, of which earliest are those of Anip&re, and the most accurate those of Weber,' tliut the magnetic action of a small plane circuit at distances which are great comp«re<) with the dimemiions of the circuit is tbe same as that of a magnet whose axis is norma) to the plane of tbe circui and whoee magnetic moment is equal to tbe area of the cireail inulti])lied by the strength of the current. If the circuit be supposed to be tilled up by a surface bound by the circuit and thus forming a diuphtagm, and if a moguetio shell of strength i coinciding with this surface be substituted for tbe electric current, then the mngnetlo action of the shell on all distant points will he identical witli that of the current. 483.] Hitherto wo have supposed the dimensions of the circoi' to be smalt compared with the dist»iKe of any part of il fi the part of Uie field examined. We shall now suppoee the circui to 1*0 of any form and si&a n hatever, and examine its action at naf point P not in the conducting wire itself. Tbe following method, which has im[H>rlnnt goomctrini) applications, was introduced by Aiu|>^ for thin pur)>o«e. Cuncvive any Kurfare S botindcd by the circuit and not passing tlirough tJie point P. On this surface draw two sertce of line» srosring each other so aa tu divide il into vhmcntary poKinns, ih , Ul^ 484.] WAOKETIC SnELL IN PLACE OF CDRBEKT. 133 I dimensioaB of which arc small compared with thc-ir iliHtonou from P, Aod with the radii of ourvature of the curfAco. Round eax\i of thaw elemfntK ooDCciv« a current of Bt.i«n{*th i to flow, Um direction of oiroubtioii hciii^ the samo in all tiie elements as it is in the orig;inal circuit. vMoDfT ev«iT line fonniiig; tlii> diviiiinn between two conUguous elements two equal currents uf Hlixnglh (' flow in opponte direc* tioite. Tlic effect of two oqiul and oppoaife cuirenftt in the same place is ahHilutely zero, in whatpver asjieet we consider the currentii. Hcncv tlteir ma^etic f.S\xil \s xero. I'he only portions of the elcnicntttry circuits which are not neutralized in this way are those which coincide with the original circuit. The ixital eifect of the elementary oircnits is therefore equivalent to that of the original circuit. 484.] Now since each of the elementary circuits may be con- sidered as a small plane circuit whone diKtanoe from P is ^reat compari'd with its dimensions, we may substitute for it an ele- mentaty maffnetic shell of strength 1 whose hotinding edge coincides with the elemeutaiy circuit. Tlie magnetic effect of the elementary shell on i* is equivalent to that of the elementaty circuit. The whole of the elementary shells constitute a maffiictic shell of strength t, coinciding with the surface S and bounded by the original circuit., and the magnetic action of the whole shell on P is eqaivalcnt to that of the circuit. It is manifest that the action of the oircoit is independent of the fonn of the surface S. which was drawn in a perfectly arbitTsry manner bo as to fill it up. We see from thin that the action of a magnetic shell depends only on the form of its edge and not on the form of the shell itself. This tcnuU we obtained before, in Art. 410, but it is instructive to see how it may be deduced from clectToma^nctic consiilcrntioiix. 'fbe ms^ctic foroo due to the circuit at any point is therefore identical in magnitude and direction with that due to a magnetic shell bounded by the circuit and not passing through the pointy the strength of tla- shell being numeriually equal to that of the eumnt. The direction of the current in the circuit is related to the direction of miignetixation of tlie shell, so that if a man were to stand with his ftwt on that side of the shell which wc call the positive side, and which tends to point to the north, the current in Iroab of him would be from right to left. 134 ELECTROMAONETIC FORCB. [485. 485.] The m&t^otic [wt^ntiitl of th« circait, howcrer, difl«n from that of tho maj^otie shi-ll for thowc ])omt» wliiuli are in the anl^stftnco of the mngnotic »h«\\. ^^ If at is the solid Angle isiibtoiulotl st th« point P by t1>« nwgoetid^^ ehcil, rockonci] jionitivv wlien tliv jxiHitire or auxtrul «)<lv of the tihell ie nest to /', thvn the inngnftic |)Ot«titiat at any |ioint not in tlie shvll it««lf is loif), wht're <t> a lh« rtmnglli of the ithell. At any ^^ point ill tlie inibttjinco of the shell itwlf vie may nippofte thn «helt^| <livi<!<Hl intn two pnrt« ivlioiie stKiigths are 4k, and 0,, where ^1 ^, -f ^j = 41, suHi that the )>oint is on the ]M»iitive eide of ^ and on tJie negative side of 0j. The potential at thin point is On the negatiye side of the shell the potential becomeH ^(w— -I r).j In this case therefore the potential is eontinunns, and at ever point hati a Htn^le d<>terminat« value. In the case of the electrit vireuit, on the other hand, (lie magnetic potential at everr point not in the conducting nire itself is eqoal to la, where i ie the strength of the current, and 10 is the solid angle uihlended by the circuit at the point, and is reckoned positive when the current, as seen from P, circulat«B in the directionopposite to that of the hands of a watch. The quantity Jw is a function having an inlinite aeries of valae whose common difFereaoc is 4ni. Tlie diftV'reutial cueflBcients iw with reeprct to the coordinates, ]uve, however, single and de terminate valued for every point of space. 4B6.} If n long thin flexible solenoidal magnet were pUced in the neighbonriKxid of an electric clrtnit, the north and south ends of the solenoid would tend to move in opposite directions round the wire, and if they were fiee (0 obey the magnetic force the magnet would fiually become wound round the wire in a close ooil. If it WfTT' powibh^ to obtain a magnet having only one pole, or poU'n of untinittl strength, soch a niagtii-t would be moved round and round the win^ continiadly in one direction, but sinoe the poles of ever_> magmii are i-qunl and opposite, this result can never occur. Faraday, however, hiut Klien-n how to produce the continuous rota- tion of one pole <^ a magnet round an eleetrie current by malting it powiiblc for one pole to go round and round the ourretit whttd the other pole docs not. That thin process may be repeatMl ttJ definitely, the body of the magnet mnsi be transferred from nam sitif of the ciirrciil to tin- iitlier ouce in each rcvohitioQ. IV) tM this nithout interrupting the Aow of electricity, llui current is aplS HBVOLTIWO MAOSKT. 1S5 I Nli U»l into two bnin«hc», » tlwit when one branch is opened to let thft taagoei poxs tho current continues to flow thron^h the otJier. Vami»y utcil fnr this puqmso a circuliir trough of mercury, as eheivn in Vig. 33, Art. 491. The current enters the troug'h through llie vfirc AH, it id divided at B, und after flowing through the ares BQP and BKP it uiiitc« at P, und leaves the trough through the wire PO, the eiip of mercury 0. and a vertical wire beneath 0, down which the eiirrent flows. The mngnet (not fhewii in the figure), is mounted so as to be capable of revolving about a verticul a>:is thtDUgh 0, and the wire OP revolvea with it. The body of the inngniit. pnsisie-ii through the aperture of the trough, one pole, say the north pole, Iteing benenth the plane of the trough, and the other above it. As the magnet and the wire OP revolve about the vertical axis, the cnrrent it gradually transferred from the .branch of the trough which lies in front of the magnet to that which lies behind it, so that in every complet« revolution the ntognet passes from one side of the current to the othiT. The north pole of the magnet revolves jibout the deeeending current in the dirx^-tion N.G.S.W. and if w, w' are the wild angle* (irre«|)ectivc of sign) subtend»>d by the circular trough at the two poles, the work done by the electromagnetic force in a complete revolution ix m({47r— (0— «'), where n is the stJ'ength of either pole, and i the strength of the CUTTVDt ♦. 487.] L«t OS now endeavour to form a notion of the state of the magnetic field near a linear electric circuit. Let the value of w, the solid angle subtended by the circuit, be found for every point of space, and let the surfaces for wlueh ■ [TU* prolilou may be diwuand u {uUoar*- Referring tu Wig. SS, Art, 491, lat u t»ke OP in uit ■mailioD uui inlndurc Jmagiiuu'y bafaiiciog piiit<<dU i aloii^ BQ Kad a, y klaug OB. Am Ihe mngno( ntuicbotl to OP i* carried chroufh ■ oenploU nvolutlon nn wotk U Anixv on Ihi- inuth pule bj^ tbu cummt (. tuppoMnl to |ian timtiK AHOX, tli«l uola ilwc^ribini;' n i:ti)»'<1 i^iirr* which dfiw nol tinbnc* l]i« cuamnt. Tbr north pola howsver clHuribiui n uluMd ourra whioh don emhooci llio cvmmti iuhI the nrk Aowt u^od it in 4 vmi, Vfa hare now ta estimate the •ttwU at tbo cuTTVDta i in the circuit BFO and V in the oinmic BSPO. The pnlewllal uf th« tuirUi pula whicli ta bolciw the planon of thoui circuit* trill be — niv^'Mnytv— Bf) ami, iif tliu »">ulh, — mia', — ra)r(— •' t b'j). wImm t>f wkI ttf' donoto the nlid anglcn >ubteni3o] at the two |>iil«a by BOF, anil ». ■* ikiaa mibtonied by the drculor trough. The rceultant poUmtUl !■ my (■> + !«')— (It ( (a>j+ ir'j). H«nM aa OF rcrvlvM from OF in tile dirwtion NEt^W back lu OP ■c*'* <''* poMntlail arlU cban^n by —(«((* + ■')• The work dcniB by the inuTWiU i* ihersfute Uial |{<min la th« xvxt,] VOt. II. ELRCTROMAOSETIC FOBCB. M is oonsfant he deitoribed. Tliese aurfaces will Iw tlio ixjiiipott-ntial surfaces. Each of tUeee surfncea will bo boiiniiwl hy the eirrmi and nny two surfacea, wj and <»,, will mii-t in the oircait at Fiifuiw X\Tn. at the ond of thin vi>lume, represents a. Bcction of tbo eqnipotential eurfacM diip to n circular current The Rmall rarclu rt'jiresenta n section of the conducting wire, and the hori Mnt«1 line nt ihc bottom of the fi^ur^ ik tlie per|>endien1ar to th plane of the circular current through ito centre. Tlie eqnipotential siir&cca, 24 of which are drawn corresponding to a series of values arc surfocfw of revolntion, having this tine (o\ of «e dilfcrini* br ■ , o their oouimon axis. ;ial lit^d 1 inn ^ I ■4 They are eridcntly oblate Ggtirev, being Bat- tened in the direction of the »xi». T1>ev meet nob other iii the lint of the circuit at angles of 1 S°, The force acting on a magnetic pole place«I at any point of anj eqnipoloiitinl MiHacw ia perpendienUr to this surface, and variMJ invoracly a** the di^nce between consecutive surfaces. The cln«ed eurvea surronndiog the section of the wire in Fig. X^^II are the lines of foree. They are copinl from Sir W. Thom«»i*s Paper on ' Vortex Motion t-' See also Art. 702. Actum of am Elrrtrie GntiU om irny Magnetic ^H^m. 48S.] Wfl u« DOW able to tleduoe the action of an eU^■tnl' circuit^ on my niagnrtio aystem in it« neighlKKirhood from \\w theory of | magnetie ahelbi. For if we eonstnict a magotfic i^>dl. whoM* | strength is nnmerically equal to the stmngth of the cunvnl, and whoae edge coincides in ]>oeitiiin with tha ououil, white the rheW itedf dow not pass thron^h any part of the magnetic system, the ' action of the shell on the magnetic ayvtem will be ideetieal with that of the electric cnrrent. Am«M iftXt MfmHe &f»l«m m Ika Bitirit CiftmU, 489.] Kmm this, applyiag the prineiplfi that action and mction ara n|iHU and opiwsite^ w« Hiooliide that the nMdianical actieo «f 1 TORCE OIT THX CIRCUTT. B th« mftgnetio tytAeia on the electric circuit is identical witli ite ~ acttoti on a ma^octic sbell havinsr the circuit for its edge. Th« potential energy of a magnetic ehell of sttength ^ placed in ft field of magnetic force of wbich the potential is V, is, by ml, m. H ua the direction-cosines of the normal drawn rrom the potutive ud« of the element dS of the shell, and the intogrntion is extended over the surface of the shell. H Now the suriitce-intcgral A'=JJ{ta + mit + nc)dS. where a, (, e are the components of the magoetic induction, re. presents the (|uantity of tnagaetio induction through the shell, or. in the Ituigoagc of FAradaf, the tiumher of lines of ma^ctic in- ^daction, rackoood al^braieally, which pass throKph the shell from ^■thtt negative to the positive side, lines which pnss through the shell in the oppowto direction boioff reckoned negative. Kememheriiig that the shell does not helong to the magnetic syittein to which the pot<minl V is due, and that the mngneUc force is tlierefore equal to the magnetic induction, we have a= — i- — c--'^^ 6- ''^ ^L^ dx' df' '~ it ^^^^Vc may write the value of JA, If ftjp, represents any displacement of the shell, and .V, the force acting on the shell so as to aid the displacement, then by the principle of couEerrntion of energy, x=^ dx We have now det«rmined the nature of the force which cor- responds to any given displacement of the shell. It aids or reaista that displacemeot aocordinj;ly as the displacement increaeea or diminishes A', the nnmber of lines of induction which pass through the shell. Tbo same ia true of the equivalent electric circuit. Any diit- ^jjutmeDt of the circuit will be aided or resisted accordingly an it id 138 ELHCntOMAOSmC POSCB. inoreases or diminiehcs tho Dtimber of lines of iDduction wiiich j*u through the oircuit Id tho positive iliroctioa. We must rciDcmWr thiiL the positiri; dircct4on of a lint* o^_ magnetic induction is thi; ilrrection in which thu polo of n magne^^ which pointE north tends to movt? along thv line, and that a litte of induction puK»<s through thu circuit in the po«itiTC dirfctioD|^_ when tLe direction of the lino of induction i» related to tl>^| direction of the current of vitreou* electricity in the circuit as the loDgittidinal to the rotational Diotioii of a right-banded wcrcw^H ScttArf. 23. fl 490.] It i« matiifM tliat the force corresponding to any di*- placement of the circuit as a whole may be deduced at once fron the tlicoiy of the magnetic shell. Dut this is not all. If a portic of the circuit iti flexible, so that it may be displaced independent!] of the rest, we may make the edge of the shell eapable of the sami kind of displaceracDt by cutting up the surface of the shell int a snfficieDt number of portions connected by flexible joints. Henc we conclude that if by the displacement of any portion of the circiiii in a given direction the number of lines of induction which through the circuit can be inereu^cd, this displacement will be aided by the electromagnetic forct> mtiug on the circtiil. Erery portion of the cinuit tben-fore i* acted on by a force urging it acroiw the tinea of magnetic induction m> a« lo incltidc a greater number of the«e lines within the embrace of the circuit and the work done by the force during this displacement namerically equal to the number of the additional line* of duction multiplied by the etrcngth of the current. Let the element d» of a circuit, in which a current of strengttf t is Sowing, be moved parallel to iteelf tlirough a space bx, it will sweep gut an area in tho form uf ii iiarallclogram whoso sides arc parallel and equal to </« and b» reiti>ect)vely. If the magnetic induction is denoted by Q, and if ita directiog inakcH an angle t with the normal to tlte parallelogram, the vali; of the increment of jV corresponding to the displacement is found^ by multiplying the are* of the parallel<^ram by 39 cos «. The result of this opemtion ia represented geometrically by the volume of a pnmllelopiped whoee edges represent in magnitude and direction 6x, di, and 9, and it is to be reckoned positive If when we point in the«c three directions in the order here given tho pointer moves ronnd the diagonal of the panllelopiped in the dirccUo;! of thf huDd« of a watch. The volume at this pciralleIopi{<ed is equal to Xia arc 491] FOKCB OS AS ELEMEST OP CIRCUIT. 139 I; ¥ If 6 is the an^le Iwtwecn tit and $, the arw of th? par!i11elo'>rain is tls.^ ein 0, And if i; is the »n^lp nhit-h thi.' disphicemcnt Sx makes with the normal to this paruJIulogrkm, the volamo of the paralk-lopi^ IB I rf* . S sin S.ixixs r} = iy. I Now Xix = iJ.V= fV/.SdudSJTCOSi), I and X = iJa . S sin fl cos jj ' |» th« foree which nrgee d», resolved in the direction ix. The dirwtion of this force is therefore perpendicular to the paral- lelogram, and is equal to i .ds.^Q sin 6, Tbis is the area of a parallelogram n-hose sides represent in maj[- nitude and direction idt and ®- The force acting on </» is therefore represented in magnitude hy the area of this parallelogram, and ID direction by a normal to its plane drawn in the direction of the loDgitudinal motion of a rig'ht-handed screw, the faimdie of which is turned from the dii'ection of the current icit to that of thv nuignetic induction ©. Wo may express in the language of Quaternions, both the direction and the magnitude of this force by saving \fitt that it is the vector part of the result of multiplying the vector ids, the viemmt of the current, by the vector ^B 9, the mn^ctic induction. S*ua 491,] Wc hnvc thus completely de- ■^ termiiied tlic f'lrce which acts on any ^P |)ortio» of an electric circuit placed in ~ a magnetic field. If the circuit is moved in any way fo that, after osetiming various forms and jioaitions, it retuniH to its original place, the strength of the eument remaining ooniitjtnt during the motion, the wliolc amount of work done by tbe elcctm magnetic forces will he zero. Since this is tine of any cycle of motion* of the circuit, it follows that jjH it IB impossible to maintain by t-U'cironingiielic forces a motion ^^ of eontiniioua rotation in any purt of a linear circuit of constant strength against the resistance of friction, &c. It is possible, however, to produce continuous rotation provided that at some part of the courwe of the electric current the current in, passes from one oonducl/ir whieb slides or glides over another. ^m When in a circuit there is sliding contact of a conductor over ^K the surface of a smooth solid or a fluid, the eincuit caa no longer KtirIA firl Znt Fig. 32. KLECTBOMACSOTIC FOBOf. mt I bo considered afi a single linear circuit of constant etr«n^h, but miixt be rcK'snled as a system of two or of some ^ft^ater niimVicr circuits of variable strength, tkc current beinj; so dtstnbut ftmoii|r tlium that those for which A' is incrensinf; have ciirrcn' in the [luKitivu direction, while those for which iVis iliminishi have ciirreuLd iu the negative direetion. Thus, in the apparatuA represenUHl in Fig. 23, OP is a mov<-abl6 conductor, one end of which reels in a cup of mercury 0, while the other dips into a circular trough of mercury concentric wiUi O. The current t enters alon^ AS, and divides in the circular trough into two p&rts, ono of which, X, flows along the wc BQP, while the other, y, Bowb along BKP. Th«eo ctirrcnts, uniting nt P, flow along the moveable conductor PO an the elcctrwle 0^ to the zil end of the Wteiy. strength of the current ale Fi8.83. OP and OZ is z+y or i. Here we have two circuita, AMQPOZ, the strcnglh of the ourre in which is x, flowing in Uie poallve direction, and ABU POX, tli strength of the current in which is y, flowing in the negatii direction. Lot 9 be the magnetic induction, and let it be in an upwa (lircclion, normal to the plane of the circle. While Op mov«M through an angle $ in the direction oppoaitj to tltut of the liande of a watob, the area of the first circuit inci by {0P*.$, and that of the second diminishes by the Esmc quanti^. Since the strength of the current in the first circuit is «, the work done by it is | ;r . OP^. . ®, and since the strength of the second is — /, th« work done by it is ky.OP'.O^. The whole work done is therefore ^^ do])cnding only on the strength of the current in PO. Hence, if i is maintained constant, the arm OP will l»e carried round and round the circle with a uniform force whose moment ift \ i .OP') If, as in oorthem latitudes, $ acts downwards, and if the curreig is inward*, the n'tatioa wiU be in the DCgative direction, that w7 in the direction I'QUS- d 493] ACTTON BRTWEBS TWO CITRTIESTS. t41 ^1 492.] V!e are now able in pass from tlie mutnsl aclion of ^nnognels and currentii to the action of one current on another. ^Kpor we know ttuit the magnetic properties of nn electric circuit (?,, ^prith respect to any magnetic BTBtero J/,, are identical with those of ft ma^etic efaell iS,, ivfaose edge coincidee with the circuit, and Hwhose Btrength is mimerically equal to lliat of the electric cnn-ent, Hliet the tna^etic system J/^ be a magnetic etell S^, then the mntual action between S, and S. ie identical with that between 8^ utd a circuit (7,, eoinciJinf* with the ed^ of S.^ and eqiiiO in nomencal BtrcD<rth, and this hitter action is identical with that between C, and C,. Hence tb« mut^inl action between two circuit)!. C, and Cg is iilenlioiil with that between tbo corresponding niajfucltc shells ^ hnd S^. Viv hare alrcjuly invi'Hltgnted, in Art. -123, the mutuiil aetioo 'two iiiagnctie shcllit whose edges are tlie cloved curvus *, and »^. If we make Jn . In coae .'o .'0 <f'l^f where ( if the an^lc between tbe directions of the olementH /'#, and dt^, and r is the distance between them, the integration Ining extended onco round a, and once round «,, and if wc call M the potential of the two closeil curves «, and 4., then the potential encrj^y duo to the mutual a<^!tiun of two muj^nctic sbclU whose ■trengths aro ij and i^ bouudctl by the two c-ircutts Is and tlie force X, whteb aids any displacement ix, is I ''•»^- The whole theory of the force acting on any portion of an electric circuit due to the action of another electric circuit may bo deduced ^^Aom this result. ^B 493.] Thi> method which we have followed io this chapter is ^nbat of Faraday. Tnxlejid of beginning, as wc shall do, following ^T Ampere, in the next chiiplcr, with the direct action of a portion of one circuit nn a portion of another, wc shew, firet, that a circuit produces llie same effect on a magnet a« a magnetic shell, or, in other wordic, wc determine the ualnrc of the magnetic field due to Uie circuit. Wc chew, Rceondly, that • circuit when plai^ in af magni^c field exjuirii-nccs tbe mme force ek a nuignctic ehell. Te thus determine the forec acting on tbe circuit placed in any 142 BLECrBOMAOSRTfC POBCl!. .494- muj^ncttc field. Lustlj, 1>y snpiKiKing the in>gi)eti« field to be to » «'corni electric circuit, we dftt>nnine the action of ouc ci CD the whole or iiiiy iinrtion of the olhfir. 4{^.] Let iiM iipply thix method to the caee of a Rtraight current of inlinite length acting on a portion of a parallel straight oon- dtiutor. I^et us tFiipposi- ttiut a current t in the finl conductor is flowing vcrticnily downwnrdo. In this case the en<! of a magnet vrbi poiiitu north will point to tlie rigbt-huid of a man looking at from the axis of the current. Tb« lines of magnetic induction are therefore liorizontat circl haTing their oentreti in the axis of the current, and their po«iti direction is north, ea^t, south, west. Let another descending vertical current be placed due west of tbe first. The line^ of magnetic induction dne to the first current are here directed towards the north. Tbe direction of the forci acting on the second current is to be determined by turning t handle of a right-handed screw from tbe nndir, the direction tli« cnrrent, to the north, the direction of the magnetic induction. The screw will then move towurdx the cast, that is, the force acting on the Gccond cnrrent is directed towards the first curr«-nt, or, in general, sinou the phenomcnou dcpvnds only on the relative poniti of the currents, two iwunUel currents in Uk snnie din?ution att «ach otber. la the xame way we may shew that two parallel currents opposite directions repel one another. 499.] The intcntiity of the magnetic induction at a diiitance from a stiaighl current of strength i is, as we have shuwn Art. 479, i r Honc», a portion of a second conductor parallel to the first, oarr^'ing a current i" in the sntne direction, will be attracted town the first with a form „ . „« r« 2»r-i r where m is the length of the portion considered, and r is its dist from the first conductor. SiDM the ratio of n t<t r iii a nnmerieal qnantity indepcndedf the olwolute \'«lue of eittu-r nf thnto Hues, the piuduct of twt) currents measured in the elect roniagnetic srstcm most be nf dioacsuiou of a forve, hence tbe diiiiennions of the unit euncnt are [0 = [/'»]=[j/u»r-]. * 497-] DnnWTlOK OP FORCB OH CTHCmfT 143 ^P mS.] Another method of dotormintiig the direction of the force " whioli acta on a current is to consider tli»> tclfttion of the ning^netic ' action of the current to that of other currents and magnets. ^m If on one aide of the wire which oarrieM Uie current the magnetic action due to the carrent ie in the same or nearly the same direction as that due to other currents, then, on tlie other aide of the wire, tlieM forces will be in opposite or nearly opposite directions, and the force acting on the wire wili be from the aide on which the »£jn!ea strengthen each other to the side on which they oppoee each other. Thus, if a descending current is placed in a field of magnetic foroe directed ton-ards the north, its magnetic action will bo to the Dortli on the vroit side, and to the south on the east side. Hence Pthe forces strcn^hcn each other ou the west side and op]>ose each other on the eiut Hide, and the current will then^forc be acted on bjr a force from woift to «ii»it. See Fig 22, p. 139. ^H In Pig. XVtl uttlicend of tbie volume the small circle repn-si'iits ^^« Roction of the wire carrying a descending current, and placed in a uniform fivld of inagnclic force acting towards the left-buiid »of the figure. Tlie magnetic force i« greater below the win' than above it. It will therefore be urged from the bottom towards the top of the figure. Hf 487.] If two currents are in the name plane but not parallel, we may apply this prineii)le. Let one of the oondnctors be an ^^ iDBnit« straight wire in the plane of the paper, supposed horizontal. ^M On the right side of the current the magnetic force acta downwards ^B and on the left side it acta upwards. The same is true of the mag- " tutie force due to any short portion of a second current in tlie same plane. If the second current is on the right side of the first, the magnetic forces will stren^^hen each other on its right side and op|)Osc each other on its left side. Hence the second current will be acted on by u force urging it from ita right side to its left side. The magnitude of this furce depends only on the position of the ^_ aecond current and not on its direction. If the second current ia ^poo the left side of the fimt it will be urged from left to right. ' Uence, if the second current is in the same direction as the first it is attracted ; if in the opposite direction it is repelled ; if it 6ow8 at right angles to the first and away from it, it is urged in the direction of the first current ; and if it flows toward the first eiinenfc, it ia urged in the din-ctioD opposite to that in which the first 11, cnnent flows. ELECTBOKAOJranC TORCB. [498. In coDsiilorin^ the mnttial action or two ourreota it in not ncces- aar}- to b4.-ar in inini3 t.Iic roUtioiiM betwesD el«otricity and ina^cticm-^B wliicb wclmvv (>n<luuvourvd to iUiutrat«bymeanBot'ariglit~luuidoi^P Bcrew. Even if wc have forgotten tlieae relations we tbM anivo at vorrvcl n*itultfi, provided we adhere consiBtenUy to one of the two posiiiblo fnrni» of the relation. 408.] Let us now bring- together the ma^etic phenomena 0: the electric circuit so far aa we have inveetirrated them. We may conceive the electric circuit to consist of a voltaie battery, and a wire connectin^^ its oxtromities, or of a therinoelectrio urrKngcmcnt, or of a charged Leydcn jar with a wire connecting ita poNitive and ne^tive coatings, or of any other arrangement for producing an electric current along a definite path. The current produces magnetic phenomena in it* Deighbonrbood. If any closod curve be drawn, and the line-int<rgral of the magnetic force t^kcu completely round it, then, if the clo«od curva is not linked with the circuit, the linisintGgml ix I'.ero, but if it IE linked with the dreuit, so that llwi current > flows through tint IUImSm bMnw tU alaetiw cnntrat u<l Ika Una ef nanu tic (nducUon ] oUwwl cam, tlw lut^integal ts 4»i, an«l » positive if the dinwtiaa' of integrmtion ronnd ihr elowd curre woqU coincide whh tbati of the hands of a watch u «M>n by a {wnvn pusiiig throogh in the dirw'twB in which the ^Wtric cnm-nt Bow*. To a peiw muTiag along Ihe clwscd curve ia the direction ^-t inUsgntiaa, i RBCAPmTLATlOS. 145 I pnsEinf; tlirou^ the elwtric circuits Uie direction of tlie carrcnt would appear to be th«t of the hands of a watoh. We may oxprvnt thi» in another wjty by ea,yiag that the relation between thi- liiR-c- tioD8 of the two closed curves may be expressed by dcsoriliiii^ a nght*ltund«(l screw round the electric circuit and a right-hamlwi •crew round tlw clo«>ed curve. If the direction of rotation of the tlin-iid of either, aa wc pass alon$; it, coincides with the ])08itive direction in the other, then the )ine>iutc^tal will be positive, and in th« opposite case it will Iw ncjpitive. 49!).] AWtf. — The liui^-iutc^^l 4»t depends solely on the quan- tity of tJtc current, and Dot on any other thing whatever. It doea not dcjiend on the nature of the conductor throug'h which the currtrnt in pitsiin^, as, for instance, whether it he a metal Of an electrolyte, or an imperfvel- conductor. Wc have n-iwon for b^lievin^ that even when there i» no proper conduction, but merely a variation of electric diaplacenient, ax in the f^loxs of a Leyden jar during char^ or diwliurge, the magnetic ofliect of tlie electric movement is preoiseiy the name. Af^in, the value of the line-integral 4;rt docs not depend on the nature of the mediutn in which tito closed curve is drawn. It is the ame whether the cloMtd curve is drawn entirely through air, or pasMis through a mag-net, or itofl iron, or any other sub- stance, whether pammagnc-tio or diamagnetie. 500] When a cinniit ia placed in a magnetic field the mutual action t>etween the current and the other constituents of the (ield dc]*ends on the surface-integral of the magnetic induction through my surface bounded by that circuit. If W any given motion of the cin-uit. or of part of it, this surface- in (^^■gral i-au be irec/rdwrf, there will be a mechanical force tending to movis the conductor or the portion of the conductor in the given mitnner. The kind of motion of the cotiduetor which increases the surfncc- tatc^t^ >B motion of the conductor [>erpondicular to the direction (^the current and across the lines of induction. If a parallelogram be drawn, whose sides are parallel and ])ro- portioual to the strength of the current at any point, and to the magnetic indnetion at the same point, then the force on unit of length of the eonduetor is nuraericully equal to the arw* of this HI parallelogrum, and is {terpen diculnr lo it> pliiue, and aL'tn iu the ^1 direction in which the motion of turning the handle of a right- ^f handed acrew from the direclii>n of the current to the direction ^ft of the magnetic induction would cause the screw to move. ^M VOL. n. L us BLECTSOMAOHBTIC FOfiCE. [501- Hviice WA have • new eluctromngnetic definitioD of a line magnetio iaduction. It is that line to which the force on conductor is aUvays pi^qiendicular. It may also he dflined as a line along which, if an electric od be transmitted, the conductor carrying it will experience no foroe. 601.] It mustbeearefiilly rem^mhered, that the mechanical force which urges a conductor carrying a current across the lines of magnetic force, acts, Dot on the electrio current, but on the ood- duotor which cjirrics it. If the conductor be a rotating disk or a fluid it will move in obedieuce to this force, and this motion may or may not bu ni.'<'om{ianicd with a change of position of the electrio current which it carric*. But if the current itsEtlf bu tne to choose any path through a Rxed m>lid conductor or a network of wires, then, wlirn a eonalant miignetic foive is made to act on the system, the lutth of the current through the conductors is not pormancntl] altertid, but after certaiu transient [ihenomcna, called inditetic currents, have subsided, the diiitribtitii>a of tlie current will be fou to be the same as if no magnetic force were in action. The only force which acta on electric currents is eleciromotii force, which must be distinguished from the mechanical force whic is the Huhject of this chapter. rt( u. thMS riihMsMM Mm CHAPTER II. AlirERE's INVESTIGATION OP THE MUTUAL ACTION OP ELECTKIC CURItENTS. 502] Wa have conBidered in the !aBt chapter the nature of the magnetic fitld produood hy an electric curreDt> and the mcehaaical action on a conductor carrying^ an electric current pUoed in a mag- iwtic fifld. From this we went on to couaider the artioa of one electric circuit upon aDother, by determioin^ the action on the first diK to the ma^riic field produced hy the second. But the action of one circuit u]H)n another was oriirinnlly investigated in a direct maniwr liy Amjidrtf almost immcdialcly aftt-r the publication of Orated's discovery. We shall thtTcforu give iin outline of Ampere's method, resuming the method of this treatise in the uest chapter. The iden« which guided Ampiirc belong to the eystem which ftdmita direct action at » distnncc, and we shall find that a ivmark- able course of tpcouhition and invCKtigstion founded on tbo»e ideiis has been oarried on by Qau^ ^Vcl>er, J. Neumann, Itirmanii, Bettt, C. Neumann, Lorenz, and other*, with very remarkable results both in the diecovery of new facts and in the formation of » theorj- of electricity. See Aria, 8(6-866. The ideas which I have attempted to follow oat are those of action through a medium from one portion to the contiguous portion. These ideaa were much employed by Faraday, and the development of them in a mathematical form, and the comparisi.>n of the results with known fact«, have been my aim in several publitlicd papers. The comparison, from a philosophical point of view, of the resalts of two methods so completely opposed in their fmt prin- dplefl must lead to valuabli} data for the study of the conditions of scientific Bpcctilation. 503.] Ampere's theory of the mutual action of electric currents is founded on four experimental fucts and ono assumption. L a 148 AMPfellE'S TBEOBT. ts-H. Ampftro's fundamfntal experiment are nil of tliem oxamplea of vlixt has been callt-d ihv null method of e(>ni[i«riiij> foroca. Sec Art. 214. losimd of m<'UKuring Uk^ forec liy th« dynamicml oITm^h nf commnninling motion to a body, or tlte Rt«tioal method c«^| placing it in equilibrium with the freight of a body or the elaslicitj of a librv, in the null method two forces, due to the same »ourc4>, arc nuide to act simultaneously on a body already in e4ui)ibnuro, iind 110 effect is produced, which shews that these forces are tbem- sclves in equilitiriiim. This method 19 peculiarly valuable for comparing the etlects of the «loctric current when it passe« tbrouj^h circuits of diSbreiit forms. By connecting all the conductors one oontinuous wncw, we ensure that the ftreiigth of the correntl is the Fame at every point of ita oourHC, and »ince the L*urT«ntj begins everywhere throughout its conrxc almost at the nme instiint,| wu may prove thitt tho forces due to its aelion on a sujipend^d body are in etjtiilihiiiim by observing that the body is not at all affected by the starting or the stopping of the current. 804.] Anii»4re's balance consiBts of it light frame capable of reTolving about a vertical axis, and carrying a wire which forma two circuits of equal »r««, in the same plane or in parallel plauea, in which the current Bows in opposite directions. The object of J this arrangement is to get rid of the effects of terrestrial nutgDetisiti [ on the conducting wire. When an electric circuit is free to moreJ it tends to plac« it#elf so n« to embrace the largest possible number of the lines of indiu-tioo. If these lines are due to terrestrial magnetism, this |>o«itii>n, for a circuit in a vertical plane, will Im when the plane nf the ein-nit is Mst and west, and when tlie direction of the ctirrcnt is opposed to the apparent coutm of the^J ily rigidly connecting two circuits of eqnnl am in panillel phines, in which equal iiirrents mn in opposite directions, a eoinbinutioaj is formed whieh is nnaffi-ctcd by terrestrial magnetism, and isT Iherelbre called an Aitatio Combination, see Fig. 26. It is acted on, however, by forces ariwng fmax owrrents or magnets wtucfa so near it that they net dilTer^'ntly on the two circuits. 506.] Aup^rs's Grsl ex)>enment is on the vfivcl of two cqinl Fwrents close lugx-lhtT in i^pposite tlin\-tions. A win> rovetvd withj iwutating matetiat is douhlt^d on itwlf, and pUced near one of the' eitvuits of tlw aslalii' halamv. When a current is made to pun thr\>u;;h the wirv ami Ute hatsnw, the cqnil>bnum of the Imli nuuiUDs untlistnttied, ahevrinit that two ninnl ourTVDtaeltM>t(^«tlier] I 1 :4 507-J TOUE EXPEBIMEKT9. 149 in opfKMite directioDs neuLriiliRo umih otlter. If, uutMd of two wires tiido br side, a wire he iiwiilul^ i» tlie midillc of a rooUl tube, and if the curreut paas through tlie wtte and back by the tube, llifl action oatside the tube i» nnt only approximately but accurnlcly null. This principle is of great impiirtance in tJie COO- structioD of i-Wtric apparatus, as it atTordti the means of convoying ibo current to and from any f^alvanomctor or other iniilrumeDt in fiiclj a WHy that no clcctroma^ctic effect is proilm-c-il by the current on it« pagitagt! to and from the inEtrumcDt. In praclice it is gv&e* I Fig-M. rally iculTicitnt to bind the wires tojjfetber, care bein^ tak«n that they are kept perfectly insulated from each other, but where they mu»t pa« ncM any sensitive part of the apparatus it is better to make one of the conductors a tube and the other a win; inside H. See ArL 6S3. 506.] In Ampere'* s»M>nd experiment one of the wires is bent and crooked with a number of small Kinuosities, but so Uiat in every part of its courae it remains wry nt-ar the straif^ht wire. A current, flowing through the crooked wire and back again through iJie straight wire, is found to be without iutlu^uce on tbe ■static balaooc. This proves that the effect of the current running through any crooked part of the wire is equivalent to the same current running in the straight lino joining its extremities, pro- vided the crooked line is in no part of its course far from the straight one. Hence any small element of a circuit is equivalent to two or nion^ component clcmcute, the rclulion brtwo^n tbe component elcouenu and the resultant clement being tbe same as tliat between eouipunent and n^sultaiit diK]>Uiccmeiit« or velocities. fiOT.] In the tliinl exx>eriuienl a conductor capable of moving IBO A«P^BR*S TnBORT. only in thft direction of itx len^h is Rnbslitiiled for tb« astatio baUiioe. Tlie ctirrent eaten the conductor and leaves it at (iied points of ipatfc, and it if round thut no clnit^ circuit pla(.<ed in ■, the neighbourhood U able to moTC the cundactor. Rj.W. The conductor in this oxpcrimcnt is a viro in the form of a ciroolar arc euspvndt-d on a fnimo which is c>i]>ab1e of rotation about n vcrtiwd axis. Tb* circular arc it. hori/.ontnl, atwl it« centre i-oincidw with tho vertical axis, Twi-o Muall trough* arc filled with ^ nicrcur}' till the convex surlace of the mercury riiceK above t]iS^| level of ihe trouffhs. The tronghB ar^ placed under tlic cirwihir^^ arc and n4^uat(>d till th«> mer<?nry tdiiches the wire, which is of copper well auui1j^nuite<l. The current is made to enter on« of ihcM tionj^ha, to IravcrBe the port of th« circnlar an* lietweeo the troafHu, and to esMpc by the other trough. Thus part of the circular ant is traversed by tho Rurrfnt, and the arc ia at the now time capable of moving with eonxidcrmhle freedom in the dlree- tion of its length. Any closed ourrents or ma^^ncts may ttow be BMwle to approach tho moveable condw-lor without producing* the ■Ughlvist tendency to more it in the direction of its length. soft.] In the l^^urth cxpmment with the astatie hakaea two rireoita are employed, each joiniUr to one of those in the but one of them, C. hannit dimensioDS ■ timn greater, other, J, m time* less. Theee an plan<J oo opposite sides of th«' oin'uit of the balaoKi, whtch we ehall all B, so that thrr are ^ similaHy pUml with rvtftvi to it, the distaoee of f from S hdl|ri| UMW tWO^I balana^H aod th^ >s or til* V « tiniM gnatvr than the distauoe of if frow A. Tb« drtedaaa' J 508.] 151 ■treoglh of tiie. <nirT«nt is tbo snmo in A and C. Its direction in B may be the samo or op]>osil«. I'ndcr the$o (;irciitnstanci>6 it is foDud that Ji iaiu <^uilibrium under thr action o( A iind C, wbatpver be ihe forms and diatanoea of Uie throe circuits, provided thoy Iioto the relations igrivon above. Since the actions between the complete circuits may be conxiOcrcd to be dne to actions between the elementH of the oircniU, wt may use the following: method of determininj^ the law of these actions. Let J,, 5,, C,, Fig. 28, be corresponding elements of the three drctiits, and let J^, R,, C^ bo nlso corresponding elements in an- otlicr part of the cirenit«. Then the dttmtion of 5, with respect to Af is similar to tite situation of C^ with respect to B.f, but the Sig.se. distance and dimensions of (7, itnd 1i^ are it times the distitnce and diroenxions of B^ and A^. roxiH'Ctivoly. If th<> Uw of electromag- netic action is a function of the distance, then the action, what- ever bo it« form or quality, between H^ and A„, may be written and that between C, and B.^ wbere a. b, e are the strengths of the currents in A, B, C. But nB^ = C-i, nAj = B^, hBi Ag= C, B^, and a = c. Hence r^ n*Bi.A,f{nS^)ab, and this is equal to F by esperiment, so that wo havo n>/{HAtS,)=f{Aj,): tkefartt tvrta ineerteiy aa ike aqitttre of IA« dUiaHce. 1S2 AMPKBES THEOBT. [509. 609.] It may be observed with reference to these experitaents that ever}' ele^ric curreDt forms a closed ci'miit. Tlie currents usttl by Ampere, bt-inf* producc^l by the voltaic buttery, were of oonrso id closed circuits. It might bo supposed that in the ess* of the current of dtsi-hurgv of a conductor by a spark we mif^ht have X current forming nn open 6nitc line, but ncconling to lb« news of this book even this caiti; is that of a closed circait. No^f (.'xiKriments on thi- mulual action of unclosed ciimrnts liavo bevg n1all(^ H<-n<'e no sintement about the mutual uelion of two ele- ment* of circuit* can he «aid (o resA on purely exjwrimental gmiinds.! Tt is true ne may render a portion of a circuit nioreabh-, so as to ' ascertjiiii the action of the other currents upon it, but thc«e eur>J rentH, together with that in the moveable portion, necessarily for closed circuits, eo that the ultimate result of the experiment is t\\t action of one or more closed currents upon the whole or a part of 1 closed current. 510.] In the analysis of the phenomena, however, we may re- g»td the act'ion of u closed (.-ireiiit on an i-lcmcnt of ilaelf or of another circuit as the resultant of a numl>er of separate forces, depending on the t>ei>uniti- }<urt.ii into which the fir«t ciroutt may be conceived, for mathematical purpowes, to be divided. This is a men^y mathematical analysis of tbe action, and ia therefore perfectly legitimate, whether tbe&e finoea can really ao^H separately or not^ ^ 511.] We shall beRin by considering the purely geometrical relations between two lines in space representing the circuits, and betweeu elementary portions of these lines. Let there be two curves in space in each of which a Bsed point is lalien, from which the arcs are mcuMirvd in a defined direction along the curve. Let J, /f ht these poiut«. Let PQ and i^^ be elements of Ihe two curves. ^A Let AP^t. J-p-=^/, ) , ■ F%.W. r=i.\ *" PQ = rfi, P'Q' and let the distance PP' be de- noted by r. Let tl»e angle P'PQ be denoted by f. and WQ* by tf*. and let the angle between the planes of these angles be denotisl by ij. The relative position of Ihe two elcroenia i* suffii their distance r and the three angles 0, jf, and ^ CEOSJETBICAL SPKCIPl CATIONS. 153 given their relative position is «» eonipletely determinnd as if tliey ForKKHJ part of" th« same rigid body. 512.] If ue use rectangular coordinates and make if, g, t tlie fioordinatea of P, and jp*, /, / lliose of 7**, and if we deDote by /, Ml, n and by f, n^, »' the direetion-eosines of i*Q, and of P<^ respect ivuly, thi-n dx = 1, <y 4a = tii, -7- = », U9 = «, , = «, ■ and /(a^_a) + «(/_,)+i,(-'_j)= rcos^, . ■ '•(*'-') + «'(/-^) + «'(--'-^) = -'■cos^'. \ ^B W + ntts' + tin = coat, * B whcr« 1 18 tlie angle between the directions of the elements thom- mItm, and Crts « = — co« 9 COS I?* -1^ sin sin ^ cos 7. (2) (3) 0) (5) -Sr=-(''-')5"(^-^)if-(''-')i' = — rcoBtf. = — reo«(f ; (6) rfr mod diflerrntJuting t-^ witti respect to /, d*T dr dr ^dtd^'^ 4t dt' did/ dtdi"' in djt duf dy dy did7~d^j7' = — COB t. We COD therefore eiprees the three angles 6, 6', and t;, and the auxiliary un^lv < in terms of the ditferential eoellicients of r with respect to I and / as follows, dr di' dr ooefl = — COB^s — d/' coat •= — r nn (? 6Jn fl* COB 17 = - r dsd^ d'r dtdd' d»d^' ifi) 154 ahpIese's theort. 613.] AVc Bbnli next coiuiidor id whiit way il i* matlieinntinllf . eoncdvmblc llutt the vli-nutntM PQ nnd P'Q' might uct on «acl!^f other, and in doing so wt? ulinll not at first XMume that th«ir mutoaf^^ actiiin is nvcf^Hiril}- in the lino joining them. Vfe have seen that we may ttappoae each elenaeot Ksolved into other elements, proWded that these componente, when combined according to the rule of addition of vectors, produ<^e the original _ element as their reealtant. We bhall therefore consider dt as reeolred into ooe Odt = a in tK«1 direction of r, and ein 0Jt = fi S^ ^ r in a direction peqiendicular to r in the plane F'Pq. W« shall «l*o conxidcT d. the direetioD of r reversed 1 I FSk- 30. aa reeoWed into ta%& di =a m ttie direetioo ol r sin 0'coB !)<//= ^ in s direction parallel to that in whieh was measured, and &in ff'sin i)(^/= / in a direction perpendicular to «' and j3'. Let us consider the action 1>etween the components a and ^ oi the one hand, and o', f(, y' on the other. (1) a and a'are in the same straight line. The force beiwoeD them must therefore bo in this lino. Wc shall suppose it to SD attiractton = j^ao' ii, where jf ifl a function of r, and i, V arc the intensities of tlM current* in da am) di{ rcspcetiveljr. This expression satisBes condition of changing sign with J and with %. (2) )3 and are parallel to each other and pcrpcndienlar to line joining them, llie action between them may be written This force is evidently in the line joining j3 and f(, for it most be in the plane in which tJiey both lie, and if we were to measure /3 and 3* '« the reversed direction, the value of this expression would r«-miuii thv same, whii-h elicvrs that, if it repreHeots a force, that foren Itiia no component in the direction of ^, and must tliere- fure be directed along r. Let ua assume that this expression, wbeo positive, repTesenta au attnu'lion. (3) j3 and / ore perpendicular to each other and to the lino joining them. The only action poasihle betwM>n elrtiu-ntii ho related \* a couple whose axis is parallel to f. We are al prt-Bt-nt eng with forces, so we shall leave this oat of accoant. (4) The action of a and /?*, if they act on each other, most \m~- ignifOi^ not lut^^ exproBsed by C^0 »i . FORCES HE-nVEES TWO ELEMENTS. 155 Tiiff fdgn of thi> cxprc^ion is ruvcrecd if «c rcverw the direction |in whicli we mvaKura j^. It niii»t tlivrcforc ropresent oitlicr a (one 'io the direction of ff', or n «r>u]>k! in the jilniie of a and ff. As we are not iDvestignl ing enapU-M, vc nliiill tulcc it lu a force acting on a in the direction of ff. I There is of course an c<iual force acting on ^ in tbe opposite direction. We have for the same reason a force j Cay it acting; oa a in the direction of /, and a force Cfia'ii' ting on ff in tb« opposite direction. 514.] Collecting our results, we find that the action on d» is ampoundcd of the following forces, X = (Aaa'+ B^a^) ii' in the direction of r, ) r= C(oj3'-y;3)»i' in thciiirectionof/J, | (9) £ = Cay'iV in the direction of /. ) ■ u» cuppote that this action on d» is the rc«n1tant of three reee, Si/iitd/ acting in the direction of r, 8i\d*<l/ acting in direction of ds, nnd Sii'dtda' acting in the direction of dt', hen in tenox of 0, C, and 7, R = A cimDcosd'+^sinlJain^cosi}, 1 8 =-Cco»fi', 5'=CeoBtf. i In terms of tlic diScrontial coi-fiicicnts of r (10) ^~ ^d.d/-^'d^'' S=.C%. as 00 In temu ot i,m,n, and /', «', m', = -(^ + 5ji(/f+Mii + iiO(/'f4-<a'.j + ti'0 + ^(i/' + «w' + «n')/ c!(/'f+«', + i,'f), S'=ci(/f+«i,+«<). whore f, ij, C arc written for a^—x, y—y, and /s respeetiTely- 515.] We have next to calculate the fiirce witli which the finite current / actit on the Ruitfl current «. Tlie current a extends from (, whrre *=0, to f, where it hiw the value *. The current / extendit from A", where «'= 0, to P", where it has the value /. (12) 166 AMpfeKE'a THEORY. [516- The (>oonliiiat«B of points on either current are functions of « or of #'. If F JB »ny function of th« position of a points then we BhaU n the stib-teript (, „] to denote the eice«« of iU valu« at P over that at J, thus i',..rt = >V-/'^. SikU fiinciions nec«Bsarily disappear when the cirrtiit is closed. Let llie conipoDenta of tbe total force with which A'P' acts AP be i/X, ii'T, and tiZ. Theo the component i>ani)lel to X of 1^ lat J the forcD with whidt da' acta on itn will be tt' Maue £^=*^ *'+''■■ 4*S dt4/. Suhetitutin^ the values of S, S, and S" from (12), mnnnbering that ^ . dr r^ + m'n+H'C^r d/' and arian^g the t«ntis with respect to /, in, 11, we find lUik' Siaee A, B, aod C are ftmcUona of r, we may write P=[(A + B)^d,, q^Tcir, tbe int«gT«tkHi b«ing tak«B betveeo r and ac becatue A, S, vanish when r s oo. Itcnoe 1 iP and Iff- I 0-) 51 61] Now WB know, by Amp^'s third otae of equiUbriuin, that when / is a ckwed circuit, the fom aotinfj: on Jt is perpradicular to tlM diroetiaa of d*. or, in other worda. the oomponent of the fi>re» ia tbe direetiM of dt itself is zem. Let as therefore aasome tht direction of theaxi» of x so as to be {larallel to dt by making' f= I, « M 0, a s 0. Equation ( I i) thm bvcomn To Gad .- , tbe focw on dt nbtnd to mut of tength, wc muiL I. [gi7.] icnoF or a closed cntcmr os as elemebt. I I tal«{>rate this exprf«sion with ri?«]wct to /, Jnttgratiag the first ternj by farta, we find ^:^=iPe-Qh.0-Jj^P''-^-c)^^^. (19) "Wlien y U » cloaed circuit this eiprossion must be zero. The first term will disappear of itself. Th« Rcond tprm. however, will not in general liiaappenr iu the case of a clowd circuit nnleae the quantity under the taixn of intcfjrntion is alwny« zero. Hence, to satisfy Ampere's condition, wc must put P = rr^^+C). (20) 617.] We can now eliminate P, and 6nd the genoial value of dX ( -S + Cf ... ,, -) ■ ml - ~c m'S~rn d/ -.r? -crc-M's d*'. (21) 2 r .'o 2 r When »' is a closed circuit the first term of this espression vaniuhes, and if we make 2 r -c rc-n'( 2 d/, d^. d/, (22) 2 r ' where the integration !a extended round the closed circuit /, we I nay write jx ^=1-/-.^.! Similarly dV d* = »a'-//, ^f=.^-«a'.J (23) The qiiantiti^ a, ^, •/ are somctimos called the di'tvrmSnanls of (1m! oirciiit *' referred to the point P. Their ru«iiIUint i» called l>y Ampiirc the directrix of the eleetrodynamic action. It id f^vidcnt' from the eqnation, that the force whose components are -^i -y-, and -j- is perpendicular both to dt and to this dircHrix, and is repri'4mt«d nun>cri«ally by the area of the parallel- , c^niffl wboJie sides are da and the directrix. AVPERRS THEORT. [518- In th« langujige of qnaternioiui, the resultant force on i/« is tlie vector juart of the product of the directrix multiplied by dt. Since we already know that the directrix is the eame thing the nmgtietic force due to a uoit carreot id the circuit ^, we : b«nceforth eiieak of the directrix as the nu^eUc force due to th circuit. 518.] We diall now complete the cuIcuUtion of the compoaeati of the force acting botncon two finite currcuttt, whether clo«d open. Let f) be a new function of r, such that p=ij'\B-C)4r, tbeDbjr(l7)MKl(20) And equjitJou (1 1) become i» JE = -J««c+r~,«24..), S's dt With thcee valaes of the component forces, equation (13) becomes rf»X dpi ^Q^r'fQ _^ = -co..^j+f3-5,(Q+.W^+r-^. m 00* 519.] Let F^f'ipdt, •i'-^-^^-'^'-S- m I'm pdt, cr= rm'pd/. ff=f\pd4, w=jyp4i. (29) T^ptp qnaatitiM ban definite mines for any given point oT <|mm. Wti«ii live nrcuite are etowd, tbey cofw s pond to the oompotten to of the Yoctor-potcntials of tbo cinuits. Let £ be a new ftumtMn of r, fuch that i=rr(«+,)*. ami let Jtf be tli« «]«ubl« int«^ntt /"T fOMtdtif, [52*] POBCB BETWEBS TWO niHTE nTBRESTg. wfaicli, wh«n the oircaita are closed, beootnes tlieir mutual pot«ntia], . then (27) may be written 520.] Integrating, with respect to i aiul /, bvtwcoo the gives limittf, wo find IX= -j^ — ^{Ipr—ZAi'—L^'e + J'Ajr), ^ +n~FU-F^+FA,, (33) whi>T« 1h« suWripts of L iuditalc the di«tsncP, r, of tvbioh tlio quantity £ is a function, and tlxt etibscrijitv of F and /" indicate ith« pointH at which their values are to be taken. Tlie expressions for Y and Z may be written down from thi*. Mnltiplying the tlirce conij>oiientfi by i/x, dy, and di respectively, we obtain Xdr+Yds + Zd: = J)M-I){Lrr-Ljp-ruf+L^j-), ^ -{Fdx + Gdy + IId.'\^^A^, (34) wncre D U the eymbol of a complct* ditlbrcntial. Since Fdx -t-Gdy i^ Udz is not in general a complete differential of afttnction a{x,y, z, Jrf»+ Ydy + Zds is not a complete differential for currents cither of which is not closed. 521.] If, however, both currents are closed, the temu in L, F, .. 0, H, r, G', 11' disappear, and ■ X<ix+ Ydff + Zrh = DM, (35) ^ where M is the mtitnal potential of two closed circuits carrying unit currents. The quantity M expresses the work done by the electro- magnetic forces on cither conducting circuit when it is moved parallel to itself from an infinite distance to its actual position. Any alteration of its pcMttion, by which M \» increased, will lie axtitted by the electromagnetic forces. jB It may be shewn, as in Arts, 490, 596, that when the motion of ^"thfl circuit is not pamtlel to itself the forc(« acting on it are still <let«rmin«d by the variation of M, the potential of the one circuit on tfae other. 522.] The only cxprimental fact which we have made use of in Uiis investigation is the fact cstaliUshcd by Am[)ftre that the action of a closed current on any portion of another current is perpcndicnUr to the direction of the latter, Every other part of 160 AKPfcRK's THEORT. tbe invnti^tion depends on purely mathematical considoratione depending on the properties of lines in space. The retuonin? there- fore limy be presented in a much more condensed snd apprupriAte furm by the use of the Ideas and Inngnage of tbe mathematial mctliod speciiiUy adapted to the cspression of such geometrical relations — the QHatemwm of Hamilton. This liaa been done by Professor Tait in the Qaart^fy Mati«-_ KUtHeal /ournat, 1860, and in his treatise on Qaairrnioiu, ^ 399, fyi Ampere's original investigation, and the titudent can t^ly adap the same method to tbe wmenliat mora gvnerul investigation givt hero. 623.] Hitherto vt« have madi- no ajsnmption with respect to tl qiiantiticK A, It, C, excejil that they are functions of r, the distAoe between the elements. We have next to ascertain the form the«e fuuetions, and for this purpose we make use of Amp&re'a" fourth case of equilibrium. Art, 508, in which it b shewn that if all the linear dimensions and dtstanoes of « s^'stem of two circoitt be altered in the same proportion, tbe ctirrente remaining th« suoh, the force between the two circuits will remain the same. Now the force between the circoita for unit cnrrents is -j— , MB nns^l Binee this i« independent of the dimennons of the system, it m be a numerical quantity. Hciiee M itself, tlie coefficient of the mutual potential of the eirciiitfi. mwil be a quantity of tbe dim •iouB of a line. It follows, from equation (31), tJtat p must )>c reeiprocal of a line, and therefore by (24), ii— Cmnst be the invei square of a line. But since B and C are both functions of r, B— must be tbe inverse square of r or some numerical multiple of it. 524.] The multiple we adopt depends on our system of meadure- ment. If we adopt the electromagnetic system, so called hecauae it agrees with the system already established for magnetic measore- menls, tbe value of J/ ought to coincide with that of tbe potential of two magnetic m1mi11« of strength unity whose boundaries are tbe two circuit* respectively. The value of if in that case is, by A'^-*". M=(j^d,4r, (36) the integration Uting {lerforined round both circuits in the positive direction. Adopting this as the numcrieal value of if, and paring with (31), we find 1 and B-C=p,. (37)' J t525-l AMPBBraPOBMUtA. 525.] We may now express tlie components of the force on ti» ^tnsias fpotn the action of r// in the most general form consistent ^with oxpcrimcntiil Satds. The force on ds is compounded of an Attraction the direction of r. 8=: — -^ i'/iUd/ ill the dircctJoii of 44, tnd S'^ irhere Q : —Wdfil/ in the direction of <Ai', m -f: Cdr, and since C is an unknown function of r, we luiow only that Q is some function of r. 536.] The quantity Q cannot he determined, without osEtitnp- tions of some kind, fruro exjieriments in which the active current Ibnns a cloited eirciiit. If we suppose with Ampere that tho action en the cU'menta d» ami d* is in the liuf joining them, then <9' must diaapjii'ar, and Q must be coiistaut, or Ecro. Tho ree is thon ix-duced to an attraction whose value is ^i^dsd^. (39) ^-^\d»d? ^' Ampere, who made this investigation long before the magnetic tryvtem of units had hvt-.n estublishcd, uees ti formula having a nnmerictil value half of tiiis, naniidy Here the strength of the current is meaitured in what is colled fleetrodrnamic measure. If i, t'art- the strength)! of the currents in leetro magnetic measure. and_;'.y the same in olcCtrodj'DUmic meu- ore, then it is pl&in that y/=2ir, or >=^/2;. (41) Hence the nnit current adopted in electromngnetic measure b greater than that adopted in electrodynamic muisurc in the ratio of v'S to 1. The only title of tho elcctrodynamic unit to consideration is that it was originally wlopted by .\mpiire, \\w dixcovervr of th« law of action between eitrn-nts, Tlic continual recurrence of •/^ in calculation* foundcil on it is inconvniicut, and the electro- magnetic system has W\e great udvuutugeof coioeidiug numerically VOL. ri. u AUPESSS TRBORT. [527. villi all our na^«lic rormiiloe. As it is tlifficnlt for the stadetil to bear in mind whether he is to multiply or to divide by \^2, we ehall hf^nccforth uso only the ek-ctromagnetic system, as adopted b Weber nnd most other writers. Since thi- Torm itnd vhIuu of Q have do effect 00 any of the eicperimente hithorto made, in which the active current at least is always a cIokviI one, we muy, if wc please, adopt any value of wbicli a|ipc3r> to us to simplify the formulae. Tlius Ampere u^iMiiDieE that the force between two elementa the lin« joining llu-m. Tliia gives Q = 0, Gmssmann * sssumes that two elements in the nm« straight lini have no mutual action. This givc« laSt I -i .=. 3 dh ^-A£-<^^j We might, if we pleased, oienime that the attraction between t' elements at a given diatanee is proportional to the ooeine of angle between them. In thin case .-, J?=^eo9.. S = -;i^. S = -^^ Finally, wc mig^it assume that tlte attraction and the obli forces depend only on tbe angles which the elements make with the line joining them, and then we should have «=-r' t. S = -3 I dr rfr S- 2^^ ^*--^il^' *=^£- (»: 527.] Of tltme four diffrn'ttt a»stimption« that of Aii>{>^ro ii andoubtnlly the l*rst, since it is the only one wbioh makvs tli forom on the two elements not only txjual and oppoaiW but in tli •tnight line nhieh joins them. * P<«B.. Mm. lxl>. p. 1 (IMS). ClfAI^ER III. THE INDCCnON OF BLECTBIC CCBBEKTS. 5S8.] Thi: <li»ci>v(>ry by Onttt^ of the magnetic notion of an electric cnrrent led by a direot prix-esa of i-easoninjj to tliat of ma^DetizatioD by elwrtric ourrents, and of the mechanical action between electric currents. It wo* not, however, till 1831 that Famday, who had been for some lime endeavouring to produce electric currents by magnetic or electric action, discovered the con- ditions of mngoeto-eleciric induction. The metbod which Faraday employed in hia researches consisted in a constant appeal to ci- imcnt as a means of testing the truth of his ideas, and a constant iltiration of ideas under the direct influence of experimenl. In his pnblished researehea we find these ideas expressed in language which is all the better fitted for a nascent science, because it is ■oni«whnt alien from the style of physicists who have been aocus- tomvd to cstAblislied mathematical forms of tliought. , The trxpcrimmtal investiffation by which AmpJrre establivhnl the laws of the mechanical action between electric curn'ots is one of the most brilliant achievements in science. The whole, theory and experiment, seems as if it hJld leaped, full grown and full armed, from the hr:iin of the ' Newton of elec- tricity.' It is perfect in form, and nnassailable in accunu^y, and it is eammed up in a formula from which all the phenomena may bo deduced, and which must always remain the cardinal formula of el ectro-dynam ica, Till- method of Amp^^e, however, though cast into an iiultictive form, does not allow ns to trace the formation of the idotx which guided Urn "We can scnroely bvliere thai Ampdre mlly diiicovered tin' law of action by means of the experiments which he describes. We ore led to susp4'G4i what, indttd, ho tells uk liimself •, that he • TiMu dt§ PJIoMmAMi Bntndynamlfutt, p. 0. K Z Vki [5»g- I (liitcoTcred Ike law by aomc ])rooe68 which be bsB not shewo uii, And that when bo had afVerwanls bniU up a perrcct lU-moD- •tntion he removed all tntces of the ecafTolding by which h« raiaed it. Faraday, on the other band, shews ua his nnsacoesaful aa^ aa Ilia Hiocewtibl experiotenta, and his erude ideas as well as liii developed ones, and the reader, however inferior to him in inductive (lower, feela sympatliy even more than admiration, and is tempted to believe that, if be had the opportunity, he too would be a dis- coverer. Every student therefore should read AmpJrcB rwearch as a splendid esnmplo of scientific style in the statement of a di covery, but he should also study Faraday for the cultivation of scientific spirit, by means of the action and reaction which tnko plaoo between the newly discovered facts as introduced to him by Faraday and the nnscent ideas in his own mind. It was perhaps for the advantage of science that Faraday, though thoroughly con«cioiis of tltc riindnmental forms of spaccj time, and force, wiu not a profesited mntliematiciim. Ho was not tcmpt«d to enter into the many iatcre»tinf> researdies in pure motbematM* which his discoverifs would have suggested if tlicy had exhibited in a mathematienl form, and he did not feel eallcd u eillier to rorc« hi» results into a Kha|ie acce|>tible to the null matical taste of the time, or to express Ihem in a form wb matbematiciAns might attack. He was thus left at letsore do his projK-r worV, to coonlinate his idea« with bia facta, cxprttw them in natural, untevhnical langnagek It is mainly with the hope of makini; the-te ideas the basts matbentaticiil mvtht^l that I have uudertakea this treatise. 52f>.] We are uceuKtomed to consider the universe as made up of parts, and matbemalieiann ukudIIv he^in bv eonsiderin^ a si particle, and then conceiving iU ivbtion to another particle, and on. This has generally been sup)>cMctl the most natural melb To eoneeivc of a particle, however, nHjnirex a pn>ct»i of absttactii nnoe all our percepltuns ore rebted U> ejitt^ndoil bodies, so tlw Mm of the 41// tluit is ra our HmacioudneM) at a given is perhaps as primitive an idea as that of any indi\-idual Henee there may be a utatlieinatical method in which we pri<e from the whoht to the parts itistltad of from Ihe patt« to the who' For example, Kwlid, in his first hook, oonMives a liiw as traevd out by a p^int, a fMrfiK.'e as swept out by a line, and a volid m generated by a suifacr. Dut he alwi defines a snr&oi 530-] BI.BMENTABY niENOMHNA. 165 boundary of a aaMi, * line m the iHlg« vf a KurfaiM, and a point I tbe extremity of a line. In like manner we may coneeive th« potential of a material ■ystem as a function found by a certain prooe^ of intcg^ration with Lreepevt to th« maaees of the bodies in the tield, or we may Hup^Kiw lliMO masses themselves to have do other mathematical meaning Ihan Ute volnme-intO|rntlK of — V**, where * is the potential. In electrical investigations we may use formulae in which the ]aantit!c« involved arc the dielauces of certain bodies, and the ^electrifications or currents in these bodies, or we may use formulae which involve otlior quantities, each of which is continuous through all epoco. The mathematical process employed in tJie first method is in- tegration along lines, over snrfaees, and throughout finite spaeea, thow employed in the aceond methoi:l are partial dilTcroutial equa< _tiona and integrations throughout all Ktinoe. The method of Faraday scema to be intimately related to the ad of these modes of treatment. lie never oonsidera bodies existing with nothing between them but their distauoe, and acting on one another according to some function of that distance. He eonccivc* all spiicc ns a field of force, the lines of force being io general curved, arnl those due to nny body extending from it on all sides, their dirretionn being modified by the presence of other bodies. He even m{ks1>'h * of the linen of force belonging to a body M in »ORM> tteiiMt part of itiielf, sen that in iU action on distant bodies it cannot be ^id to act where it is not. Tliiii, however, not ■ dominant idea with Faiaday. 1 think he would rather bavc mid- that the field of space is full of lines of force, whotv I arrangement depends on that of the bodies in the field, aud that taie mechanical and electrical action on each boily is determined by tiona 1 ^^neconc ^98 exi ^tia PHBXOUEXl. OF MAOKIITO-BLIXTRIC IKDUCTION t< 580.] 1. Muciioit hi/ Farialion of the Primary Ciirrmt. Let there he two conducting circuits, the Primary and the idary circuit. The primarj* circuit is connected with a voltai« • iSip. A>., ii p. 903 ; iii. p. 417 t Bud Pandky'* JSK/nrimatal Jtofordtn, mtim I ud IL 160 MAGSETO-ELECTEIC HTOUOTIOS. [53c bntlety hy wliich thn primaiy carrent may be produced, tiiamtauiL><l, Ht«{>]wd, or Tovoraed. The wcondaiy circuit includea a fpil^no- m«Uir to indicate any ciiri«ntd which msy W formed in it. This galviinoRi«UT iR placed at such a dietance from all jtarts ot the I>riinury circuit that the primaty current hs8 no sensible direct iiitliwnw OH ita indications. Let part of the primary circiiit consist of a stTaifjlit wire, and part of the secotidury circuit of n strai(;ht wire near, and pnmUel to the first', the otlicr parts of Ibu circuit* bein^ at a ^^Jitor difitanoc from each other. It is found that at the instant of Miidin); a current through the straiffht wire of the primary circuit the galii'aDonH-lcr of the secondary circuit imlicut^-Jt ■ ciirreut in the fivondary straight wire iu the (tppotUe direction. Hiis in culled Uie induced current. If the primary cnrrenl '\* maintained L'onatant, the induced current soon ditapjieam, and tlio primary current appears to produce no elTeet on the Mi-coiiilary eiivuiu If now the primary current is stopped, a aeoondary current is observed, which ta in the an>« direction as the primary cnrreut Every variation of the primary currvnt produecA electromotive force in the seoondary circuit. When the primary current inerawco, the electromotive foroe is in tite opposite direction to the current. When it diminishes, the electromotive force is in the same direction as the current. When the primaiy aurrent is conttunt, there is do electromotive force. IImmo effectn of indudion are inereasvd by brini^ing tlte two wires ncanr ton^llier. They are alw iocrvoeed by forming them into two oirouUr or spiral coils placed doeu together, and still more by placing an iron rod or a buntUe of iron wires inside tfaa ooilsi. 3. Imimetiom if Motam V Ur Primaty CinaU. We ba^e seen that wbm the primary enrrent is maiBtained vonatant and at rest tbe aeeotkdary •.-urrmt rapidly disappears. Now let tbe primary cum>nt be maintained constaiDt, bat let tbe primary' slrnight nire be made tin approach tbe seeoadary ittaigfat wire. IHiriiig the approach there will be a smiodaiT comnt in tbe vfpMtt* dit«vlii4i tkvm tbe pfiiaary. If tbe |>riuiary eiivall bo moved away from the svcoDdaiT', tbd* will be a secondary ourrvnt in tbe mmr dirvetiua as tbe prnnarv If tbe senudary eimni W mvn^ tba saooadb^ ettmat m 531.] ELKUBXTABT PHKNOUE.V^. 167 opposite to tha priinar3r when the E««ondar)' wire U approKohin^ the primary wire, nad in the giimu (lircctiou when it is rocodiag Ktrora it. In nil cofca the dircc-tioii of the sccondury current is ttiie)) that the mechanical action between the two conductors is oppoKite to the direction of motion, iming u repiilHion when the wtre« are ajt- prouehintf, and an attmctiou when tliey nro receding. This wry importBiil fact vmit eetJihlixhed by Leiix*. I 4. JnduelioH by tie Behlire Motion <^ a Magnet and Ike Stcondary Circuit. If we inih«t1itute for the primary ctrcait a ma^etic shell, whose ke(]g« coincides with the circuit, nhose strength is Qumfricutly c(|[uU ho that of the current in the circuit, and nlioae auHtml fui-o oor- KspoMlM \o the positive face of the circuit, then the phpnomena produced hy the relative motion of this shell and the gecoiidary circuit are the same as those observed in the cose of the primary' eirouit. 531.] The whole of these phenomena may be summeil up in one ||>iw. When the nnmher of lines of magnetic induction which puss fUiroug'h the seoondary circuit in the positive direction is altered, in el<'elrnmotive force aet« round the circuit, which ix measured ]ty the rate iX decreikse of the magnetic induction throu{>^h tb« ^rcuit. 532.] For inalanee, let the raits of a railway tn* insulated from ^hc earth, hut connected at one terniitius tlu'ough s galvanometer, id let the circuit he oampleteil by the wheels and axle of a rail- ray carriage at a distance x from the terminus. Neglei-ting the eight of the axle above the level of the rails, the induction lirough the secondary circuit is due to the vertical component of the earth's magnetic Ibrcc. which in northern latitudes is directed lownwards. Hence, if i is the Bffluge of the railway, the horizontal area of the circuit is bx, and the surface-integral of the mugnetia iaductioD through it is Zbx, where Z is the vertical component of the magnetic force of the earth. Since Z i« downwanls. the tower face of the circuit is to be reckoned positive, nnd the positive direction of the circuit it*-clf \t> north, vnA-, iwuth, wcstv thai in, in direction of the sun's apparent diurnal mur^e. S'ow let tlie carriage be set in motion, then x will vary, and " P«|g., Am. uL 403 <1834)l 168 MAGNETO-ELKCTRIO ISDUCTIOIT. tiiere tvill be an electromotiTe force in tlie circuit whose valoe u -<■ n If « is increa^Dg, that is, if the earring \» movingr away from the tcnninufl, this electromotive force is in the negative direetinn, or north, weet, south, east. Hence the direction of this force through the ax]e ia froro ri^ht to left. If x were diminishing;, tJte absolute direction of the force would be reversed, but since the direction of the motion of the carriage is also reversed, the eleotto- motivc force on the axle is still from rig-ht to leR, the observer io thff carriage being always supposed to move face forwards. In southern Intitudcs, where the south end of the needle dips, the elcctromotivu force on a moving body is from left to right. Hence we have the following rule for determining the eleottx>- motivc force on n wire moving through a field of magnetic force, Ftaoc, in imagination, your head and feet in tbe position occupied \>y Uie emlH of a coropuss needle which point north and south respec- tively ; turn yrmr fuet- in the fonvard direction of motion, tlicn tho electromotive I'oree due to the motion will be from left to right* S33.] As thnra dircctionu) ivlatioits are important, let us tike another illuKtrntion. Suppose s metal ginllc laid round the earth at thu «|utttor, and n mi-tiil wire laid along the meridian of Clrccn- wich from the eijtwtor to tlic north pole. Let a gmt quadrantnl arch of metal be constructed, of which one extremity is pivoted on tlio north pole, while the other is durieJ round tho equator, sliding on tJio great girdle of the earth, and following the sun in his duily ooiir»e. Tht^re will then be an electromotive fc along the moving qnadranl, actio, fn<ni the pole towards the equator. The electromotive ror<« will be the saaie whether we Eoppooe the earlh at re*it and the ijuadnint niovpd from cant to we§t, or whetWr we «ippo»c llie iiuadnml ul rnst and the earth tnmed from west to east. If we suppose the earth to rotnte, the elcctromotiv force will be tlie same whatovcr \\v the form uf the part of lb circuit fixed in space of which 004 end touches od0 of Iho po here «j roTo4^| tio^ tor. ^ 5S^-1 EXPEKIMESTS OP 0OMPABI8OS. 169 Band ^^BDi the other the equator. The cnrront in this {Hirt of tho circuit is from the pole to the equator. The other part of the circuit, which is (iscd with respect ta the eartli, may abo he of noy form, snd eitht-r withiu or uithoiit the earth. In this part the current ib from thf i-quntor bo cither pole. 534.] The int^neityof the cktitromntivc forco of maffncto-cli^ctrio iutlactioD is entirely imk-peudi'nt of the nntiirv of tli4i KiiliKlance of th« ooDiiuctor in which it acts, nnil also of tlie nature of the condiK-tor which carries tho indiicin^^ current. To shew this> Faradsy * made a conductor of two wircK of different letals in«nlat«d from one another hy a silk vovurinj'', but twiiftiMl together, and solileied together at one end. Tlic other cods of the wirvs were connected with a ffalvanomctiT. In this way the wires were similariy situated with rcspi-et to the primary circuit, hut if the clet'trwmotirc ibrco were ptronffcr in the one wire than in tho other it would produce u current wliich woidd he indicated hy the gialra no meter. He fonnd, howcvir, tluit *uch a eomhination may be ex|>osed to th« most powerful elwtromotive forces due to in- ducliiin without tha galvauomcter Wing iilfccted. He also found that whether the two branches of the eomjiound eonductior conMsted of two metals, or of a niotal and an electrolyte, the galvanometer ,»«• not aflVcted t. Hence the clectromotiTe force on any conductor de|iend8 only on tlie fonn and tlie motion of that conductor, together with the Htrength, form, and niotioit of the electric currentn in the field. r»3o.] Another negative property of electromotive force i» that ha» of itxelf no tendency to caui«e the niccliauioal motion of any ly, hut only to cause a cunynt of electricity within it. If it aetnaJly produces a current in the body, there will be mechanical action due to that current, but if we prevent the current from beio^ formed, there will be no mechanical action on e body itself. If the body is electrified, however, the electro- JTO force will move the body, as we have described in Electro- statics. bSQ.] Tlie experimental investif^tionof the laws of the induction of electric currents in fixed circuits may be conducted with oonaiderabic accuracy by methods in which the electromotive force, and therefore the current, in the ffalvanometcr circuit is rendered zero. instance, if we wish to shew that the induelioit of the coil 1 mec cun r^ Ua.. 19S. t W>., WO. 170 MAGXETO-SLECTRIC ISDUCTION. A on the «oil X is equal to tliat of B upon T, we place the fintl ]«iir of coiU A and X at a suffic-k-nt distance from the seeoDd pair fig. 3S. ^ B and Y. We then coniwct A and 5 wHb a voIt*ic Taltwry, ao that W(! CAD make the E&nu: primary current flow through A in liie powitivL- direction nnd thi'D tlii'oii{;h B in th*.- m-jpitivt: diri.H-tion. We aim connect X and }'willi ngalvanomi-lci', ho tluit tl^; Mvondary onrrvDt, if it eitsts, ehsll flow in the >umc direction tkrxiugb Xaad 1' in series. Then, if tho induction of A on A' \* equal lo Uiat of .ff on }', the f^vani>nirt^M- \\\\\ tiidkatj^ no taductioD current whi-n tl batlprjf circuit it) «Ui«cd or bniljMi. The accunK-y of this method iiicn.'ai>e« with tlie ftren^tli of the primnry furnMit and tbi' nennilivcneja of the gmlvanomcter to iij- stunlaneous currcuts, and the ex|>eriin<:utN are much ninrv easily pcrfomied thau those relating to elect roma^tetic attrnction*, wltero tho conduflor ilKrlf haa to ke delicatelv nuRpendi-d. A very instructive series of well-devivcd esperimenta of this kind is de0eril(i-d by Professor l-'elici of W.^a '. I shall only indicate brielly some of tlic Uws vi tiicb may be proratl in this way. ^^ (1) The electromotive force of the loduotioa of one eircuit on^| auothcr is ittdcpendent of tlw area of the section of the conductora and of the malcrial of nliioli tht^y nrc made. ^l For wc cjin excbangv any one of tlic circuits id the experimental for aiiotlHT of a difTcrent section and roalcnal, but of the same fnrni, without altering the resall. • Aa*ak* it CUmU, xxiir. p. M (t8»), and Kmtn OfaiinMi ix. p. 34i (IfvV). S37-] rBLICIS EXPERIUEN'TS. 171 I I I » (2) The indudion of the cireait A on the circuit X is equal to that of X upon A. For if we (mt .1 in llu- cr»IvaDOinet«r cireuit. and X in the battery circuit, Itie 4^(11 iijlirium of flovtromotiri.' force is not disturbed. (3) Tlte indu«tiuo is propnrtional to the inducing current. For if we Have aswrtaincd t}i;it the induction of J on X is cqnfti to that of H on }', and nliio to thiit of C on Z, wo mny make the battery current Unt How through .4, and then divide ittrvlf in unjr proportion bL-ttveeu /i and C. Tiit^n if viv. connect A' rcvcnted, Y and /direct, nil in «erie«, nith the galvanometer, tUo cli;ctroniotive force in X will balance the sum of the electromotive forces in Y and J?. (4) In pairs of ciretiitit forming systems geometrically niinilur the iodoctioD is proportional to their linear diuienaiona. For if the three pairs of circuits above mentioned are all itimilnr, but if the linear dimension of the first pair is the mim of th« oorrespondiot; linear dimensions of the second and third pairs, then, if A, £, and C are connected in series with the battery, and .V reversed, faud H are in series with the galvanometer, there will be equilibrium. (5) ITms electromotive force produced in a ooil of n windings by • current in a ooil of m windingrs is proportional to the product mn. 637.] For ezi^riments of the kind we have been considering the galvanometer should be as sensitive as possible, and its needle as lif^t as poasibte, so as to give a sensible indication of a very small transient current. The experiments on induction due to motion require the needle to have a somewhat longer ]i«r!o(l of vibration, so that there may be time to efTccl certain motions of the condnctors while the needle is not far from ilK position of Ci(|nilibrium. la the former expi-rimoiit*, the eloctromoUvo forcM in the galvanometer circuit wore in equilibrium during the whole time, so that no currtnt panwd tliroufjh tho gnlvano- meter coil. In those now to be described, the elctrnmotive forces act lintt in one direction and then in the other, so as to produce in Kucccjwion two currents in oppnditc directions through the ^1- vanonietvr, and we have U) shew Unit the iuipulBes on the galvano- meter needle due to these successive currents are in ecTtaiu cases equal and opposite. llie theory- of the application of the galvanometer to the mcoMaremcnt of trannient currents will be considered more at IcDgith in Art. 7-18. At prvveDb it a sufficient for oar purpoi>« to 172 MAQSETO-ELECTBIC INDVCTJOS. [53S. 4 observe that as lon^ se the galvanometer needle is Dear its position of oquilibrinm the deflecting force of the current is proportiooal to the current itself, and if the whole time of action of tlie current is small compared with the period of vibration of the needle, the final velocity of the magnet vUl be proportional t« th« total quantity of electricity In tlie current. Hence, if two currents pass in rapid succession, conveyinq; equal quantities of electricity in opposite directions, the oeoillc will be left without any final velocity. Tlius, to shew Mint tive induction-currents in the secondary circuit due to the closing nni] the breaking of the primary circuit. »re eqiiul in totiil ijimTitily hul' oppoxiU' in direction, wc may arriinjpt the priniuiy cin,'uit in connexion with the battery, »o tliat by touching tt ]n}y the current may be m.'ut through the primary circuit, or by ruitioviiig the finger tito contact may be broken at pleasure. If the key id preaiscd down for some time, the ^Ivanometcr in the Kccondary circuit indicates, at the time of making contact, a tr.intiient current in the opponitt? direction to the primary current. Ifcaiitat-t be maintained, the induction current simply pai^ses and disappears. If wc now break contact, another transient current passes in the opi>oaite direction through the eeoondary circuit, and the galvanometer needle receives an impulse in the opfioaite direction. But if we make contact only for an instant, and then break contact, the two induced currents pass through the gnlvnnomete^H in Bueh rapid succession that the needle, when acted on by the firs^H current, has not time tt> move a sensilile distance from its positioo of equilibrium before it is stopped by the second, and, on account of the exact equality between the quantities of these transient^ cnrrents, the needle is stopped dead. If the needle is watcheil carefully, it appears bo ha jerked suddc froin one position of rest to another )io«ition of rest very the first. Ill this way wo prove that the quantity of electricity in th^ induction current, when contact is broken, is exactly i-qiinl an opposite to that in the induction current when oonlaet is ina<lK. 538.] .\iiotlier application of (hisnu-thcNl in the following, wUk is given by Fclici in the second veries of his lle*ftreh/Ji. II is always possible lo find mnny dilTervDl positions of U secondary coil it, such tliat the making or the breaking of contoi in the primary coil A produces no induction currvnl in li. 'It 539-] 173 I positions of tbo two coils arc in such cases said to bo em^Kgal* to cnch othor. Let Sj asd S, be two of these positions. If the eoil B be sud- denly moved from the position ^| to the position B.j, Uk al^braicnl sum of the transient currents iti ttio eoil fl is exactly zero, eo that the galvanometer needle is left at rest whuii the motion of £ is completed. This is true in whatever way the coil Ji is mov«d from B, tn B.,, and also n-hcthcr the current in the pnmiiry coil A bo continued Iconstant, or mmlo to vary during the motion. A^in, let ]f \k any other pofition of B not ooujiigati* to A^ BO that the makiiif* or brvakin^ of eonfjict in A produces an in- duction ctirront when B a in thi.- position 7f . H Let the contact be mnile wht-n B is in the eonjiigale [wsition j?,, , there will l)c no induction current. Movp B Ut B", theio will be an induetJon current due to the motion, hut if U is moved nipidly ^rto B^. and the primary contact then broken, the induction cu^-c^ut ■ due to breaking con1iu;t will exactly nnnnl the viFeet of that due to the motii'ii, ho that Uie galvanometer needle will be \vh at rest. » Hence Ihe current due to the motion from a conjugate position to any other position iii ci)ua1 and opposite to the current due to breaking contact in the latter position. Since ihe effect of making cootact is equal and opposite to that of breaking it, it follows that the effect of making rontact when the coil B is in any position J? is equal to that of bringing the coil from any conjugat« position Bi lo £" while the current is flowing through A. If the change of the relative position of the coile Is made by moving the primary circuit instead of the secondary, the result is found to be the same. 639.] It follows from these experiments that thv total induction current in fidnn'ng thesimultimcous motion of /^ from Aj to ^,,, and of B from Bf to B.J, while the current in A cliungM from y, to y,, dci>om]K only on the initial i;tatc A,, B^, y,, and the final state Aj, B., yj, and not at all on Die nature of the intGnnediate statea tiirough which the Kyxt<-m may jMss. Hence tJie value tif tho total induction current uiujtt be of the form hy^. B„ y,)->'('^i. A- Vi)- where /* ia a function of A, B, aud y. [h With reicpcct to tlie form of tins function, we know, by Art. 53il, ^wthat when Utere is no motion, and therefore Ai= A^ and if, = //,, 174 re ISDUC the iodueLion ctirroat » |>roportional to the primary' cunrnt HcDc*- y enter* simply iw a furtor, th« olh«r fnrtor l»cing a fuoc tion of the form and pnitition of the c-ircuito A mul B. Vio iilso know thnt Wvit Tulue of this ftinction <I«pcn<Is on the relative nod not on the nlisohito povitions of A imd B, so that it must be cHpiiblc of bciti^f cxprcsMd as a fonction of tttc dutAtice* of th« diliercnt cl«in«nt« of which the circuits nrv composed, and of the aii<>:l(.-)t wliieh these rlements malie with eaoh other. Let M be this fuiiction, thon the total induction current may written C{J/, y.-i^y,}, where C is the conductirity of the secondary circnit, and J/,, arc the oriffinal, and ,1/,, y^ the final \-*Iue8 of Jf and y. These ciprrimenU, therefore, shew that the total current induction depends on the chan^ which takes p1ac« in a certain quantity, My, and that this change may arise either from variatioa of the primnTy current y, or from any motion of the primary or^^ eecondiiry circuit which alters M. ^^ 540.] TliceonoeptionofHiich acjtiantify. on the changes of which, and not on its absolute magnitude, the indtiction current depi-nitii, occurred to Fnnulay at an early sfugw of his iwearohcs*. He . ohserved that the t^ccondary circuit, when at rest in an e1«etfo*^| magiiclio field which rentainD of conittant intennity, doc« not shew^^ itny eIi-etrio»l efleet, whereas, if the Miinc xliite of the field hud V-en suddenly produced, there would have been a current. Again, if the primarj' circuit \a removed from the field, or the magnetic furees abolished, there is a current of the opj>o«ite hind. He therefore ^i recognised in the secondary circuit, when in the electroinagiietia^| field, a 'peculiar electrical condition of matter,' to which he gave^^ the name of the Electrotonic State. He ufterwaids found that he could difipensG with this idea by means of considerations founiled on the !inf> of mn^'netic force f, but even in his latest researched he snyK, ' Agiiin imd iigain Uie idea of an tltctrolome state § been Ibrced upon my mind.' The whole history of this idea in Uic mind ofT^raday, as shewn in his published researches, is welt worthy of study. By a course of experiments, guided by intense application of thnnght, but without the aid of mathematical calculations, he n-as ImI to recog^ nine the existence of something which we now know to be a matb&- taalical quantity, and which may even be called the fandameatal Hip. lUt, Hriei i. CO. t H>„ U. {.U2). t lU, 32<9. I lb., SO, 1114, 1<«1, 17», 17S3. S4(l LI! 175 quantity in the theory of e1ecfroina);;nettem. But ua he was led up to lliia conwptioD l>y a purely experimental path, he awribed to it a physical existence, and euppoaed it to bo a peculiar con- dition of matter, thotigh ho was ready to shaodon this theory at soon as be could explain the phenomena hy any more familiar forms of thought I Other iiivestijiratore were lonp afterwards led up to the same idea hy a pun?ly m:ithematio:il path, but, so far as I know, none of them recn(;ni6cd, in the roRnod mathcmntical iden of the polentiat f two circuits, FaTaday'e hold hypothesis of nu uloctrotonic state. Th<«e. therefore, who have approached this subject in the way pointed out by thow eminent in vest if^a tors who first reduced ita laws to it mathematical form, have sometimes found it diSicult to sppnctato the sricntific accuracy of the statements of laws which Faisdiiy, in the Gist two series of his Hr-tcarcien, has given trilh Buch wonderfnl oomplcti-ness. Ute seientilic value of Faniday'* conception of an electrotonic c consists in its direelintf the mind to lay hoW of a certnin (jiuintity, on the changY'^ of which the a<'tuiil [ihcimmcna depend. Without a much gn»ter degree of devel(>|iment than Kiirailay jfavo it, this conticptinii di>e« not easily lend jtaelf to the explanation of (ho !ie»om<nta We .ihall return to this eiubjeet again in Art. 594. S4l.] A method which, in Faraday's hands, was far more powerful u tliat in which he makes use of those lines of magnetic force which were always in hie mind's eye when contemplating hia magoeta or electric currents, and the delineation of which by means of iron filinge he rightly regai-ded * as a most valuable aid to the xperiineataliet. Ftiraday looked on these lines as expressing, not only by their direction that of the magnetic force, but by their number and concentration the intensity of that force, and in his later re- se«rchc6t he shews how to conceive of unit lines of force, I have explained in various parts of this treatise the relation between the properties which Faraday recognised in the linejg of force and the matlieniatical conditions of electric and magnetic forces, and how Faraday'e notion of unit lines and of the number of lines within certain limits may be made mathematically precise. See Arte. 62, 404, 490. In the 6rst series of his ScManiei % he shows clearly how the direction of the current in a conducting cirouiti part of which ia BUCi ■ qiui W it. • e*p. ttm..sai. t IK 31«. : lb., lu 176 fAOSBTO-Br-ECTBIC ISDPCTIOK. moveable, dopendti on the mode in ivliich tlie moving parL cute tUrougl) the linen of tnagnetio force. In th« seeond Rones* he sbewa how the phenomena |>rodaMd by variation of the strength of a enrrent or a magnet may be explained, by enpposin^ the system of lines of force to exjKind from or contract towards the wire or magnet as its power rises or fallM. I am not certain with what degree of clearness he then held the doctrine afterwards so difilinetly laid down by himf, that the moving conductor, as it cuts the lines of force, sums up tJie action due to an area or section of the lines of force. This, however, appears no new view of the cose after the inTeetigrntions of the •ocond series J liave hcen taken into account, Tlic conception which Faniday had of the continuity of the lines of force prcchides the jirts^ibility of their suddenly startin;* into existence iii a place where there wore none before. If, tborcfor^H the number of lines which pusB throii;;h a conducting circuit ^^ made to vary, it can only he by the circuit moving iivroKs the lines of force, or elite by the line* of forre moving iicroos the rarcuit. In either i'usf a (.tirrent in generated in the circuit. The niimberortlieline«of'foree which nt any instaat patw through the cireuit is mathematttuilly e<iuivulent to Fanidfty'K earlier con- cci)tion of the eleetrotonic stat*.' of that circHil, and it is repreMrnted by the quantity .ffy. ^M It is only ainoe the definilioiw of eleetromotive force, Arta. 69. 274, and itii mefisuretnt-nt have I>»en made more precise, that we am enunciate completely the true law of magneto-ele«tric induction in the folh)wing terms :■ — The total electromotive force acting iwmd a circuit at any instant i» mi.-«sured by the rate of decrejise of the number of lines of magnetic forire which pu« through it. When integrated with respect to the lime this statement comes : — • The tinw-integral of Oie total electromotive force acting roui any circuit, together with the number of lines of miig^etic for which pass through tlie circuit, is a constaat qnantity. Instead of speaking of the number of lines of magnetic force, may speak of the magnetic induction through Ihu circuit, or ll BUrface>integtal of ma;[^ctic induction extended over any anrfiicc bounded by the circuit. ■ ■ £n. ifM^ S>8. t Ik, S0S3, 3037, 31U. ^H : lb, SIT, ftCL ^H HBLSnOLTZ ARD TnOHSOlT. I » I We dUall rt-tuni again to this method of Kanda/. In the mean time we must enumerate tlie theories of induction which are founds! on other considerations. Leas'* Law. 542.] In 1834, I^ienz* cuuueiatcd the followinf* remarkable niUtioo Ixrtwevn the ptieiiomcnii of the mechanical action of eloctiie current*. M defined by Ampferv's formula, and the induction of electric ciirrenls hy the n-Iiitivi- motion of conductors. An vurlicr utlcmpt at a statement of KU(th a rulaiion vim given by Ritchie in ill* PkUoMjiiieal Mugasine for January of iho Nimc year, but the direct ion of the induced current, viva in every cn«e stated wrongly. Ijtvx'* taw is as follows.— If a conttaxl current JfoKt in fhe primary circuit A, and i/\ bg tie notion of A, or of Ihf lecniidiirij/ circuit h, a current in iniluceiliu Jl,fie direelioH of thii induced curreitt tuUl be tac/i Hot, bjf its eleetromaguetie aeiioH on A, it fends to oppose tie retatice motion of the eircuita. On this law J. Neumann t founded bia mathematical theory of induction, in which he established the mathematical laws of the induced currents due to the motion of the primary or secondary conductor. He shewed that the quantity M, which we have called the |>ot«nt!a1 of the one circuit on tho other, is the same aa the eleclromaj>nctic potential of the one circuit on the other, which -we have already investi^tod in connexion with Ampere's formula. We may regard J. Neumann, therefore, as having completed for the induction of currents the mathemiktical treatment which Am[>£re applied to their meehanicul action. .] A ftcp of still greater Btientific important was soon nPtcr le by rielnihoitz in his Kitay on fie Conaervaliim oj Force ;. and Sir W. Thomson ^ working somewhat later, but iiidepeudently elRibollK. They nlH-tNX'd that thy induction of elect ric current* vcred by Faraday could be mathemattcally deduced from the ekctroougnctic actions discovered by Orstcd and Ampere by the appliration of tho principle of the CoiiBcrvation of Energy. Uelmholtx takes the cosc of a conducting circuit of reststanco R, in which an electiomotivo force A, arisb^ from a voltaic or thcrrao- • P(« , Jm. KiKi. leS (18811. t BaiiB Af«d.. Mih and tSI7. * K«ad Ului« Ui» fhyitaJ HneldT xf B^tIId, JuIj 3S, 1S47. TmuUtoa in Tft)ti>'> ' 8ciMitif{^ Mnmrin^* fttU ii. p. I14>. I Trim. lint. ia.. 184«. uia Fhit. Mtn.. Dm. ISSl. Sn klto LU jutpw on "IVaaiiimt Ehetrie ConcM*,' fkit Has-, IbCi- VOL. It. K 178 MaoiraTO-ELKTrRio isDtrcriOK, vl«ctric An-angomcnt, nets. The current in the circuit at any insUiDt i* /. He Ktipposes that u magnet is rn motion in t)i« n«ighbourliood ol' Uie cirouit, ivtid Umt it« potontial wiUi respect to the conductor \» V, bo tliat, during any xuiaII inUnal or tiioe dt, tho encr^- communicated to the magnet \>y the eleetromitgDetic action Thv work done in generating heat in llie circnit is, by Jouh law, Art. 242, 1-Jtdt, and the work Hjicnt by the eteotrontoti force /f, in ntaiutaiiiing the current / during Uie time dt, ia Aldt. Hence, eioce the total work done must be ecjual to the work spent. I ncc we find Uie intensity of the current rff di A- 1 = R Now the value of d may bo what wo pleoso. Let, therefor A = 0, and then . \dV ^^-li-di' or. there will be a current due to the motion of the ma^et, equal jzr to that due to an electromotive force ~-j7' The whole iudticcd current during tlw motion of tltv magD< from a place where its potential is T, to a place where it« potei IB a* IB ^r/r /^'''=-i/^'" = i<^'-^*)- antl therefore the total current is independent of (he velocity or tli« path of the magnet, and depends only on its initial and final positions. Ilelmholtz in hia original investigation adopted a system of units founded on tlie mcnsuremcnt of the heat generated in the conductor by the current. Con6i<Ienng the unit of current as arbitrary, tho unit of resistance is that of a conductor in which this unit carrcnt gnnentea unit of heat in unit of time. The ooit of electromotive force in (his Kviitem im (hat required to produce the unit of current in the oondiietor of unit resistance. The ndoptioo of this lyrtem of units nevcMitates the introduction into the equa- tions of a quantity a, which is the mcchauioal t-fjui viiU'nt of t unit of heat. As wc invariably adopt eitlicr the elect roxtutic WEBEB. 179 I the elcctromngnctic system of units, this factor docs not occur in tho <y]u»lions here giveu. 541.] Holinholtz also deduces the current of induction when n condtK'ting circuit and a circuit cArryiog a conittanl current are made to move relntivel}' to one another. Let It,, /^, bs the rcaist^nceH, /,, I^ the currents, v/,, J^ th« external electromotive furces, and F the potential of the one circuit OD the other doe to unit current in each, then we have, ae before. If wc suppose /| to be the primniy current, and /^ so much lens ftlian /|, that it does not by it« induction produce sny sensible -^ [•Iteration in /j, so that we may put /, = ^ then h = II, reiiult which may be interpreted exactly as in the case of the /,= A -T we siippouc /; to be the ]>rimary current, and /, to bu very Ifiiuch MUiullcr than /,, we get for /,, Tliis fhcwe that for eqtial currents the electromotive force of the ' first circuit on the second is equal to that of the second on the first, . whatever be the forms of the circuits. Hclmholtz does not in this memoir discuss the case of induction due to the strcngtbeninp or weakening of the primnn,- current, or the induction of a cnrrent on itself. Thomson * applied the same principle to the determination of the mcchaniml value of a current, ind pointed ont that when work is done by the niuluul action of two constant currents, their mechanical value is iuen-titxt by the nme amount, fc that the battery hax to supply tioulle that smonnt of work, in addition to tliat required to maintain tJie currents Bgainat the resistance of the circuiu f, 645.] The introduction, by W, \Vcber, of » system of absolute * MochiMkal TIifotj of Elprtnilnit. Phil. Um., Deo. 18S1. t NkUuT* <'ycioiHiAV>Vi «/ riiytlnil Scinet, A IMO, ArUi^ ' UsgoetUm. T>yiw K a 180 MAOSBTO-ELECTRIC INDUCTION. [545, unite for the meaBnremeut of electrical qoantitiee is one of the moEt important steps in the progress of the Eoience. Having already, in conjunction with Qauss, placed the measarement of magnetic quan- tities in the first rank of methods of precision, Weber proceeded in his Electrodynamic Metuuremenft not only to lay down sound principles for fixing the units to be employed, but to make de- terminations of particular electrical quantities in terms of these nnits, with a degree of accuracy previously unattempted. Both the electromsgaetic and the electrostatic systems of aoits owe theii development and practical application to these researches. Weber has also formed a general theory of electric action from which he deduces both electrostatic and electromagnetic force, and also the induction of electric currents. We shall consider this theory, with some of its more recent developments, in a separate chapter. See Art, 846. CHAPTER IV. OV THE IKDVCTION OP A CDBRBST ON ITSELF. >.J Fabaday )ifl£ devoted thv ninth G«ri(>e of his Resfarciet to the inv«8li»»tion of a class of phenomena exhibited by the current in a wire which forms the coil of an eh-clixiTiiagnet. Mr. Jeiikin had ob«crvLHl that, Blthoii{;h it Js impossiblv to prn- duoe a nenxiblc shock by the direct action of a voltaic system teonsiftliiig^ of only one pair of plat(4, yet, if the current is made lo poM through the coil of an elect romagnet, and if contact is then broken between the extremitieif »f two wires htld one in ' ach hand, a smart xhoek will be telt. No «ucb nhock is felt ou making: t contact. Faraday shewed that this and other phenomena, friiich he de- scribee, are due to the i>ame inductive action which he had ulrendy observed the cnrrent to exert on neig'hbouring eomlactors. In thin oaae, however, the indtictive action is rxurtcd on the same uouihiclor which carries the current, and it is so niueh the more powerful ad the wire it«elf i* nearer to the diOerent elements of the current thaa any other wire can he. ■ 647.] Ue obserres, however*, that ' the (IrMt thought that arises in the mind is that tlie electricity circulates with something like momentum or inertia in Uie wire.' Indeed, when we consider one partieuUir wire »nly, the phenomena are exactly analo^us to those of a pipe full of water (lowing in a continued stream. If while the stream la flowing we suddenly close the end of tlie tube, the iiHnnentam of the water produces a sudden pressure, which is mtieti greater than that due to the bead of water, and may ho sufGcicnt to burst the pipe, If the water has the means of escaping tbrongh a narrow jet • Etg. Jtm., Iffr. ^ 182 SELF-ITOOCTIOS. nhen the principal aperture is closed, it will be pR>jwt«d witb n velocity much ({reater than tbat iluu to th« head of wator, nnd if it can escape Uiroug:b a valvo into » chamber, it will do », even when the pressure in the chamber is greater thui that due to the bead of water. It is on this principle that tlie hydraulic rem \» constructed, by which » stRBll quantity of water may be niiiEicd to a great height by means of a lar|^ quantity flowing down from a much low«r level. 548.] Titoc ellWts of the inertia of the flnid in ilie tube depend solely on the quantity of fluid running throogh the tube, on its length, and on ita section in diOerent parts of its length. They do not doiH-nd on anytliing outride the tube, nor on the form into which the tube may be bent, provided its len^h remains tlie same. With a wire conveying a current this is not the case, for if a long wire is doubled on itself tlie effect ia very small, if tlie two parts are separated from each other it i* preater, if it IK coiled up into a helix it is still greater, and greatest of all i^ wlien so coiled, a piece of soft iron is placed inside the coil. j Again, if a second wire is coiled ap with the 6tsi, but insulated from it, then, if the second wire docs not form a closed circuit, the phenomena arc as before, but if the second wire forms a closed circuit, an induction current i« formed in the second wire, and the ciTects of nelf-indocliou in tlic first wtiv are retarded. 549.] Tlteie results shew dnHy that, if the pbeDomena arc dite to momentum, ihe momentum it ceHainly not that of the eleclrieityi in the wire, beeauw the same wire, cenvenng the nme current,' rxhibits eflecis which difler according to its form ; and erea when its form remains the Esme, the pneeence of other bodies, sncb we a piece of iron or a closed metallic ciieait, affects the romlt. 530.] It is difficult, bowerer, for the mind nhich has onea recognised the analogy Wtwren the pbenonena of telf-inductioii and thocc of the motion of material bodies, to abandon aitogcther the help of this analogy, or to admit that it ts mtiidy mpofieU and misleading. The fundamental dynamical idea of matter, ae capable by its auitiuo of becoming the reetptent of momentum anil of energy, b so interwoven with oar fonna of thoogfat that, whrn- rver we catch a gUmp*e of it in any part uf natiuv, we fed that a path is beforv m leading, wooer or l^er. to tha complete under- vtandiaf of the aal(i«et. BLECTSOKlilOTIC EyHROT. [83 H 551,] In the case of the elwtric carrent, we fiod that, when the eleciromotiTC force begins to act, it does not at once produce t!ie ^_full current, btit tbut thu current rites ^nduitUy. What is the ^ntlcctromotive force doicig (luring- the Lime thnt the oppo&iujtr ^- ^Bsistancc is not able to Iialunce it? It is incn^ming the electric ^kca Trent. ^g Now an ordinary force, acting on n body in the direction of it6 motion, increase* it» momentum, and comQiuuieiites to it kinetic » energy, or the power of doin^ work on account of its motion. In like manner the unresisted part of tbe electronKitive force hsia lieen employi,-d in inorea^ing^ thfi electric current. Has the elcctrio current, when tlnui ]irmluoed, either momentum or kinetic energy ? We have already ahewn that it has something very like mo- ^knentum, that it resists being suddt^nly stopped, and that it can " exert, for a nhort time, a great electromotive Ibice. But a conducting circuit in which a earreut has been set up has the power of doing work in virtue of tlits current^ and this power cannot be said to be Bometliing very like energy, for it i^j« really and truly energy. ^B Thus, if the current be left to itself, it will continue to circulate till it is stopped by the resietanec of the circuit. Before it in stopped, liowever, it will have generated » certain ijuantity of heat, and the amount of this heat in dynamical measure i* equal to the energy originally existing in the current. Again, when the current i« left to itself, it may be made to do mechanical work by moving magnets, and thu inductive effeet of these motions will, by Lonz's law, stop the current eooner than tbe resistance of the circuit alone would have stopped it. In this way part of the energy of th« current may he transformed into mechanical work inst«ad of heat. 552.] It appears, therefore, that a system containing an electric current is a seat of eoerg)- of some kind ; and since we can form ^■no conception of an electric current except as a kinetic pheno> ^^menon*, its energy must be kinetic energy, that is to say, the ^^ energy which a moving Iwdy has in virtue of its motion. ^P We have alrcAdy shewn tliat the elfctricity in the wire cannot fiT»e considci-ed ns the moving body in which we art^ to find this energy, for the energy of a moving body doi-s not dei>cnd on anything external to ititclf, whoreas U»e pK«enec of oUier bodie« near the cuircnt alters its energy. 11^ • ITjkntUjr, Sjrp. Bo. {iSi}. tSRhT-IVDVCnOTS. We are th«rerore led to enquire whetber tber« may Dot be some motion jKoin^ oa in the space ontaide the vrire, wliich is not occnpic hy the electric current, but in which the electrom»gi>ctic effects the current are manifested. I Kliall nnt lit present enter on the reasons for looking in onal place mtlit^r tliiin another for such motions, or for regarding th< ■DOtiotiH tut of one kind rather than anotJier. What I propose now to do is to examine the oon»eqaenc«a of tlic asaamption that the phenomena of the electric current are those of » moving system, the motion bein^ communicated from one part ttf the system to another by forces, the nature and laws of whiol^l we do not yet even attempt to define, because we can eliminate™ these forces from the tsjuations of motion by the method given by Lagrange for any connected system. In tjie next five chapters of this treatise I propose to dcdti' tile main strticture of the theory of t-lcctricity from a dynamical hypotlicsis of this kind, insind of following the path which has led Weber ami othor iuv<i«tignU>rs to many rumarkahle di««ovcri and vxperintent^i, nud to conct-ptions, some of which are aa bciiulii'i u tbey are bold. 1 hare chosen this method because I wixh to ■how that there are other ways of viewing the plienomena wh appear to me more sati^bctory, and at the xanie time are moi consistent with the methodit followed in the preceding partjt of thi| book than those which proceed on the hypothous of direct acU' at a distance. 4 »lfl CHAPTER V. OK THK EQUATIONS OP MOTION OF A CONSECTKD SYSTEM. i 653.] Ik tlip fourth section of tlie second part of bi« JSfAattljue daalyltqnc, Lagran^ has ffiven a method of reducing the ordi'nury dynmniral equations of tlie motion of fbo parts of a connccti-d ij9t«m to a number equal to that of the degrees of freedom of be ^if«m, Tbe c<)uations of motion of a connected sj'stem have been (riven in a diirerent form by Hamilton, and have k-d to a great cxt«.>usioQ ^^of tbe hightT pnrt of [iiiro dj-naroics*. As »c iibrtll find it neecHwry, in our cndoavonrs to bring eleetrieal phenomena within the province of dynumicii, to haro our dynamical idnnn in a stxit* fit for direct nppbeation to physical qiiestioos, we ■halt devote this chapter to an exposition of those dynamical ideas from a phmcal point of view. 554.] The aim of liUgi-nn^ waa to bring dynamics under the power of the ealunluK. lie brgnn by csprcssiug the elementary dynamical relation.i in ti-rniN of the corrcKponding relutionit of pure algebniienl ipiantities, luid from the equation* thus obtained be diduct^ hi.i final e<)iiaUon« by a purely nlgebniical proeess. Certain i)uantiti4>» (exprwwing the reat-tionv between the parts of the .*y«t«m ^■Called into play by '\i» phyitioni connexione) appear in the eijtiations ^^f motion of the component parts of the svotcm, and Lagrange's investigation, bm mxju from a inatiieniatical point of view, i« a method of eliminating tkeae (|uantitie)i from tbe final cquatiou.t. In following the stepii of lliiH tilimituttion tbe mind is cxerei-tMl in calculation, and should therefore hi? kept free from the intrusion or dynamical xA^as. Oar aim, on tbe other hand, is to cultivate . • See VtvXrmar't Cajl^'i ' Ueoan on ThdONtical Djnunic*.* BrtliA Auotiittiva, ■SAT i and TbooiMti Mid lUt'* AiMunt FUloioplkf. m KISETIC3. onr dynamical ideas. We therefore avail ouraiJves of tlie labour of the mathematiciani), and retranslate their rcfultti Troin the Ian-* ^iig« of the calculus into the langiuf^ of dynamtcit, »o tliitt our words m^ call up the mental image, not of some algcbnucat process, but of some property of moving bodies. The language of dynamics has been considerably eactftnded bjr those who have expounded in popular t«nns the doctrine of the Conservation of Energ}-, and it will be seen that much of tfa« following statement is su^^sted by the investigation in Titoraaon and Tait's Natural PiUosofAj/, especially the method of beginnin g , with the theory of impnlaive forces, ^H I have a|>ptied this method so as to avoid the explicit oon-^1 sidcration of the motion of any part of the syetcm except the coordinates or variables, on nhich the motion of the whole depends. It is doubtless important that the student should be able to trace the connexion of the motion of each part of the system with tJiat of the variables, but it is by no means necessary to do this >^H the process of obtaining the final equations, which are independent^^ of the particular form of these connexions. J to u uiese 1 itself^^ Tie Fariallet. 555.] The number of degrees of freedom of n system ts number of data which mast be given in order completely to detirrminc it« position. Dilferent forms may be given to these data, but thvir number depends on the nature of the xyHtem itself and cannot be altered. To Gs our ideas wc may conceive the system connected by of suitable mechanism with a nnmbvr of moveable pieee», caeb capable of motion along a straight line, and of no other kind of motion. The imaginary mechanism which coimeda each of th pieces with the vystem must be conceived to Iw free from frictioi destitote of inertia, and iucajathle of being strained by the artJo: of the applied foroos. The use of this roecliauiitm is merely ' Bssbt the imagination in ascribing iwwitiou, velocity, and moinentn to what appear, in Lagrange 'm investigation, as pure algi-braical quantities. Let ^denote the position of one of the moveable pieces a» define by its distance from a fixed point in ila line of motion. We k1: <li^inguish the values of ; corresponding to the diflercnt pi< bv the suffixes ., ,, &c. When we are dealing with a set !■ 3> quantities belonging to one piece only wc may omit the suCBz. ss* iMPFiJtE AND MomnrrcM. ^P When the taIu^s of all the varinbles (</) are given, the position ~ of each of the moveable pieces is known, and, in virtue of the ima^innry mcvlioiUBiii, the coufiguratioa of the entire ayatem is determined. H Jie VeloeifitB. H 556.] During tho motion of the system the configuration changes " in «>nK' tU-finiUr miinner. and BJncc the confiiiHration iit each instant I i« fully d<^riii>-<l \>y till- values of the varinhk'S [q), the velocity of ^■«very part of the eystem, ns well ns its confii^uration, will he com- " pl«t«ly defined if we know the values of the rariablea {q), together with tJieir veiocitiM (^, or, according to Newton's notation,^) ■ I t t The Force*. 537.] By a proper regulation of the motion of the variables, any motion of tlio Ky«U>m, consistent with the nature of the connexions, nuiy ho produced. Id order to produce thiM motion by moving lie variable pieoen, forces must he apiilied to these pieees. We shall denote the force which must he applied to any variable J, by F,. The system of forees {F) ia meehanically equivalent (in virtue of the connexions of the system) to the system of forces, whatever it may be, which really produces the motion. (The Momenta. 558.] When a body mores in such a way that its configuration, 0^ respect to the foroo which acts on it, remains always the same, BR, for iiiKtam^, in tJiu cut^e of n force acting on a single particle in the line of ito motion,) the moving furce is measured by the rato ^br tncreoKC of the momentum. If F tg the moving force, and p the f'lnomcnttun. F = dp 4i vncnee =/ Pdi. The time^integrat of a force is called the Impulw of the force ; ao that we may assert that the momentum is the impulse of the force which would bring the body from a state of rest into the given Btat« of motion. In the caw of a connected system in motion, the configuration ix continually changing; at a rale depending on the v«locitic» (^), to KINBTTCa. th»t vrv ean no longer astmnie thitt tlie mom^-ntuin iii the time- integral of the force whioli acts on it. Dul tlio increment 2} of any vari&ble cannot be greater thin ^it, wtierc hi in tite time duriog nhich the increment takes place, and ^ is the greatest value of the velocit}' during that time. In the cue of a system moving from rest under the action of forces ilnraj^J in the same direction, this is evidently the tiDal velocity. ^M If tbe final veloeity and configumtion of the system are given, we may conceive the velocity to be communicated to the system io a very small time tt, the origioal coaiij^uration diflcring from the final conOgnration by quantities 8 j,, i^^, kc, which aro leas than ^,B^ j,&', &c,, respectively. The smaller we suppose the iucrement of time it, the grenbeT must be the impressed forces, but the timc^integnil. or impulse, of each force will remain finit«. The limiting value of the iinpitliw, when the time is diminished and ultimately vantahos, is <Ii;linvd as tbe inatantaticoKi impulse, and the momentiun p, corrcfponding to any variable q, is dcGucd nit the impulse corresponding to that variable, when the system is brought instantaneously from a state of rest into the given state of motion. This conception, that the momenta are capable of being produood by instantaneous impulses on the systMu at rest, is introduced only as a method of defining the magnitude of the momenta, for the momenta of (he system depend only on the tnstantancotu stnt*- of motion of the system, and not on tlie prooe^ by which that atate was produced. In a oonncet«d syrtcm the momentum corresponding to anf variable is in geneml a linear function of tJie velocities of all the varinbicK, instead of being, as in the dynamics of a particle, simply proportional to the velocity. The impulses required to change the velocities of the system suddenly from ^p j,. &e. to yi', j/, &c. are evidently equal to Pi'—pi, fia~Pt' ^^^ changes of momentum of tho scversi variablos. Wori done by a Small ImpuUe. B30.] The work done by the force F^ during the impulse is Iha ftpac«-intcgral of tlie force, or =jP^i^4t. 56ol mcREirauT OP KXTxvnc enrhqt. If ^,' U the givatesi and ^," tbe l«st value of tbo velocity ji luring tbe action of tbo force, ^' must be less tban ad greater tbaa 4i"J- Fdt or Fit or % If we Doiv Etippose the impulse / ¥df lo be diminii^ed without limit, the values of ^,' and ^," will approach and ultimateiy coiucide with that of <y,, and we nay write /»i'— jo, = ijoj, so tbat tie work done is ultimately g ^-^ = ?, 8ft . ir, W« »ey>*-it ifoii* fiy a triy *OTaW impiihe it uUimaittjf (At prvduel qftic mjinltt and the veiocily. ^M iHerement ^ tAe Kinetic Energy. ^ 560.] When work ie done in settiuj* a conservative sj-stcm in jnotioii, energy ii; coramimicuted to it, iind the system hccomes cupshlt! of doing an eqmd iimonnt of work ugainst resistanoes L'bcfore it is reduced to rest. The energy which n syst«Ri poE6«'ssGH in virtue of its motion cullitd its Kinetic Encrg}', and is communicated to it in tlie form |of ihe work dontr hy the forces which Bct it in motion. If T Vmj thu kinetic eni-rgj' of the system, and if it hecomei ^+ir, on account of the action of an inliniteKimnl iinpulte who»o [eomponenttt are hjp^, ip^, &e., the increment hT muit he tlie Mum of the <]aantilica of work done hy the components of the impulse, >r in symbols, ^j.^ ^,ip,^j,bp,+ko., = 2{?M- 0) The instantaneous state of the system is completely defined if the variables and tbe momenta ar« given. Hence tbo kinetic energy, which depends on the iastantaoeous state of the system, CIO he ex])ressed in tenns of the variables (^), and tbe momenta (p). This is the mode of expreesing T inti-oduced hy Hamilton. When T is expressed in this way we shall distinguish it by the suffix ,, thus. r,. lite complete variation of T^ is »n = S(^'e;>)HS(J'5,). dp ^dq (2) 1 90 KIKBTIC8. Tltc Ust term mny be written = (S*»0' [561- 4 which dimiDiBh^ with 6 1, and uUimately vanishes with it when ih^ impulee becomes iDBtantaneoiis. Hence, cqoating the cottfiicients of bp la equations (1) and (s1 wre obtain rfT", ,. or, tie veioeily eorretpondit^ fo fAe tnricMe j it tie different efitjicient of T^ wi/i retpevt to tAe carr^tponding tnomenfum p. We hare arrived at this result by the consideration nf im]iuln« forcce. By this method we have avoided the oon.tideration of tt chan^ of configuration during the aefion of the forces. But th^ instantaneous state of the system is in all respects tlie same, whctli< the ^atem was brought from a state of rest to the gircD at of motion by the transient application of impulsive force*, wbctlier it arrived at that state in any manner, however gradual In other words, tlw variables, and the corresponding velociti4 and momenta, depend on the actual state of motion of the system it the given instaut, and not on its previous history. ^A Hence, the equation (3) is equally valid, whether the atate <^| motion of the syst«m is supposed doe to impulsive foroe^ or t« forces aeting in any maniter w1uite%'er. We may now therefore dismiss the consideration of impulsive forces, together with the limitationii imposed on their tJm« actioD, and on the chnngn of coufiguratton during their action. HamiliiM't Eqnafiont ^ Motto*, &61.] We have already shewn that LiCt the system move in any arbitrary way, subject to the ditious imposed by it« connexions, then the variations of^ and q are rf/. hp=-H, hq^qht Henoe Hi dT. (») <*?. «^=:f*H di if dt '?. 562-3 BAUILTOn's BQITATtOXS. 191 and the complete variation of T^ is '^- = M^*^ + 5»0- =^((: ,ip (n h~ ~ \'<d( ^ dq at the increment of ihc kinetic energy arises from Uio work Idoae by Uie impreBsed forces, or hT, = S(Fh^). (8) In these two espreesions the Tariaticn» ig are all independent of eooh oilier, so that we are entitled to c<|uatc the cocffioicnts of each of them iu the two expressions (7) and (;g). Wo thuit obtain ' '. = #^'^' (») where the momentum p, and th« forci; F, belong to the variahle ^,, There are as many equations of thin form as there are variablcv. These eqnations were given by Ilnmilton. They shew that the force corresponding to any variable is the eum of two partsi. The first part is the rate of increase of the momentum of that Vkriuble with respect to the time. The second piirt is the rate of increuK; it the kinetic energy per unit of increment of the variable, ths other variables and all the momenta bciog constant. Tie Kinetic Sneryy cxj/rrxted in Tenas of the Mom^nla and I'tiacilien. 562.] Let p^, Pg, &c. be the momenta, and j,, y,, &c. the rdoeitiea at a given instant, and lot i\. p^, &c., q,, q^, &c. be Ftnother system of momenta and velocitiex, itueh that Pi =«Pu qi = «y,.&«. (10) It i)> manifeit that the systems p, <[ will be consistent with each other if the systems p, q are so. H Now let « vary by In, The work done by the force F^ is I F^h<\^=i^^h■[>^ = 4^p,nhn. (IIJ 1^ Let n incnase from to 1, then the system ts brought fmm A itate of rest into the stotc of motion (^qp), and the whole work eipOMled in prodncing tliis motion is Bnt /'*"'" = 1> aiiil tli« work fpciit in producing the motion is eqairalent to tlv kiuetic cni»igy. Ilcnco wUero 7^ cIeDot«« the kinetic cnei^ expressed in terms nf tJiu motnmtu and velocities, llic \-anab)cs ;, , ;,, &g. do nofc enter inl') UiiK cxjireftiion. 'Die kinetic enorjfjr is tlierefore half tJie sum of ibo product* of till' mnmi'-nta into their corresponding velocities. When the kinetio energy is expressed in this way we shall denotf it hy UiL> symbol T^. It Is a function of the momeota aod vr)n- cities only, and docs not involve the vartalilcs thetnsclvett. DGfl.] Tht'i-o is H third method of expr(>!i!iiDg the kinetic energy, which \» generally, indeed, regiinhtl lui the fundaunental one. By wJving the (K{tutionB (3) w« may cxprtM the momenta in terins ofthe velooitiea^and thfn, introducing theM vnlne* in (13), wv Khali have au expTvoaion fur T invoK-ing only tlie velocities and the nriahles. When T m expr^j^ in this form vt shall indicate it hf tlM aymhol T^. This is the form in which the kinetic energy b Mtprtncd in the equations of lAgrang^ SGi] It is roanifftti that, sim-e T^, 7||, and T^ are three diflerfjit wrpnwions for the same thing, or i;+'*-Aft-Af,-ftc-o. (H) titwt. if all the tjiiaatitira ^, f, ud ^ vary. Ttw wriMtKws A^ »n net t^ •» tiat wv cKBnol at vac* wriatim in IW njwbiw W kva. IV wi — i»>> f VMnAana tf »lUl wv tM^ttf •^MtMhgt* B«t w« knov. boa (15) If ud ^*<u t«) mobakoe's equations. 193 or, tie eompweiti* ^ mcmenluim are t^ diffe/entiat coej^ienit <^ T^ with resfiect to He corresj^ndimg vticeiliet, Agaiiij by equating to i«ro tli« ooefficients of S^i, &c., dT, dT. „ ^1 'hi (18) or, tie differential ccejficieat of the kinetic enrrgi/ witA retpecf to any tarialte 7, M equal in ma/^niiuile but opposite in lign when T it atpreued at a Jnnetion of He velociliea iniiead <if a« a /imctictt of tie momenta. In virtnc ofequktion (18) wc may write the equktion of motion (9), rf/., dT^ ^'--dT'df,' ' dl dq^ dq^ (19) (20) v ^■wbJch is the fonn in whicb tbe equations of motion wore gireo by HliBgrange. ^^ ^^-] ^° ^B procedioff iDvestigntion we have avoided the con- eideratiou of the form of tho function which expresses th« kinetic (enerjry in terms cither of the \-clocities or of the momenta. The only explicit form which we have assigned to it is in which it is expressed as half tho mim of the produottt of the momenta each into its corresponding velocity. We may expresa the v«Iocttien in terms of the difTcrential co^ ofiicients of 7*, with iesi>ect to the momenta, as in equation (3), n=*(/.S4.p/^..&c.) (22) This shews thut T^ is a homogeneous function of the second ' of the momenta /J, ,/^, &c. We may also txprcM the moinenUi in terms of T^, and wc find ^-=*o/4';V4f^H (23) [which shews that 7|| is a homogeneous function of the second degree nth rrspcct to the velocities q^, ;,, &c. If wc write P., for 44^ , &«. P„ for -^-i. u ^1 YOI.. II. For - j ~ / , &c. i CnfBTICS. then, siooe both T^ and T^ are functionE of tlie second dc{;re« ^ and of p respect! vol}*, both the P'e :ind tbo Q'e will bv funrtion* of the variables q only, and indcp-ndcnt of the Tclocitii;* uotl t tnomenta. We thus obtain the pxprcsdous for T, ^ {2i) (26) llic motrwDt* are exprosiwd in terms -of the velocities by linmri-imtioM ^, = />„,-, + p,^^^+&c., ('^ and the relocitJes aru expressed in t«nua of the (oomcnta hy linear equations ^^ ^ Q„;..+ fi^+fcc. (2 In treatises on the dynumin of a r!(>id body, the oovSivieDi «orrvKpiindt»g ta P„, iu whioh the HuS!s«(i tkn tlto same, arc called MoiDCiitH of Inertia, and Uio»o corri'»iioiidin{r to P^, in whii-li the Kiiffixeit are ditltTeut, arc called I'roiinetM of Inertia. \Ye may extend lliese names to the more gentinil problvin which in now before u^, in which these quantities are not, as in the cdk of a rigiil I'l-xly, absolute constants, but art; fuuetions of the vari In like n)iiim<T we may call the «oe(Bcknt« of the fonD Momenta of Mobility, and thote of the form 9,j, Product* Mobility. It w not oAen, however, that wc shall haw occowion to H[)fal; of the coefficients of mobility. S6(>.] TLie kinetic energy of the system is a quantity msential po«itive or zero. Hence, whether it be expnwsed in terms of TOlocitiw, or in terms of the momenta, the (Mxrfficit'nts mast vnch that no rwd valnes of the variableif can make T ncgativv. There are thus a wt of ncceKKury conditions which tlic rnlom the eooflicicnt« /' must xatisfy. TheM! conditions are a« follows: Thu (jnantities /'„, i'^, &c. must nil be pointivc. ! 01 a ■iJjlM Tbti M — 1 detemiinantfl formed jninant iu mieccvsion from tho detci^ 111 u> Pi. a- »*• 13,. ' 3i' p.. ■ A. A- A by tlie omission of terms with suffix l.dien of terma with either or 2 in tbeir suSii, and bo on, must< all be positive. The number of conditiotts for a variables is therefore 3 b— 1. 567-1 irOUESTS IKD PBODDCTS OF ISBBTIA. 195 I The cocfDrlrnls Q arc 8ul>j<>ct to oonditioDs of the Rime titid. 667.] In tliia outline of Iltf runcluniontnl principles of ihe Ay- imic* of a oonnrt'ted nj-sU-m, we Iwvo kept out of vi«w the ecliaiiiiim by which the pari« of tlw syrtt^m arc connected. We itav« not even written doivn a »ei of eijimtions to indicate how this motion of any part of the xystem depends on the variation of the variables We have confined our attoution to the vnriahlce, their Telocities and momenta, and the forono whidi act on the pieoes representing the variables. Our only a9«iim|)tioiiK arc, that llie connexions of the system are eueh that the time in not explicitly contuined in the equations of condition, and that the principle of the conservation of energy is applicable to the system. Such a description of the metbo«U of pure dynamics is not nn- oeoeesary, because LajirrRDge and most of his followers, to whom we are indebted for tbeHc methods, have in general cou6ned tbem> elves to a demoDstnition of them, and, is order to devote their Attention to the symbols before them, they have endeavoured to all ideas except thoise of pore qnantity, so as not only to 'dispense with diagrams, but even to get rid of the ideas of velocity, momentum, and energy, after they have been once for all eop- planted by symbols in the original equations. In order to be able to refer to Uie results of this analysis in ordinary dynamical hin- II g uage, we have endeavoured to retranslate the principal equntions ^fef tlte method into lan^a^ which may be intelligible without the ^T»e of symliols. Ah the development of the ideas and methods of pore mathe- matics has rendered it possible, by forming a uatbemntieal thcoiy of dynamics, to bring to light many truths which could not have been diseovered withotit mathematical training, eo. if vk are to fofin dynamieal Uteorica of other sciences, we must have our minds unboed with these dynamical truOis as well as with mathematical Kpetbods. la forming the ideas and words relating to any science, which, ike electricity, deals witli forces and their effects, we must keep ^^noatantly in mind the idea* appropriate to the fundamental Kienoe ^■f dynamics, ao that we may, during the first development of the ^Bcienoe, avoid inconiiHleiicy with u-hat is already establish itl, and ^^ftlio that when our views become clearer, the tankage we have adopted may be a help to us and not a liindrance. CIIAPTEE VI. DYNAMICAL THEORY OF ELECTBOMXGNETISM. 568.] We liavo shewn, in Art. GS3, lliat, when an elcctrio omrenl Kiiets in a conducting; circuit, it Iihs ii i.-apacity for doin;^ b cei lUDOunt of mvchniiical work, unil this imicjiendeiitly of any cxtc elMtroinolivv foroc muintuiiiin;; tlie ctirrent. Now cakpocity ft performin;; work i^ nothing eivt; thun onorgy, in wbateTer wa] it arises, and all eactgy is the same in kind, however it may difler in fonu. The energy of an electric current is either of that form which eonsi«ta in t!ie aotual motion of matter, or of that which consist* in the capacity for being set in motion, arising from forces ■ding between bodies placed in certain positions relative to each other. Tlie first kind of energy, that of motion, is caJled Kinetic eoer^, And when once understood it appears so Ibndaineiita] a fact of nature that we can hardly conceive the poutibility of resolving it into anything elne. The second kind of energy, that depending on po-iition, is called Potential energy, and is due to the action of what we call foccos, that is to eay, tendencies towards change of relativo position. With re*pcct to these forces, tbougli we may aoocpt their vxiElcnce as a diinoostrated fact, yet we always feel that every explanation of the mechanism by which bodies are set in motion forms a rent addition to oar knowlcd^. 569.] The electric current cannot bo conceived except as a kineii phenomenon. Even Faraday, who constantly endeavoured to ei cipkle hii( mirul from (ho influonc« of those suggestioDS which words 'electric current' nnd 'electric fluid' ace too apt to carr witli them, speaks of tl>c electric current as * something prc^ivssi' and Dot a mere arrangement.' * • £ip. Am., 2SI. ineii^l nuu^l h th" KINETIC ENEROT. 197 H The eBecis of the current such as eIi>otroly«i8, and the translet of electrification from one body to anothiT, ur« all progressive actions which require time for their accom^ilishment, and xre there* fore of the nature of motion§. H As to the velocity of the current, we liiive shown that we know HliothinK' shout it, it m»y be the t«ntb of an inch in nn hour, or ^K^undrvd thousand miles in a second *. So f«r arc we from HScnowtn;* its absolute value in any case, that we do not even know whether what we call the positive diiectioD is the actual direction of the motion or the reverse. But all that wo assume here is that the electric current involvCM 1 of some kind. Tliat which is the cause of electric ciirrentu I'lMb Mllcd Electromotive Force. This name has* long been n»d with frrcat advanta]>e, and has never led to any inconsistency in the IaTi!;uft(ji' of science. Electromotive force is always to be •understood to act on electricity only, not on the bodies in which bbe electricity lesidee. It is never to be confounded with ordinary mechanical force, which acts on bodies only, not on the electricity in them. If wc ever come to know the formal relation between electricity and ordinary matter, we shall probably also know the fetation between cl<,-ctromottvo force and ordinary force. 570.] When ordinary force acts on a body, and when the body yields to the force, the work done by the force is mco^tired by the product of the force into the amount by which the body yields. BThus, in the caso of water forced through a pi|>c, tlic work done Bat any section is nwisunH) by the Quid pressure iit the «cvtioa ntuUipIted into the quantity of water which crosses the xoction, In the sime way the work done by an clcetromotivc force is _faeaaure<l hy the product of the electromotive force into the quantity ^f electricity which croxxes a section of the conductor under the 'action of the cleclromotive force. The work done hy an electromotive force iM of exactly llie same kind as the work done by an ordinary force, and both are meamred hy the same Ktondards or units. Part of the work done by an eh-ctromotive force acting on a conducting circuit is spent in overcoming the resistance of the circuit, and Ihic part of the work ib thereby converted into heat. Another part of the work is spent in producing tJie electromag- netic phenomena observed by Ampere, in which conductors are modo to move by electromagnetic forces. The rest of the work • £fr- Hf-. >«ie- 198 ELECTROKlSFTtCS. it spent in incmsing tlm kiD«U« energy of the carrent. and efteats of this [)art of th« action are ahewn in tbe pbenomoui of L indoction of cuminbi whwrved by Faraday. Wc thi'irforv know ctKXigb about electric coirents to in a sTEbptn of outerial conductors carrying currents, a dynami system which i* the eeat of eaetgy, part of which may be kin nnJ part {totcntial. Ilic Datura of th« connexions of the parts of tliis system nnknown to tw, but an wc have dynamical methods of investigation which do not require a knowledge of the mechanism of the i^atcm, we shall apply them to this case. We shall Grst examine the consequences of ossnmin^ the Ui ^neral form for the function which expresses the kinetic eaergy the systcni. 571.] Let the system oottsist of a number of conducting circuits^ the form aDd position of whidi arc ilctmnineal by the raluea a system of vflriabU>s ar, , r,, &c-, the nombcr of which is equal to the number of drgrvcs of fRHtlom of the system. If the whole kiiu-tic energy of the systi>m were llial doe to tbi motion of tbcso conductors, it would be expressMl in the form r = 1 (*, «,) i,' + &c. + (x^ jg xj ^j 4 &c., wlicre the symbols (a*, Xi), &c. denote tlio quantities which we ha' oalled moments of inertia, and (2, j-J, kc denote the pro<luct« of inertia. If X' is the impmsod force, tcndin^f to increase the coordinate whkb i» m|uircd to produeo the actual motion, then, by Lagrangv' equation, ^H_il— r' (/( di dx ~ When T denotes the energy' due to the visible motion only, w« shall indicate it by the su0ix ^i thux, T^. But in a Kyittcm of conductors carrying «)i-ctric currents, part the kinetic energy is due to the existenec of Oicw currents. the motion of the electricity, and of anything wliuso motitm goTemed by that of the electricity, b« detcrmim-d by another of coordinates jr, , y^, &c., then T will be a homo^uwius Uinctioi of squares and products of all the velocritics of the two «ct« of coordinate*. We may therefore divide 7' into three portions, in tJ: first of which, T^, the veli:idlies of the CMirdiiuilcs « only ocei white in the second, T„ the velocities of the ooortUnates j onl;. oceur, and in the third, T^, each term eoutains the pnMlitel of the ^locities of two ooortlinutes of which ouu is x and the other >. Jl of , I of I J BLECTHOMAONKTIC FORCE. 109 We have therefor« T = T +T i-TH,, n, where 7, = i(»,a!,) i,* + &c.+(a,jEjjf,jrj+&c, B ?'* = ('.5'i)j^yi+&«- ^ ST^-] I" tbp CPiieral ilvnaiiiiciil tUeon', tho cneffioienU of every I t<Tm msy be functions of all tlie coordinatcis, both j: and y. Id ^kthfl case of electric currents, however, it is easy to see that the coordinates of the cln« y do not enter into the coeflicienta. ^_ For, if all the electric ciirrentii are mnintiiin«d consliuit, and the ^poondiictors at mt, the whole state of the field will remain constant. Bnt in this ease the coordinates y are variable, tliough the velocities jr arc constant. Hence the coordinates y cannot enter into the expression for T, or into any other exjirMuiou of what actually takes place. Besides this, in virtoB of the equation of oontinnity, if the con- ductors are of the nature of linear circuits, only one variable is Pe(|uired to express the strength of the current in each conductor, jet the velocitieA y, , ^,, &c. represent the strm^hs of the currents n the several eonduetors. All this would 1>e true, if, inatoad of electric currents, we had currents of an inoonijirestiible fluid running in flexible tube*;. In tills CTJsc the velocities of tliwc eiim-nto woidd enter into the expre«i(ion for T, but the coefficients would dcjwnd only on the vanablr« i, which detenninc the form and position of tlic tubes. In Ihe case of the fluid, the motion of the fluid iu one tube does not directly HlTeet that of any other ttibe, or of the llnid in it. Hence, in the value of 7"^, only the xquares of the velocities^, and not their products, oocur, and in 7'm„ any velocity y is associated ouly with those velocities of the form i^ which belong to its own tul>e. In the case of electrical currents we know that this restriction does not hold> for (ho currents in different eircnits act on each other. Ilt'Oce we most admit the existence of terms involving- products of the form ^i.r,, and this invotvt« the existence of something in SH>tion, whose motion depends on the strength of both electric cartmts ^, and y,. This movinff niatler, whatever it is, is not confini<<I to the interior of the condiM^tors currying the two current* but prol»I)ly extcnils thnHighout the wImIc space surronndinf; them. S7$.] Let us next consider tlie form which I^agraoge's equations of tnotiott a«sume in this cose, het X' be the impressad force 200 I0K1HBTICB. [573- corr«]}onding to tlie coordinate x, one of those which determine the form and position of the coinln<?tin;y drcuits. Tina ia a force in the ordinary ^nse, a tendency uivards change of |>oeitioD. It it> &:iveti by Uie equation ' lit dx ^ We may consider this force as the stun of three parte, corte- Bpondin^ to tbe three parts into which we divided the kinetic energy of tbe system, and we may distingaieh them by tbe aame suffixes. Thofl i'= JC'^ ^ X',+X'^ . The part A"„ is that whieh depends on ordinary dynamical con- nderations, and we need tint attend to it. Since 7*, does not contain x, the fint term of the expreciioil for X', is xero, and its value ia reduced to * dx This i« the expression for the raechanica] force which mnst be applied to a conductor to liabinoc tbe elect romngnetie force, and it imertit that it is measured by Ihe rate of dimtnatitm of the purely elect roki net ic energy duo to the variation of tbe coordinate «. Tbe electromagnetic force, X^, which bring« this extcnial mi-clianical force into play, is c<]iial and opposite to A"^ , and is therefore mejMured by (he rate of encreate of the electrokinctio energy corre«pontling to an increase of tbe coordinate x. Tbe value of .V^, »ince it depends on squares and produeta of the currents, reuairu the same if we reverse the directions of all the eanenls. The third part of AT' is I " <ff "35" dig The qaantity T^i contains only products of the form x^, so that ■ j^ » a linear function of tbe sf lengths of tbe currenta j>. TTie first term, therefore, depends on the rate of variation of the elrcni;ths of the currents, and iodicntcs a mechanical force on the conductor, which is zero when the ctirrents are constant, and whieb is positive or negative according as the currents are in- creasing or d<.'Cii-a»ing in strength. Tbe second tt^rm depends, not on the variation of the canents, but ou their actoal strcngtlm. As it is a linear IVinction with respect to these currents, il chiing<-s si-^n wbi-n the cnm-ntj? chnng rign. Since every term involvs » v.!"citr i, it is zero when 5 74-] WAS AN ET-ECTBTC CTRllimT TOIJE MOMKNTCM ? 201 f »: coitductors are at re*t. Then; iirc aUo tvmiH arising; from the tunc ilT variations of the coefficients of/ in -y™ : these remarks apply tiso to t)i«n). Wc may th«Teforc iiivp«t.ig»tc Ihcse t*nat Beparately. If the conductora are »t rt'itt, we bnvc only the first term to deal with, f the currents uro condtitnt, «c liuve only tlie second. 574.] A* it IK of groat imporlnnce to dftorniine whether any part of the kini-ltP rtiergy in of the form ?*,«, eonnijiting- of products of or- dinary velocities and strt-nf^hs of vlcetric currents, it is desimhle that ex[icriinciitM should W made on thin «ul}ject with ^reat cnre. The dctenni nation of the forces noting on bodies in rapid motion ia diffionlt. Let u* therefore attend to th<^ lirnt term, which depends on the variation of the strength of the current. If any part of tlio kinetic energy depends on the product of •n ordinary velocity and the strength of a current, it will pri.lmbly be moBt easily oI>- Mrved when tlic velocity and the current are in the same or in uppo»ite dinctions. We therefore lake a circular coil of a great many windings, an<l wispi-nd it hy n fine vcrticid wire, ao that its windingit arc horizontul, and the coil is capable of rotating about a vcrticid axis, either in the same dircetion on the current in the coil, or in tbe opposite direction. i We shall suppow the current to be conveyed into the coil by means of the eusjK-Dding wire, and, after paxsing round the windings to com- pleto its circuit by piwxing downwards through wire in the jame line with the 8UK])ending wire and dipping into » eup of inerenn-. Since tJte action of the horizontal component of terrestrial magnctiKm would tend to turn this coil round a horizontal axis when the current flows througli it. we ^hiill suppose that the horiEootal component of terreatrin) niugnrtism is exactly neulralized by means of fixed vatgaeU, or that the exj>erimi,-nt is made at the majjnclic pole, A vertical mirrorLiuttach^lotheooil to detect any motion in azimuth. Now let a current be made to pafs through the coil in the direction X.ES.W. If eloctricity were a fluid like water, flowing the wire, then, at the moment of etartiitg tlie eurrcut, and as R».W. S02 ElECTROKINKTlCa. [574- lonar OM its velocity in increaiting, » force n-otikl rcqaiiie to ht supplied U> produce ihn nngiilar mnmcntiiin of tlio fliti<i in passin^f roiiDd the coil, and lU thi.i niiuil W Htiji|tlio<] by tlie elasticity of the BUBpendin^ wire, tlie coil would at firrt rotate in tbe o]>(weit4^ direction orW.S.E.N., and this would be detec-tod hy mcnnti of the mirror. On Htoppiiig the current t^crc wduld be anotbei inoveiuent of tbe mirror, this time in tJic mmc direction as thai of tbe current. No phenomenon of this kind hatt yet l>ocn oWn'od. Such nn action, if it existed, might be eonily diittin^ti«hcd from the alrouly known actions of the current hy the following peculiaritieii. (1) It would occur only when the ntrcn^lh of the current raries, a8 when contact b made or broken, and not when tlie current is constant. All the known mfekanieat actions of the otirrent depend on the strength of the currents, and not on tite rate of vsriation. Itie electromotive action in tbe case of inditced currcntd canbot be confounded with this electroougnetie action. (2) The direction of this action would be reversed when that of all the currents in the Geld is reversed. All the known mechanical aetioiui of the curri>nt remain the same when »ll the aiirrentd are reverwed, siuce tliey dejiend on Kiiuares and products of these current*. If any action of this kind were (li-tcovered, we should be able to regard one of tbe so-catled kinds of ctectricity, either the ]>o«itivo or tlie negative kind, as a real xubiitance, and we should be able to dcAcribe the electric cum^it ii» k true motion of this substance in a particular direction. In fact, if elMtrical motions were in any way comiJarable with the motions of ordinary matter, terms of the form 7^ would exist, and Ihcir cxisteaoc would be manifested by the mechanical force X^. Aecording to Fechner's hypotb<«is, that an electric corrent con- sists of two equal currents of positive and n^ati^e electricity, flowing in opposite directions through the same conductor, the terms of tbe second class T„ would vanish, each term belonging to the positive current being accompanied by an equal term opposite sign belonging to the negative current, and tbe pbN Bomeoa depending on these terms would have uo existence. It appears to me, however, that while we derive great advan(s»^ from the recognition of the many analogies between the i'IpcItm current and a current of a material lluid, we must carefully M 575-] BXPKRIMEJTT Oy BOTATIOK. 203 makiti(> any aasuinptioD not warrant^tl hy cxporinionlal oviilenco, »Rn<l that there is, aa yet, no pxperimeutat t.-vi<i«iict> i/a shew whetlier thf clcvtric current is really a current of a matorbl substance, or In double current^ or whether ita velocity is gn^t or amM aa mea- ■ured in feet per second. A know]cd};r of theee thin;^ vrould amount to at least the be^D- DiD^ of a complete dj^amical theory of electricity, in which we vbould rc^rd electrical action, not, as in this treatise, as a phe- nomenon due to an unknown c&use, subject only to the (general \&vi» of dynamics, hut as the result of known motions of known portion* of mutter, in which not only the total efTeuts and final rcAilts, but tht' whok- intermediate mcclianism and details of th« ^^ motion, aro taken ■» the objects of study, ^B 57S.] I^iv experimental inveetigation of the second term of X,^, ^^HlKly —j^, is more diflieult, a» it involves the observation of the eiTvet of foroeit on a body in rapid motion. The apparatus shewn in Vig. 3(, which I had eonrtmctod in ■ iSfil, is intended to t««l the existence of » force of this kind. BLECTKOKISBTICS. 31 I ia The el«ctitHiiBgnet A is capable of rotating abont tbe horizontal axis Blf, within a ring which itaelf revolves about a vortical axis. Let J, B, C he the moments of inertia of the electromagnet abont the axis of the coil, the horizontal axis BB". and a third axis CC pespeclivply. Let bo the angle which CC' makes with the Tcrti<«l. ^ tbf azimuth of Die axis BB, and ^ a variable on which the motion of electricity in the coil depends. Then the kinetic cnei^ of the electromagnet may be written 2T = J4'' sin' S+B4' + C4>*coaH + S{4> an 0+^)*, where £ is a ([uuntity which may be called tiio m»m»)t of inertia of the eWtricity in the coil, If & i» the moment of the impa-^cd force tending to locreaM we have, by the cquationg of dynamics, & = B'^-[(A-C)4fi»ia0coBO+£^txn$(4>f\n0+'if)\. By making 4', the imprc^xed force tending to increase ^, eq ml ,, to xero, we obliun ^H ^8ind4^ = y, ^^ a oonxtant, which wc may consider as representing the strength o^ the current in the coil. If C is somewhat greater than A, €t will be zero, and the lihrium abont the axis BIf will be stable when This value of depends on that of y, the electric cnrrenf,' is positive or negative according to the direction of the current. The current is passed throngh the ooil by its bearings at S and B', which are connected witJi the battery by means of sprin^H rubbing on metal rings placed on the vertical axis. ^^ To determine the value of 0, a di«k of {uper is placed at C, divided by a diameter parallel to BB" into two parts, one of which lA pniiit<-d red and the other gra-n. \^')u-n the instrument is in motion a red cirde is seen at C when $ is positive, the radius of which iitilioate« roughly the valne of S. When is negntivc. a green circle w *een at C. By mcanic of niit« working on ecrews attached to the dec! magnet, the axis CC' is adjust«d to be a principal axis liavin its moment of inertia jnst exceeding that round tlie axis A, to sin 9 = ZLECTEOMOTIVE FORCE. inuk« tlic instrament very Kcuiiilile to the Action of the force if it itxiNtM. ^ Tlic chief difficulty in the ex])i;rim<!iit8 arose from the ilietiiTbing ^kction of the earth'^ ma|<;tietic f<irce, whiob caused th<- ek-ctro- ^unagnct to itct liki> a dip-iiet'dle. 'Vha rt^ulla obtained were on this 'iwcounl very rough, but no evideuce of any ehange in d could be obtaiocd even when an iron core ivas ioverted in the eoti, so as tu make it a |)Owitrful electromagnet. If, tlierefore, a m^net containii matter in rapid rotation, the angular momentum nf tliiii ixitatiDQ must be very umall compared Iwilh any <)uaDtitieii which we citn measure, and ne have as yet no lence of the existence of the terms 2*^ derived from tiieir aanical action. 9T6.} Let us next consider the forces acting on the currents fof electricity, that is, the electromotive farces; Let }' bo the eSeetive electromotive force due to induction, the [cleetromotive force wliich must act on the circuit from without to balanoe it is y = — }', and, bv Lngrangc-'a cquntion, r=-r =---+-• ttl </^ dy Since there are no terms in T involving the coordinntti y, th« lod terra is zero, and }' is reduced U) its firxt t«rm. Hence, romotive force caonot exiet in a itystcm at rcKt, and with cou- flt«nt currents. Again, if we divide I' into tlireo part«, I'„, f,, and T^, cor- ie»l>onding to the thmc parte of T, wo 6ad that» since 7", does not contain y, r_ = 0. Weilsofind r = -~'^fi. ' at dv dT Herw -5.- is a linear function of the currents, and this part of Uw eJectromotive force is etjuul to the rata of chan^ of this fiinelion. This is the electromotive force of induetiim discovered by I'liraday. Wc shall conifidcr it more at lt>iigtli afterwards. 577.] From the i)art of 7', depending on velocities multiplied by currents, we find i_ = — I I I Now dt .Jj, is a liocnr function of the velocities of the conductors. If, thcrirforc, any terms of T^ have an actual exiatenee, it woidd be postible to produce an electromotive force independently of all existiDg carreitte by aimply altering the velocities of the conducton. 206 ELECTHOKnJETICa. [57/. For inetance, in t,h« ca«c of the «ii«pcui]c<l coil at ArU A59, if, wIicd the ooil is at rest, w« suddenly met it in rotntioa nboiit the vcrticail lucis, nn «1l"cI roiiiotivc force woiiW U>«ill<^ into action proportional to tlic n«Ci^leniLi»ii of thiit motioo. It vroiild vanish whim lite Diotion becume tiDifonn, and be rovenod when the motion waa ruturdtKl. Now few a?ientilio oWrvationg can be maile with greater pre- cision than thai which determine* the existence or noifCxiitten<-c of a etirrcut by ineans nf & galvanometer. Thedelicacy of thiit method far i'xi^e<l.<< tluit of luoot of tlte arrangemenlK for mi-JiMiiriiig the mochaiiical foi-ci^ acting on a body. If, Uii>rcfore, any currents eould be produced in tliis way they would be detected, even if liiey were very feeble. They would be distinguished from ordinary carreota of induction by tbe following clinracterictics. (1) Tlii'v would depend entirely on the motions of the conductore, and ill no degive on the strength of currents or tnagnetic fo already in the tietd. (2) They would depend not on the abjioloto velocities of the co ductors, but on tlieir accelerations, and nn squares and products velocities, and they would change sign wlien tbe acceleration be- comes a retardation, though tbe absolute velocity is the same. Now in all the canea actually observed, the Induced currenta depend altogether on the strength and the variation of current* in tbe field, and cannot be excited in a field devoid of magnetic force and of cunentB. In so far as thoy depend on the motion of coa- doctors, the)- depend on tlie absolute velocity, and not on the chango of velodty of these motions. We have thus three methods of detecting the existence of the tenns of tbe torm T^,, none of which have hitherto led to any positive result. 1 have pointed them out with the greater care because it ap[>enr« to mr important that we should attain the greatest amount of certitude within our reach on a point eo strongly on the true theory of electricity. Sincf, however, no evidence has yet been obtained of such 1 Ehall now proceed on the assumption that they do not cxmi or at least that thoy produce no sensible 00*001, an assuniptioa whii will oonstdembly simplifj- our dynamical theory. AVe nliall ha occasion, howi;ver, >n discussing the relation of ninguetism to Ugl to shew tlukt the motion which constitutes light may enter u« a lactoT into tcnni invoiring the motion which oooatitnles mu^ nctism' 1 CHAPl'ER VII. THEORY OF ELECTEIC CIUCKIT8. 578.] Wr may now oon6ne our attention to that part of the EiDetic energy of the system which deponcU on wiuaree and products of the strenglhii of the electric currents. Wo may call tbia the Elcctrokinetic Knei^ of the system. The part depending on the motion of the conductors belongs to ordinary dynamics, and we have shewn thai the part depending on products of velocities Mid teorrents docs not exist, IjCt yf,, Jj, &c, denote the different conducting circaits. Let their form and relative position be expressed in terms of the variables 4*1, x^, &c., the number of which is equal to the number of degrees of &eedom of the mechanical sj-stem. We shall call these the Geometrical Variables. Let y, denote the quantity of electricity which has CTO6ae«I a given , section of the conductor .(, ainoe the beginning of the time /. The trength of the current will be denoted byj/,, the fluxion of this ^quantity. We shall call jr, the actual omrent, and y, the integral current. There ifr one variable of this kind for each circuit in tho system. Let T denote the electrokinetic energy of the eyetem. It is a Itom^^-neous function of the second degree with respect to the ,.^ rixe ngths of the currents, and is of the form ^Brnere the ooeflicients A, M, &c. are functions of the geometrical l^tarlables Xi, r., &c. The electrical variables y,, y^ do not enter into the exprcMtion. We may call A,, Z.,, &c. the elnctrie moments of inertia of the circait« /fj, J.^, &c., and ifj^ the electric product of inertia of tlic two circuits J, and .4^, When wc wish to avoid tho languugi- of l9K 4 aHM LIITBAB CIRCUTT8. the dynamical theory, we shall call J/, Uie coefllcient orsalf-inducli of the circuit y/j, and J/^, the coefficient of mutual induction of i circuits ^, and A^. M^^ is also called the jiotenUal of lite cirLiili ^j with ree])ect to Aj^. These qaantitics' degiend only on the form Aiitl relative position of the circuits. We shall find that in the electro magnetic system of measurement they arc quantities of the dimension of a line. Sec Art. 627. By diiTercntiatinf: Z" with respect to j^, we obtain the quantity />,, whieh, in the dj'naroical theory, may be called the momentum corresponding to y,. In the electric theory we shall call pi the clectrokinotie momvnlum of the circuit A, . Its ralue is The electrotdnetic rocroentum of the circuit J, is therefore mode up of the product of its own current into ita coefficient of self- induction, together nith ttie sum of the products of the currents in the othor circuits, each into the coefficient of mutual induction of Aj and that other circuit. Electromotive Force. 579.] Let fi' l)c the impressed clectromotiTe force in the circuit A, an»in{> from some caoac, such as a voltaic or tliermoelectric b«tteiy, which would produce a current independently of mngneto-electric induction. Let R be the resi^noe of the circuit, then, by Ohm's law, an electromotive force 7lf is required to overcome tlw resistance, leaving an electromotive force £—Jly available for changing the tnomeulnm of the circuit. Calling this force J", wc have, by the general equations, ^^ Jp JT but einoc T'doos not involve y, the laxt temi disappoors. Hence, the equation of clectromolivo force is TTie impressed electromotive force /' is then'fore the sum of ' parts. The first, /iyr, is required to maintain the current jr ugaii the redstenoe R, The second part is rcquin-ti to increaw the i ttomagnetio momeDtum ji. This is the electromotive fom* whict must bo supplied from sources independent of magtwto-elect TWO CI1ICUIT8. )9 induc-tion. The electromotiTe-force arising from magDCto-vleutric Dduction aloae is evid^Dtly — -n < or, tAe rate of decrease of the fUctrokinttic momentHst of the circuit. (I f Efeetroma^netic Foret. 580.] Lrt X' be the impreBsed mecluoical fwrne arising from kxtvrDiil cauBC», tuid tending to increase tJie variable z. Hy the gencnil equations ^ 4 dT dT ~ dt dx dx Sin<-e the expression for the clectrokinetic oncrgy does not contain th« velocity [i), the iirst t«rm of thu second member disappears, und wc find ^ ^f Here X' h lite fxt^-rnal force required to balance the forces arising from electrical i-itusc-s. It ie nsiial to eoimider this force us the resclion against the elect to tna^etic force, which we shall call A', ^^ad which is equal and opposite to X', I Hence dT Win, ; tie elftiroinaffnelie /orcf tt-nding 'o incrfaM any rarialln iV equal io fit ratt of ihCTfase of the tleelrokinelic energy jtcr uiH increase of tiat variaiU, He cwrratte being mainlained exmttant. If the cnrrenta are tnaintained consljinl by a liattery during a displacement in which a (juantity, W, of work is done by electro- motive force, the eleolrokinetic energy of the system will be at tho Mine time increased by W. Hence the battery will be drawn apon for a double quantity of energy, or 2 W, in addition to that which it ■pent in generating beat in the circuit. This was first pointed out ly Sir W. Thomson*. Compare this residt with the olcctroirtatio perty in Art. 93. Catt of Tko Circttite. 681.] Let Ji he called the Primary Circuit, and A.j the Secondary Circuit Tbo dectrokinetic energy of the system may be written where L and .V arc the coo<Iicicnt« of »eir-indncUon of the primary > N>d>..l'« Crtop<Hili» 1/ Njoinil Stitntt, (d. 1840, Artldo, ' MignMim. Dy- ■tmkal livUliuna «f.' k TOL. II. \AufM USB4H CIRCUITS. and Eecon^MTy circuits rccpt'clivcly, iind 3f is tbfi coefTicicnt of mutual induction. Lot as suppose that no i-lcctromotive foroo *ci» oa th« second, dicuit except that duo to the inductioQ of the primal}* curreDl- We have then . Integrating^ this equation with reepeot to /, ire have If,^a + Mfj + Nf^ = C, A constant, vrhrrc y, i« the integral current in the secondary' circuit. 'Hie mvttiod of nitiasuring an integral current of sliort d will be dviMsribed in Art. 746, and it is easy in moat cnaoii to «i that the duration of the secondary cmreDt shall bo very short. Let the values of the variable i^uantities in the equation at end of the time i be acccnti-d. then, if y^ is the integral current,' or the whole quantity of electricity which flows through a section of the secondary circuit during the time t. If the secondary current arisirs entirely from induction, its initial value jTf must be zero if the primary current is oonstsot, aod the con'luctors are at rest before the bej^innin^ of the time i. If the time / is sufBcieat to allovr the secondary current to dis awav, y/, its final value, is also Kcro, so that the equation bcvomo The integral current of the secondary circuit dDpends io on the iuitiul and (iiuil values of J/y,. Indueei CmrrenU. S83.] Let us begin by supposing the primary circuit or y, = and let a current y,' be eiUbli«hed io it when oont is made. The eijuutioa which delermioes the BMondary integral current i When the eireoib «tt pUoed side by side, and in the same direct tion, J/ is a positive quantity. Hence, when contact is made io the primary circuit, a negative current is induced in the soooodaty circuit. When the contact is broken in the primary circuit, the ; current ceases, and tlie induced current isy, where Hie Moondary enrrent is in this ease positive. fe TWO CIBCUITB. 211 I If the primary current U muintaimid coostant, and the form or relative position of the drcuils altered so that M becomes M', the int^ral secondary current in^^, wh<-re In Div cn«c of two circuits placed side by side and in the same diruotion M dimiiuahes as the distance lH'tw«cn the circuits in- creases. Hence, the iiiducoit current is piwitive when this distuooc ia increased and ne^tivc when it is diminiehud. t These are the elementary ca«es of induced ciirreiite described in :t. 530. 6S cl I I I • I Mechanieat Action heltcerit Ihc Two Cireuiti. 583.) Let X be any one of the geometrical variables on which i« Ibrm and relative position of the circuits depend, the clectro- nuignotiv foreu tending to increase x is dL dM , ,dN If the motion of the system corresponding to the variation of x IS Fiich that each circuit moves as a rigid body, L and .V will be indt^pcndvnt of x, and the equation will bo reduced to the form dM J^=i>>''^+^./.^4U/' ■i" = hh dx Heooej if (he primary and secondary ourrents are of the same sign, the force X, which acts between the circuits, will tend to move them so as to increase M. If the circuits are placed side by side, and the cnrrcnts Dow in the same direction, M will be increased by their being brought DeiU¥r together. Hence the force X is in this eai<e an attraction. 684.] TIio whole of the phenomena of the mutual notion of two CircuiU, whether the induction of currents or the mechanical fore*? betwMn them, dejicnd on the quantify -V, which we have called the eoeflieient of mutuiil induction. The method of calculating this itity from the geometrical relations of the drenita is given in 524, but in the inveKtigations of the next chapter we shall not me a knotvledge of the mathematical form of this quantity, sliall consider it as deduced from experiments on induction, for instance, by observing , the integral cuiTcnt when tlie secondary circuit is suddenly moved from a t,'iven poKition t« an lite distaote, or to any position in which wo know tliat J/s 0. F a CHAPTER Vin. bxpijORation of thr field by ubaxs of thb secokdakt CIBCUIT. vnit I :tn>- hese My «(] in ■M 585.] We have proved in Arts. 582, 583, 5B1 that the eleotro- augnetic uction botwoen the primary and the accondary oircait depends on tlie quantity deuoted by if, which is a fuactioa of form and relative position of the two circuits. Although this quantity ,V is in fact the same as the pot«ntii of the two circuits, the msthcmattcal form and proportice of whici we deduced in Arts. 423, 492, 521, 639 from magnetic and electro- mag-notic plienomena, ne shall faorc make no reference to thew r«Bu]tB, but begin again from ■ new foundation, without any assumptions except thoM of the dynamical thc<ory aa slutcd in Chnptcr VII. Tlic clei;tr»kinetic momentum of the secondary' circuit conna of two part* (Art, 578), one, J/<i, depending on the primary eur »',, while the other, ?i\, depends on tJie ttecondaiy current i',. We arc now to investigate the first of theae parts, which we alialt denote by 7», wbiTC /> = .l/t,. (l) We idiall alao suppose the primary circuit fixed, and the primary current coitxtant. The quantity jb, the electrokinetJo momentum of tJie secondary circuit, will in thiti case depend only on tJie form •nd position of the veeondary circuit, so tliat if any dosed cunre he taken for the iteooodaty circuit, and if the direction along this ctirTO, which is to be reckoned positive, be chosen, the valiM of ^h for this closed curve is determinate. If the opposite direction aloo^l the curve bad been chosen aa the positive direction, the sign of the quantity /i would have been reversed. S8€.] Sinc« the quantity p depcnda on the form and positiii of the cireuit, wu may sappo^ that each portioD nf the circnt 588.] ADDITION OP CIECUIT3. 213 ■jlribi] oontributas somethinji^ to the value of p. and tHat the part con- bated by each portion of the cireuit depends on the form and ntion of that portion only, and not on the position of other parte of the circuit. This asstimpUon is legitimate, hucause we are not now considcrinf^ current, the pari* of which niuy, and indeed do, act on oiw an- other, but a mere arcntt, that ix, a closed curve alon^ wliieli a eurreDt nnr^ How, and thie is a puiyly ^somutrical fifj^ure, the |iart« of which cannot be conceived to have any physical uetton on eoelt _other. We may therefore aMume that the part contributed by tlie element */* of the circuit i* Jd*. whi-rr / in a fjtiuntily dc))ending on the portion and direction of the cli-nicnt J». Henee, tbe valne of J) may be expTt««ed as a line-iiit«gnil fp=jJd>. (2) 1ier« the integration ix to be extended once round the circuit. 887.] We hav(^ next to determine the form of the quantity J, a the finti, place, if tta is reversed in direction, J is reversed in uga. Hence, if two cireuita ABCE and AKCD hare tbe arc AEC common, hut reckoned in opposite directions in the two circuits, the sum tof the values of ^ for the two circnits ABCE and AECIf will be e(iuat io the value of yi for the circuit ABCD, which is made up of the two circuits. For the parts of the linc-intcgrul deppndinff on the arc AEC ai« equal but of opposite sign in the twd partial circuits, ao that they destroy cndi other when the sum is taken, leaving only those {torts of the lino-integral which depend on the external boundary of ABCD. ^P In the fame way we may shew that if a surface hmindcd by a eloeol curve be divided into any number of part.N, and if the bonndary of each nf thc^e jtarts be coti»idered u« ii ein^iiit, the positive direction round every circuit I>eing the «arae an that round the external closed curve, then the value of'jO for the cUibo<1 curve in ^^qual to the sum of the values of /t for all the circuits. See .\rt. 483. ^P 68B.1 Let ns now consider a portion of a aurfaee, the dimensions "of which are so small with respect tolheprinci|ial radii of curvature of the sarliu« that the variutioii of the direction of tbe normal within this portion may be neglected. We shall also suppose tha', if any very small circuit be earned parallel to itself from one part if this surface to another, the value of p for the small circuit in 1 If Fig. 36. I 2U fiLfiflHOMAOSETIC ?IEI.R. not aenaibly altered. This will cviilcntly be tbc cam U* tlm diuen aiona of the portiou of surfacu arc ridhH onoug^i eomiured wit ita distance from the priman- circuit, f arty closed carve be drawn on Mi* jiorfhit of He tvrfaee, tie raltre of p will 6e proportional to tV* arta. For the nrt-as of any two circuit* may be divided into sma elements all of tbo :^tmc dimenxioni*, and having the same valv of p. The area* of the two circuit» are as the nurobera of theM elements which they contftin, and the values of p for the two eircoiu are aL-ui in t.)ic same proportion. Hence, the value of ji for the circntt which hounds any elemental lis of a iiurface is of the form IdS, where / is a quantity dei>ending; on the poeition of d$ and on the direction of its normal. We have therefore a new expreasiOQ for jo, =1 tie aalH tlaeV . =jjld8, 1« Kg. S8. where th« double integral is extended over any surftce bounded b; the circuit. 589.] Ijct AliCD be a circuit, of which AC is an i-lonientary portion, m> small tbnt it may he conHHlen^ Htruigh Let APH and CQB \x nmall ctjual areuit in tl same plane, then tbc value of p will be the MOM for the small circuits APB and C<i,lt, or p{APIi) = p{CqB). IIcDce p{APBQCI/) =p{ABQCD) + p{APiri, =p{Asqcii)+p(cqs), = p{A£CD), or the value of p is not altered by the substitution of the crool line APQC for the straight line AC, provided the area of the circuit is not sensibly altered. l"hia. in fact, is the principle established by Ampin's second experiment (Art. 506), in which a crooked portion of a circuit i* shewn to be equivalent to a etrsigbt portion providod no ]>art of the Crookeil portion is at a sensihle dist>n< from the straight portion. If therefore wc substitute for the element i/t three small elemeni lie, Jjf, and tl:, drawn in succession, so as to form a c«nlinuoi patli from the be^nning to the end of the element i/#, and Fdje, Gdjf, and ffdx denote the elements of tlic line-integral eo: tesponiling to dx, djr, and dg respeetively, then Jdt = Fdx-i-GdyiIIde, ILTCTBOKISETIC KOyCEVrtUV. 590.] Wc «re now able to determine the mode in which tbe quantity / (Icxwnds on the direction of the elemont ds. For, by (4J. 4* <lf da (5) This is the cxpnsdon Tor the resolvetl i>art', in tho direction oft/*, of a vector, thv components of which, rceolved in thti directions of the ttXCT of », ^, nnd i, mv f, (r, nud tf rcepuctirely, Tf this vi-ctor be denoted l>y 91, nnd the visctor from the orif^Jn to a point of the eirciiit by p, the (element of the circnit wiil be dft, ^—»ud the quitemion expreaaion for Jd* will be ■ -S3ldp. H Wc may now write eqiiatioD (2) ia the form -M-oh^sy- or fi =-fsndp. lie vector ?[ and its conptitiiont^ F, C TT depend on the position I of dt in the field, and not on the direction in whieh it is drawn, rhcy arfl therefore functions of «, y, t, the coordinates of d», and not of I, m, n, its direction-cosines. Th« vector 91 represents in direction and magnitude the time- hitcgnd of the electromotive force which a pitrticio placed at th« point (*, y, ;) would cuperiencc if the primary current were sud- denly stopped. We shall therefore call it the Electrokinetic Mo- mentum at tif point, {x, y, t). It is identical with the quantity which we investigated in Art. 403 tinder the name of the veetor- potvntiul of mof^ietic induction. The elootrokinetic momentum of any finito line or eirouit is th« iine-inlegnl, extended along th« line or circuit, of the resolved port of the eloctrokinctic momentum at faeb point of the some. 501.] Let U!i next dt-termino the value of p for the elementary rectangle ABCD, of which the sides are Jy and dz, the positive direction bein;; from the direction of the axis of ^ to that of t. Let the coordinates of 0, the centre of gravity of the clement, be J'ofJ'u. -o- ""d '"^ On, Hn be the values of G and of //at thia point. Fig. W. lie coordinates of A, thn middle point of the first side of Uie BI^ECTROMAOIi [59* rectangle, are y, and t^— - dz. Tlie correeponding value of Q '» ( M and the part of the value of p which arises from the side A u .pproxin.ut.ly G^J^-'/PyJz. (9) Similarly, fra JS^ B^dx^ - -r- dgdz. For 5, -H^dz^Y^dgdz. Adding these four quantities, we flod tlie valne of j> for rectanel»^, viz: ^if dG. , , , If we now Asciime three new <|aantitie8, a, 6, e, such that ^d/l_dO rff dc dF di dG dU (^ dz dj_ dm djf ' ', ^g^ conitidfr these um the eonstitucnt* of a ncn- vcdor 9, tlien, TiieorDin IV, Art. 2-1, we may express the lim-int^'gral of SI roue any circuit iu the form of the 8tirEaec>iut£^7i>l of 3 over a »u ft'acc hounded by the circitit, thus p=JT%cta*dt=jJT'^cMiidS, wbere t a the angle between ?l iind dt, and ij that between © and the norma] to dS, wlio»e <liri>ctii)n-«otine* aru /, m, n, and T% 79 denote the numerical vatuea nf 'Jl awd '8. CompunD{> this result with equation (3), it is evident that the qunntity / in that equation ia equal to 9 coa >}, or the resolved part of $ nonual to dS. 592.] We have alimdy seen (Arts. 'I'JO, 54 1 ) that, acoordiny : l-Vraday's thcoiy, the phenomena of elcctroma^etio fbrea. or MAOSCTIC INDOCTIOJT. indoctioa in a circuit depend on Utu vaHattoa of tlio number of lines of ma^etic indactioD which pass ihrouf^h the circuit. Now the number of these lines is exprcfsi.'d mathemHtically by the sarbcc-inte^ral of the maf^nctic induction throug'h any eurfnco bounded by tbo circuit, tlencc, wc mii«t rrfjard the vector S and its components a, 6, c ok representing what wc arc ulrcudy ucquuintcd with ue the mngtictic induction and it* components. In llie present investijfation we propone to deduce thi- projierties [of thiM VL-ctor fn)ni iho dynrtmieul principles stated in the Xust, |chapl«r, with -.i* few nppoiilx to experiment as possible. In klentifyiiig this vector, whinli has iip|}eared as the result of 'a matliematical inveati^ution, with the magnetic iniluclion, the Ipropertien of which we learned from experiments on magnets, we do not depart from this method, for wc introduce no new lact into the theory, we only give a name to a mathematical quantity, and the propriety of bo doing is to be judged by the ngreement of the relations of the mathematical quantity with those of the physical quantity indicated by the name. nic vwtor 33, since it occurs in a surfaee-integml, belongs I evidently to the eatepory of Huxes described in Art. 13. Tho I vector SI, en the other huud^ belongs to the category of forces, ^nnc« it appears in a line-integral. 593.] Wc roust here recall to mind the conventions about positive ' and negative qiuintities and diicctions, «otne of which were stated in Art. 23. We adopt the rigiit-humled syslero of axes, so that if a right-handed screw is placed in the direction of the axis of a>, aad A nut on tliis screw is turned in the positive direction of rotation, tliat i^ from the direction of y to that of :, it will more alon^ the screw in the positive direction of a*. ■ We also consider vitreous electricity and austral magnetism as positive. The ]>0Bitivc direction of an electric current, or of a line of electric iuduction, is the direction in which positive electricity Imoves or tends to move, and the positive direction of a line of 'magnetie induction is the direction in which a compass needle points with the end which turns to the north. See l-"ig, 24, ^Vrt. «8. and Fig. 25, Art. 501. jl^ The student is recommended to select whatever method appears ^Kto him most elfcctual in order to fii these conventions securely in ^Bbts memory, for it ts far more diflicult to remember a rule which ^^det«rmincs in which of two previously indiflerent ways a stotcmecit ia to be made, than a rule which selects one way out of many. »i IIJCTROMAONETIC FIELD. [594- 594.] We have next to dcduoe from d^amical principle* pxpn-sMiona for llie elct;tri>magn<)tio force actiu^ on a coadac' currjing an electric cuirciit tbroufi^li the mai^netic ficM, and the electromotive force acting on the electricity within • body moving in the magnetic field. The mathemiitimi method which we flhall adopt may be compared with the cxiwriinenlal metliod used hy Faraday * in explorinfr the field by meanit of a wire, and with what we have already done in Art. 490, by a method founded on experiments. What wo have now to do is to determine the effect on the value of p, the electrokinetie momentum of the secondary circuit, due to given alterations of the form of that circuit. Let AA', BB" be two parallel straiglit conductors coaneotcd I^ the conducting arc C, which may he of any form, and by a strai^t ■^ ong tio9 F<(. Da. conductor AB, which is capable of sliding parallel to itself along the conducting rails A A' and B&. Let the circuit thus formed he considered as the secondary cuit, and let the din^rtion ABC be assumed as the poativc directioi round it. Let the sliding piece more parallel to itKcIf from the postion AB to the position ^Jf, Wc have to determine the variation of /r, t1 cicctrohinetic momentum of the circuit, due to this dieplaci-mcn of the gliding piece. Tlie aecondary circuit i« dian^i^ed from ABC to jtlfC, hence, b; Art. 587, j>{ArS'C)-p{A£C)= piAA^BTB). (i.h) We have tlierefore to determine the value of p for the |iaralU-I*M ogram AJ^ffB. If this puralielogram ia so small that wc mayH neglect the variations of the direction and magnitude of the mag- netic induction at difiV-rent points of itd pluiic, tJje value of p by Art. 59 1 , © 00a ij . AA'IfB, where S is the tnagnolio inductio • Bip. ft*, SOte lusr. 3lW. 596.] SLinixo PiKCE. 2X9 I ^bnd >) the angle <iThich it ranker with the positive direction of the ^piormal to the parallelogram AA'lfB. ^ We may represent the result georoetrioallj- by the volume of the pamllelcpiped, whose base is the parallelogram AA'JfB, and one of whose edges is the line AM, which ropresenta in direction and ^maivnttiido the magnetic induction ^8. If the parallelogram is in Ijo plane of the paper, and \i AM is drawn upwards from the paper, tie Tolumc of the parallelepiped ia to be taken positively, or mora cnorall}'. if the direotions of the circuit AJi, of the magnetic in- ^duetion AM, and of the displacement AA', form a right-handed syBtem when taken in this cyclical order. The volume of this parallelepiped represents the increment of the ^-alne of /> for the secondary circuit due to the displacvment >f the eliding pieci! from AB to d'iJ'. Sleetromolive F(»ve acting on tie Sliding Piece. 595.] The electromotive force jirotluccd in tlie Mconduy circuit by the motion of the sliding piece is, by Art. 879, If we suppose AJ' to be the displacement in unit of time, Uiea iA' will represent the velocity, and the parallelepiped will rojiresent R^, and therefore, by equation (H), the electromotive force in tlia negative direction BA. I^L Honco, the electromotive force acting on tha sliding piece AB, ^Kn consMitienoe of it^ motiuu through the mAgnetio Geld, is rcprw- ^Bent«d by the volume of the pnrnlK-Iepiped, whose edges represent ^^in direction and magnitude — the velocity, the magnetic induction, and the aliding piece itaelf, and is ponitive when these three direc- tions are in right-handed cyclical order. ^ EUeli^ftagnelie Force aH'mg on the Slliiing Pieee. S96.] Let f, denote the current in the ecconilury circuit in the positive direction ABC, then the work done by the electromagnetic forre oo AB while it slides from the position AB to the position A'W is (Jf'— J/) /, 1^, where M and M' are the values of 3/,^ in the initial and final positions of AB. But (.V— J/)'| is e(|ual to p'~p, and this is rcprcwntwl by the volume of the parallclepti>ed AB, AM, and AA'. Hcnoe, if wc dmw a line par»Il«l to AB EtBCTBOMAOlTEnC nB!.T>. to rc])n»eot the qaantity JB.ij, th« {laralldcpiped contained this lino, by AM, llic mnf^oliv indnotion, SDi] by AA', Ihu displiwt- ment, will represent the work done during this diHpbct'invnt. For ii ^von dietunci' of difplaocrovnt this will be gn-ulxict when tlio di^placfmont i« perpendicidsr t» the pnrBllcIcigmm whom? «dw an- AB iwwi AAf. The electroiuagiwtic force in ihcrvfoTv rc{>n.>»fiilnl by the arcu of thi> pHivllelognun on Alt and A-V multiplied by ^, Knd i« in the direction of th« normal to thin parallelogram, drawn >o tliHt AS, AM, and the aonuat are in rtght-han<led cyclical order. Four DffinUiona of a Line of Mapietie iHduelioH. 697.] If the direction AA^, in which the motion of the sliding piece takes place, coincides with AJif, the direction of the magiietfc indaelion, the motion of the sliding pioec will not call electmmotive force into action, whatever be the direction of AB, and if ^// carries an i-lcctric current there will be no tendency to slide along Ait. Again, if Ali, the sliding piece, coincides in direction with AM, the direction of magnetio induction, there will be no electromotive forru ciillt^ into action by any motion otAli, und a current through All will not cause AB to be acted on by mechiuiical force. We may therefore define a line of magnetic induction ia four different ways. It is a line mich that (1) If a oonductor be moved along it paraUe) to iuelf it will experienoe no electromotive force. (2) If a conductor carrying a current be free to move aloni* a line of magnetic induction it will experience no tendency to do so. (3) If a linear conductor coincide in direcUon with a lin« of magnetic induction, and l>c moved parallel to itGl^ll' in any direction, it will experience no electromotive force in the direction of it« length. (4) If a linear conductor carryinff an electric current coincide in direction with a liitc of magnetic induction it will not cxiwrtenoe any mechanical force. OenVTtil EquatiMt g/" Elteir9m«Utt Force. &9B.] Wo have Been that M, the electromotive foiro duo to in- duction acting on the secondary circuit, ia equal to — ~, where dt J ELECTHOMOTITE FOBCB. 221 IN) determine the value of E, let us (liiterentiat« the quantity fnnder the integral sign with respect to I, remembering that if the |«ecoD(laT}- circuit is in motion, x, if, and s are functions of the time. 1 "We obtain ,rf^^ rfO^ dUdz^ ^dt th'*' di ds"^ de d»^ ^ = ~ J Ut di'*' dl t,"^ M T»)'^ _ \ c^^ ^'k. 'i^^\ ^j J^dx di '*' dJ dt '*' d9 ds^ di -M dy dt'^ ds^'^ dy <U' dl J Us tU"^ dt'di^ d: dt^dt ■/c: dsdl -'^^s^ + ^.^.+'^^>- dJdl^ f«) Now consider the second tcnn of the integral, and siibotitute ocn equations (A), Art. 501, the values of j- and -^ . This term |th«n becomes, flFdy dFdx^dx -/(' of 1 "' dF dx iir Of ar ai\ ax , "di' di'*"^di'^d^di'*"didi^Tf'^' TOiMic© we may write ^df , dz dFx dst , Treating the tJiird and fourth terms in the same way, and col- dx df , dz ikcUng the terxns in -j-i ~, and ^*, remembering that J(didi'^^didt)'^' = ^di' (3) therefore that the integral, when taken round the closed vantahca, f, dz dx dG\dy, f,,dx dy dir.dz . {*) EI.ECTBOMAOKCTIC FIELD. We nifty writ« tli» expression in tike Torm w where „ <fy .''' d^ *'* q = a ds das dt~^4l g_.dx d^ dif d^ dt "di dt di BquUeAaof Blrctnaiiotiv* Furo*. I The tenns involving tbe new quantity * are introiluced for tin sake of giving generality to tbe expressions for i*, Q, S. Tbey disappoar from the integral when extended round the closed circuit. The <)uaiitity "^ ie therefore indet<irmniutc as fur aK reg-^nls tli« prublem now before as, in which tbe totul electromotive force round tbo circuit is to be determined. Wc shall tinil, howevor, that when we know all the circumstancca of the prohlcm, wc can asvigu a dcfiniU vnltio to 4*, and that it reprcM-nU, ncoording to a certain defmitigti, the electric potential at the pmnt (r, jr, .-). The ijrmntit)' under the integral nign i» etjuation (."S) r«prcsenta the electromotive force jicr unit length acttn|f on the element da of the circuit. If we denote by T(S, tlie numerical value of tlie resultant of P, Q, and li, and by *, the angle between the direction of tbia re> sultant and that of tbe element lU, wc may write equation (5), E^jmwttdt. (6) The vector ff is the electromotive force at tbe moving element d». Its direction and magnitude depend on the podtion and molioD of da, and on the variation of the magnvtic field, but not on the direction of dt. Hence we may now disregard the circum- stance that da forms part of a circuit, and consider it nimply aa a portion of a moving body, acted on by the electromotive force tf. The electromotive force at a jmint ha* already been dt^fiucd in Art. G8. It ia also called tbe reaullant elrctriea) force, Wiug th« _ force which would be experienced by a unit of positive electricity^ pluu^I at that point, We have now obtained the moet general value of this quantity in the ca« of a body moving in a magnetic^ Geld due to a variable electric Hy»tcm. ^| If the body is a conductor, the electromotive force will produce a ' current ; if it is a dielectric, the electromotive force will produce only electnc diq>Ucem«nt. d 599] ASALTSia OF F.I.ECTBOMOTITE POItCE. 223 ^P 'ni« «)ectromotiv« force nt a point, or on a patiiclt;, must be carefully dUtinguUlied from the electromotive force nlon» an arc of a curre, the latt«r ijuftBtity being the line-intogral of the former. I 81* Art. 69. ^p 599.] The electromotive force, the components of which are defined \>y equationH (K), depends on tliree circumstances. The firet I of U>ese ii the motion of the particle through the magnetic Held. H^e put of the force depending on this motion \e expressed by the first two terms on the right of each equation. It depends on this t Telocity of the particle transverse to the lines of magnetic iaduction. If IB is a vector representing the velocity, and © another repre- senting the magnetic induction, then if 6, is the part of the clec- ^ tromotivo force depending on the motioo, P e,= r.®©, (7) OFi t])e electromotive force is tJie vector part of the product of the magnetic indiKtion multiplied by the velocity, that iti to my, the ^ magnitude of the electromotive force is represented by the urea B of the parallelogram, whose sides represent the velocity and the magnetic induction, and its direction is the normal to this pamllel- ogram, drawn so that the velocity, tho magnetic induction, and the electromotive force are in right-handed cyclical order. I The third term in each of the equations (B) depends on the timo- variation of the magnetic field. Tliii< may bo due either to the time-variation of the electric current in the primary circuit, or to motion of the primary circuit. Let (S^ !>« the part of the eleetro- , motive force which depends ou these term*. Its components are 4F dl' dG dt' and — dll JSI ^B mod' ^Faad tlictte are the components of the vector, — "^ or —SI. Hence, ^^ @, = -9I. («} ^M The last term of rach equation (B) w due to the variation of the ^ fanctiott 4" in diScrvnt iiar1,» ijf thi; field. \V« may write the third part of the electromotive force, which is due to this cause, m fs, — V*. (9) Tlie electromotive force, as defined by equations ( B), may therefore be written in the quaternion form. 224 EtEarEOMAOKETlC FIELD. [600. Om tht JfodificaticH o/tie BqaafUnt of Eketfom^t'irt FQret lehm tie Am* to nkich tiey are nferred are moctHg in Space. 600.] Let a<, /, / be the ooonliQates of a point referrwi systeiQ of rectangular axes moving: in space, and let x, y, z be ODoixIinatee of the same point referred to fised axee. Let the componente of the velocity of the ori(;in of the roo' eystem be u, v, k, and those of it« angular veloc-ily w,, <■,, mj referred to the fixed syet^-m of axes, and let us choose the fixed ax«s so a» to coinoido ul the f^iren instant with the movioff odM) then the only quantities whieh will be different for the two syBteKl of axes will be those diir«rcutiitt«d witJi respect to thv time. ftjr i r— denotes a component vciocitr of a point moTinff in rigid Ci nexion with the moving axes, and -j- and -y- those of any mnvin^f point, having the same iostantaneous position, referred to the fixed and the moving axes respectively, tbca fl with similar VfoHaui for the other components. By the theory of tl>e motion of a body of invariable form, 8* 8? = •+-»*--» J"' Since .F is a com]>onent of a direct«d ({aaDtity parallel to x,^ if -jT- be the value of -r: referred to the moving axes, it may he dt shewn tiint dt dr dVhx ^ dFhg . d¥hx -„<//■ ,,. dV dV Substituting for j- and -^ their valuM as deduced &&m the dz eqnatione (A) of magnetic induction, and reroeubering that, by {2) rf B* d h}f d it _ df dt ~ dxbt'^ dxbt * 'dx it* dyit "•" S" 8« ■'' rf» U -^87 + M.dF ^7 + 37* (*>- If we now pat ELKCTROMAOIfETlC FORCB. _^=^»£,<,»J,^-, 225 («) ^ The eqwttion for P, the component of the electrorootire forw parallel to «, is, by (B), ■ ''' '^ '* is) i>=/^_i--_:^.-^, dl dt dt dc referred to the lixed axes. Siib»1ittitinfr the tuIucs of the quanti ^^in tis n-fernid to tJie moving axvo, uo hjivtt (9) ^ W the valac of P referred to the movin^r axes. 601,"! It api^eara from this that the eUctroniotiTe force is c-x- iressed by a formula of the same type, whether the motions of tlie Gondactoni be referred to fixed ases or to axea movin)^ in apace, the only ditference bctneen the formala bein^ that in (he case of moving axe« the electric potential 4' must be changvd into 'i' + 'V. In all cat«s in vrbich a cuTreDt is produced in a condnctiu^ eir- loit, th« eJevtromotire forve is the line-integral '^-M-<^t-'>' (10) Icen round the curve. The value of 4' disappenra (Vom this' ntegrai, so tJtat the introduction of 4^ ha« no influence on its Lvalue. Id all phenomena, therefore, relating to oIim^ eircuit« and he currents in them, it ia indilferent whether the axes to which we Er the sy«tem be at rest or in motioD. See Art. 668. ^B O* tJie Eiairotaagndic Force aeititg on a (hndnelor kUcA cttrriet ^B a» EUcirie Otrreni ikrough a ifagHttie FieU. ^^^^pS.j We have aecn in the gt^nornl invc»l.i(^(irin. Art. .Sfl3, that if ^HP> one of tlie Tariablex wbici) delf rmine tlie [M>!(ition and form of ^Hlie secomlar}- drcnit, and if .V, is thfi force acting on the necoudary ^circuit tending to increOM this variable, then X.= dM d^ 'ih- Sine* fi u independent ofj*], wo may write Hi 'P =/( 'fy dt ^iJ*' (1) (2) TOLII. II.BCTROMAOKl FIELD. [6o| and wfl have for tbe value or J '. = --.i,/(' p fix -» .A (>) Now IH 1)8 «uppo»! that the displaccmcDt conni4« in movins every point of the circuit through n distance bx to tlie dirccti of X, ir bcinff any oontiiiuous function of #, eo that tim difle parts of th<! circuit move independently of each other, while t ciretiit n^RiuinK continuoUH and closed. AW let .V h<^ the total force in the direction of dl vsting the part of the eirrait from * = to * = », then the (MiTt aponding to Uie element dt will be - Jt. We shall then have followinff expression for tbe work done by the force during th^^ dieplACi'ment, ^H wher^ the integration is to he extended round the closed cutvi^j remembering that tr is an arbitrary fanotion of*. Wo may then^^ fore perform the differentiation with respect to hx in the mmr way that we dilfercDtiatcd with respect to V in Art. 598, romcm- beriug that dx d^ , dt . = '. j'\~ = 0, and ^7^3 = 0. (j dbx Wc thus find dix The la«t terra vnniehee when the intoftration is extended round the closed curve, and since the equation mnet bold for all fonnt of tbe function bx, wc must have -di^'-i'di-^d;)* ' nil ijijuntion which gives the force panllel to « on any clement the circuit. The forces parallel to y and ; am dY . , da dx\ dZ . /, dx dv^ llie reaultant force on tJie element » given in direction and magnitude by the ijualcmion rxpre*Hon i^Fdp^. whore f, is the nDn»ric*l mcauirc of the current, and dp and S are reoi ( ioc^ ILKTBOMAOITBTIC FORCB. trG])rc(MMit.ing th« etomcnt of tlie cireuJt and tlie magnetic in> duotion, and tlw mtiltiptication is to be nndorrtood m Uic Hamil- jtooian wn«e. r 603.] If the conductor in to bn treated not as n line but tus a bod}', we must express the force on tlie el<^mcnt of li'iigth, iintl tlie ■ current through the complete section, in terms of K)'mbiils denoting 'tbe force per unit of volume, and the current per unit of area. I^t X, }', H now represent the oomponenta of the force referred to unit of volume, and u, r, la those of the current referred to unit of area. Hien, if S represents the section of the conductor, which wi- |shaU suppose small, the volume of the element dt will be Sd», and I, dx Hence, cqtuition (7) will become XSd* = S{vc-k6), {Equaclont nf Foroc.) (10) (C) ■ similarly T = ua— «c, and ^ = «{ — ffl. ^ Here X, Y, Z are the components of the eleetromngnetie foroo on ^fun element of a conductor diviiUtl by the volume of that element ; >, r, v arc the oomponents of the electric current throngh Uie element referred to unit of nren, and a, i, t arc the oomponents of the magnetic induction at the element, which arc itleo referred to unit of area. If the vector % reprosenbt in magnitude and direction the force < acting on unit of volumt- of thv couductor, and if (£ reprueents the ^■electric current flowing through it, B g = r.(£S. (11) ^^^^ft« •quMtmw (B) of An. DflS nuiy ba jinrfod by Ihe following m*thwl, derived ftttn ProfoMir M>i<t«S'4 HMBoir nn A [tynkmicol Theory of tbc ElcotrDiiuigtigtic VMd. Fkit Tnn^ ISS4. Tbe lime rartaUan of — ii nujr be Ukvii in two (]»rt>, imn of wlildi dopMiiU aiiil th« «lliv doa Bot depend on Uic motion of the irirouit. The latMr fon la deul; -/' ■if, ifO. ifff . I find the former let ua coniider >n tre It fonuins put of » dTciilt. *ni] let u* i-" tbie •!« t« move klong rula, which inav Iw uVnn u ;ianll*l, with velodcj * hoae oon^Mmcoit* m i, jf, k, the rart vF i)im i:in:ii>i livioK meanwhile (uppiieMl [■tolJiinBr;. We may then euppoeo Uut > mull pBnUc'ogrBin ia gUMnted bjr tb> [inovn^ an, Ui» ilinnlloa'Ctwioea oTtbe niunuJ to which are fly-mf U-n* n.J^-tfr *■'*''■•* nin»"" eirtn*' ».ift# a, Handle dfaectiau-«o*ianiifJlf and 6 Ulh«M>sUl'oliroon e uul >«. ELECTBOMAGiTETIO FIELD. ±0 B «ign« of X, ;i, >■ wo may put m — —1, r = t ; thejr then became 0. 0, - ..v'j oiryht to do with a riaht-luiidwi syalcin of alee. Now Itit a, h, c be the componenU of magDCbio inJitclJon we then hare, due to tbc motluD of Ii in time Sc If we tuppoee eftcb put of the circuit to move in a eimilu' manner the reaultaiit Mt vi'lU be the uiutiun of the circuit aa a whulu, the cuireats in tlie rails forming » I in each case of two adjacent arcs. The tiinu varialioQ of — p due tu tlu of the circuit is therefore — y{a(nj— Biil + twosimilai tenni} dt taken round the tnrcuit —J'iry — M) dx *■ two rimilar tenni. The result* in Art. fl02 for tbe componente of eledramagnetic force m»y bo dedims! li-oni the above exprestion for tp; viz. let the arc 3( b« dinpluced in the dJreetim I', m', n' through a diatance ti', then Sji -= {^(_cla — bB)^ > similar terms} tiS»'. Now let £ be the :e-camponeat of tbe farce pon the arc I, tliea for unit ouirenl we d bj Art, fifiS, rfx dp dt dx •« cai — bn,'] CHAPTER IX. GESBRAL EQUATIONS OP THE ELECTKO MAGNETIC FIELD. k 6OI1.] Xx our UiGOTOtical tlt^cussion of «Iectro(lynamios wo lic^n hy assuming that a ejiiiem of circuits cnrrj-ing vItK;tnc currcnU is a dynamic^] syatem, in which tlie currents mtiv l>e ro^mlvd as velocitiea, and in which tJifi coordiuaU-s corresponding lo thvse velocities do not them!U>]v«« appear in the oquationti. It followi; irora thin that tlie kinetic energy of UKiRVHtcra, ineofarasitdopemlc on the currento, is a homogeneous quadratic function of the currcnUi, in which the coeSicients ilciwiid only on the form and relative position of the cireuit.H. Assuming these coefliaents to be known, by experiment or otherwise, we deduced, by purely dynamical rca- sooing, the bws of the induction of currents, and of electromagnetic attraction. In this investigation we introduced the conceptions of the elect rokinetio energy of a system of currents, of the electro- magnetic momentum of a circuit, and of the mutual potential of two cirouits. We then proceeded to explore the field by mcansof various con- figurations of the eeeondary circuit, aud were thus led to the couception of a vector 91. having a determiuato magnitude and direction at any given point of the field. We called t.Iiit! vector the electromagnetic momi'ntum at that jKiint. This qtmntity may be conHidercd as the timo-intt^gral of the t-lcctro motive force wliich would ho produced at that point by the utiddcn removal of all the currentfi from the Sold. It it! idviiticnt with the quantity alrcjidy tnTeetigated in Art. 405 as the vector-potential of magnetic in- duction. Ibt components parallel to ic, y, and z are F, 0, and //. The electroma^ctic momentum of « drcuit i« the Iine>integral of tl round tlie circuit. W« then, by meant) of Theorem IV, Art. 24, Irausformei tlie liMI 230 GE.VEilAL EtiUATIOSS, [605. I he M »i1 linc-iotegral of 91 into tbe sarTace-intogral of aootber vector, ^1 vihoBe compoDbnta are a, i, c, and we found ihat the ]>henoineiu of induction Hue to motion of a coodtivtor, and tfaoee of elrttro- ma^ctic force can be expressed in t^'nus of >£. We gsre ta IB tiie namo of tbe Majfnctic induction, einoe it« properties are iden- tical witb those of tbe lines of magnetic induction as iaveetigat«d \>y Faraday. We aUo estnljIislRtl three eets of equations : tbe first set, (A), are those of inaguetic induction, «x])rc»8inn: it in terms of tbe elei> tromagnetic raoRientum. Tlio second set, (1)). aro those of electro- motive force, expressing it in terms of tbo motimi of the condoctor ucrnea the lines of magnetic induction, and of thu latv ofTariationj of the electrtjmagnetic moni«ntuni. The tbinl set, (C), are tbr cc|iiations of electromagnetic force, wcprtwsing it in terms of tbe current nnil the magnetic induetiott. i Tbe current in till tlicvc nu-cs is to be nudcrstootl as (he sctiul current, which includen not only the current of conductiou, but the_ crurrent due to variation of the cleetric diifjilacement. The magnetic induction 9 i^ the quantity nbiefa ire have Blreadji considered in Art. 400. In an unmngnet.ixed body it in identical with the force on a unit magnetic pole, but if the l>ody is ma^ m-tized, cither permanently or by induction, it io tin* force which wt^uld be exerted on a unit pole, ii' plaoeil in a narrow cTvvasse in the body, the walls of which are perpendicular to the direction of magnetization. Tlie components of ® are a, b, e. It follows from the equations (A), by which a, 6, c mw defined, that da di tk Thi* WW riuwB nt Art. 403 to be a property of tbe imiuction. c 606.] We have deiined the inaguetic force within a magnet, >^| distinguished from tbe magnetic induction, to be the force on a" unit polo placed in a narrow crevasse cut parallel to the direction of UMignetizntion. Tliis quantity is denoted by ^, and ita compoDGOta by 0, fi, y. Sec Art. 398. If 3 ia tbe intensity of magnetization, and A, S, C its com- poDcntd, then, by Art. 400, fi = ^ + 4sJ9, J (BiuatloM of M>««Mli«Uon.} (D)' ^ = '»- r defined, nugmo^i I H W« iwiy cull these the equations of magrnctization, and they HiniltcuiU- (hat iu the electxomagiietic Bvetem the matr'ietic indwction V S), ciniMdiTed tts a vector, is the eum, in the [I;imiltonun sense, of two viH-tow, the ma^netifi force ^, and the magnetiKatioQ 3 multi- plied by 4it, or S9 = ^ + 4ii3. la oertaiu «iib»1ance«, the magnetization depends on the mai^nclie force, and this is expressed hy the system of equations of indtiood ma^iHism frivco at Arts. 42G and 435. —^ GOfi] Up lo this point of oar investigution we have deduced Be\'iTything from purely dynnmicul eouaiderations, without any n-feroiici; to qiiantitittive experiments in electricity or ma^ctism. The only nst- we have miidi.' of experimental knowledge is to rc- cognifc, in the nhstrjicl quantities deduced from the theory, the t^nctete quantities discovered by experiment, and to donoti- tliem I by namex whieh imliaik: their physictil relations rather than their [ntatbemattcal ^-ncMition, In this way we have painted out the exidtonee of the electro- momentum 31 as a voetor who:^e dinntion and magnitude Dm one part of njuiec lo another, and froni thin we have deduced, by a mathematical proce-ttt, the magnetic induction, j*, as a derived veetor. We have not, however, olitained any data for _detennining either SI or © from the distribution of currents in the Por this purpoi»e we must find the mathematical connexion rfweiMt tbne qniuitities and the currents. We begin by admitting the existence of permanent magnets, the lutiml action of which satisfies the principle of the conservation of energy. We make no assumption with respect to the Inws of magnetic force except that which follows from this principle, nftinely, that the force acting on a magnetic pole must be capable of being derived from a potential. We then observe the action between currents and magnets, and ^■we iind that a cnrrent acts on a magnet in a manner apparently the same as another magnet would act if its strength, form, and position were properly adjusted, and that the magnet iict« on tlte etirrcnt in the same w^y as another current. These observations need not be supposed to be necompanied with actual m<.-ai«iireroenU of the foTOOB. They are not therefore to bo conBiilcn'<l a* furnishing ntunericsl data, but are useful only in suggesting queations for our consideration. He question these observations su^^'st is, whether the magnetic field produced by electric curreuta, as it is ainiUar to that [iroduced :we^ 232 MNEBAL EQUATIOm. [607. " Ity permanent magneU in many respects, rettembles it abo in beinjH related to a ])vt«nliAl ? ~ The. cvkiJ«Duc tli«t aa electric cireait produces, in tlie Kpooe cur* rouiiiling it, magnctiv cflV-ct« precisoly the name u tfaoae produced by a niiign<-tic mIivII buundetl by Uie circuit, baa been »tatcd In Art*. 482 *85. W« know that in the ca»e of th« nu^iietic Bh«U there is • potential, which has a dt-tcrtninato value for all jioinlii oulitide tl «ubBtttUC« of the shell, but that th« valueit of the ]x>teutjal at twtr' nvi^hbouriug points, on opi>oiiit« eidea of the shell, difler by a flniu quantity. If the macnetic field in the neig-hbourbood of nn electric current rcftemliles that in the Dei^bbourbood of a tnagnetio shell, the tnftgnetic potential, as fooud by a line-iate^iatioD of tJie magnetic force, will be the same for any two Unes of integration, provided one of these lines can be trangformLii into the other by continuous motion without cutting the oleetric current. ■ If, however, one lino of inttf-ration cannot be transformed into the other without ciittinfj the current, the linc'intcf^.kl of thij_ magnetic force along the one line will ditfer from that along tli otliL-r by a quantity dependitifj wti the strength of the current, Th^ magnetic potential due to au electric curn.>ut is therefore a fuuctic having an infinite Hcrics of values with a common difTereacei, particular vuIik' depending on the course of the line of integratioaj ^'ithin the siibntancc of the condiwtor, then is no such thing a ntngnetic jwleiitial. 607.] Asniming tliat the magnetic action of a current has a mo^iiotic |)otentJal of this kind, we proceed to exprvas this rwal^^ tnathematically. ^M In the itnrt place, the line-integral of the magnetic force round any closed curve is zero, provided the cloised curve does not suiTonnd the electric cnrront. In the next place, if the current paeees once, and only once, through the closed ciirvo in the positive direction, the line*tntegnl has a determinate value, which may he used as a meaaure of the strength of the ctirrt-nt. For if Utu closed curve altera its form in any continuous manner without cutting the current^ the lin^ integral will remain the same. In electromagnetic measure, the line-iutegml of tlw mAgoe foroc round a olnsttd curve is numerically e<iual to tb« cuiraol tlirough the closed curve mulliplicd by iv. ET.ECTBIC CURRENTS. If WO take for the olostHl curvu tho psrallelo^m vrho«c niim are tljf and ilz, iha liue'inU-g-nil of the mag-iietic force ronud th« ad \{ u, e, «t aro Uie comjioncntB of the flow of electricity, the irrent through thp {Uiralk-lo^rram is Mii!ti(i]ymg this hy 4ir, and equating the result to the line- ategral, we ohtain the er|iintion dy dz ith tbi' similnr oquations 4a dy at dx d& da ■\-nw= J --i dx dy (BquBtion* of Bloetrio Cumntf.) (E) Jwhco Icondi Fhioh determine tho mafiiiitutle and direction of the electric cnrrente tJiu mu^nctic force at every point \» given. When tliere is no current, these equations are equivaleDt to the lition that adx\^ds->rydz=-hQ., >r (hat the magnetic force is derivable from a magnetic potential I all (mints of the lield wht-re there are no currents. By dilTeretitiating the equations (E) with n-spect to a-, j, and * rcsp«:tive)y, and adding the results, wv obtain the e([uation iu dv ^"^ _n di'*'d^'*"dl~^' i» that the current whose cdmpoiieiits are u, v, le is the eondition of motion of an iiici'mprcssihle fluid, and that it must nect«»irily How in closed circuitM. Thia equation is true only if wc take u, v, and ir as the com- ta of that oledfie flow which is due to the variation of electric displacement as well n* ti> true conduction. We have verj* little experimental evidence relating to the direct electromagnetic action of cumcDta due to the variation of electric diiplaoement in dielectriea, hut the extreme difliculty of reconciling the laws of elect romagoetjsm with the existence of electric currents which are not clowed i» one ncaM^n among many why we mu«t admit the exi.-'teQce of transient currviit» due to the variation of displuce- menl. Itieir importance will be seen when we come to the olectro- ^ Tht ^^Msnen ^magai etic theory of light. GEKEnAT, EQFATIOyS. f6o& 608.] We have now d«t«nnin«J the n-Utioni of th* princiial (|uantitiefl cx>iicemed in fhc pliouomctiu dii^ovcreil hy Orctod, Am- pirc, Kod Faraday. To coiinwt tlicw" with the [ilK>iinniL''na <U«cribctl in tlic former parts of thi» treatiNe, Home additioiuil rclutions ■« jwcvesary. When electromotive force sc-Ik od a material body, it gimduce* in it two electric&l effects, called by Faraday Induction and Con- duction, the 6rst being moat conspicuous in dielectiics, and tlie socoiid in conductors. In this treatise, static electric induction is mea&ured by what we iutvv cuUvd the electric <!i8plaoement, a directed quantity or vector which wc IiavD denoted by X, and its components by J", g, h. In isotrupic suliatance^ the displacement is in the same direction 0.1 the electromotive force which produces it, and is proportional to it, at leiut for small values of this force. This nay be expressed by the vquatioa , ^ . , _. . iT> «'fe (Eqn»U"n of Eleotno /p. where A' i« the dielectric capacity of the snbstanoe. See Art. 6!>. In «ubstanccs which ore not iiotropio, the components/;^, k of tlic electric displacement £ are liiieur functions of the comjwnents P, (2> R of the electromotive force ^. The form of the (^nations uf electric diaptacement is simiUr to thiit of the (spiationx of conduction lut given in Art. 296. Thene relations may be cxprevscd by xaying that K is, in isotmpic bodies, a scalar quautity, but in other bodies it is a linear and v function, operating' on the vector (S. 609.] The other dfcct of electromotive force is oondnotion. laws of conduction as the result of electromotive force were esta-i blished by Ohm, and an> explained in the second part of Uii treatise, Art. 2-11. They may be summed up in the equation where S is the intensity of the electromotive force at the poin j[ is the density of tJie current of wind iicl ion, tlie oomj^neuts which are p, j, r , and C is the conductivity of the Bubst-inco, wbich^ in the case of isotropic subefances, ta a simple scalar quantity, but in other mbstanecs becomes s linear and vectAf function operatin]^ on the vector 9. The form of this function is given in eoordinatcs in Art. 29B. 610,] One of the chief poculiaritien of thi» treatise is the doctri which it Bseeits, that the true electric currvnt (5i thnt on which I I ^ 614] CUREESTS OP niSPLACEMEKT. 235 lectromngDetio phenomena dctpend, ut not the same thing- as j^, the current of conduction, but tluit th« time-Tariution of 'S>, the elet^trio displacement, must he taken into aocount in estimating tlie total oiovement of electricity, so that we miiKt write, 6 = ft + S), (BiumW uTTVub Cumiilt.) (H) r, in terms of the components, do ("*j ill,] Since toth St and [D deiwnil on Uie electromotive force 6, twe may express tlie true current 6 in terms of the electromotive force, thus , 1 j ^ (I*) »r, in the coeo in which C and A' are constants, Alt dl 612.] The volume-dcneity of the free electricity at nny point u fon»d from the components of electric displacement liy the ,«l'»t'«i> df da . dA P-di + d^ + S- fJ) 618. The surface-density of electricity is <- = f/+ «,!f + nJi + lT + *«'/ 4- n'A', (K) (where i, m, n are the directioa-eosinea of the normal drawn from the xiirfnce into the mcdinm ia which f,g,h are the eomponenta of I dinpliiecnient, and V, m', n' are those of the normal drawn from I Burfiict- into the medium in which they arc f, y', A', 614.] When the m^netizaticn of the medium is entirely induced liy the magnetic force acting on it, we may write tlie equation of 'ioiluced magTM-tiuttion, @ = uJS, (L) where fi IK the coefficient of ms^etic pennouhility, which may bo considered a scalar quantity, or a linear and vector fnnetioa ittn^ on ^, according ns the medium is itotropio or not. 236 GENERAL EQCATIOKS. [615- < 615.] Tliiise may be rcgatded aa the principal reUtioos anions the (juanUties we have bvcn oonsidering. T1i«y may b« combined so a» to eliminate Aomo of tboM quantities, but our oltjcct at prwral is Dot to oI>t«in oompftctAeM in tbe mathenuit iuil Torniutac, \M to ezpren every relation of which wc have any knowledge. To eliminittv a quantity which ex]irc»tc:t a uovftil idea woulii b« ntber u lo9S ilian a gain in this stugv of our cni]uiry. There is one remit, however, which we may obtain by cnmhining equations (A) and (K), and which is of ver}' jfreat importance. Mi If we suppcute that no nutlets exist in the Beid except in the^ form of electric circuits, the distinction which wc have hitherto maintained between the magnetic force and the ma^E^^etic inductioa vauiehea. because it is only in magnetized mattor that these quan- tities tlifier from each other. According to Ampere's hypothesis, which will be esplained in Art. 633, tbe properties of what we call magnetized mattor are dne to motec-ular electric circaits, so that it is only when we regard th* Hubstaiico in large masses that oar tiieory of ma^etixation ii upplictibtc, and if our matlit'matical methods arc supposed capable of taking account of what goes on within the individual molocolcK, they will discover nothing but electric circuits, and wc ehntl find the magnetic force and the magnetic induction everywhere identical. In order, however, to be able to make uec of the olectrOKtatic or of the elect romagnetiu system of meneurcmcnt at pleasure wc shall retain the coolTtcicnt fi, remembering that its v«luc is unity in Uie| electromagnetic n'stom. 616.] The components of the ma^etic induction are by e^po* taoDS^A), Art. 691, The oomponents of tho electric current are by equations (E) Art. 607, given by Jy 4^ Ja dy m dO is' ' Hx dP dff 'di' dx dO dF Ji' "^ ivm 4vv dfi *»" = ^-^ da If wo WTit« I may write equation (1), imilarl}'. If we write \\iien T '\% the ilJEtance of the given ptiint from the element (jJ,^,*) tb« intcgratioiu arc to be extemktl over ull space, then (ix The qnnntitj x di«tp])onrs from the eqiiatioiis (A), anil it \» not t\»iti to any phvsiotl phenomenon. If we Eiipjtojte it to he ten rywhcre, .' will also bft wro cvetywhoro, an<i equatioDi (5), litting the accmtit, will give the true ralnn of tho component* ■81. ' lb* nogitir* flifn I* vaplojtA )i«r# in nttlar t» duJ(o oiu cxpttaMMU cgmiBleat *i libcw in vhicb (juatoniMH m vaifiojfL 617.] Wo may thcrorore adopt, as a dofinitioa of $(, ihat, is the vector-potential of the electric current, standing: in the sam^ relation to the eWtric current that the scalar potential staods ta tJie matter of which it is the pot«Dtial, and obtained by a sinuhr process of integration, which may be thus deecrJbcd: — 1-Vom a given point let a vector be drawn, rcprcsontinj^ in nttg- nitiide and direction a given element of an electric current, dividrd by the namerical value of the distance of tlie clement From t ^ven point. Let this b« done for «very olcmont of the i current. Tlie rcsnltant of all the vectors thus found is the tial of th« wliole current. Since the current is a Tcctor qnaotit it* priteiittal is also n v<«tor. Sec Art. 122. When the distribution of electric currants is given, there is one, and only one, diiitribution of the values of 91, sncb Uiat !l iit ereiy- where finite and eontinuouf, and mtiitfieG the c<|nattODs V»91 = *s/*g, S.r9l = 0. and vnniHhe» at on infinite distance from the electric system. Tbii value i» that given by equations (5), which may be written I ^ = p.jjj^dxdfdt. QuatfrnioK Erpre«tloiu/or the EUciromagnetie EfHatioMt. 618.] In this tnatise we have endeavoured to avoid any procew demanding from the reader a knowledge of the Calculus of Una- lernions. At the tame time we have not scrupled to introduce the idea of a vector when it was noceteary to do so. When vre have had occasion to denote a vector by a symbol, we have taetl a German letter, the number of dilTerant vectois being so ^reat that Hamilton's favourite synibots would liavc been cxhangted at ont^'. Whenever theraforo, a Oermao letter is used it denotes a Ilamil- tonian vector, and indicates not only its magnitude hut its direction. The constituents of a vector are denoted by Roman or Greek letters. The principal vectors which we have to conntler are Vcctw OcAiiitiiifntei The radius vector of a point p * f * The electromagnetic momentum at a pMnt M P G B Tlic magnetic induction B air The (total) electric ctirrent Q ■ v k The electric displacement J) f f k 6 1 9.] oWtbhsi^^Spr^ioss^^^^^ 238 The cWtromotivo force II P Q R The mectiaiiical forou % XTZ Tbc v«lovitj' of a point ........................ & ov p x g i The mu|>nctie forci<! ......iii i... ^ a j3 y The infousit^- of nmsnctiJiTttioo 3 ABC The current of oomiuclKiii ft P J ^ Wv have also the foIluwiDg t>t:alur fundions ; The electric {iot«nliiil +. ] Tbv tnagTietic iioteiitinl (wbovc it exiet«) 11. ■Hie I'tcctric dciigity e. The <]<!nsitf of mujfiielie 'mutter' m. Beei<]cs these wc hnve the foIiowiD)> qiiftntities, indicating; physical tropertiee of tho meditim at each point : — C, the conductivity for electric currents. K, the dielectric inductive capacity. fi, the magnetic inductive capacity. Tlieee quaotitieB are, in isotropic media, mere scalar functions f p, but in general they are linear and vector operat«» on the rector functions to which they are applied. A' and n are certainly . always self>conjiif^te, and C ie probably bo also. 619.] The equaUons (A) of magnetic induction, of which the ~ dg dz hfl written 9J = TvM, Fwhero V \% the operator .d .d , d 'T^-^^dy-'^dB' V indi«at«s tliat the vector part of the result of this operation ' is to be taken. Since 91 \* i^nbject to the condition SV9I = 0, V% is a pur« TCetor, and the symbol V is unneccesary. ^TIw equations (B) of electromotive force, of which the first in o . ,. dF d* ^u e = r«s-«i-v+. no oquationa (C) of mechanical force, of which the first is V . J* dil ax dx ^become g = FfS^-eV'i'-mVa. GEITERAL EQUATIONS. Ifl Ions (D) of magnetization, of which the first is a = a +AvA, tecome S =^ + 4Tr3. The equations (E) of electric currents, of which the first is become 47iS=: W^. The equation of the current of conduction is, by Ohm's Law, St=Cik. That of electric displaeemi The equation of the total cnrrci , arising from the variatioD of the electric dieplacement as well as rom conduction, is When the magnetization arises frcm magnetic induction. We have also, to determine the electric volume-density, To determine the magnetic volume-density, m = SV% When the magnetic force can be derived from a potential CHAPTER X. DIMENSIONS OP ELECTRIC UNITS. S20.] EvKnT clcetromajjnetic quantity may be ilelincd with reference to the fundiimcDtul units of Length, Mass. nnd Tirav. If we be^n with Uic Ht-finition of (In? unit of electricity, « {jiven in Art. 65, we tnny obtain definitions of tho units of every other electromagnetic quantity, in virtue of the equations into wliich they enter alon^ with quiintitie^ of olettncity. The syvtem of onitB thns obtained i» culk-0 the EK>ctrost»tic SyKtein. If, on the other hand, we hei^:in with the definition of iJie unit mn^etic pole, as given in Art. 374, we obtain a different syetcta of units of the suue set of quantities, 'litis system of units is not consistent with the former 8yBt«ni, and is called the Electro- I nweoetic System, ■ We shall bc^^in by stating those relations between the different units which arc common to both systems, and we shull then form ■a table of the (limensione of the units according to each system. ^1.] We shall arrange the primary quantities which we have to conxider in jiairs. In the first three pairs, the product of tlie ^Iwo quantities in each pair is a quantity of cncr^ or work. In Bihe vecoiid thn'c pairs, the product of each pair ie a quautity of ene^y referred to unit of volunw. FUUT TURKB PaiB8. Eleetrottalie Pair. Symbol . (1) Quantity of electricity ...... e (2) Line-integral of electromotive forco, or electric po- tential S rouu. B 242 DINIKSIONS 0? tflfira [623 (3) Quafitity or free nugoeU*!!), or strength or a pole . ■> (4) Mkgnctic potential EltrfrokiKctU Pair. (5) Ei«otrokmet)c momeDtuot of a circuit . (6) Electric current a f c StxoKD Tnwa Paibs. EUftrOttatic Pair. (7) EU^nc displacement (mvararcd by KurraceMleDsity). 2) (8) Electromotive force at a point . . . -Si Maynttie Pair. (9) Ma^etic indiietion ■ Q (10) Magnetic forcx; , ^ ElectrokimHie Pair. (11) Intensity of electric current at a point . . S (12) Vector potential of electric currenU . . . fl 622.] The following relations exist betveea tbe«e qi In Uie Gr^ place, since the dimension* of mtxgy mm ~J^ I , and [M ~[ -jnli *vfl Iuiv« the [*i] =[-.a] = [fC] = [^]. [i;s] = t«*] = [«a] = [-^]. Secondly, since e, p and 9( arc the time-integnk <^ C, £, and G "^'"''' ra=[S]=[l]=tn (« nirOly, since i^ Q, and j» are the lin^^.il»tegralH of (J, ^, aod M reepectively, rj?-] rO-i r^-i Fioally. since e, C, and m are tlte sorfaee-integnb or 3>, Q, and TRB TWO HTSTBHS OF UNITS. 243 6!!3.] These fift««D eqnationH are not independent, and in order I to deduce the dimeDBions of the twelve nnita involved, we require one additional equation. If, however, we take either ^ or >» as an independent unit, we can deduce the dimenEions of tlie rest in r terms of either of theee. « ^-1 =[f]=B]- (3) ™i (5) [,] = W = [^] = [«]. (9) (10) (IS) m -m-{ir\ 624.] The relations of the 6r3t ten of these qnantitiea may be exhibited by means of the following arniDgement : — <f, 3), ■§, C and a. m and p, ^ d, £ Jf IS, ©, Mand;5. C and li, ^, 21, c. The quantitiex in the first line are dcriv<'d from e by the same oiKTations aa the eorren ponding quniilitleM in the second line are derived from m. It will be seen that the order o( the quantities in the first line is exactly the reverse of the order in the second line. The Grat four of each line have the first symbol in the nnmerator. The second four in each line have it in the dciio- K minator. H All tho relations given above are troe whatever system of unit« Hw« adopt. ^^■025.] The only eyet^ma of any scientific value are the electro- ^HRk40 and the elL-etrontu^elic system. The clectrosbitic system is H 244 IJIMESSIOSS OP DKlTa. [626.1 founded on the definition of tlw unit of electricity, Arts. 41, 49, | and niAy bt- doduoed Irani the equation whieh enpresaes that the reBiiltant fopcu S «t any point, d«e to tlie action of a quantity of electricity ^ at a dintance L, is found l>y dividing e by L*. Suhittitutiii^ tho oqtnition* of dimension (I) Bii<I (8), wcfind rUfi r e -\ r m T r If] in Uie elcctro^Utic system. The electrom^netic system is founded on a preiriKcIy fiiniilir definition of the unit of utrengih of a magnetic potc, Art. 371, leading to tho equation ta whence [^] = [^]- [fr^J^M" and [f] = [i'i/l]. W = [i*-3f *r-'], in the electrama^ictic system. From these results we find the dimcn!«i«ns of tlie other quaatittca. 626.J Ta&U ^ Dimatnoiu, Diamuiant in Quantity of electricity .... * [f*,!/'!!'-'] £Z.*ifl]. Quantity of magnHism j Electrokinctic momentum [ . j"! [L^M^] [i^J/tr-'J. of a circuit ) " Electric current > iC ) ,,i„i„., ,. ...,„.. Magnetic potential} U] [i'^'?"'] [i*.W*r-']. Electric dieplaceroentl ~ 7Ti,,>-. ., .. ■ .. SnrfaccHlcn-ity } * ' " ' ® [i"*^*?"'] t^-«i/»l- Eleclroiootivo force at a point S [A-LV*?"-'] [Z*Jf*J^. Magnetic induction 89 [J^''JW^'] [-£-t.l/*r-'].| Magnelic force ^ [UMiT-'] [i.-*ir*r-'].l Strength of current at a poiat fi [^-iJ/Jr-'J [£"'jtf*7'-'l Vector potential V [Z-*Jtf*] [iiifir-']. 6fl8.] TABLE OP DIMENSIONS. 246 627.] We hftve strtttdf Mnsidored llic products of the pairs of tbeae quantities in the order ia which tliey .it:iiid, Tlieir mtios are I in certain cases of scientific importance. Thns GlMbwUtio EteotTCma4(iiDtia l-j^ = cnpacity of an accumulator ' cocAiciont of eelf-induction of a circuit, or oloctro magnetic capacity _ t spocific inductive capacit}* l Fe ~ i of dielectric J ■ ^=1 :( e = magnetic inductive capacity . j^ = resistance of a conductor . d _^ ( spccilic rGMstiinoe of a 1 Symbol, ■ ? Bjitom. .yj. L [LI . K [«] • f* to]- , R ■T- ■L' r [7] ttaic ...I, ' 63S.] If the units of length, mass, and time are the xanie in the two systems, the number of ek-ctrogtatic units of electricity con- uDed in one clcctromafjnetic unit is numerically equal to a certain '^velociiy, the alwoUilc value of which docs not depend on the magnitude of Uio rundamental units employed. This velocity is f important physical tjuaiitity, which wc shall denote by the mbol r. Nnmier of EUctntttatie Unitt in one Meetromagnetie Vnit. ¥tyte,C.Q;t>,^.^ p. T9fm,p,E,^,f&,% .V For electrostatic capacity, dielectric inductive capacity, and con- ductivity, H. For electromagnetic capacity, magnetic inductive capacity, and siftance, -f Several methods of determining the velocity v will be given in 768-780. In the electrostatie cyiiteni tlio KpeeiRc dielectric inductive capa- city of air IN amumcd e<]ual to unity. This quantity is therefore ented by -j in the eleotromagnetic system. 246 DIMENSIOSS OP WXIT9. [629. In th« elect romaguotio Ej'stem the spcdfic nuigtictio inductin capacity of air is aseumoil equal to unity. Thin (iiianttty is tbcn^ fore represented by -^ in the electrostatic ttystem. M < Practical Sj/atem of EUctrie UmU. 629. 1 or the two syxtems of units, tlie electromagnetic iti of the greater use to those practical electrioiaus who are oooupied with electromagnetic telegraphs. Jf, however, the units of length, time, and ma&s are those commonly used in other scientific work, soch ■a the m^tre or the centimetre, the second, and the •^tamme, the units of rcsietancc and of oWtromotire foitw will Iw eu small thut to express the (jiiuntiticH occurrlu); in practice enormouii numluT* must he used, and thct units of qimntity and cttiiacity will be W large that only cxcwdinfily xmall fmetioiiii of them can ever occur in practici?. Priictiail vleotrioiaiis have tlierefore tuloptetl « «ct of eU^ctrical unit* deduced by the electromagnetic flynteni from u large unit of length and a small unit of ma:is. The unit of length used for this purpose is ten million of m^lnt^' or approximately the length of a qoadrant of a meridian of earth. The unit of time is. as before, one second. The unit of mass is 10'" gramme, or one hundred millionth part of a millipnimmc. Tlie elci'.trimi units derived Tioni these fundamental units hav beCD named after eminent elt-etricul discovemri. Thus the praetif unit of renistance is called the Ohm, and is ivpmented by the rcsintanoe-coil issued by the British Association, and dencribed in Art. 340. It is expressed in the electromBgDetic ayKtcm by a velocity of 10,000,000 metres per second. fl The practical miit of electromotive forc« is oalled the A'^pII, and is not very diflcrvnt from that of a Daniell's odL Mr. Latimer. Clark liaa recently invented a very constant cell, whose eU motive force is almost exactly ].(57 Voltw. The practical unit of capaeity is called the Famd. The qraintitj of clcetrioity which flows through one Ohm under the electrorooti^-e force of line Volt during om; second, is equal to the charge produci-d in » condenser whoee cajwoity is ono Farad by an cleetnamotive, force of one Volt. The us« of these names i^ f<mnd lo be more «oDTeBient in practiii than the coDstant repetition of the words ' dMttonuguetic units,' ^-4 atimer « uintit^ 629] PBACTICAL UKITS. 247 with the ftdditional statement of the ptuticular fundamental unite on which they ore founded. When yery large quantitieH are to be measured, a large unit U formed hy multiplying the original unit by one million, and placing before its name the prefix mega. In like manner by prefixing micro a small unit is formed, one millionth of the original unit. The following table gives the values of these practical units in the different systems which have been at various times adopted. FViniAIOU'TlL UVITB. FsAoncAL Ststu. B.A. RsroBT, 16SS. Thohson. Vises. Ill Eartk'i Quadruitt, Stetmd, 10-" OramiM. Mart. Setond, Gmmme, Cudinulrf, Second, Oramvie, MiUimare, Second, MiUignimnie. Bemitmnoe Ckpacitj Qumntlty Ohm Volt Fankd F»nd (ohkrgedtokVtdt.} 10' 10-' lO-* lO" 10* 10-' 10-' 10" 10" lO-" 10 CHAPTER Xr. ON BSB80V AH1> STRESS IN THE ELECTROIlAOSBriC FIBLD. EleetroHatic Enetyy. 630.] The «acrgy of the ejrsteco mxy be divided into Uie Potential Energy and the Kinetic Kncrgy. The |>ot(>ntial energy due to electrification haa been ilre&dy coB-l sidored in Art. 85. It may bo written r=}s(«*). (I) where e is the charge of electrieity at a p?aee where the clcctrio' jKitentiiil is ^, and the Kiimmution '\t to bo oxteaded to evcij y\»et where there is electrifi cation. If y. S' ^ 1*^'' ^^'^ components of the elwilric di^lncemeDtj tbaj quantity of electricity in the element of volnme dxdydi is in itnd -'^dxdj/dt, where the integration is to be extended throughout all spaee. 631.] Integrating this expression by parts, and remembering that when the distance, r, from a given point of a &aite eWtrified )<y)itein becomes infinite, the potential '!> becomes an infinitely small quantity of the order r"', and that/, </, i bwomc infinitely small quantities of the older r^', the expreiwion in reiluccd to where tbi^ integration is to be extended throughout all space. If we now write P, Q, R for the oomponeuts of the clectromotiTe i/4' ti"^ force, iastood of — ^ , — <tt 'ty and — -i- , we find r = \fjf{P/+Qfi + RA)dt<fy'tt. m 3-] 240 [Hence, the electrostatic oncrgy of the whole 6eW will l>e the «ame we Biippose thjit it reeiJcs in every part of the field where fiW- bncnl forci." nnd ck'ctrical (lis placement occur, instead of being conliiictl 1o the places where free cicctricitj' is found. Ttw! encrpry in unit of volume is half the product of the electro- motive feroi- And the eU-ctric dieplaectncnt, multiplied by the cosine of the un^lo which these vectors incltide. tin Quaternion languag'u it is — 16'@£. 1 Magnetic Energy. * 632.] We may treat the energy due to magnetization in a w»y ilar to that pursued in the case of electrification. iVrt. 88. If B, C an the components of maj^Detization and «, ft y the componetits of ma^etic forc«, the potential eoei^y of the eyet«m of magnets is then, by Art. 389, I ~\Jff(Aa+Sff+Cy) d3>dyds. (»} iC integration being extended over the apace occupied by mag- netized niatt«r. This part of the energy, however, will be included in the kinetic energy in the form in which we ahall presently obtain it. 633.] We may transform this expression when there are no elec- tric currents by the following method. We know that ia db de Henoc, by Art.97, if do. ^ da. da ... »=-.u--' ^=--df' ^=-df ('> if alwayB the case in magnetic phenomena where there are no currents, rrr , , i jJJ{aa + 6^+cy)d^dydz = Q. (9) the integral being extended tlirooghout all space, or ^ffJ{{a + i^A)a■^(8 + ivJi)fi + {y+4l,C)y)dxJyd^^^). (10) Hence, the energy due to a magnetic aystem ^kJJfi^«+^P + CY)drdyd: = ~^JjJ(<,*i-^^ + y^)djcdfds, • Sm Appandix 1 •! tbe and of tbij Chnfltt. j S60 SXBBOr AND STRESS. [634, ^K SlfCiroJkiiutie Snerpg. fl F 6S4.] We have already, in Art. 578, expressed the kinetic eiMfgf I of a sj'fitcin of mrrenta in the form T^nipi), (12) nlivrc p is the electromagnetic motnentnni of a circuit, and i ii the strength of the curpcnt Jton-ing rouD<l it, nod the BOmaiatiaa extends to nil the circuits. But w'l; have proved, in Art. 690, that p majr be expressed « a linc-integral of the form where F, G, U arc the componente of the elect ramof^etio mo- I meutum, 91, at the point {xt/i), and the int«^mti<in is to be ex- tended round the closed circuit *. We therefore find I If H, t>, IP are the components of th« densily of the cnrrent at any point of the conducting: circuit, and if S 'i» the tniti«Tcr«e ^^ Motion of the circuit, then we miy- write " 'S = "* '% = "- '%=•"■ ('.) and wc niny utso writ« the volume Sdi = dxdgdz, and we now find r= iJJJ(FM+Gv+fftf)dxtlfd:. (16) where the integration is to be extended to every part of apac* where there are i>lecirio currents. 635.] Let us now substitute for u, r, v tiuAr values an given by the equations of electric currents (E), Art. 007, in terms of the compoueuts o, ^, y of the ms^ctic force. We then have where the integration is extended over a portion of space including all the currents. If we integrate this by part«j and remcinber that, at * gn-aL distance r from the system, a, 0, and y are of the order of mug- nitude r~*, we find that when the inta'^ntiou is ext^ndi-d tbro ngby oat all Bpoeci, the cxgiresston is rvdoocd to BLBCTROKINETtC ESEROT. 251 I I I By tlie equations (A), Art, &91, of mo^ctiv induction, we may flubstitate for the quiintiticii in »mall bnickets tlic cotnponeots of tna^etic induction a, b, c, so tliut tlic kinetic cni>r^y may be written . ^,-j- ^'^ {'^jjj{«^'rb&-\-cy)d;cdsdt, (19) wliero tbe inte^n^tioa is to be extended throuf^hout every part of >pac« in which the magnetic furce and mu^netic intlnction have va1u4-H differing from zero. 'File quantity within bmckete in this «ipre:j!9ion is th« product of the magnetic induction into the resolved part of the mn^rnvtic focct^ Id ita own diix'cUon. In the laiiguagL' of qiiiitirnionM this may be written more simply, where 35 is the magnetic induction, whoee components are ft, h, e, and ^ is the magnclic force, whose components are o, ^. y. 63&] The eleotrokinetic energy of the sy&tem may therefore lie •xpnssed either aa an inte^^l to he taken where there are electric currents, or aa an integral to be taken over every part of the field in which magnetic force exists. The first int^'gral, however, is the natural expression of the theory which mipjiOBcs the currents to act upon each other directly at a distnnecj while Ihe Bwy)nd is appro- priate to the theory which endeavours to explain the action between (he Gurrente by means of some intermediate action in th« space between them. As in this treatise wc have adopted the latter method of investigation, we naturally adopt the second exprt^ion as giving the most signihcant form to the kinetjc energy. According to our hypothesis, we assume the kinetic energy to exist wherever there is magnetic force, that is, in gt'nvral, in every part of the field. The amount of this cnergj' per unit of volume ig — — 5®$, and this energy exists in the form of some kind of motion of the matter in every portion ufupace. When we come to consider Fumday'it discovery of the cfTcet of msgnetisnt on polarized light, we shall jioint onl reasons for be- lieving that wherever there are lines of magnetic force, there i» n rotfttory motion of matter round those lines. Sec Art. 821. Maffvffic and Kleclroiiitetie KHtrffj/ compared. 6S7.] We found in Art. 423 that the mutual potential enei^ m ETTEBOT AKD STRESS. [63». <^ two mnguctic shells, <^ stivn^lhs ^ anJ ^', imd boiiDded hy tht doeod curves « and »' wepcctively, is -**'//" C08« ■forf/. where < is the angle between the directions of dt and A", and r , IB the distance between them. fl We alao found in Art. 521 that the mutual en«i^ of two cipcnit^^ « and /, in which current* i and i' flow, is •■'-//^ tbd/. le vor>1 If i, I are equal to ^, 0' respeotivelj-, the ni«haniea! between the magnetic shells is equal to that between the res|)on<ling cleotrio circuits, and in the same direction. In the oue of the magnetic eliells, the force t«nds to diminish their mutual potential enerjjy. in the cfflsfl of the circuits it tends to increase i mutual energ'y, because this energy is kinetic. It is impossible, by any arrangement of mngnctized matter, to produce a system corresponding iu a,\\ respects t« an electric circuit for the potential of the magnetic system is single valitod at eve point of space, whercus that of the eU-etric s}'stcm is many-valued. But it is always ]>o8siblc, by n proper urningrrocnt of iutinitety small electric circuits, to produce a syifteni corresponding in all rcHi>cctH (o any magnetic sytitcm, provided the line of integration which we follow in calculating the pot^^ntiat iit prevented fmrn^ passing through any of thet« amall circuits. This will be mc fuliy explained in Art. 833. The action of maguel* at a distance i* perfectly identical with? that of electric currente. We tht^rcfore endeavour to trace both to the same cause, and since we cannot explain electrto curreat by means of magnets, we must adopt the other alternative, and ' explain magnets by means of molecular electric currents. 638.] In our investigation of magnetic phenomena, in Part III of thia treatise, wo made no attempt to account for magnetic action at a distance, but tn-atcd this action as a fundamental fact of experience. We tlicreforo a«5iimcd that the energy of a mngnelio ■ fVKtcni is |>Dtential energy, and tluit thi« energy is dmin'uked whCB the parts of the system yield to the magnetic forces which aot on them. If, however, we rt^rd magnets as deriving their propertiea froai* eteetric currents circttlating within their moleooles, their AUPRRirs TnitonY op maosets. I kinetic, and the foitie between tliem is such that it tendii (o move them in a direction such that it' the strengtha of the currcntii ^^were miiintained constant the kinetic energy would incrf.a.*e. ^R This mode of explainin;> mat^etism requires us also to ahandon ^Vtlio method followed in Purt III, in which we rejrarded the magnet ^ as a continuous tmd humogx;ncous Lody, the minut4.'et pact of which has Diabetic projHTties of the eamu kind as the whole. Wo mn»t now rctjiini a itm-jrict as containing n Einit<^, tbon^h ^ very grent, number of electric eircnit«, so that it has essentially HiB molecular, ax diiiting^uixluH] from a contiuiiuiis Gti'ucturc. If we cuppose our mathematical machinery to he so coarw; that Aur line of integration cannot thread a molecubr eirouit, and tliat •n immense number of magnetic molecules are contained in our element of volume, we shall still arrive at results simiUr to thoso of Part HI, but if we suppose our machinery of a Gner order, and capable of inveatigating all that goes on in the interior of the molecules, we must give up the old theory of magnetism, and adopt that of Ampfcre, which admits of no magnets except those which coastst of electric currents. H Wit must also regard both magnetic and electromagiie^c energy Has kinetic enci^, and wo must attribute to it the proper sign, Hm given in Art. G35. ^P In whAt follows, though we may occasionally, as in Art. 639, &c., ■ttcmpt to carry out the old thcorj' of ma^notitim, wc kIiiiII find •that we obtain a perfectly consist*-ut ByMt<;m only when wo abandon that theory and adopt Ampere's tJtcory of molecular current*, as in Art 1544. The energy of the field therefore consists of two parts only, the eleotroetatic or potential energy and the electromagnetic or kinetic energy ' = sV//< 4 + J ,8 + c y) rfx 1^ th. I OH TUB PORCeS Wllica ACT OS AK KI.F.MRNT QV A BOOT PI-AOKD IN THB ELBCTROMdGSCTIC HF.IJ*. Forces actiitif oti a Magnetic Element. *630.] Tte potential enfliify of th« element itxtlfd: of a lx>dy ma^Detized with an intensity wliovc components arc A, B, C, and ■ Sm ApfwaiUx II kt tli« and of tlib CbipUr. EVEBOT A>-D STRZ8K. [64a plnc«(] in n field of msgnotio force whoae components arc a, Honce, If the force urging the element to move withoat rotat .in tJie direction off is Xjdjrdj/dz, -^-■*Si + ^^ + ^^' (I) and if the moment of the couple tending to turn the elemoot about the axis of k from^ towards z is Ldjtiydt, Z = Sy-C0. (!) The forces and the momenta corresponding to the axes of f and t may be written down by making the proper siihstitutions. 640.] If tlie magnefiBed body carries an elrctric current, of which the components are u, v, ie, tixn, by etiuntions C, Art. 603, there will be an additional electromagnetic force whovv components are J^, I'„ /j, of which S^ is ^1 H<fi0^ the total force, X, arising from tlie nMfglMttiBlQ of tie mol^ale, as well as the current passing through it, it da ax dx ,.^ The quantities a, h, e are the componeDta of mogDOtic inductioii, and are related to a, fi, y, the components of Boagnetic force, by the ex^natioDs given in Art. 40O, A = a + 4 nA, b = p+ivB,\ (5) c = y + inC. The components of the current, m, r. k, can be eipreased in tcrmi ofa, 0, Y hy tJic equations of Art. 007, dy M Hence da at ^» A-aie = -f — ax di' da ,] THEORY OF STRRS8. 265 By Art. m. | + | + g = o. (8) IklnlUplyin^ tliijt liquation, (8), hy a, and dividing- by 4 v, we may Ittdd the result to (7), and we find atso. by (2). I = ^((i-^) y-(c-y)i3). (10) = J_(8j,_.e), (11) fbere X ia the force rrfen(.'d to unit of volume i» llie directioa of p, iml L ill the moment of the I'ui-ces iiliout this axix. Om tkt Bxplanalion ofihete Foreea hi/ the Jlj/polkesU of a Medium in a Stale of Stress. 64] .] Let us denote a stress of any kind rofciTed to unit nf area [by a vymhol of th« form /*„, where the first suffix, ^, indicate)) that tlie tionnnl to the enrfaec on which the stress is EUpj)i]se<I to act is pwrsllcl to the axtit of k, and the second stiQix, ^, indicat«s that the direction of the stress with which the jiart of the body od the positive iiidt! of the surface acts on tlie part un the negative nde is parallel to the flxta of k. The directions of h and k may bo the same, in which case the ' stress ia a normal stress. Tliey may be oblique to eaeli other, in which case the stress is au oblifjue stress, or they niay be perpen- dicalar to each other, in which case the stress ia a taiigx^ntial ■tress. ^ The condition that the stresses sliall not produce any tendency ^■to rotation in the etemcnbiry portions of the body in t tl>c< »: f» = -P». In the case of a majjpietized body, however, there is such a lency to rotation, and therefore this condition, which holds in tl>c ordinar)' theory of stress, is not fulfilled. Let OS consider the effect of the stre^see on the six sides of the flementary portion of tlic body dxdgds, taking the ori^n of eoor<iinati-« at its centre of gravity. On the positive face liyih, for which the value of .r is \ dx, the ar»^ SlfEBQT AITD STftieS- [641 I'amllel to x, Fiirellel to y, Parallel to s, <iP. {P«+hA^'lx)d,ds = X,„' 4P (la) The forces acting on the opposit* side, — 1'_„ — J'_„ and —Z.,. may be found from tbeii« by chiiiging \\k fign of dx. We may exproat in the same way t!ie xyitt4'mit of thrre forces acting on e»di of the other fnces of the filuincnt, the dirMtion of tlic force being indicated by the capital Ictl^r, and the face on which it actx hy the KuQtx. If Xdxdgd: is the whole force parallel to x acting on the eleineat. Xdxdgd^ = Jr,.+.v^,+A-,.+x^+x^+aL,. ,dP„ , dP„ ^ dP„.. . . wheuce J = ^i*«+^P^+^i>.. de (13) If Ldxdj/dx is tlio moment of the forces alouf the axie of a tending to tnrn the element from y to ;, Ldxdyd: = Kds{Z^,-Z.,)-\ds{T,,-Y.,). = {P„-PJdxdyd^, whcnee L = P,,~P^. (M) Comparing the ruhies of X and L given by equations {fl) and (11) with those given by (13) and (14), we find that, if we make '*- = o*" the force arising from a Hviitem of stress of which these am the' coiDpODeDts will be suticalty equivalent, in its eflbcts on H2.] MACNTTIC 251 Blcmcnt of tti« body, witb the f<irc«8 ariaiDg Trom tltc mngnctization > doctric eum-nte. 'MS.'] Thu nature of the streas of whieh theec arc the camporXTDts i»y b(! rttMly found, by making the axia of j biiwct the angle Iwtwoen the directions of the magnetic foreo and the miij^netic intiiivtton, ftnd takin<; the axis a(y in the jthine of these dinictions, and na>a«tirc<l towards the side of tlie magnetic force. If we put J^ for the numerical value of thi? magnetic force, 3 for (hat of ibe magnetic indactioD, and 2« for the angle hutweea thoir directions, a=.ftco«<, j3= ^sin*, y = 0, 1 , . a = ©«»(, B =-S9einf, e = 0;/ ^'^ P^ss — I&Jq cos ( sin e, (17) P = — -— ©^ 009 e siu f. 4 IT Hence, the state of rtress may he considered as compounded of — (I) A prwsure equal in all directions = - ^'. (3) A tension along the line bisecting the angle Wtwccn the directions of the mi^netic force and the magnetic induction L=l^«^cos^c. (3) AprcMurcalong the line biitecting the exterior angle between' in directions = — © fi sin' «. r iv ^ (!) A couple tending to turn every element of the gubstanca in the plane of the two directions /><?»• the direction of magoetio indnctioD to Uie direction of magneUc force = —SQ^aa2t. VThcn the magnetic induction is in the same direction as the magnetic force, as it always is in Ihiidit and non-magnetized solids, then ( =: 0, and making the axis of « coincide with the direction of IUic magnetic forcc^ TOt. II. S p^-- and the tan^ntial streesce disappear. The stress in thia case is therefore a hydrostatic pressure — ^*, I combined with a longitudinal tension j- © •& ^ong the linos of force. 643.] When there is no magnetization, S = ^, and the stress is itill further simplified, being a tension along the lines of force equal to — ■ ^^, combined with a pretwi"-" in all directions at right angles 8T7 to the lines of force, nume poaents of stress in this impoiM 1 1 ,al also to -^ 6*. The com- 8 T dse are P„= ^{y'-^^-^^ P». = K = i-^^y< I (19) The 2!- component of the force arising from these Btresses on an element of the medium referred to unit of volume ia I I da ^dfi dy) 1 ( d$ ^da) 1 ( rfy da) I ,da (f/3 dy\ 1 ,da rfy\ 1 a/^^' ''"X ~4n'^^dx'^dy'^dz'ili^^dz~di^~'ilr^yd^~dy'' Now da da dy da dy -d}-di='^''' rfj8 da dx wh«re « is the density of austral magnetic matter referred to unit dx dy 645-1 TKKStOir ALONG LINES OP FOIICE. 259 of Tolume, and v and w are the componenU of electric curronfs rcfi'iTcd to uoit of area perpendicular to y and z respectively. Hence, X= aw+ t'y— W/S, 1 f (Equitloui of Simuariy r= fita+iea~uyA Elcetrommnello (20) P Z = ylW + KjS— CO. ) **"* 644.] If we adopt the theories of Amii^re and Weber aw to the nntiirr of ma^etic and diamagnetic bodies, and assume that toAg- netic and diamafn^ctio polarity are due to molecular electric currents, we get rill of imaginary ma^otJc matter, and find that everywhere r = 0. ""d rfa dS dy J- + J + J ■ ='0, ax ay dc (21) (22) so that the otjiiations of oWtromagnctic force become ^^ 7=«a-uy,\ ^W 2=ii^-va.' P TIkm are the components of the mechanical force rtferrcd to unit of volume of the substance. Tlie components of the mafjnctio force ■re o, 0, y, and those of tlie electric current arc u, r, w. Theae ecjiiationit arc identical n-ith those alroudy eetabliifhed. (K({Uiitiona (C). Art, 603.) 645.] In explaininfT the olectromagnHic force by means of a state of stresM in a nicdiiira, we are only following out the con- ception of Faraday*, that the lines of magnetic force t«nd to ehortcn themselves, and that tliey rcpttl ea<?li other when placed side by side. All that we have done is to express the value of the tension along the lines, and the pressure at right angles to tbem, in mathematical language, and to prove that the state of ■tress thns assumed to exist in the medium will actually produce tiui observed forces on the conductors which carry electric currents. Wc have aseerted nothing as yet with rc8p«:t to the mode in whiuh this state of atrees is originated and miiintuiu«-d in the m^am. Wc have merely shewn that it is poK^ihle to conceive the mutuiil action of electric currents t" drpcud on a particular kind of fIreiH! in the surrounding medium, instead ol' being a direct aod immediate action at a dietaoce. Any further explanation of the slate of stress, by means of the potion of t;he medium or otherwise, muitt be regarded as a separate fed indepeudeiit part of the theory, which may stand or fall without aflecting our prcMMit position. See Art. B32. • jffjy. tia.. !!8e. 8M7. 32$S. m EITEBOT AKD STBXSS. Iq first part of this treatise, Art, 1 08, we shewed that lie obeervGd ek'ctrostatic forces may be conceived us operating- throagh the intervention of a state of stri^ss in the surrounding' meiJium, Wo have now done the same for the electromagnetic forces, ami it remains to be seen whether the conception of a medium capable of supporting these states of stress is consistent with other known phenomena, or whether we Ghall have to put it aside as unfruitful, In a field in which electrostatic as well as electromagnetic actioo is taking place, we must suppose the electrostatic stress described in Part I to he superposed on the electromagnetic stresa which we have been considering. 646.J If we suppose the total terrestrial magnetic force to be 10 British units (grain, foot, second), as it is nearly in Britain, then the tension along the linos of force is 0.1 2S grains weight per square foot. The greatest magnetic tension produced by Jonlc* by means of electromagnets was about HO pounds weight on square inch. ^ • SttirgBoii'« AhiuiU of Slretrlcll;/. vd. v. p. 1B7 (1S4CI) ; or Philaiopkical Xagatlmt. Dec. ISGL APPENDIX I. [The roUoving note, derived horn a latter written by PrafHnr Clark Maxwell to I n«ft«Bar Clu^vla]. u iicpoiiuit in coneexlaii with Arta^ 3S9 and 891 : — Id Art. 389 the cnci^ <Iue to the |>rMi^nc«< of a in&f;itet whoM msg- DOtixntion cotnponcntu arc .f,, B,, C,, placed in a field whOM nUifiieUo foroe compoiK-nU are a,, ^,, j-,, ii -//A^.<^ + S,^ + C,y^)<ix'Ivdz, then the intefpvlioD is couSDed to the niugii^ in rirtue ot A^, B,, C, ^bdug MPO evtiywhcrc else. Bnt ihc vholc pncrgv is of the form Itiie intc^mtion extending; to every part of apaee wlirre tliero an mag- taftimd bodiee, and A,, B,, C, denotiuj;; the components of mgpMtiutiou at AD}' point (otlorior to the mngiKit. The whole energy tlius coniiistA of four part* ;— -i//AA,a, + &^.)<Lrdydz. (I) vbicb IB constAnt if the nia^'netization of the inognet b rigid ; ~\///i.Ka, + &c.)<Ixd,jd^ (2) wliieh is equal, by Qruen'M Tlicomn, tu -\///(A,a, + Ai:.)dxdi,d:, (8) awl -i///{A^a,+&c.)dxdyds: (4) wliich lost we Dia; 8ui>pose to arise from rigid magnetisations and therm- ion cooRtrat. BeiMc tlie rnrialilc part of Ute encrgj of tbc moveable magnet, a.s ri^dly il, b tlie aura of the ezpreasions (2) and (3), tik.. Remembering tliHt the diaplaccmunt of the mi^ict aliens the value* of S> i^i' Vk ^^^ "^^ IhoM of A^, B„ C„ we Cod for the component of llie force on the magnet in aQy direction 0— I If] (".t^^'t^'^'t""**- I I If instead of a rangnct wc hare a body maguetiiicd tiy induction, th* exprtnion for ibe for« must be the same, vi*., writing A, = ka, Ac., we have fi'f , dn, , ,J,^, . dy,. , , , In Ihi* expression a ia pnt for n, + o,, Ac, but if either the magnet ixe<! body be smnll or k bo small we may neglect a, in compariaoti with a,, and the exprenian for the furee beeumea, as in Art. 440, The worii done by the magnetic forces while a body of ainuU inductile ca|>ucily, magiietiied inductively, ia carried uff Ui infinity ic only half of tliat for the aiinie body rigidly nia;^etieed to the wime uriginal •Ireogtli, for n« tbe induced magnet ia carried off it loica iU atrength.] APPENDIX n. {OliJMitJuii hail liMU bikoii lu di« cxprenBion omittunnl io Art 639 ftr the putetitiail energy per uiiH volume of the medium arising from nwE- ueUc forces, for the reason that in finding that vspressioo in Art. 389 «« Assumed the force coinponcnta a, |9, y lo }>o di:riviibl« from a iiot^otid. vhcreu in Art*. 639, 6-10 tliiH is not tlie case. Tfain objectiwD extemit to th<! exjtrcti^Dn fur the force JT, which it the tipaoe rarititioii uf the tntTgy. The puqtose of this note is to bring fonrsriJ gome CMtddernliom tending to confirm the kccunic}' of th<^ t«xt. We oonmder tlicu tb<> uleuitml dxdydz isolated twvm Uic rat of tht medium with u current of electricity- (u, t', te) flowing tliroiigli H. 13ia fortna uoUn^ on the element ari»e fium two causes, vix., first, in rirtna of tbe element being roagnntisird, next, in virtue of ib orrjring h curtent The force dno to the hitter CflU»e is (X,, I',, Z^dasdt/dt, giren in (3), Art. 6-10, viz. X, = re— wb, etc. To entimate the other we oliacrve thai, if no currt-iit wen; flowtn)( tlirougU the eleuicDt. the tnngnetic force on th« eletni-nt would (« (X', 1", Z') dxdyds, where ax dy efa 4 Jltit in widiiiou to this there will be forcea from the current of eleotriot)* neting on tbi- ntngncfism of tlie fjices of the element. Kuw it Is sheim in Art. 653 that in pasting from the negative to the poaitire side of * current sheet the conipoumt of th« magnetic UfK» nOTiBal to tbc shcel experiences no discontinuit}- ; neither doe* Uw component in th« diroctiuo of the current, but the componeut paniliol to the tihcet aikd nornud to (br current changes by 4itr, where i is the strength of the cuTreat per nnit Uugtb. Let us then suppose that the faces of our eleraeut which ait perpendicular to the axis of y in a right- handed system of ftxee an portions of an infinite current sheet in which there is flowing a cumnt parallel to tbc axis of e equal to tcdAf. There will tlicn be a nmgnctie force BAvtedxdyia over and itboye the force /I- - Jxdffdx previotudy reckoned in tbc estinmte of th« forcn* pnmllcl to dx and ndiog on the dxda faces. In like m&uner we ihould find an MlditiMml (breo —Civvdifdyd* acting on the dxdy faces. Vr'e have thus to add to X' • term X"= ii:B«>-4iTCv. and the total ^T'Componeiit of tlie magnetic force will 1m My ''^*^i'+%- ^dy tlMttt, tl wUl be observed tbot tlM force (X' Talaeor{£,if,^).] r'.Z'^iixttsolatlam CHAPTER XII. CURRENI^HEBTS. I Gi?.] A cVKBEirT'SQiiitT b an iofinitely thin stratum of con> ductin;;: matter, bounded on both sides by iosulatinfr media, eo that filectric currents mny 6ow in the ahcct, but CHnnot e6ctk[K> from it «xccpt at (wrtnin points called Electrodes, where curreats are made to «nt«T or to Icavo the sheets In order to conduct n finite cleetric current, a no.] sheet must have a Unite thickness, and oug'lit thurc^fore t-o be considered m DOndador of throe dimensions. In many cases, however, it is practically convvnicnl to deduce tlie electric properticn of a real conducting tihcetj or of a thin layer of ooiled wire, from tliow! of a ourrent-shet-t it* dcltned above. We may therefore r^'ganl u surface of any form a« a curreut-fhcct. Having selected one side of this surl'ncc aii the positive side, we shall always eiipj>of!e any linvH drawn on the tiiirliico to be looked at from the positive side of the surface. In the case of a closed surface wo shall eoUKidrr the outttide as positive. See Art. 294, where, however, the diroutiun of the current ie defined a« seen from the negative aide of the sheet. Th« CttrrtHt-ZuHclion. 6-18.] Ijet a fixed point A on the surface be chosen as origin, and let a line he-drawn on the surface from A to another point P. Let the quantity of electricity which in unit of time crosses this line from left to right be <^, then i^ is calUd the Current- function at the point i*. The current-function depends only on the ])osition «f Uie point P and is the same for any two forma of tlte Hno AP, provided this : line MO be trnn^fonnod hy continiiouf motion from one (nna to tlii* ollter witliout. passing: Uiroag'h an cleotroile. For the two forms of tlic line uill enclose an arcu witliin wliJcli tliere in no electrode, utd tbererore the Game quantity of electricity wliicli cntent Ike ares iicre« one of the lines must issue aoroi^ the other. If t denote the length of the line AP, the current acrow d» from left to right will be -^ dt. If is oonstsnt for any curve, there ia no current across it, Suob a enrve is called a Current-line or a Streara-line. 649.] Let yp be the electric potential at any point of the sheet, then the electromotive force along any ctom«nt tJ> of a curve will bo provided no electromotive force exists except that which arises from difTereDCeB of potential. If ^ is constant for any curve, the curve is called an Eqni- potential Line. 660.] Wc may now Biippose that the position of a point on Iha Rheot is defined by the values of ^ and ^j/ at th.it point. Let rf^ be the length of the element of the ei]uipot«Dtiul line >j/ int«rc«pted between the two current lines ^ and ^ + d^ and K-t dji^ he the length of the clement of the current line ^ intercepted between the two eqaipot«ntial linc« ^ nnd ip + ii^. Wc niay consider f/^, und</<,. as the sides of the element dipiiyj/ of the slieet- The eli>utromotive force —dyfi in the direction of tt», produce* the current rf-^ acroM f/i,.. Let thi.' Tcttistance of a portion of tlie sheet nhoae length is tU^, and who«c breadth i* da^, be dt, (fa, where a m the spocific resistance of the sheet referred to unit <if area, then , , di, , ^ 1 ds, dL whence 4 = "^- 651.] If the slieet is of a substance which eonduots oqnally weli in all directions, dtj ia perpendicular to d»^. In tlie ease of a she of uniform rceistanoe a in constant, and if we make ^ « o*^, wc sJiall have A, d^t and the stream-tines and cquipoteutial Uneawill ent tlu suriaaeiat little t>4»aro(; i MAGNETIC POTEITTIAI,. 265 I) tit foUoirs from tUia tliat if 0, and t^i' arc conjii^nie funotinns (Art. I$3) or and yfi', tho curvett <^ may be strx-um-'Iinoit in the lebeet for which tlic curves ij'j'are the ooiivapondiii^ I'tiuipotontial lines, One case, of coume, in that in which 0, = ^' and t^,'= —0. ' In tJiis case the «>ijui potential lines become curreiit-liiie^ and the kcuiTeDt-tines etjiii potential lines *. If ne have obtained the solution of the distribution of electric ctineDts in a nniform sheet of any form for any particular case, we may deduce the distribution in any other case by a proper trans- I formation of the conjugate funotions, according to the method given in Art. 100. 65a,] We have next to determine the magnetic action of a cnrrcnt-«hcet in which the current is entirely confined to the sheet, thorn bring no electrodes to convey the current to or from the sheet. In this case tho current-fanction i^ has a determinate value at evorj' point, and the stream-lines are closed curves which do not intfrrsect «Bcli other, though any one stxeam-lioe may intersect itwlf. B Consider the annular portion of the eJieet between the stream- ^■1inc« nnd i^+hif). This [lart of the sheet is ft conducting circuit Hin which a current of strength &</> circulates in the positive direction ■.round that part of the sheirt i'or which tp is greater than the given H Taluo, Tho nmgDetio elfoct of this circuit is the sumo as that of H« magnetic shell of strength S^ at any point not included in tiko B lubHtimoc of tho sbell. Let us suppose Utat the shell coincides with that part of the current-sheet for which ^ has a greater value than B'it has at tho given stream-line. H By drawing all the successive stream-lines, beginning with that ^■for which has the greatest value, and ending with that for which ^■it« value is least, wo slmll divide the eurrent-eheet into a Kcries ^^of circuits. Substituting for each circuit its eorr^eponding mag- ^Bnetio Hboll. wo find that the magnetic ciTect of the current-xhoct ^■■t any point not included in the thickness of the sheet is the >^ne ^L aa that of B complex magnetic shell, whose strength at any (loint ^Bia C+<p, where Cis a constant. If tlie currcnt-shoet is bounded, then we must make C+i}> = at Ui« bounding cnrvo. If the sheet forms a clowd or an inlinite Mirfaoc, there is nothing to determine the value of the constant C. ■ 8m Tlioaucin, Can^. and Dub. Math. /auni„ voL Hi. p. 280. 206 [653- 053.] The magnetio potential at any poiot on either side of tbo current-Ehcct is given, aa in Art. 4 1 5, by the expression "=//i vra dJS, wbere r is the distance of the g^ven point from the olemeni sariaoe dS, and is the angle between the direction of r, and th: of the normal drawn from tlie poeitive sido of if^. This expreagion gives the magnetic potential for oil poinU inelutled in the thickness of the cument-slMirt, and we know for points within a conductor carrying a current there is no «ich thing ax a magnetic potonfial. The value of it is discontinuous at the cnrrent-aheet, for if Qi is its value at a point just within the current-shwl, and il^ ita Tolue at a point close to the first but jost ootaido the curreot'sbeet, where is the current-faoction «t that point of the sh«et. The value of the component of magnetic foroe normal to sheet is continuous, being the same on bi>tli sides of the sheet The component of the magnetic force parallel to the current-liMS is also continuous, but the tangential coin{>onent perpendicular 1« tlic current-lines is discontinuous at the slieet. If * is the length of a curve dran-n on the sheet, the component of magnetic force ot9 rfft, and for the in the direction of </« is, for the negative side, — - . , — .-. — ^^ positive side. -^ = - ^-4- ^ . ^ The component of the magnetic force on the positive ride tliere- fore exceeds that on the negative side by — 4 it --7- ■ At a given potut this quantity will be a maximum when da b perpendicular to the Gurrent-linea. On He Indaedon of Blecirie Currenlt in a Shttt e/ Infiiiilt Conduellrity, 654.] It was shewn in Art^ S79 that in any circuit where E is the impn-iwe«l t-lectrorootive force, p the e1«ctro1nB momentum of Uie cirtriiit, li the reiustanec of the (lircuit, and i current round it. If tltere i« no impreascd electromotive force 1 DO resistance, then -^ = 0, or pia ootutoai. at " 656.] - PLANE SHEET. 267 I Now />, ihe electrokinetic momentum of the circuit, wan dicwa in Art, 388 to be measured by tlie surface- integral of magnetic induction through the circuit. Hence, in tlio cuite of a ounent- she«t of no rcaietance, the surface-integral of magnetic induction through any closed curve drawn on the surfiice must be constant, ami this implies that the normal component of magnetic induction nimains oonstant at every point of the current •sheet. 1655.] If, therefore, by the motion of magneto or variations of cunvnte in the neighbourhood, the magnetic field ia in any way alt4.Te(I, electric currents will be set up in the current-sheet, such that their magnetic effect, combined with that of the magnets or cuTTcntx in the 6eld, will mointain the normal component of mag- ■ nrtic induction at every point of the sheet unchanged. If at first there is no magnetic action, and no currents in tlie sheet, then the normal component of magnetic induction will always be zero N at every point of the sheet, B The sheet may therefore be regarded as impervious t« magnetic induction, and the lines of magnetic induction will be deflected by tlic sheet exactly in the same way as the lines of flow of an ok-ctric current in an infinite and uniform conducting mass would bo ectfd by the introduction of a sheet of the same form made if A suhetnnw of infinite reai^ance. If the sheet forms a closed or an infinite surface, no magnetic actioiiH which may take place on one aide of the sheet will produce any magnetic efieot on the other side. TAeor^ of a Plane Cirrfnl'sieei. 656.] We have seen that the external magnetic action of a cuiTent>sheetia equivalent to that of a msignetic shell whose strength at any point ix numerically e<iual to if>, the current-function. When the sheet is a plane one.n'e may exproe* all the quantities required for the determination of tiWtroiiiagiiclic effects in terms of a single funHion, P, which \a the potential due to a sheet of imaginary matter spread over the plane with a surface-density ^. The value of P is of course rrA, irbere r is the distance from tli« point (x, y, 2) for which P is calculated, to the point («'',/, 0) in the plane of the sheet, at which the element li^il^ is taken. To find the magnetic potential, «c may regard the magnetic I k at» CntESin^SBEBtS. [65-. Rbcll as oonKirtinf* of two enr&cee parallel to the plane of tf, tbe firfit, whose equation is « = )«, having the Rurfstcc^Dritj ~, and the Kcoond, whose eciitation in z =: — \e, having tbe sarfacesli e TIm) potentials doc to thcM turiaoes wiD be J'H) and --P, e ■(-i) e , respecttvelr, where the soffises indicate that t—- a pat in the first expression, and £ ■!■ - Tor ^ in Uic i«cond. Expandio^ these expressions hy Tnvlor's TlK-orein, addinff thtrm, and tbtn making c infinitely small, nc obtain for the mitgaiAic potential do^^ to the sheet at any point external to it, ^| fl= — dP d:' (2) 657.] The quantity P is synintctrical with respect to the plane of tbe ahect, and is thcretorv the same when — ^ is substituted Tor £■ n, the magnetic potential, changes sign when —x is pnt for t. At the positive surface of tbe sheet " = -?="'♦• (= At the negative snrfaoe of the sheet ^j Within tbe sheet, if its mafrnctic effects arise from the magnet- ization of its substance, thi? maj;nctic potential raries ooDtinn- ously from 2t^ at tlie positive surface to — 2s^ at the negative Burface. If the sheet contains electric currents, the magnetic foiee within it. does not satisfy the condition of having a potential The magnetic force within the sheet is, however, perfectly deter- minate, ^i Tbe normal component, ^M da d'P ... is the same on both ndea of the sheet and throaghont ita aafi stance. If a and p be the componentu of the magnetic force pantllel VECTOR-rOTENTIAL. I and to y at tbe positive suriace, and a', ^ those on tlie negative flarfaw, ^ „ - - ' (6) « = — $=-- (?) Tho iMjuutioiiti ^ ds dx iF dU da it dx ~ if JG dF dil («) Within th« she«t the components vary continuously from a and ,^ to a' and 3'. ^^^^^P da di/ ~ dz which connc-ct the components F, 6, U of the vector-potential due Ito the curn-nt-Blieot with the sualar potential it, are satisfied if Itrv make Ap ,IP F=~, G = -^, /f=0. .(9) ay dx * ' We may also ebtnio these values by direct iDt'?^ratioii, thus for F^ Since tiie integration is to be estimated over the infinite plane tflbeet^ and since the liriit term vitiiit^lics at infinity, the cxprescion \» [reduced to the secund t«rm; and by substituting d \ . rf 1 y~- for — T-'-' "S T ay r \*aA remomberin^ that ^ dq)enda on of and /, and not on x, jf, e, U obtain F=^fJ^^d^dy, If Q' is the niBj>netic potcntiiil due to any mng'netic or electrie sysbem €xt«raal to the sheet, we may write '=~fsi'dz. and wo sball tlien have &=- dy dx ' U'=0. (10) for the component* of the Toctor<potential dae to this system. 270 CUSREtfT-SnEBTS. [658. 658.] Lot ii» now determine tJie electromotive Toree at w»y pointfl of the shfct, supposing the shwrt fixed. ~ 4 , »uppoaing Let X and f !« the compoiienU of the eiectronwtive force paralle! to J and toy respectively, then, by Art 598, w« have '-iio.'n-t If the electric resistance of the sheet is uniTorm and equal to <r, X=aM, r=<Tr, (H) whore «i and c are the components of the current, and if ^ ia thai current- function, J^ ^^ But, by equation (3), (15) Tlencv, eqtiatioas (12) itt the positive siirraco of the current-sheet, and (13) may be written 2vdydz~ dydt^ ^ ' dx 2itda><h dxfU^ '*' ' off' where the values of the expressions are those corresponding tu tbt Iiositive surface of the sheet. If we differentiate the first of these equations vrith rrspect to and the second witli respect to y, ond add the rc«ultti, we obtain (16) The only value of ^ which satisliM this equation, and is finite and continuous at every point of tlie plane, and vanisheA at si inGnite dietftsce, is \b = 0. (I9l Hence the induotion of electric currents in an infinite plane she of uniform conductivity is not uecompanied with diBcrcnoc* electric potential in different parts of the tsheet. Substituting this value of <^, and integrating equations {iGJj (17), weobuin „ dP dP dF ,, ,. .„„( Since the values of the currents in Uie nheet are found diffen^ntiutin^ with respect to * or y, the arliitrary runctinn ol and / will disfl])i>c3ir. We shall iherufore Iciive it ont of accounk 660-3 DBCAT OF CITRRESTS IK THE SHEET. in ir we dbo write for — , the single symWl ff, wbich repmenls L oeitain rolocity, ttie equation bebwcca P and i^ becomea !>.] Let o» first suppose that there is no external mAj^etio actings on the current sUcot, We nrny therefore suppose : 0. Tiie case then becomes that of u sy«teiQ of etei^tric ciirreuta ih tlie sbei-t left to thcmsulves, but nirting on one another by their il induL-tioD, and nt the suine time losing their energy on aunt ol' the resistance of the sheet. The result is expressed by the equKtJon ^rfP dP the solution of which is P =/{*, y. (c + ^0)- (23) * llencej the value of P at uny point on the positive side of tha rTfbsM coord inn te.« are x, y, x, and at a time /, is equal to itwP tit the point *, y, (;+ Rt) at the iniitiint when ^ = 0. If therefore ft system of ctirrent« is excited in a uniform plane sheet of infinite extt-nt and then lelV to itself, its magnetic effect at any point on the pogittve wide of the sheet will he the same aa if the syntem of currents had been maiutnineil constant in the sheet, and the sheet moved in the direction of a normiil from ila negative nido with the constant velocity I{. The diminutiun of the electromagnetic forces, which arises from a decay uf the currents in the real case, is accurately represented by the diminution of the t force on account of tlie increasing distance in the imaginary case. P-i-F -M di. (24) If we suppose that at first P and P" are loth zero, and that a magnet or electromagnet is suddenly magnetised or brought from an infiuite distance, so as to <>liange the value of P" suddenly from into to P', then, since the time-integral in the second member of (24) vanishes with the time, we must have at the finrt instant P = —P" at the surface of the sheet. * rnWBqiuU(au(lQ)Bnd(S31aF«' provHltabstrueoDljriit the surhavof tlteilucl fcc whicib t — 0. lio csjiroMan (33) latinfln <S2) genL-nlly, uid tberdoro hIio at tha inrbM «f tka tbeet. It alia MiUin the iiUitv coiiditluiu uf iLo problem, uul U IbmGirD » aalutlon. ' Any ctbtr tuluUun mud iltlTar (run Uite by ft i^«l«in of olacad cimanlf, ilmndinii! ou tbe tnittkl tMa al the ilwel, dM iIu« to *i>7 aitanMl otUM Mid wLich tWnlore niuit iIucbv rapidly, fienn (iiMv wa wtnim* ut vtunitr or put tino tlin b i&c only wlutioo u( lb« problm.' Set Ftahmor CScric Muwell ■ Papct, lUfiU ioe. PttK., XI. pp. I90-lliS.l 272 CntHgaWBj^l^B, Hence, the system of tarrent* excited in the sliect hy the suiUeii introduction of the system to whieb P' is due is such that at tht surface of the sheet it exactly nentmliw-* Uie mng-aetic effect of this system. At the surface of the sheet, Ihorcfore, and consequpntly at «II paints on the ncf^tivc dde of it, thv initial fvxtem of cuTTnil* produces SLti cfftxt exiictly equal and op[>o»itc tci thiU of Uie magnetic system on the positive sid«. We itwy exprcM tht< by saying that the eflt-ct of the c-urront« \m equivalent to tliat of la ivutge of the mofpietic system, coiueiding in position vritli thnt system, hut opposite as rrgurds the direetion of its magtietiutJOD and of it^ electric currents. Such an imi^ is called a ntffative image. The clleot of the currents in the sheet at a {oint on the podtin side of it is equivalent to that of a positive ima^ of the ma^iMtic sysU'm on the npfjativc side of the «hcet, the lines joioiD^ cone- spondin^ points bein^ bisected at right angles hy the sheet. The action at a point on cither Ride of the sheet, due to the currents in thi; sheet, may ihert-fore he reg^flfdcd as due to an image of the iiiiignotic sj'stcni on tlie side of the sheet opposite to the poiut, this image being a positive or a negative image according as the point is on the poattive or the negative side offl tlie sheet. ™ 661.] If the sheet is of inRnite oonductivity, 2t = 0, and the second term of (24) is zero, so that the image wiU represeat the effect of the currents in the sihect at any time. ^ In the case of a real Nheet, the resistance H has some fiuite v»1u«. | The image juat deaerihed will therefore represent the effect of tho currents only during the first instant after the sudden introduction of the magnetic system. The ourrents will inimediatfly begin to decay, and the effect of this decay will he accurately represented if we suppose the tno images to move from their original positions, in the direction of normals drawn from tho sheet, n-ith the oonitant velocity R. 662.] We are now prepared to investigate the system of Mirrrntw mduced in the sheet by any system. J/, of mngneta or electro- nagnets on the positive »de of the sheet, the position and stnogth of nhich vary in any manner. Let P", a« before, he the function from which the direct actJoa^ of this system is to be dedneed hy the equations (3), (d), ftc., th«D -nr- it will be the function oomhtpottdiDg to the system 6640 UOVUfO TSA1L OP tUAGBS. m presented by -jt- S'- Tliis quantity, ivhicli ia the increment of 3f in the time 9^ nwy be regimlLil as ititelf rqimcoDUog n maguetio system. t If we supjioEC tbut at tlto time f » positive image of the system ait -jfit ifi formed on tho nogatiro dde of the shoot, tho mo^Dt-tie action at any point on the positive adi- of the sheet due to this ima^ ^vill be equivalent to that due to tho currents in tho sheet excited by the change in At dnriog tho liret instant after the ch&Dgo, and the imitgc >vill eontiiitie to bo equivalent to the curreatfi in the sheet, if, lui soon as it is formed, it begins to move in the negative dircetion of s with the coiistnnt velocity R. I If we BHpposc that in every auecessive element of the time an inuigc of this kind is formed, and that as snnn as it is formed it Ijcgins to move away from the sheet with velocity E, we shall obtain the conception of a trail of images, the last of which is in process of formation, while all the rest are moving like • rigid body away from the sheet with velocity li. I 668.] If i^ di'jintes any function whatever arisiing from the action of the magnetic system, we may find P, the corresponding function arising from the currents in the sheet, by the following nrooew, which is merely the symbolical expression for the theory of the trail of images. . Let /', denote the value of P (the function jurising from the pnrrents in the sheet) at the point {a, y, i + Rt), and at the time t—T, and let i^ denote the value of P" (the function arising from jthe magnetic system) at the point («, ^, —{s+Bt]), and at the (5.6) iet^T. Then dr 4ts (it ' d e<iuatlon (21) becomes d7=-sr' **'' id we obtain by integrating with reepoct to r from r=: tor = ec, tbe value of the function P, whence we obtain all the propertiM rtfafl eorrent sheet by ditferentiation, as in eciuations (3), (9), &c> 664i.] As an example of the proccta here indicated, let us take e case of a single magnetic pole of strength unity, moving with miform velocity in a straight tine, VOL, II. T m m CntHEKT-SHEETS. Lcl the coordinates of the pole at the time i be f = u/, 1? = 0, C=: c + vol. TIic coordinates of the image of the pole formed at the ti t~T are € = M{e-r}. n = (», f=-(ff+»(^-r)+i!r), and if r is the distanee of thia image from the point (x, g, t). To ohtain the potential due to the Imil of image* we have calenlate ^ r" dr ~dTJ, T" If we write Q» = u» + (S— »)', _^y = -glog{Cr+u{*-uO + (fi-w)(«+*' + »01 + a term infmitely f^rent which however will disBf^iear on diSi entiation with regard to /, the value of r in thifl expreesion \m\ found by making r = in the expression for r fpven above. DifreKntinting this expression with reepect to /, and poi f K 0, ne obtain the magnatie potential due to the trail of i: ^^ n>f.- + .)-u. _^,_^,^^^ ^ Q Qr+ii»+(i?-»)(s+c) By differentiating this expression with respect to 2 or obtain the components parallel to ir or « leepcctively of the mag- netic force at any point, and by puttin<> r = 0, z = c, and r => 2r in these expressions, wc obtain the following valne« of tho CODI^ poneats of the force acting on tho moving pole it«clf, J = - 1 i'^Q+S ks!' Q QiQ J>! J, ^-Mh ( + -K-W)) 665.] In these expressions we must remember that the moSo is aapposed to have been g<»ng on for an infinite time before time coiuictered. Uenoe we must not take tv a positive qoantil; for in that case the pole must hare passed through the sheet within a 6nite time. If we make u = 0, and tp negative, X =0, and or the pole as it approaches the sheet is r«p«Ue<l from it 668.] FORCE OS MOTINO POLS. 275 If we make » = 0. we fin^ Q^ = u' +^, X = - uR and Z= --. The oomponent X repa'sent* n retarding force nctJD^ on the pole in the dirccUon opposiUr to that of its own motion. For r given valae of K. X is a masiinum when u = 1.27 Jt, When the sheet is a nou-couductor, R=ys and X = 0. When the sh«et x* a perfect conductor, R = and A' = 0. The component Z represents a reimlsion of the jmle from the [ elMet. It increases as the velocity increases, and tdtiniat«ly becomes -—-^ when the velocity is iotiuite. It has the Bame vnltie when a is zero. 666.] When the magnetic pole moves in a curve parallel to the sheet, the calculation becomes more complicated, hut it is easy to see that the effect of the nearest portion of the trail of images is to produce a force acting on the pole in the direction opposite to that of its motion. The eScct of the jxirtion of the trail im- mediately behind this is of the t^mv kind as that of a magnet with ita axis parallel to the direction of motion of the pol« at I some time before. Since the nearest pole of this magnet is of the ' nun« name with the moving pole, the force will consist partly of a repiiUion, and partly of a force parallel to the former direction of motion, but backwards. This may be resolved inbi a retarding force, and a force towards the concave side of the path of the nioTing pole. 667.] Our investigation does not enable ua to solve the case ID which the sjBtcm of currents cannot ho completely formed, on aoeonnt of a discontinuity or boundary of the conducting sheet. It is easy to see, however, that if the polo is moving parallel to tfae edge of the sheet, the currents on the side next the edge will bo enfeebled. Hence the forces due to these currenl^t will be lees, and there will not only l>e a smaller retarding force, but, since the repulsive force is Icaxt on the side next the edge, the pole will be attracted towards tbe edge. Tieory <^ Araga't Sotating Pist. Arago discovered * that a magnet placed near a rotating metallic disk experiences a force tending to make it follow the T 2 270 [66E motion of the disk, altbongh when the diek is at rest there is no nction between it and the ma^i^npt. j^m This action of a rutatinf; disk was nttrihated to a new kin^^ of induced m^>netization, till Faraday ■* expluned it by means «f thtt electric cinrcntii induced iu thv disk on account of ite motion throogh the field of muf^netic f<irce. To dctcitninc Uie diiitrlbution of these induced correntd, and thi-ir i-fTrat on the magnet, we might mske uw of the results nlready found fur a conducting sh^t at rest aot^d oil hy a moving magnet, Bvatling ourselvM of the method given tn Art. 600 for trcatin<^ th* electromagnetic equatiouH when referred to mnring Gystcms of axes- Aa this case, however, has a special importance, wu shall treat it in a direct manner, beginning by assuming that the [Kilos of the magnet are so far from the ed^ of the disk that the effect of tb limitation of the conducting sheet may be neglected. Making use of the same notation as in Uio [ii«ceding article (556-667), we lind for the components of the electromotive parallel to x and y respectively, a« = dx J<(f where y in the reeolred part of the nugnetic force normal to tkc disk.] If we now express « and c in terms of ^, the current-functioB, aod if the disk is rotating about the axis of t with the angular «lodty w, ^_„. ^ ^=„,, ^ = -«jr. Substituting these values in e<]uation8 (1 ), we (ind dd, d^ ''ds=>'''-di' Moltipl^'ing (4) by e and (5) by jr, and adding, we obtsiii Multiplying (4) by jr and (5) by —*, and adding, we obtain d^ _ d^ diff il<fi 1 / dib (lA. d<b d<li df^ dy • £r^ Ra, gl. >68.] ARAG0 3 DISK. 2rr ar~- = It we DOW express tbeae equatioQs ja tetats of r and 6, where 2! = r COB 9, ^ = r sin $, (6) hey become a -^= ymr'—r -^, (9) Eqaation (10) is sntisfied if we nseumo imy arbitrary function ^ br r and 9, and make j. __ ^X ^1 tibetitatin^ these values in eqimtion (9), it becomes Dividing by vr', and restoring the coordinates « and y, tliit |>eoome8 ''"x . ''^X _ " /,^i This is the fundamental equation of the theory, and expresses thu elatjon between the function, x- and the component, y, of the mag- letic force resolved normal to the disk. Let Q be the potential, at any point on the positive ade of the iflk, doe to imaginary matter distribut«d over the disk with the ar&ce^ennty x> At the positive stirfacc of the disk f = -2.x. (15) HflOOe tlic first m«-mber of equation ( 1 4) becomes di^^ df~ 2-niiz War* df-) ^"' Bab Mice Q ssitisfieii Iia])Iace'a equatioa at all points external |o the disk, d^,'PQ__d^ .... dx* ^ dg' ~ -'^ ' ^"f lud equation (H) become* i/j* 2l^^=">- 0«) Again, eince Q i» the potential due to the distribution Xi the otentiol due to the distribution A, or -}~,w)Ube -^. From this (19 off obtain for the magnetic potential due to the currents in the disk, ill- dedt' m ttf . ..wortw force uoimal to tbc diik (80) ■ M>(Aiti*' due to external inag^et«, and .!*?"'■ tAP */>(* :-yv (ai) . oaMCtic force norina] t« the disk due to 00*' _ -j^^oHion (18), remembering that y = Yi + yi' iiri« '"''' "^T*^ *o '' *'*^ writinj J for — . fto*" L^ of /" a"^ Q ^"^ expr«scd in tcnns of r i If '''^^^^jjwdiak, f and f two new vnriablca buc^ t^ J? the dis sueb tlwt .f =.-?., . j«it becotoMi '»y integration with reep«ct to f, ■ IV form of this cxprcwton taken in coi^unetion with lb« ^Tj nf Art- fi^^ Bbrn'H fJiat Uic nmgnetio action of the currcnl^l "^ ilUk )B ^oi'">'^'>^ to thai of a trail of images of the " fid errteitt in the form of a helix. i*"^. _ggneti« iTBtem connete of « single magnetic pole of " . ynitr, '1"^ beWi n-ill lie on the cylinder whoso axis is %f bcli tjve eofl» of tte helix, will I>e 2w — . The magnetic effect of jl „iil be the Mune a« if thi» bclix had been magnetized uhew '° '''^ direction of a tangent to the cylinder perpen- ^ ' Ur to it" '^^^ ^'^^ "^ intensity such tltat the magnetic moment '*■ j^ftll jmriion ia namerically equal to the length of its - ]^M,ontbedist. - -rttem in the form of a helix. 1 sti«' tbat ""the disk. The dixtanoe, parallel to the axis between oon- . unitV '"'' uciii ntti iiv vu lUB uj iiuutT miHSJu aj.ta i ***** f the di*It, ""d which parses through the magnetic pole^ ""^ ■■ ^ij] begin at the pMition of the optical tmaiire of tli^J >7o.] 279 The ealeulntion of the efft-ct on the magnetic pole would be jmplicateJ, but it is easy to eeo that it will consist of — (1) A (inigging force, piirallcl to the direction of motioD of the disk. (2) A repulsive ffTce Bcting from the disk. (3) A force towards the axis of the disk. Wbon the pole is near the edge of the disk, the third of theae jrvGS niny be overcome by the force towards the edge of the disk, adicatcd in Art. 667. All these forces were observed byArago, and described by him in Annales ile CA'imk for 1826. See aUo Folici, io Tortolini's fyna/ji, iv, p. 173 (1853), and v, p. 35; and E. Jochmann, in Cretlta \jounial, Ixiii, pp. 158 and 320; and Pogg. Ann. cxiij, p. 2U [1864). In the latter paper the enuations necessary for deter- mining the induction of the currents on themselves arc given, but Itliis part of the action is omitted in tlie snhsequent calculution uf suite. The method of images given here was published in the ^roceedinga of the Rogal Socictj/ for Feb. 15, 1872. 1^ »; Sjiheritat Cun«nt-Sieet, 670.] Let ^ be the current-function at any point Q of n spherical cnrrent-sheet, and let P be the po- tential at a given point, due to a sheet of imoginarj' matter distributed over the sphere with BDrfacc-density ^, it is required to find the magnetic potential and the vcctor-iiotontial of the enrrcnt-shect in terms of P. Let a denote the radius of the sphere, r the distance of the given int from the centre, and jn the reciprocal of the distance of the given point from the point § on the sphere at which the current-function is 0. I The action of the current-sheet at any point not in its substance is identical witli that of a magnetic shell whose strength at any point is numerically equal to the current-function. The mutual ])otential of tlie magnetic shell and a unit pole placed. at the point P is, by Art 410, Fig. S». -/M as. Since/' is a homogeneous function of the degree — 1 in r aad it, da dp dp id , . Since r nod a are constant throughout the surfivce-iDtc^nttion, But if P is the potential du« to » sheet of imaginaiy mat of surface-demity A, _ rr and £1, the mn<>nciie [)otcntia1 of the cnrrcnt-sheet, maj be < in terms of i' in the form 671.] We may determine F, the e-componont of the rectc potential, from the expressioa given in Art. 116, '=//♦(-'/,-•!)'* where f, •?. C are the coordinates of the element dS, and l,m,nue the direction-coBines of the normal. Since the aheet is a sphere, the direcUoa<oo«aes of tlie notmal an '=!■ BI = a But and so that ~ adf a3t' multiplying hy ^dS, and integrating over the aurfaoe of the sphere, we find F-i^^y^^. HELD OF USIPORM FORCE. 281 iimiliiHy adx a dy The vector SI, whose componentfi arc F, 0, U, ie cnJontly pcr- cular to the rntliue vector r, and to the vector whoso oom- jpoaents *^ j~ > -r- > ^i^ -7- ■ If we determine the lines of inter- [SectioDs of the sphericAl surraco whoso nidiuB is r, with tho strict of [ equipotontiul surfuees corrvspondJDf* to Tnlues of P tu arithmcttoiil tprof^rcssion, these lines will iaclicat'i by their diniction tlie direction of 91, and by their proximity the mnjpiitudc of this vector. Id the language of Quateniions, 81 = i FpVP. 672,] If wc assume as the value of /' within the ephere \ where I'f i< a Bpherical harmoQic of degree i, then out«ide the sphere "Die curreni-fuiiction ^ is The magnetic potential n-ithia the sphere is example, lot it be required to produce, by means of a wire coiled into the form of a spherical shell, a uniform magnetic force a within the shell. The magnetio potential within the shell is, in Khia CMO, a solid harmonic of the first d^ree of the form a=—Mrcm9, , where Jf is the magnetic force. Ileooe A = (d'^, and <f = -— 3faooB9. ^0 current-function is therefore proportional to the distance I from the equatorial pbtne of the sphere, and therefore the number of windings of the wire between any two small circleii muHt be . proxwrtional to the distance between the planes of these circles. 282 CUnBBST-SHEBTfl. If jV is the wbolc Dumber of windings, and if y is the strei of tbc current in each winding, <f,= J.Vyooefl. Hence t]ic magnetic force witliin the coil ia 3 a 678.] Let us next find the method of coiling the wire in older to produce within the sphere a magnetic potential of the form of a solid zonal liarmonic of the second degree, Here * = ^|(Sco8"tf-J). If tho whole number of windings le iV^, the number between the pole und the polar distance 6 is \Ns\a*$, The windinirs aro closest at latitude 45°. At (he oqitator t! direction of winding changes, and in tho other hemiiipbcre t! windings arc io the contrary direction. liet y hv the strength of the current io the wire, tlicn vn the shell 4 « r3 Let us now consider a conductor in the form of a plane c1 curve placed anywhere within the shell with its plane perpendic: to the axis. To determine its coefficient of induction we have find the surface-integral of — -t- over the pbine bounded by the curve, putting y=l. Now ll=_iJi^f(^_i{^+y^), and J- = n " «• as 6a' Henoc, if ^ is tJte are* of the closed curve, its ooefScient of in- duetioo ia a. If the current in this conductor is >', there will be, by jVrt 563, a force H, urging it in tho direction of e, where ^=yy-5^ = _A^5yy, J and, since this is independent of r, y, t, the force is the some ia whatever part of the shell the circuit is placed. G74.] The method givm by Poisson, and dMcribed in Art. H UNSAS cmiREjrr-pnircTiOir. 283 may bo applied io current-slieeta by substituting: for the body supposed to be wDiformly magnetized in the direction of s with intcnBily /, u cnrrcnt-sbeut having the fonu of its surface, and for whioh the currfiiit-fiinctiori is d> = Is. (1) [ The current* in tlie whuct will bo in pianos parallel to that of jy, and tbc strength of the current round a slice of thickness dx will be Id2. The mngnctic potential due to this cnrreot-flbcct at any point i outside it will be ,dr At any point inside tbc sheet it will be as The components of the vector-potential art! dm S=0. (2) (S) (») These reeults can be applied to several cases occurring in pnotioft. 676.] (l) A plane electric circuit of any form. Let r be the potential due to a plane sheet of any form of which the surface-density is unity, then, if for this sbi-ct we sulistitute cither A magnetic sheli of strength / or an electric current of strength / round its boundary, the values of il and of F, ff, II will be tltotie given above. (2) For a solid sphere of radius a. ~ 4 »r a* , , , ^ = -^ — when r i» greater than a, and F= -5- (3 a' — r*) when r is less than a. (6) Hence, if such a sphere is magitclized parallel to ; with intensity I, the magnetic potential will he (7) (8) 4w and 11 = -^I ~iS outside the sphere, D = — - /? inside the sphi-re. 3 If, instead of being magnetized, the sphere is coiled with wire in e^nidiotAnt circles, the total strength of current between two small eirclci! whofc planes arc at unit distance being /, then outside the sphere tbc value of U is as before, but within the siihere fi=-^/,. (9) I This is the cn»e already discussed in Art. 6T2. 284 CFBREST-SIIEETS. [6? (3) The case of an ellipsoid noUormty ma^etiz«) parallel to « giv«D lin« liiu been discussed in Art. 437. If the ellipitftid is coiled tvitk wire in parallel and cqnidistul planes, tbe magoetic force within the ellipsoid will bo antform. (4) A C^linilrk Magnet ttr SoUaatd. 676.] If the body itt a cylinder hnrin^ any fonn of section ind bounded by pliincs ]>erpen(liGular to ii« generating lines, a&d if l\ ig the potential at the point («, ^, r) due to a plane area <J euriaci.'Hleusity unity coinciding with the pontive end of tliv soteuoid, and Y.^ the pot-uutial at the wanie point due to a plane am of snTface-dertsity unity coinciding with the negative end, then, if tbe cylinder is uniformly and longitudinally magnetized with in- tensity unity, the potential at Uie point (j, _jf, c) will bo n = r,-r.. (lo) If the cylinder, instt^d of being a magnetized body, ts nniformly lapped with wire, so that tbere are m windings of wire in unit of length, and if a current, y, is ma<le to (low tlirougb ttus wire, the magnetic potential outside the solenoid is as before, ii = «y(r,-r,), (II), but witbinthe space houoded by the solenoid and its plane ends ii = ny{-4iir+r,-r,). (la) Tlie magnetic potential is discontinuons at tbe plane ends of t^Hi solenoid, but the magnetic force is continuous. If r,, r,, the distanoes of the centres of inertia of the positive and negative plane end respectively from the point (*, y, r), are very great compared with the tranavene dimensions of tbe solenoid, we may wnte " \ — — . (13), where A is the area of either section. The magnetic force ontaude the solenoid is therefore vety small, I and the force inside the solenoid approximate^ to a foroe parallel to ' the axis in the positive direction and e<iual to \xny. If tbe section of the solenoid is a circle of radios a, the Talue* of I r^ and r, may be expressed in the sertea of splierioal harmonics given in Thomson and Tait's }!edural PkUoto^/tg, Art. 546, £x. 11^ 677.] SOI-BNOID. 285 In Uiwe expTcsnoQs r » tlie distance of the point (x, y, z) fr&m ae centre of one of ttie circular endn of the solenoid, and the uioul irmonica, /*, , P^ , &c., are those corresponding to the an^le $ which mftlccs frith the nxie of the cylinder. The lint of these espressions is diseontinnous when = -^ bat ve must rcmetober that within the Golenoid wo must add to the oa^ctjo force deduced from this expression a longitudinal force 677.] Let us now consiiler a solenoid so long that in the part 'of space whicli we consider, the terms depending on the distance ^Jrom the ends may ho neglected. H[ The magnetic induction through any closed curve drawn within ^Rhe solenoid is ■i unyA', where A' is the area of the projection of ^■Ihe curve on a plnne noruinl to the axis of the solenoid, ^1 If the closed curve is outside the solenoid, then, if it encloses the ^bolenoid, the magnetic induction through it is \TsiiyA, where A is ^M;he area of the section of the solenoid. If the closed curve does not surround the solenoid, tJie magnetic induction through it is zero. If a wire be wound n' times round the solenoid, the ooefficient of ioductioQ between it and the solenoid is M= \isnn'A. (!6) By supposing these windings to coincide with n windings of the we find that, the coefiicient of self- induction of unit of th of the soleuoid. taken at a sufQcieut distanoc from it« eix- emities, is L = 4ir»M. (17) Near the ends of a solenoid we must take into account the t«rms Jepending on the imaipnar^' dixtribution of magnetism on the plane ends of the solenoid. The efFuct of these terms is to muke the co- efficient of induction between the i^olcnoid and a circuit which sur- rounds it le«>< than the value 4 n n J, which it has when t)ie circnit sumnndw a very long solenoid at a great distance from eitlier end. H Let as take the case of two circulur and coasal solenoids of the ^^Binie length /. Let the rudius of the outer solenoid Ite c-^, and let it be wound with wire k» a? to hnvc i] windings in unit of length, let the radius of llic inner solenoid be r,, and let the number of indings in unit of length be n^, then the coelTicient of induction between the solenoids, if we neglect the eflcct of the ends, is U=Gg, (18) 'wh«Te = lirn,, (I9) (20) y = nc//»j. 286 CTTBREST-SHEETS. [67S. 'J 678.] To (Ictcniune th« oflect of the positn-c «nd of the soleooiib we mu»t caloulatu the cucfTicicnt of mduction on the outur sol«no«l due to the oircular disk which forni* the wivd of Uie iuo»r solenoi For thisi purpose ne take the tecond expression for F, as givei in ^Illation (15), and diOerentiale it with reapect to r, tb'u ^ the magnetic forco in the dirootiou of the radius. We then muJtipljr this Gxpressioa by ivr^dfi, and inti^rate it with reepeot to/i from u = 1 to u = - , _ ■ — . Tins {fives th« coefllcieDt of induetioa with T«spcct to a single winding of the oater eolenotd ut a distsncr t from the positive end. We then multiply this by ^h acid integn>t« with respect to t from i = I to « = 0. Rnally, we multiply tho result by Wjij, and so find the elTect of one of the «nd« in diminishinfj the coefficient of induction. We thii!! find for tlie vidno of the coefficient of mutual indue betwocu the two eylinder», c,+/— r_ .where a = 4 2.4 2.3 c,*'- "t'^ 1.S.5 1 c,*/ 1 „p,* . 6V\ 0. ISS where r is put, for brevity, instead of •/t^^ c,*. It appciars from this, that in calculating the mutual induction two coaxal solenoids, wo must use in the expression (20) instead of the true length I the corrected length ^— 2e,<i, in which a portion eqtial to aOj is supposed to bo cut olT at each end. When the solenoid is very long compared with it« external nulius, n «=i- t~ii< (23 679.] When a solenoid consists of a number of layers of wire of 8uch a diameter that there are n layers in unit of length, thi number of layers in the thiokoess Jr is ndr, and we have P 0=4-aJn''dr, and y = ■wtjn'r*4r. , tii^ If the thickness of the wire is constant, and if the induction take place between an external coil whose outer and inner radii are » and V respectively, and an inner coil whose outer and tooor radii are f and r, then, neglecting the edect of the ends, (?, = j^/V«.'('-f)U*-'')- (Ml 68o.] iKOtrcnoif ooiL. 287 ^TI>at thi« may bo ft mtucimiini, x nod s I>eing given, and y riabl«, , -» • = 0-4p- (20) This (-qHation gives the beat rclitUon between the depths of the pritnary and M->ci>n<Iury coil for SD induction- machine without an iron core. If there is an iron core of mdios ;, llien Q remains oe before, but ,.„«(^ (27) (28) I I f ^ is 'giren, the value of t which gives the iDaximnis value of ^ is Vhen, as in the case of iron, « U a hirge number, z = ij, nearly. If we BOW make x constant, and y and : variable, we obtain the maximum value of Gy whea z : f : g : : i : 3 : 2. (30) The coefficient of )>clf-in<Iuction of a long solenoid wbottc outer Rod inner radii arc « and y, and having a long iron core whoM nkdins is f , is per nnii length 4ir/"'j«;/'«'(p« + 4nKa«)rf7-+»//*V(^-f4ff«f')rfrj«»rfp, = §7i''/«*(j--f)»(«" + 2«y + 3/ + 2<jr«.-*). (31) 680.] We have hitherto supposed the wire to bo of nnirorm thkkncxti. Wo shall now determine the law according to which the thiekn»N muRt vary in the dintTent layers in order that, for a given value of the resistance of the primary or the twcondar^' coil, tlie value of tiie coefficient of mtitunl induction may be a maximum. Let the leaistAnce of unit of lengtli ofa wire, ttuch that h windings occupy unit of length of the solenoid, be p»'. The reoEtoDce of tJie whole solenoid i* R^%itlj«*rdr. (82) Thecondition that^ with a given value of A, Onuiy boa maximum dG 3? iG ^dR , ^. IS -I- = C -T-, where C is some constant. This givM ■* proportional to -, or the diameter of the wire of the exterior coil must be proportional to tJio square root of tlw radius. IH 288 CDB C BST-snEETS. In order that, for a g'iven ralue of S, g may be a maxinaam «. = C(r+i^'). (35, Hence, ir there k no iron core, the diameter of the wire of th« interior coil thould be inversely as ike sqnare root of tho ntdiu*, but if ttiere is a core of iron having: a hig^h capacity for magnet^d izalion, tho diameter of the wire should he more nearly dirvdlji projwrtional to the square root of the radius of the layer. An EndltM Soknoid. 681,] If a solid he generated by the revolution of « plane ana . about an axis in ibs own phino, not cutting it, it will hare tlie fovm of a Ting. If this rinj* be coiled with wiro, tio that tho winding of thv coil are in planes jutesing: through the axiv of the ring^, then, if B is the whole number of wLndings, the current-function of the 4 layer of wire is ^ = -— nyd, where $ is the angle of aximuth about the axis of the Hog. If £1 is the magnetic potential inside the ring and ii' that ou side, then £l-a'=-iint>+C = —2nye + C. Outside the Hng, A' must satisfy Laplace's equation, and must vanish at an in6nite distance. From the nature of tlie problem it must be a fanotion of only. Tlie only value of 12' which folGls these couditioDS is xcro. Hence C=0, U = -2ay^+C. The magnetic force at any point within the ring i* perpendicular to the plane passing through the axis, and is equal to 2 ay-, where r ie the distance from the axis. Outside the ring there la DO magnetic force. If the form of a closed curve be given by the coordinatee z, r, and of its tracing point as functions of t, its length &ODt a fixed point, the magnetic induction through the closed curve may be found by integration round it of the vector potential, the com- ponents of which are i t 1 I ,< We thus find 2Ny f'sdr^ I ' J-* .'ft f as taken round Uie curve, provided the curve is wholly inside the ring. Vlg.m EITDLESS BOONOID. ^■f the curve lies wholly without the ring', bat embraces it, the HvDig7>eiio iDdnction through U is ^■rh«n I' tt aini ^"•'11?%^=^'^''^ rherc the accpnted ooordinntes rofer not to the closed carve, but to . singte wiiKling of the solenoid. Tlie magnetic induction through tmy closed curve embrftcing the ring is therefore the same, and equal to 2nya, where a is the linear rif .' J/ qoantitj i ^ -7-, A*. If the closed curve does not embrace Uie ring, the magnetic induction through it is zero. Let a second wire be coiled in any manner round the ring, not necessarily in contact with itv so as to embrace it m' times. The induction through this wire is 2n it'ya, and thereibre 3t. the coefficient of induction of the one coil on the other, 10 ^=2Hn'a. Since this is <|U)te independent of the particular form or position of the second wire, the wires, if traversed by electric curreDte, will rienc« DO mechanical force acting between them. By making fthc second wire coincide with the first, we obtjtia for the coefficient Wf-induction of the ring-coil Z = 2a*(i. CHAPTER Xin. PABALLSL CUBBBKI& Cylindrkat Omduetori. 682.] In a very importimt cla&s of eltctriciil arrang«menta th« current is conducted tliroU):fti round wires of nearly uniform aection, Mild fillier straight, or mch that the radius of curvature of the axi< of the n ire is ver\' ^real coin|iarnd with the radius of the trstuveisA Mwtion of the nire. In order to be prepared to deal mathematically with such arrangements, we shall begin with the case in which the circuit couBists of two very long parallel conductors, with two pioM* joining their end^ and we slial) confine our attention to a part of the circuit which ia so lar from the ends of tlie conductors that the fiict of their not being infinitely long does not introduce any sensible change in the distribution of force. We shall take the axis of i parallel to the direction of the con- ductors, then, from the syrometr}' of the arraugements in the part of the field oonitiderod, everything will depend on U, the comi>oaent of the vector- potential parallel to :. The components of magnetic indactioo become, by equattona (A), .4J. <■) For tlie sake of generality we vhnT) f^ppose the co^cient t mugnetie induction to be fi, so that o =^a, i = p,fi, where a and nre the compoucnts of the magnetic force. The equations (E) of de«tri« carrents. Art. 607, give I a 4 STRAIOnT vritB. 688.] irUie outTvnt ix a fiinotiua of*-, the di»t»Doc from tho axU ^of ;, «ud if we wrilv x = rcosO, and y^raaff, (4) I And ;S for the magiietie force, iu the direction in wliicli $ is nica»urcd 1 pvqicudiculur to the plani; througt tLe axis of ;, we have --^.^. = ^^(^0. (») If C is tho whole current flowing tlirougli a section bounded by [s circle in the plunc xy, whose centre is tlic origin and whoso Tiiditis is r, f C= I 2Trrii)dr = \^r. (ii) It appears, thcreforo, thut the magnetic force iit a given point due lo a current nrnmged iu cyliiulrieid »trula, whose common axis i» the axis of i, deiwnds only oii the tuUil slreiigtli of the uiirreut flowing through the strata which lie between the given point and the axia, and not on the diiitribution of the current among the » different cyliudrical strata. For instance, let the conductor le a uniform wire of radius a, and let the total current through it be C, then, if the current is Dnifortnly distributed tJtrough all parts of the section, w will be oooBtant, and 6' = u len*. (") llie current flowing through a circular section of radius r, r being |l(t8B than a, is C'= itior-. Ilcnce at any point within the wire, lOatside the wire /5 = 2 (8) (fl) In the subetanc« of the wire there ie no magnetic potential, for Ewttliin a conductor carrying an electric current tho magnetic force- not fulfil the condition of having a potential. Outeido tho wire the magnetic potential is il=~2C0. (10) Let ns guppoBC that instead of a wire the conductor is a metal tub« whose externa] and internal radii are tij and a^, then, if C is Khe current through the tubular conductor, %« magnetic force within the tube is zero, tube, where r is between a, and ■>,, (11) In the metal of tli« (12) \ 1 29S vxnkthsjr BBESTS. and oataide the tube, ^ (" the same as wheo the ciirrent flows thronffh n solid wire. 684.] The magnetic induction at any point i« ^ = n^, and siacv, by equation (2), rf/f Tb« Taloe of 11 onteide the tube is J A~2^,C\ogr, (uy nhcre ^ i^ the value of^L in tlif space outnide the tub4>, and Ai» t constant, tho value of which dopeuds on the pontioD of the return carwnt, In Uic tubstanee of the tube, Ib the space within the tube // i* constant, and II=A-2^Chgc,+^C{l+^^,\os^^- (IB) 685.] Let the circuit be completed by a return curront, flowing in a tube or wire {nrallol to the first, the axes of the tn-o currents b«ing at a dieitanei' i. To determine the kinetic cnci]gy of tht^ i>yst«m we have to catcuUU; the integral T= \jjJH«dsdyd:. (19) If we confine our attention to that pari of the system which lies IwtwMUi two planes perpendicular to the axes of the conductors, and distant / fron each other, the expression becomM ^ T=\lfJHKd^dg. (80) If we diKtinj^ish by an accent the quantities belonging to the return current, we may write this ^ ^ =JJirti^d^dy+JJja'icdrjf + jj Bwd^dg+jJH'm'djr^/. (sip Since the action of the current on any point outside the tabe is the tame ait if Uie Kiinie current had been concentrated at th« axis of the tube, the mwiii value of ff for the section of the return current w A—'2iJyC\ogb, a»d the mi.-«n valoe of H' for the sectioa of the iKisitivt' current i« A'—2^„C\vgh. LOSGITCJDINAL TEN8I0K. 293 llence, io the expression for T, the first two terniB may be writt«o InteiiTatiii^ the two lattvr terms in the onliunry way. nn^l lulding tlie resulte, remembfritijy that C+C'= 0, wo obuiii t!i« valuo of the kiiiotic enei^gy T. Writing: this k^C^, whoru /- u th« co- L-fBvit-nt of solf-imluetion of the syBtem of two cmduuton, we find lU Die value of £ for unit of letigtb of the eyctem + 1*^17^=^ + 2fl/* iog?L. 1; (22) (23) '1— "«- «^-<')^ If tbe eondtictors ftre solid wires, a.^ and ^/ arc zero, nod r /a -j- = 2^ol"8 7V' +i(M + f')- * a, a, It is only in the case of iron wires that we need take account of the monetae induction in calcuUtinpf thi-ir gclF-induction. In other casce we may make ^, ft, nud (i nil oqual to unity. The smaller the radii of the win^s, und tlic greater the diatttnce between them, the greater is the self-induction. To/nd (ht Sepnl^oH, X, betioeen the Two Portions o/ Win. 08C.] By Art. S80 we obtain for the force tending to increase b, = 2f*ojC», (24) wbic^ Sf^rees with AmpireV formnta, when >/, = 1, as in air. <J87.] If the k-u^lh of the wires is great compared with the distance between tbem, we may use the coefficient of self-induction to ilt.it<;rmi&c (h« tension of the wires arising from the action of the current. If ^ M tlii< tension, ,rf£ z=\~c^. = C»j^.^^ + f (25) In one of Ampere's experiments the pAnillcl conductors coDsiBt of two troughs of mercury connected with each other by a fioating bridge of wire. When a current is made to enter at the extremity of one of the troughs, to flow along it till it reaches one extremity 294 PABALLRI, CtTBRENTS. [688. of the floating wire, to pnat ioto the otb«r trou'^h through liic floatincf bridge, ttod »o to return tioog tfae second trough, Ibc flontiag bridge moves a\nng the troaghs so as to leogthfio tb« part of th« mercoiy traversed by the cnrrent, Profcasor Tmit tios simplified the electrical conditionft of thii cxpcnmviit hy substituting for the vrirt- a Hooting siptinn of gls» filled wilh morcury, eo Uiat the current flows in mercury thron otft ita course. This experiment ie sometimes adduced to prove that two clemcntA of A current in the same Btraight line rejicl one another, and Uidb to shew that Ampere's formula, which indicates snob a repulrion of colIincfirelemcDtd, is more correct lliunthat of Orasamann, which gives no action between two eiemcnltt in the tame straight line Art. 526. Bnt it is manifo^ that since the formulae both of Anipire and of Gmsbmann give the same results for closed cireuitii, and since n* have in the experiment only a closed circuit, no result of tie experiment can favour one more than the other of these theoriet. In fact', both formulae lead to the very same value of the repulnon as that alrouly given, in which it appears that i, tli« diMtiince between the parallel conductors, U an imiwrtant element. When the length of the conductors i« not very gn-at com with their divlance ajiart., the form of tlie valia- of li beooi tomewhat more complicated. 688.] As the distance between the conductors is diminished, value of L diminisheis. The limit to this diminution is when the wires are in contact, or when It » a,+a^'. In this case 4 J 6S<>0 SnOMPM SBLF-IJIDPCnOX. 295 I fThia IB a minimum wh«n Hj = u,', and Uicn K L=2l(\og4 + \). ^^^ =2/(1.8863), HH 3.7726^. (27) V^rtiiB is the smallest \a\ac of the Reir-iDduclion of n round wire Hdoubled on itself, the whole len^h of the wire bein^ 2 /. f Since the two parts of tJie wire must he inxulal^d from each other, the Gelf-indnetiou can never actually reach thin limiting value. Ry using hrood tkt strips of metal instejul of round ww» the self-induction msty be diminished inde&nitely. Oh tie EUeiromoiive Force required to produee a Current <f Faryiny Intfiui^ ahiig a Cylindrkal Condactor. 68D.] When the current in a wire is of varying intensity, the ive force arifiing from the induction of the current on ' is different in different part« of the section of the wire, being' general a function of the distance from the axis of tlie wire I as well as of the time. If vre suppose the cylindrical conductor Lto consist of a bundle of wire« all forming part of the same circuit, ISO tliat Uic current it compelled to be of uniform strength in every part of the si^cticn of the bundle, the method of calculation which _ we have hitherto ui>cd would be strictly applicable. If, howcwr, f we consider the cylindricnl conductor as a solid ma«i« in which electric currents arc free U> flow in obetlience to electromotive force, tlie mtcnaity of the current will not be the same wt different didtances from the axis of the cylinder, and the clectrom<itivo fonrCK tlicmselvos will dt^pend on the diittribution of the current in the different cyiiodric strata of the wire. The vector-potential //, the density of the current », and the electromotive force at any point, must be considered as fiinctions of the time and of the distance from the axis of the wire. n Tlie total current, C, through the section of the wire, and the total BiJcctromotive force, E, acting round the circuit, are to be regarded Vm the variables, the relation between which we have to find. ^^^Irtt US assume as the value of //, ■K n=$+T,+ T,r' + kc. + T,r'', (I) '^^Sere S, T^, T„ &c. are functions of tb6 time. Then, Ihmi the equation d*H IHH , ,„, _. + j-^=_4,«. (2) we find -UK = 7', + &c-|-»'r.H-'. (31 Bwef 296 P^SJILLKL CUneEKTS. If p deDotw tlie specifio rciii»taDoo of the 8uh«tatM!e per mit of volume, the elect roniotive for«e at any jo^int in p t, und this quj- Ik ex)>re«iH>d in U'rmij of the «l«otric poteiitiul oiid Uie vector pot«a^ 7/ bv cquutioDs (B), Art. 598, or (*) Comparing the cocffiine&ta of like powers of r in eqnatioos {5} and (5), it ^ P * as dT„. ^^- P dt ' ^" pn' dl Hence wc mnj' write <f* ^^-'pdt'- dz 7.= I ^r (9) (ID) p-{n\f3F' 600.] To find the toUt current (7, we muet integrate to over tbe ^ twclion of the wire whose radius is a, 1 Jo Krdr, (II) Suhstitnting th« v&lue of v lo front eqnation {9}, we obtain C = -(7',a<+&o. + »7',fl"*). (12) 1'^ie value of H lit any point outside iJie wire dcpendtt only oB the total currpnt C, nnd not on the mode in which it is Uislrihuted within the wire. lU'Dce we may assume that the vxliie of //at the nurlace of the wire is AC, where A ia ^ constant to be det«nniDed liy calculation from the general form of the circuit. Vatimgtt = AC when r = B, wc obt^ti ^ AC=Si■T^ + T^a* + &c. + T,a^^. (13^ If we now write = «, a is Uie value of tlie conductivity of_ anit of length of the wire, and we have dT 2b* rf>7" «o" rfT +&«.), AC-S=T^J^^^,^ d'T . a' dT , . M ('«) VARIABLE CCBBE8T. 297 To eliminato T from thcae cqiuitioue wc mast Gnt reverse the ^Beriee (14). Wo thus find fit*' d*C + rfi«'»'^-fVA«'"^f&c. 4t ^'"rf/ '" ^/» W« have bJbo rroio (tl] and (15) Prom the la«t two eqiiationB we find .dC dS dC -^) + '^+*°S?-tV'^ dt dt dl* ■^^ ■ ^'^^^..^O.^ + A-V*»''' rfT* ■ If/ is the wliolu length of the circuit, R its resiiitance, and E the H electromotive force due to other cnuscs tliaa the induction of the KxorreDt on itself, dS E I F «=g' (17) The 6r9t term, RC, of the right-hand member of this eqaation expresees tlie electromotive force rei^uired to overcome the reeist- I aace according to Ohm's law. The second term, l{A + i) -^, expresses the electromotive force tit [ which nrould he employed in increasing the electrokinetae momentum the circuit, on the hypothestK that the current is of uniform th at every iKiint of the section of the wire. Tlie remaining terms express the correction of this value, arising lUw f»ct thut the current is not of uniform strength at diiferetit from tlic axis of the wire. The actual B3-«tcm of current* \h»a a greater degree of freedom than the hypothetical syrtem, in which the current is constrained to he of uniform streugth throughout the section. Hence the electromotive force required Bto produce a rapid change in tlie strength of the current is some* H what less than it would he on thix hypntlieHia. H The relation between the time-integral of tlie electromotive force H«nd the time-integral of the current is fEdl = njcdi + / ( J + i) c- A ^ ^ + &c. (1 9) IT th« cnneot before the beginiung of the time has a constant 298 PARALLBL CtTRREKTS. [fi^I. vaiuc C^, and ir daring the tiiufi it rises to Hic vnliic C, , and ny nmiDB coDNtnnt nt Ihnt vuluc, then the t«nDfi involving tite dtfler- ential coeflicivittn of C vanUh at both limits, And fEdi = R fcJl+l{A + J) {Ci-Ct), («J tlie cuat value of the eleL-tromotirc impolse u if the curreDt had been uniform Ihroujjhout th« wire. 0« Ue Geometrifal MeaH Dislaxee y Tko Ttgarea in a Plaiu* fiftl.] Til oalciilaling the plectromnfrn'^tic action of n current flowing; in a ctriiijrlit conductor of any gircn soction on the current in ft purallel conductor vrhoKc locution is also given, wv bavi; to find the intejrral CCiT where Jxiiy in an ot«nient of tlie area of tbe first sectioD, i*'^ uo element of the second section, and r tbe distance between thctie eI<!Ricnt«, the integration being extended first over every element of tbe fint Hcctioii, and then over ever^ element of tbe second. If we now determine a ttne U, sneh that tliis integral is equal to A^ A^ log R. wbere Ax and A,^ nrx^ the areas of tbe two Reetions, tbe length of fi will be the «arnc tvbat'evcr unit of length we adopt, and whatever sytttcin of logiirithme wo use. If wo supiiOKC the Mcttomt divided into elements of equal nze, then tlic Ingaritlim of B, muIUptied hy the number of pairs of cb-menls, will be ei]ual to tlie mm of the logarithms of the diittaiK'eif of all tbe pairs of elements. Here R mny be considemd a» the geometrieal idcad of all tJie distances hetwcca pnirK of clement*. It is evident that the valoe of R must be intermediate between tbe gteateet and the least values of r. If R4 and Hg are tbe geometric mean distances of two fig1lre^ A and B, from a third, C, and if Rji*a is that of tbo sum of tbe twe figures from C, then {A+B) log Ra^b = J log «^ -tSlogR". Hf means of tbis ntlalicm we can determine R for a ofHnpoond figure wheo we know R for tbe parts of tbe figure. • Trau*. S. B. RU*., l$Tl-9. 69i.] OBOMKTBIC JIBAN DISTANCB. 299 692.] IBXAUPLBS. (1) LH R be thi- ni<>-an diKtanoc from the point' to Hic lino JS. Lut OP be iwrin-iiJioiiliir lo AB, tbcn JJi{\ogIl-H)=AP\og04 + PBhs0S+0PA'0£. Fig. 41. (2) For two lines(Pig. 42) of longtlis a nnd (> dran'o perpendicu- lar to the fixtnrmities of ii line of leiij^li c and on the same eide of it. ai{2\ogR + S) = (c»— (<!—«)*) log v'c* + («-4p + c» logc — *>(« — iltan-' +act«n"' - + fctiui-' -■ ^ ' c c e Fig «. (3) For two lines, PQ and ^5 (Fig, 43), wbo»c directions inter- ect at 0. 'Q.AS(2lo»/f +-3) = \ogPR{20P.ORBia'0-PS*<!o»0) + \ogQS{20<i.OSmn'0~QS*eo90) - log PS (zOP.OSsin^ 0~PS»coaO) -\ogqR(20Q.OR^Q*0--QR'cosO) -noo [OJ^.^s-oQ^s^R-t 0R\ p'^q-os'.Ps'q}. Fig. IS. 300 IlLEL CUBRENTS. (4) For a point and » twctongle ABCD (Fig. ii). hoi OQ, OR, OSf be pcrpciidicularg on the aides, then AB.dJ){2U>gB+3)=2.0P.OQ\ogOJ+2.0Q.OItlogOB + 2. OR. OS l(^ OC-i- 2. OS. OP log OD + OI».D'd'J + OQKjd'B + OR'.Bd'C + OS'.C'd^. Fig.it (5) It is not necessary tbut tlie Iwo figures should be dilTerent, for | we may find the geomotric mean of the distanoes between every [air i of poiotfi io the soma figtire. Thus, for a ctratght line of Ivngtli a, | \oeR = loga—l, or R — aa~t, S = 0.22313 a. (6) For • reutangle whose ddc>e are a aad 6, logV? = lDg>/^?T6W^' log/Y^l + ^-i^loff/^r+Jj .J>-^*+$>n-'j-«. ^hen tlic rectangle is a square, whose side i* s, log J? = Iog«+l Iog2+ ^-H. (7) The geonuttric nicnn distancA of a point from a circular ' ie equal to the greater of the two quantities, its tlistance from ' centre of the circle, and the radiua of the circle. (8) Hence the Reometrio mean distance of any figure frol ling l>nund<-d by two concentric circles is equal to its g«ometria mean distance from the centre if it is entirely outside the ring, l)ul| if it is entirety within tJic ring 693-1 SELP-lKDUmOlT OF A COIL. where a, and a, ar« the outer and inner nulii of tLe ring. R is in this case independent of tlie form of tb« figtire within the ing. (9) The geometric roesn distance of ftll pairs of points in tlw ig is found from the equation 1 n 1 "a* 1 "i ,3a.'— a' For a circular area of radius a, this becomes ]ogi? = logffl-i, or R = a«"*, R= 0.77«ea. For A circulnr line it becomes R = o. 693.] In calculating the coefficient of Eelf-indoctioD of a coll of liform M!Ct4on, the radiiiN of curvature being grtiit c^tmparod with dimeiwiottt of the tnitiBTer«e section, wc first determine the trie mean of the distnuccs of every pair of points of the section by tlic method already de8eril>ed, and thcji wc cak-ulste the eoeffietent of mutual induction betwrn'm two linear oonducton of tlte given forn), phK'ed at thif distance apart. B This will Imi the coeffldent of Belf-induction n-ben the total cur* rent in the coil is unity, and the current is uniform at all points of ^^.he section. ^1 But if tlicre arc n windings in the coU we must multiply the ^coefficient alre-ady obtained by »", and thus we slinll obtain the 'floeffldent of self'induction on the supposition that the windings of Uie conducting wire fill the whole section of the coil. ^K But tlie wire is cylindric, and is covered with insulating material, '•o that the current, instead of being uniformly distributed over the section, is concentrated in certain parts of its and this iooreaaes the ooeffioieDt of self-induction. Be«ides tliis, the currents in the neigfabooring wires have not the same action on the current in a g iyea wire as a uniformly distributed current. ^B The oorrectiona arising from these considerations may be de- ^Termined by the melhod of the geometric mean distance. They are proportional to the length of the whole wire of the coil, and may be expressed as numerical quantities, by which we must multiply the length of the wire in order to obtaio the corroctioa ' coefficient of self-iuduotion. ^rf^ 802 PABALLEL CtTIlREKTS. [693- Let the diameter of the wina lie d. It is covered with iaffnlatinj; EDat«ri»l, ftnd wound ioto a coil. We shftll suppose that tbt eectioDs of the wires are in square order, as in Fi^. 4S, and that the distance between the axis of each wire and that of the next is i>, whether in the direction of the breadth or tiio depth of thr coil. J) ie evidently greater than d. Wc have first to dc^tcrmiiie the excess of eelf-iDductioo of unit of length of a cjlindric iviro of (liamotcr d over that of unit of len^li of o fquarc^ wire of side D, or . fl for tho square ^ Jt for the circle = 2(logT- + 0.138060fi). The inductive action of the eight oea round tvires on the wire under consideration if loss than tlint of the corrcepondiog eight equarc wires on the equaro wire in the miilil by 2x(-0197l). The corrections for the wires at « greatv distance may bo neglected, and the total correction may be written 2 (log, T- + O.I 1835). The fionl value of the self-induction is tlierefore i= ii»if+2/(log,^ + 0.Il835). ! II ID the number of windings, and i the length of the 3f tlie mutual induction of two circuits of the fonn of the mean wire of the coil placed at a dintancv R from each other, wbure £ i» the mean geometno distance between {lairt of pointa of the •eetion. J) n the distance between consecuUre wirea, and d the diameter of the wire. o o o o o o o o o III all Fig. 16. CHAPTER XIV. CIBCCLAR Cl'BKENTS. Magnetic Poienlial due to a Circular Current. ■J] The tnagnelic j>oU-ntinI nt u given point, due to a circuit fing a unit current, t.i iiuincrirally e<junl tn th<.- soliil aogU* «ul>- led hy the circuit at tliat point; see Arts. 409, 485. When the circuit is circular, the solid angle is that of a cone of the second degree, which, when the given point is on the axis of the circle, becomes a right cone. When the point is not oa the axie, the oone is an elliptic cone, and its solid angle is numerically equal to the area of the spherical ellipse which it traces on a Bphcre whose radius is unity. This area can be expressed in 6nit« terms bj' means of elliptic integrals of the third kind. Vfe shall find it more convenient to eapand it in the form of an infinite scries of ^hcrical harmonics, for the tacility with which mathematical operations may be performed on the ^ncral term of such a series more than counterbalances the trouble of calculating a nunibcr of terms guITi- ^ont to on«urc practical accuracy. ■ For the (take of generality wc shnll B^nie the origin at. any point on the sxif of the circle, that is to sny, on tlie line throogh the centre perpen- dicular to the plane of the circle. Let (Fig. ii) be the centre of the circle, C the point on the axis which wc assume as origin, II a point on the circle. I)«!*crib« a sphere with C as centre, and C/l m radius. The circle will lie Fig. IS. on thia 8ph<rrc. and will form a mnall circle of the sphere of M^hur radiuri c. [[Mgumr rau 304 CIBCULAB CUBBEXTS. [6 UA CH=e, OC = fi = ««»tt, Olf = a = enaa. Let A be the pole of the sphere, and Z xay point oa the axis, aaj | let CH = .-. Let li be any point in epace, and letCS = r, and ACJi = 9. Let P be the point where CR cute the eph«re. - The magnetic polt'ntial due to the circnlar corrent i» equal to j that due to a magnetic Dht-ll of strength unity bounded bj the I current. As the form of th« surfacn of tlie shell is indifibnal,! providi-d it i« bouuded by tlic circle^ vns may auppOM it to coincide j with thi; 8urijivc of the «plicr«. Wc have sliewn in Art. 670 that if F ia the potential dne t« a ' etratum of inalt«r of «iirfuce- density unity, sprwd over the sar&n 1 of the sphere within tlie small cirelA, the potential duv to a nu^ | Dctic shell of utrcngfth unity and bounded by the same circle is Wf liave in the first place, therefore, to find F. Lt't the g-iven point be on the axis of the circle at J?, then tbe] jiurt of the potential at Z due to an element dS of tlio spheric*!' Hurlace at /* is JS FP' Thia may be expanded in one of tlie two serioa of spberictl huA monius, AS f a ^ i or :^|7>„+P,f +&c. + i»,^+&c.j, the first scries being convergent when 2 is leas than e, and tlie second when x is greater than «, Writing dS = — c> dp.d<^, and inlcgrating with respect to ift between the limits and 2ff, and with respect to li bettreeo the limits cosa and I, we find or '"=2»~jy*i>„rfM4&o. + p/'i*<rf»i + otc.|. By the cliaracLcristic eqtiation of 7>^, 595-] SOLID AjrOLE SeBTSNDKD BT A ClftCUE. icnce £p,,. eipression £aUs when i = 0, but since Pq = 1, (2) / Ax tbe function ~j~^ occurs in every part of tUis invcstigntioD iv« l)«U denote it by the nbbrcviatcd symbol i*,'. The values of /*,' oorrespondiDg to eevnal values of i are given in Art. 698. We are now able to writ* down the value of F for nuy point S, whether on the axis or not, by substituting r for :, and mnltiplying each term by the zoniil harmonic of of the game order. For /' must be capable of expansion id a series of zonal harmonics of 9 wiih [Hoper cocfBcicnts. When $ = each of the zonal harmonies btfoomcs equal to unity, uud the jioint It lies un the axis. Hence the cocfficti-nt« are Ui« terms of the expansion of f for a point oa th« axis. We thus obtain the two series or r=2,^|l-p+&o. + i^^7r(«)7;(fl)j. (4-) 695.] We may now find m, the inagnetic potential of the circuit, Ll>y the method of Art. 670, from tlie equation We thue obtun the two aeries « = -2«|l-ooa« + &c.+ ^jJ»/(«)i',(tf)+&c.jt (6) - «2xain»«{ l5i>,'(a)/', («) + &c.+ ^^>/f«)iJ(tf}} . (6') The series (6) is converf^ent for all values of r less than c, and the series (6') is convergent for all \-a1ues of r greater than c. Al th« eurfaoe of the sphere, where r = e, the two series give t)i<* Muiie for tt when $ is greater than a, that is, for points not by the miignctic shell, but when is Jess th»D a, tliat is, I on the muj^ctic shell, o.'=M+4ir. (7) If we assume 0, the centre of the circle, as the origin of co- ordinatea, we must put a = ~ , and the eerier become VOL. II. X ^B VOL. 306 CTBCtrtAR CCRRSSTS. (696. - = -2» j • + ;-fiw+&o.+(-)'^-y^^;-'>ffp,..,,(tf)}. it) where the orders of all tlic hnmiunics are odd *. On lie Pofentiat Bnersg o/two GnuUtr Otrrenit. 096.] Let uii begin by «]p]HMnng tlic two miij^etic dicllx wirieb nruoqiiivaleiit In tJie enrrents to be portions of two concentric spbem their radii being e^ and c^, of which e, in the greater (VSg. 47). Lei us also suppoKe tlmt the axes of tlw two shells coincide, and that O] is the nng'le subtended ij the radius of the firat shell, and n, the angle subtended by the radius of tbe second shell at the centre C. Let w, be the potential due to tbe first shell at any point tvilhtn it, thai the work rcquin-dloou-ry thosocood shell to an infinite distance is tiw value of the surfnce-integn] HcncB 4«*sLn»«,v5rPi'(''i)/'^('')'^'%+*«-+^^(''i)Aw«'**,{' or, snbMtiluting the viilne of the mt«ignils from equation (2), Art. 694, J/=4«»m»a.sinS^,'ji^^A'(«.)Pi'^)+&e. + jj^^P/(aJP,'{«jj * Tlia TtloA of ttie nlid tagia aubttadot by • dnlo nujr b« obtained ia a nan* Fl«.47. extended over the second sholl. tuba tWUMfallMIA— • MBd •agjlo nibtaadad by tha ebola ai tbe point 2 in tb* kxb b «Mlly 1 . I. I — CDMB, bpwding tbi< tgc|ii«>ian Id ipliivlt^ buinanla, <t« ted for th* «ii«uiniu or w fiir jwlnu no tbo iixl* for wbMi > ■« 1«» lli«n " or fn-*— 1 tliui e tttftetiidj. TboB naalM eui sad/ b« ahewn to oounciil* witb Ibiao if toik i^B POTBSTIAt OP TWO CTOCLBS. M7 = 47l'illl*0 697.] Let Ofi next 8Uppc«e that tb« axis of on« of the sbelb itt jburned about C as a centre, so tliat it now makefi an an^e 6 witb be axis of the other shell (Fig;. 48). We have only to introduce the zonal harmonics off into this expression for M, and we find foi [the more ^-aeral value of M, This ix the value of the potential energy due to the mutual Rction of two cireulur currents of unit strciig^.h, pluoi^d so that 'th« normalf through th« eontrvu of the i>ircU-« meet in it point C in «& angle 0, the di¥t«&ci» of the oircuinferencex of th« circle* from the pciiil C heing Cj and e.^, of which f, ia the greater. If any displacement (/xaitf-'r* the value ^of M, then the force acting in the direo- H, „ rfjf I tion of the displacement is X = tlx For instance, if the axis of one of the shells is froe to l»m about the point C, m> as to cawHc tit vary, then the moment jf the force t«n<liog to increase 6 is 0, thtre dM Performing the differentiation, and remembering that rhcre P/ has the same signifioation as io the former equations, I = - 4ir" sin' o, sin* o, sin <i(* { J ^i>,'(ai,) ^/(oj) Pi'C^) + fee. fi98.] As the values of P' occur frequently in these ealnilationn 'the following ttthlc of values of the first nix degrees may bt u>»ful. In this table h stands for cos 0, and v for sin $, />,'= 3;*, X a CtBCDLAE CTOKENTS. 699.] It is BomeUtuea convenient to expreaa Uie series (or Jl ia terms of linear quantities as follows :— Let a be the radios of tbe smaller circuit, b Ute dittaoco of i plane from the origin, and a = -/a* + A*. Let A, S, and V be the corres^wnding quantities for thv circuit. Tbe series for M may then be written, Jf' jitji + 2.3.w»~«»{(coe««-isin*fl) + S.4.H' ^^^~*^^ a'{^-ia')(cos*0-|Biii'g< ;^{»§— ^^'-H- ir we cuke $=0, the two circles become pAiallel otui on mae axis. To determine the attraction between tlieuj we ma/ diiJ^Tentiate M with respect to i. Wo thns find 700.] In calculating' the effect of a coil of rectangular aecti WB have to inteffiate the expressions already found with res to A, the radius of tHe coil, and It, the distance of its plane from the origin, and to cxtond the integration over the breadth and depth of the coil. ^B Id some cases direct integration is the most convenient, boP^ there are others in which llie following method of approximation leads to more useful results. Let P be any function of « and y, and let it be roqmred to fi the value of P where In this expre«»ton F is the mean value otP witliin the limits integration. Let P^ be the value of P when x = and 5 = 0, tien, exiiandic P by Taylor's IVorem, Integrating this expreasioD between the limits, and dividing- 1 result by xy, we obtain as the vatne of i^. 701.] COIt OF BECTANOrXAH SECTIOS. 309 'Po+it{- .rf'n +j, • ^ ) tie' ■' d/ In the c«ec of the coil, liit the outer and inner radii he A + \ f , land ^— if respectively , and let the distance of the planes of the I vrindings frum the ori^ii lie between Ji+\jj and j9— J >], then the ! IjTCfldth of th« coil is ij, nnd its depth (, these quantities being [small compared with A or C. In order to culculntc the magnetic eSVct of such a ooil we may write the succcsnve terms of the scries as follows : — * _B e,= ^^{l+^. ".— g(-A(A-.4K...i^,.)' /. , /2 25 35 JS ,- . im-sA'^ c* ')■ G, = 4^^^^^"^,^^^ + ^^{C*(9S'-12A')+35A'£'(6A^-4B')} (P 24 C" 'ate. 24 C + ^l_.pj»C*(6a«-44^) + 63J«£'(4£«-J«)}, 8ec.,&c. The quantities <?„, (?,, (7^, &c. belong to the large coil. The ralue of CM at points for which r is Icvs than C is »=-2ir+2C„-CjrP,((J)-C,^PB(fl)-&c The quautititw ffy^ffi' ^°* l>c]ong to the small coil. The value of at' ut point* for which r \a greater than c is Tlie potential of the one coil with respect to the other when Oie total current through the section of each coil is unity ig M = 0,y, P, W+ G.^a /•,{«)+&<!■ To find M iy EU'iptic IntegraU. '01.} When the distance of the circumferences of the two circles I is moderate as compared with the lodii of tlic unwller, the series 310 ClBCin.AB CCRREST3. already ^ven do not omverge rapidly. In every case, bon-cri;r,l may lind Uie value of 3f for two pwsUel cin:l«a by elliptic in1«^ For let 6 be tlie length of the line joining the ocntres of the 4 and let tliis line bo perpendiovW to the jiknee of the two and let A and a be tho radii of the circles, tlien the integration bein^ extended round both curves. In this cuBo, Aacm (^— *') ii<t> r/<fr' where _ 2V^ and /"and E are oomp1et« elliptic inte^nls to modultts r. From thit, remembering that and that c is a function of &, vre End If r, and r^ denote the greatest and least valoes of r, r,* = (J + «)* + «*, r," = (^-«)« + i*. and if an angle y be taken such tliat cos y = -1 1 where /"y and fy denote tho complete dliptic integnds of the ft and second kind whose modulus i» sin y. ltA = a, w>t> = ",and -^=-2.eosyf2/;-(l+s«»y>^). The quantity ^ represent* the attraetion lielncen two panUlcl ciraalar drcnita, the current in «acfa being utity. ro..] LISE9 OF MAQXETIC POfiCE. On account of the importance of tlie qiuntity M in eteiHr»- la^Dt-tio calculatioDE tlio values of log(jI//lTrv'-irt). which is n ftmctioQ of c and tJiereforc of y only, have bwn tfthiilutH for interralB o( 6' in th« vnliio of tho tn\g\e y between 60 and 90 fdegices. The tabi* will Ir' found in un appendix to this ehnpttr. SicoHd Expratiion /or M. An expreesion for 3f, which is Honictimce more convenient, is got bv makiofr c, = J — ^ , in which case To droK tie Lines of Magnetic Force for a Circular Current. 702,] Tho linM of mag^nctic force are evidently in planes passing broug-h tbe axis of tho circle, and in each of these lines the value 'of M \» constant'. Okleulato tho vaUio of £^ = t-^ p — 7= from Lueendre'e ablee for a saffioient number of values of Q. Dmw rectangular axes of x and ; on the paper, and, witb centre 'at the point X = Ja (sin tf+ooaecfl), draw a circle with radius \a {c<iiec$—(aQ(i). For all points of tJiis circle the value of Cj will I be sin $. Uenoe, for all points of this circle, I .,^.„ ATT I ._.. ._ 1 -^'i. M =1 8iti/Aa ■ and J = Now A is the value of x for which the value of M was found. lenee, if we draw a line for which x = A,\X, will cut the drelo two points having the given value of M. * Giving M a, series of values in arithmetical pnjgression, the TaltMB of A will be as a series of squares. ]>rawing thorefore n series of lines parallt'l to !, fitr which x bus the values found for A, the points where these lines cut the circle will be the points where tbe corresponding lines of force cut the circle. f * (Tla weatoA exprcaion fat J/ iDiy b« ilvduatd from tb« fimt by mMU of Uw MuwIbk 'iHiiiP'Ptm*'""* In EU^dc IntegnU* : — U Vi-e*. W'. or e - l*^ J'W-(i*(i)^(<\>. *w- l+«» j(f,)-a-A>''(0.] 312 CIRCITLAB CCUBSKTS. £;*>> If we put m = 6sa, Jind ii = aw, then ^ = « = «»ir#a. We may call n the index of tLe line of TorM. The forme of thcw> lines nro (fiven in Fig. X^^II at Ihc end t^ this volume, Thi-y arc copied from a dnning given bj Sir W. Thomson in hi* iiajier on ' Vortex Motion *,' 708.] If the patition of a circle having a given axis is r^ardtil a« defined by i, the distance of iU centre from a fixed point od the axiii, and a, the radius of the circle, then M. the coeflSdm of induction of the cirde with respect to any ^stein whaterer of magneta or cnrrents, is subject to the following equation, da^ "*" (16* " a da ~ ■ ^** To piore this, let us consider the immbor of liooa of maguBlac force cut by the circle when a or ft is made to Tsiy. (I) Let a become a + la, i remaining constant. During this variation the circle, in exjiariding, sweeps over an unnulur sorfiwe in its own plane whose broiuUh it< ha. If r is the magnetic potential at any point, and if the axis of f he parallel to that of the circle, then the mognctic force perpen- dieular to the plane of the ring is ^ • To find the mngoetic induction throngh tbc annular sarfaoe we have to integrate p' , t?r . Jt dy ' where ff is the angular position of a point on the ring. Hut this quantity repi'csvnts the variation of M doe to the variation of a, or -;— ia. Henoe da .'(, djf m (2) Let h beoonu i+ti, while a remains constant. During thi«j variation the cittl* BWWpf over a cylindrio sorface of ladius a andi length it' The magnetic force perpendicular to this snrGuK at any point iB<i -J— , where r i* tbc distance from the axis. Uenoe DifTcrentiMinp equation (2) mtli reapcct to «, and ($) with respect to b, we get • JViM. &. S. Biiii-. y<H. xxT. p, sir (1889). 704-] TWO PABALLEL CTRCLES. da* J„ dy .'a [WcntH- da* d0. rf'Jf Jrify 313 (*) (5) (6) Coegiwnf of Induction i^Two Parallel Cirelfi icien the J)i»l,iii£i itf- tm^en He Arcs ia tmall compared with the Uadiim </ eithtr Circle. 1704-] We might deduce tlic value of M in this ciise from the ex))anBion nf the elliptic integrals already given when their modulus ia nearly unity. The following method, however, is a more direct ap{)1icat.ion of electrical principles. IFirtt Approximation. Let a and a-f it be the radii of the circles and l the distance between their plane):, then the shortest distence between tbeir circumfcreuces is given by p r = Vc" +i«. "We have to find the magnetic induction through the one circle due to a unit current in the other, ^ft We shall begin by eu]>[)OMing the two circles to be in one plane, H^Coneider a small element hi of the circle whose radius is a + c. At B a point in the plane of the cirole, <listAnt /> from the centre of ht, B measured in u direction making an angle $ with the dircotiou of it, the magnetic force i» perpendicular to the plane and equal to 1 I .mn0U. d$dp, To c*lcwl*tc the snrfaee int(^ral of tht» force over the dpoee which lies witliin the circle of radius a we must find the value of the integral /•*• j^i sin where r^, r^ are the roots of the equation H-2(a + f)Binflr+c» + 2a<?= 0, tiz. »i «■ (ii+c)8infl+\/((i + c)*sin'^— tf*— 2a<', ^1= («+c)Bind— N/(fl + f)' sm-0—<^—2ac, . ,„ «^ + 2ac CTIfCrLAB CUKBB5TS. 'When e n smalt compued to a wc may pot r, — 2a 811) 9, Intc^atiDg with regard bi p wc liavc 28*/ log( — sia'0J.an6d6= 2i«(lo(f— — 2), rnjarlj. We thus find for the whol« induction 3/„ = 4«(log^-2). Since the magnetic force at any point, the fliBlantM of wliicb from a curved wire iei small compared with the radius of curvature, is Dearly the »ame as if the wire had been etmight, n-e can calculate iha dilferencc between the induction throng'k the circle vrhote nuliufi ta a~e and the circle A by the formula ^f«A-M^= Jirff {logc-lo^rj. Hence we lind th« value of the induction between A and a to be 1 Mao = ina {log 8a— log r— 2) approximately, provided r is small compared with a. 70£i.] Since tbo mutual induction between Iwo windings of till same <,-oil is a very important (juantity in the calculatinn of (.-x-l porimental resuIU, I vhall now describe a method by which the approximation to the value of M for this caae can be carried to any required dCjErree of accum<^. We shall assume that the value of M is of the form M=iAA\osy+B\y where A = *•. ,'S^ ,.«?• and + &C. 5 = -2J + J?,«+ J?,"^ +J»j'^ + 5,^ +i?3'^ + 8te.. where a and a-\-as are the radii of the circles, and y the distance between their phinca. We hare to di*termino the value* of the ooeffioients A and li. It is manifest that only even powers of ji ean occur in these quan- thiee, because, if the ngn of g n reveraed, the value of M nia»l ntnain the same. Wo get another *et of conditioru fVom the reciprocal property of tlwj ooefEcient of induction, which remains the same wliichi'vi-r circle we take as the primary circuit. The value of hi mufit then 1 705.] ISDirCTIOS DUE TO A CIBCDI.AK CURRBNT. 315 [fore renaiD the same wh«n ve sabstitute a+x for a, and — « for a in the above npreedoii. We thus fiml the following conditionx of rociprocity by cqiialing [the (MMjfficwnU of similar coinbinatioDs of jd and y, J, =— ^ —jf, 5, = i— i^,+^ — 5g — £,, l-rA,^A,+i«~3)A,+ ^'*-^l^^-'h , + &e. + A., From the ^neral cjiiatioQ of il/, Art. 703, n-e obtain another Bet of conditions, iA^ + 2A'f + 6A, + 2A\=2A,i; {«-l){«_2K, + «(«-l)J'„,+ 2.3^. + 2.3^',^, = (II-2M/, tc.; 4^,+ .1, = 2^3 +25',- B, = 4-^3, 6^,+ 3^= 2^,+ 65a-f2/rg= eJ'a + SJ'j. = «(i.-2)£,+(»+l)«7?,^,+ I.2^.+ I.2i?',,,. Solving ilieso equations and eubstitating the values of the co- efBei«ntc, the s«rieB for .V becomes k=-^V'i-»:-^-^^-M ■i-iva m Sj?— y «*— Gayi* 4B(i^ • Sic.j. " [Tliii molt may be obuboil dlrtctly bj- the method rnggorttd in Art. 7M, *{l. by tha apuudo'n* of tlii] »t1I|>tk InlCKrali in tha eiiiiMubui fot Jf fuauri in fcAit. 70t. SMO»yU7'a£'WiJ(ic/'uiwf(o>u, All, 75] 316 CIRCm-AR CFRRENT8. [706. Jb find tie form of a coil for Kiiei ti« coefficient <^ «j^-»- duetiott 14 a naximum, tie total Ict^ih and IHeknen ^ ikt teire being given. 706.] Omitting' the corrcctionB of Art. 705, we find by Art. 67S where n is the number of windings of the wire, a is the mcBn radius of the eoil, and A is the geometrical mean distance of the transverse section of the coil from itself. See Art, 690. If Uiii' section is always similar to itself, R is proportional to its linear dimensioits, and n varies as S^, Siooe the total length of the wire is Zvan, a varies in' as n. Hence dn dR i:=^-R' and we find the condition that L may ho a maxioiiun If the transverse section of the coit is circular, of radins e, tlwD, by Art, 692, R and log — = V» , da dR and — = — 2 -=- > a R whuace ii = 3.22e; or, the mi>an radiuii of the coil should be 3,22 times the radint of the traDsvente section of the coil in order tluit HUch a coil may Iiave the greatest coefficient of self-induction. This nsult was found by Gauss*. If the cliunncl in which the coil ia wound has a square tnuisrerse section, the mcim diameter of the coil should be 3.7 times the side of the square »ectioD. Wttht, QMtiiigMi MUtion, lee?, vol T. p, tn. 1 APPENDIX I. ^ HV Taiie iff the vaXutt ^ ^l:^ ^' _ fkt* Vftl \ ^tn^Aa ' '■ 60" <y 4»v5a L« * . Log " - ^*~VJ^ T- 490 4 7 83 63" ac T-5963782 67' 0' 1.6927081 6' 1.5022651 86' 1-599132!! 6' 1.6954642 ir 1.SOSOS05 42* 1.6018871 12' 1.6982209 18* 1.6078345 48' 1-6046408 18" 1.7009782 24' 1.5106173 54' 1.6073942 24' 1.7037362 SO* 1.5133989 64' 0' 1.6101472 30' 1.7064949 36' 1.5I6170I C 1.6128998 36' 1.7092544 ir T.5I80582 12' 1.6156522 42' I.7120M6 4B' T.52173G! 18' 1-6184042 48' 1.7147756 81' T.5243128 24' 1-G211560 64' 1-7175375 61" </ T.5272883 30' 1.6239076 68° 0' 1.7203003 ff 1.5300628 36' 1.6266589 6' f.7230640 12' 1.5328361 42' I.629410I 12' 1.7258286 le- T.5S56084 48' 1.6321612 18' 1-7285942 24' T.S383796 54' 1-G349121 24' T.73 13609 30' 1-5111198 65" 0' 1.6376629 30' 1.7341287 ^^H 36' 1.5130190 6' 1.6404137 36' 1.7368976 ^^H 42' T.516G8r2 12' 1.6431648 42' 1-7396C75 ■ 48' 1.5494545 18' 1.6459153 48' 1-7424387 ■ 84' 1.5S22209 24' t. 6486660 54' T-7462II1 ■ 62' 0' 1.5549864 30* 1.6514169 69° 0' 1-7479848 C 1.8577510 36' 1-6541678 6* 1-7507597 ^^1 12' I.5G03I47 42' T.6569189 12' 1-75353GI H 18' 1.563277G 48* 1.6596701 18' 1-7563138 24' 1-5060398 54' 16624215 21' 1.7690929 ^^1 30' 1.668801 1 66" 0' 1.6651732 30' 1-7618735 ^^H 36' f.57156t8 6' 1.6679250 36' 1.7646536 ^^H 42' 1-5743217 12* f. 67067 72 42' T.7674392 ^^1 48' 1.5770809 18' 1-6734296 48' 1-7702246 ^^1 54' 1-5708394 24' i-6761824 54' 1-7730114 ^^H 63= 0' I-582fl973 30' 1.6789356 70" 0' 1.7758000 ^^1 6' i.58.'i3546 36' 1-6816891 C T-7T85903 ^^H 12' 1-5881113 42' 1-6844431 12' T.7813823 ^^1 18' 1.5908678 48' i'687ig76 18' i. 7841762 ^^1 24' 1.5936231 64' 1.6899526 24' 1-7669720 a J 1 318 ^^^H APFBKDIX 1. ^ ■ tl Leg * . L» * I^B 7^^=^ ^iwVH TO" Si/ T-7897696 75° C 1-9185141 79" SO* ■0576I3S 36' 1-7925692 6' i-9214613 36' .MO9037 42' 1 -7953709 12' 1-9244135 42' -0642054 48' 1-7981 745 18' 1-9273707 48' ■0675187 54' 1-8009803 21' 1-9303330 54' •0708441 71" 0' T-803T883 30' 1-!I333005 80' (/ •0741816 «' 1-8065983 36' 1.9362733 6' •0775318 12' T-8094107 42' 1-9392515 \^ -0808944 IS* 1-8122253 48' 1-9422352 W -0842702 84' I -8 150423 54' i -9452246 24' ■0876592 30' 1-8178017 76° 0' 1-9482196 SC ■0910619 36' i -8206836 6' 1-9512205 36' -0944784 42' 1-823A0S0 12' 1-9512272 43' ■0979091 48' 1-8263349 18' i-9572400 48' ■1013542 54' 1-8291645 24' 1-9602590 54' -1048142 72' 0' 1-8319967 30' 1-9632841 81' 0' -1082893 6' 1-8348316 36' 1-9663157 6' ■1117799 12' T ■8370093 42' 1-9693537 12* ■1152863 18' 1-810501HI 48' 1-9723983 18' ■1188089 24' 1<(1M3534 54' 1-9754497 24' ■1223(81 30' 1-8461998 77° 0' 1-9785079 30' ■1259043 3G' 1-8490493 6' 1-9815731 36' ■1294 778 42' 1-85190I8 12' 1-9846454 42' ■1330691 48' 1.85»7575 18' 1-98772*9 48* ■136C786 54' I-85761G4 21' 1-9908118 54' ■1403067 73= O' 1-8604785 30' i-9939062 82' 0' -1439539 6' r'8633440 36' 1-9970082 6* ■1476207 12' 1-8662129 42' •0001181 12' ■1513075 18' 1-8690852 43' •0032359 18' •1550149 24' 1-8719611 54' •0063618 24' ■1587434 30' 1-8748406 78' 0' -0094959 30' -1624935 36' 1-8777237 6' -0126385 36* ■1663658 42' 1-8806106 12' ■0157896 42' •1700609 48' 1-8835013 18' •0189494 48- •1738794 54' 1-8863958 24' -0221181 64' •1777219 74" 0' 1-8892943 80' •0SG2959 83' 0* •1815690 e' 1-8921939 36' -0284830 6' •18S48I8 12' t-8951036 42' -0316794 12' •1804001 18' 1-8980144 48' -0348835 18' •1933455 24' 1 ■9009298 54' -0381014 24' •1973184 30' 1-9038489 79* 0' ■0413273 30' ■2013197 36' 1-9067728 6' -0445633 36* ■2053502 42' 1-9097012 12' •04 78098 42' ■3094108 48' T-9I2634I 18' ■0510668 48' -2135026 54' 1-9155717 24' •0543347 54' •21762S9 ^^^^^1 ^^^^^H ^^^^^H H^^^HI^^^I APPEKDIX I. 319 Lo.- ^ 1 ju. Ug- — UVAa "W ;=^ "°S r 84" 0' •2217823 86" or •313t(097 88° 0' ■4 38 S 4 20 6' ■2259728 6' •3191093 6' ■4405341 12' •2301083 12' ■324384S 12' ■4548004 18" ■2344600 18' -3297387 18' -4633880 24' -2387591 24' ■33S1762 24' ■472312T SO' ■2430970 SC •8407012 30' ■4816206 36' •2474748 SC ■3463184 36' -4913595 42' •2518940 4y ■3520327 42' ■G01S870 48' ■2S63fiei 48' •3578495 48' ■6123738 64' ■2608626 54' ■3637749 54' ■5238079 85' 0' •2654152 87" 0' -3698153 89° 0' -5360007 6' •2700156 6' •3759777 6' -5490969 12' •274(1635 12' •3822700 12' •5632886 18' ■2783670 18' •3887006 18' •5788406 24' ■2841221 24' •3362792 24' -5961320 30' -2889329 30' -4020162 SO' •6157370 36' ■2938018 36' •4089234 36' ■6385ft07 42' •29873I2 42' -4160138 42' •6663883 48' ■3037238 48' -4233022 48' •7027765 54' -3087823 54' -4308053 54' -7586941 [APPENDIX ir. In the very imiMrtant case of two circular coaxal coils Lord fiayleigli has BUggeeted ia tlie uso of the foregoing tables a very coavenietit fommla of appro simatiou. The formula, applicable to auj number of vonables, occurs in ^Ir. Kerrificld's Report on Quadratures and Inter- polation to the British Association, 1880, aud is attributed to tlie lal« Mr. H. J. Purkiss. lu the present instance the number of variables is four. Let n, n' be the uoinber of windings in tbe ooUb, ~ a, a' the radii of their oentzal windings. b the distance between their oeutree. 2A, 2h' the radiftl breadths of the ooils. 2k, 2U the axial breadths. Also let /(a, o', h) be the coefficient of mutual indnction for the central windings. Then the coefficient of mutual induction of the two coils is ^f{a + h,a',b)-\-f{a~k,a',h) +/(«, »'+A', h)-{-f{a, a'^h", b) +f{a,a',b + k)+f{a,a',b-k) + f{a,a',b+k')+f(a,a',b^k') J' [APPENDIX ni. Self-induction of a eircutar coil of rectangutar aecHon, If a <1(innt« the menti rmlius of u coil of n windiujts vrhnnc nxud brnulth i« t> and radial brcndth is o, then the tclf-intluctlon, m calculated by meam of the eeiies of Art. 705, may be thrown into the form L = -lwn*{Xa + ~ + &c.). where K = log.8«-2- - ^ i\og,C- ^)-^ -(log.ft~-) 32' ' 96" '*"*' 96" ' 32' 96 A* :H«dy+^<")iog.8''-^6*+^'^-4S{iog.c-a 12' -i(3ft»+e^(lt*«-^+^t«a-^) . * «'yi '37. 1 fi* , . 137. 41)0" GO 120 P' 1 «*„ 1-17. 16',, . H7. + 240 ft»^'^"- "60">-2l0 ? tl-^*- 60-> -2^0 (^-•^'' + ^'^'4*> ('"« ^'^^~ W> *24 6 c*12 ^ « 4' 120 « b ] TOL. U. CHAPTER XV. BLECTROMAONETIO INSTRUMENTS. 6alvanomt;lert. 707.] A Galvasombter U ua instrument by mesiaH of nhich nn i electric currt^nt is indiottvil or moiieureti by its muj>nctic nvtion. When thv iustruni«nt U int«D(U-<l to indic«tv the nctMU-iieu of a fix'tiU* current, it ix called a Sensitive Galvanometer. When it !» intended to measure a current witli llie greatest aecuracy in terms of standard units, it is called a Standard Galva- nometer, All galvanometers are founded on the principle of Sdiweigger'a }ktulliplier, in which the current is made to pass thmuf^ a wire, which is ooiled so aa t« pass many times round an open space, within wliicli a magnet is suspended, so as to produce within thi» space an electromagnetic force, the intensity of which ie indicated by the magnet. In sensitive galvanometers the coil is bo arranged that windings occupy the poritionB to which their infliwuce on magnet is grcat(«t. They are therefore packed closely togvtli in order to be near the magoot. Standard galvanonictcre are constructed so that the dinH!R»ions and relative positions of oil their fixed port* may be aocurati-ly known, and that any small uncertainty about lliv podlion of the moveable part« may introduce the nnalluit possible error into the calculations. fl In conntructing a sensitive galvanometer we aim at malring tho^ field or electrooiagiietic force in which the magnet is suspended oa intense as possible. In designing a standard galvanometer wc wish to make the field of elect romagnctie farc« near the magnefr^ as unifonn as {Kigsililo, and to know its exact tntaosity in tvr of the Btrength of Uic current. J ated ..I tJiafl tlwrl MBASUREMEKT OF TH2 COIU 323 I On Stanttard GalvaHometers. 708.] In a standai'd giilvauonieter tlie etrenglh of th« current 1)Ss to be determinod from the force which it coerls oa the boa- led ma^et. Now the diHriljutioii of the mn^edsm within Ae magnet, and the position of its centre when suspended, are not capable of being determined with any great degree of accuracy. Hence it is necessary that t!ie coil should be arranged so as to produce a field of force which is very nearly uniform throughout the whole space occupied by the magnet during iU possible motion. The dimensions of the ooil muat therefore in general be much larger than those of the magnet. By a proper arrangement of several coils the field of force within them may be made much more uniform than when one coil only i» used, and the dimensions of the instrument may be thus reduced and it« sensibility increased. The errors of the linear measurements, however, introduce greater uncertainties into the values of tlie electrical cotistants for small inatnimcnts than for large ones. It is tbereforo bust to dot.erraine tlie dectricftl constants of smull instrmnenta, not by direct mr'Osuremcnt of their dimensionM, but by an electrical coinpnrisun vrith a large standard instrument, of which the dimensions are more uci^urately known; see Art. 752. In all standard galvanometers the coils are circular. The channel in which the coil ia to be wound is carefully turned. Its breadth tig. a. is made equal to some multiple, n, of the diameter of the covered WLie. A hole is bored in the aide of the channel where the wiro ia T a to enter, ood one tni of the corcrod wire is pnabed oat throngb tbit iiole to form the inner oonnexioo of the coil. Tbe channel U plaoed-on s lathe, sad a wooden axis is listened to it ; see I-^g. Of, The end of a long ttring is nailed to the wooden axis at the mma part of the circumfcrmoc as the entrance of the wire. The whole is then turned round, and the wire it emoothljr and regularly laid on tiw bottom of the channel till it is completely covered by m windings. Doring ibis procen the siring has been woaad m time* roond the wooden axis, and a nail id drivm into the string at the Nth turn. Tlie w-tudingit oT the Mtring xliould be kept exposed so that tbey can miily l>e counted. Tlie ext^mul circumference of the first layer of windings is then measured und a now layer is begun, and so on till the proper nnmber of layers has been wound on. 'Flie use of (be string is to count the number of windingK. If for any reason we hare to unwind part of the coiL the string is also unwound, so that we do not lose oar reckoning of the octnal number of wiDdtngs of tlie ooiL The nails serve to dialinguieh the nnmber of windings in each layer. llie measure of the circumference of each layer furnishes a t««t of the regularity of the winding, and ennblee ns to caleulate the electrical constants of the coil. For if we take the arithmetic mean of the circumforencee of the channel and of the outer layer, and then add to this the circumferences of all the intermediate layeTS, and divide the sum by the nuin1>er of layers, we shall obtain the moan circumference, and from this we con deduce the mean radins of the coil. The oircumference of each layer may be measured by means of n steel tape, or twtter by means of a graduated wheel which roils on the coil as the coil revolve* in the process of winding. The value of the divisaonc of the tape or wheel must be a«ccrtainod by comparison with a straight ttcale. 709.] The moment of the force with which a unit current in the coil ftct« upon the suspended apparatus may be expreasod in **» «"" ff,^, sia S+ G,9t sine P.'(0) +&c, where the ooefBcients refer to the coil, and the coefficients g to the suspended apparatus, B beiiig the nngle between the axis of the coil and that of the suspended appaiatue ; see Aii-. 700. Wlien tlic suspended apparatus is a thin uniformly and longi- tudinally mognetixed bur nuignot of length 3 /and strength onity, saspended by its middle, TANOEST OALVANOMETEK. 325 The Tiiluos of the coefficient for a ma^et of length 2/ magnetized b^in aoy other way arc smaller than when it is magnetized uni- pormly. 710.] When the apparatus is usod ae a tangent galvanometer, Ithe cot) is fixed with its plane vertical and parallel to the direction [of the earth's maffnetic force. The equation of equilibrium of the ^magnet is in this eaMC My,ffcoBfl = my8in^{(?,^i+ffjj?,;i','{()) + &c.}, l^rhflTo wj7, is the magnetic moment of the magnet, Jl the horizontal i>mpoaent of the terrestrial magnetic force, and y the strength ^of the current in the coil. When the length of the magnet is ■mall oompurcd with the radius of the coil the terms after the first ^bn G and g may he neglected, and wc find y = 7T- cot 0. I The angle usually measured is the deflexion, 8, of the magnet which is the complement of 9, so that cot 6 = tan h. The current is thus proportional to the tangent of the deviation, and the Jnittninient is therefore called a Tangent Galviinometi'r. Another mctliod is to make the whole apparatus moveable about a vertical aii^ and to tum it till the magnet is in equilibrium with its axis imnillcl to the plane of the coil. If the angle between the ,e of the coil and the miLgnctic meridian is 6, the equation of brium i« iBy,ffsin* = wy {G,y,-f C,^,+&C.}, U . , ^m Since the current is mea&ured by the sine of the deviation, the ^Binxtrument when used in this way is called a Sine Galvanometer. ^M The method of sines can be applied only when the current is Bbo steady that we can regard it as constant daring tbo time of adjusting tJie instrument and bringing the magnet to equi- librium. ^P 711.] We have next to consider the armngement of the coils of a standard gal vn no meter. The simplcjit form is that in which there is m single coil, and lio magnet ia suspended at its centre Let A \h> the menu radius of the ooil, ( its depth, ij its breadtll, n the number of windiogn, the values of the coetEcients ar« ELBCTROMAOyRTIC INSTRCllENTS. <7.= 2w» {..A^»5i' becomes 1 — 3-^=. A' 0, = 0, &c. Th« principal correction is that ansiaff from G^. The aeri«a_ The factor of correction nill differ most from unitrf when th« magnet is imifonnlj magnetized and Trben d = 0. In this cose it It vant»he!i when tail 0=2, or whvn tho de-^| flexion 18 lan'i, or 20*34'. Some obstfrvers, therefore, arrange their experiments bo as to make the ohwrvtHl deflexion as near this an^le as jioasiblo. The hest method, however, is to use a magnet so short compared with the radios of tho coil tlist com-ction may be nlUigether neglected. The suspended msg^net is carefully adjnsttd «o tliat its cent shall coincide as nearly ss posiiilile with the centre of the ooil. however, this adjustment is not perfect, and if the coordinates o! 1 ho centre of the ma^et ivlative to the centre of tlte ooil are x, y, i, t being measured parallel to the axis of the coil, Uie factor of correction is ( I + s '^ ) ■ "When the radiiiti of the coil is large, and the a<Ijui^tmcnt of tlii magnet carel'iilly made, we may assume that thi» correction is insemnble. Gatiyain's Arrangement, 712.] In order to ^t lid of the correction depending on O^ Gangain con»tnictcd a galvanometer in which this term was ren- dered 7^ro by «u«iiending the miigmet', not at the centre of tlie coil, but at a point on the axis at a disbinoc from the centre equal to half the radius of the coil. Tlie form of G^ is and, HJnce in this amngement S = iJ, G, = 0. Tliis arrangement would W an improvement on the first form if we could be sure that the centre of the suspended magovt ii >1 I s I ■ I GALVAKOHETER OP TWO COILS. 327 exactly at the point thus dotinotl. The position of the centre of the ma^et, however, is alwa};! uncertain, aud this uncortuinty iHtro duces a factor of correction of unknown amount dqtondin^ on G.^ and of the form (l — J -j), where z is the unknown excess of distance I of the centre of the msgnet from the plane of the ooiL This COTTcctioD depends on the tiret power of -j . Hence Gaugain's coil with ecccntriciitly EUgpcnded ningnct \% subject to for greater un- l^cerUinty than the old form. HelinAoUz'a Arrangement. 713.] Helmboltz converted Gaugain's galvanometer into a trust* vorthy instrument by placing a ^cond coil, equal to the first, at an equal distance on the other side of the magnet. By placing the coils symmetrically on both aides of the magnet we get rid at once of all terms of even order. Let A be the mean radius of either coil, the distance bctwcpn their mean planes is made equal to A, and the ma^^nut is sui^penilud at the middle point of their common axis. The cocfficicnte are 16,r«l, eS G- = 0.0612 ITJJ 3-s/5d[« (31 f ^-36 A G.= 0, Cj= — 0.73728 where n denotes the number of windings in hoth coils together. It appears from these results that if the section of the coils be rectangular, the depth being f and the breadth rf, the value of &,, as corrected for the 6nite size of the section, will be email, and will vanish, if (^ ie to t}' as 36 to 31. It is therefore quite unnecessary to attempt to wind the coila ! upon a conical surlitce, as has bovn done by some inetrumunt. makers, for the eonditioDH may 1>e satisfied by coils of rectansuUr Mction, which can he con«t.Tucted with far greater accuracy than ooiU wound upon an obtuse cone. The arrangement of the coila in Uelmhottz's double galvanometer repreaented in l^g. 53, Art. 723. 328 BtBCTROMAONCTIO INSPrRCMRllTO. C714. The field of force due to th« double ooil i* irprcsonted in Bection in Fig. XIX nt tlio Liid of tltis volume. Galvanometer tf Four CoiU. 714.] By combining four coiU we mfty get rid of the co«fBd«nt» 0^, &,, and 6'^. For by any symmetricAl combinations C, G. s> "ai we get rid of the coefficients of even orders. Let the four coils be parallel cirtles belonging to the Esme sphere, iMrresponding' to angles 6, 4>i ""^j ^^d it— ^. Let the nnmber of windings on tbe first and fourth coil be «, H and the number on the second and third pn. Then the condition that Cj = for the combination gives aein*^P,'(fl)+ji«an»0i's'(*) =0, (I) and the condition that G^ = o gives ■ «sin>flf;(^)+;.f.8in**P/(^)= 0. (2) Tutting sin'd = jf and ain*ip=f, (3) and expressing /*' and P^' (Art. 698) in terms of these quantities, the e<^uatious (l) and (2) become 8»- 28«' + 21 a:^ + 8/.y-28/)y + 21/^ = 0. Taking twice (4) from (5), and dividing by 3, we get Hence, from (4) and (t>), x5x-i ar'7x-6 w J> = and we obtain y»6-7y P = 32 7*— 6 Both X and y are the squares of the stnea of angles and most tbcrefbrc lie between and 1 . Henoe, either * is between and f , in which com y is between f and I, and p between 00 sod |f, or else x is between 3 and 1, iu which case y ia between and f, and p betwoon and J{. Galvanometer (f Tkrae CoUt. 715.] The most convenient arrangenHnt is that in which x =s Two of the coils then coincide and form a great circle of the ^here whose radius is C. The number of windings in this eoinpound coil is 64. The other two coils form small eiroles of the sphere. The radius of each of them is s/}C, The distance of either ul 4 OALTAirOSraTER OP THREE COIIS. 820 ' tbem From the plane of the first is </• C. The number of wmdiuge ^OD each of these coils is i<i. 120 Tho Tftlue of ff, is — ^ . c This ammgemeat of coils is represented in Fig. 51. Fig. GO. Since in this three-coiled ^Ivanometer the first term nfler G, [■which has a finite value is G, , a lar^ portion of tho sphere on ^whoso surface the coils lie forms a field of force sensibly uniform. If we could wind the wire over the whole of a Hphcrioal surface, ^M dwcrihed in Art, 672, we should obtain a field of perfectly tunifonn force. It is praetically impossible, however, to distribute rtlie windiDgi on a spherical surface with sufficient accuracy, even if saeh ai coil were not liable to the objection that it forms a closed [ aurfacc, RO thnt its interior is inaccessible. By puttinfr the middle coil out of the circuit, and making the |eurT?nt How in oppOEit« directions through the two side coils, we obtain a fa-Id of force whioh exerts a nearly uniform action in tthe direction of the nxis on a mngnet or coil suspended within it, with its axis coinciding with that of the coils ; see Art. 673. For in tbie can all the ooeffieienU' of odd orders disappear, and since Hence the exprcftnon for the majpietJc potential near the centre of the coil becomes I BMCTROMAOSETIC nraTKHMENTS. [71* On He Proper Tkieknw of lli* Wire ^ a GalvanomeUr, tke Ezftmal Besulanct being ffwm. 716.] Let the fonn of the channel in whioh the ^IvanometerJ coil is to bo wound Le f-iven, and let it be required to determiiM, whi-ttivT it ought to be lillcd with a long thin wire or with a sbor tbi<rk wiro. Let I bo the len^lli of the wire, y its radius, ^4-^ the radius of the wire when covcrod, p its speciBc resistance, g the vnloo of G for unit of leugUi of thi,- win', nnd r the part of the resistoooe whioh is in(le]iendent of the gat van o meter. The resistance of the gftlvunomctdr wire is p I It = xy' The volume of the coil is The elect romagnetio force is y 0, where y is the strength of the current and 6 = gl. ■ If £ is the eleotrotnotiTe force acting in the circuit whose resistance k R + r, E = y{R+r). The electromagnetic force due to this electromotive force is ^/- which we have to make a maximum by the variation of; aad /. Inverting the fraction, we find that is to be mode a minimum. Hence If the volume of the coil remains coostani = 0. or Eliminating dl and t/y, we obtain P y + i ' 9 T Hence the thickness of the wire of the gs]Tanomot«r should be such that the external resistance is to the misUuiw of the gal- viitiomcter coil as the diameter of the oovcrod wire to the diameter of the wiio itMir. QALVAKOUETBKS. 331 On Sensitive Galvanomeffrs, 717.] In Uie construction of a eensitive galvanometer tlie aim of erery part of the amngeinent is to produce tho grftatest pnesible deflexion of tho magiiet by means of a given siiiall electromotive I force actiuff between the electrodes of the coil. The current through the wire produces the greatest eflect when it is placed as near as possible to tho auspended magnet. The magnet, however, must be left free to iiscillnle, and therefore there is a cerlaiu spnc<' which must be left empty tntliin the coil. This dffinifs the tut«nial boundary of Uie coil. ^m Outside of this space Oiioh wiudiug must be placi'd so as to have ^^tlie greateHt possible effi'ct on tlve magnet. As the number of ^■windings incn-iises, the most advantageous positdoiiB become tilled ^^□p, so that at lust the increusiil resistance of it uew winding diminishes tine effect of the curn-nt iu the former windings more [than the new winding itself adds to it. By making the onter pindingd of thicker wire than the inner ones wo obtain the greateixt letio eiTeet from a given electromotive foree. 718.] We shall euppoue that the windings of the galvanometer are circles, the axis of the galvanometer jiassing through the centres of these circles at right angles to their planes. Let raiaO bo the radiuK of one of these circles, and rcosO the distance of its centre from the centre of the galvanometerj (hen, if / ia the length of a portion of wire coinciding with this circle, y the current which flows in it, the foree at the centre of the gal- "'Wn(MBet«r resolved in the direction of the axis is gjn & yt- If we write H = *» sin 0, 0) I this expression becomes y -^ ■ Heuec, if a surface bo constructed umtlor to those represented in section is Fig. 51, whoso polur equation is /^ = ii'naO, (2) wberc r, is any constant, a given length of W)r«? bi-nt into the form of a circular Kit 51. arc wdl produce a greater magnetic effect when it lies within this sar&oe than wh«a it lies oiit«vi)K \^. 332 ELECTEOMAOKETIC IHSTBCICBNTS. It fallows (rom this that the outer surface of any la^r of wire ought to have a conBtant vainc of ar, for if « is greater at one place thitit another a portion of wire might be traosferml from the Stst place to thu second, so as to increase the force at the centre of the galranomctor. The whole force due to the coil is yCr, where 6 fdl (3J the int(>gTation being extended oTer the whole length of the wire, X being considered as a function of /. 719.] Let 1/ be the radius of the wire, its tniDsrerso »ecttoQ will be isj^. Let p be the specillc resistance of the material of which the wire is made referred to unit of volume, then tlie reustance ofa length / is ~4r • an^ the whole resistance of the coil in n p fdl w wliere f is considered a function of i. Let r* be the area of the (luadrilateral whose angW arc tlui' BectJonfl of the axes of four neighbouring wires of the ooil by a plane through th<^ axis, then T^l is the volume occupied in the ooil by a length / of wire together with its insulating covering, and including any vacant ajtace necessarUy left between the windingt of the coil. Hence the whole volume of the coil is | r=fT^di, (5) where J' is considered a function of/. But sinee the coil i« a iigure of revolution r= 2it f J r^aaedrde, (6) or, expressing r in terms of x, by equation (2), Now 2ii / {aiifffidd ia a numerical quantity, call it^, tlwn where V^ is the volame of tlie interior apace lefl for magnet. Let us now eonaidcr a layer of tho coil contained between surfaces x and x-^dx. SEysmVB OALVATTOinCTEES. 383 The volume of this layer ia [tvliore dl is the length of wire in this layer. Thi^ giv<s us dl in tonus of dz. Substituting: this in equations f(3) anil (4), we fmd ^^ ^^dn dG^N^^, (") rhere dO and dR represent the portions of the mlucs of G and of ' due to this Uyer of the coil. Now if M be the given electromotive forcCi rliere r is the reeistanco of the external part of the oircsit, in* ei)eDdent of the giLlvunometer, niid the force at the ceotie is We have therefore to make -p — at nuudmum, by properly ad- Justinf* the section of the wire in each layer. This also neccfisarily tvolvee a variation of Y because ¥ depends on g. Lot Go and li„ be the vidues of G and of fl + r when the given flayer is excluded from thu calculation. Wc have then g _ gp + rfg R-i.r~ Ra + dR* to make thia a maximum by the variation of the value of j' for I given layer we mual have (12) I..4G ^.dR Q RTr- (IS) g.. Since da is very small and ultimately vanishes, ■— will be iibly, and ultimately exactly, the same whichever layer is ex- Eoludcd, and wo may therefore regard it as constant. We have hereforc, by(10)and(ll), (14) constant If the method of covering the wire and of winding it ti Mch that the proportioD between the space occupied by the metal of i the wire boars the same proportion to the apoice between the wires whclLer the wire is thick or thin, then y d^ _. and we must make botli jr and Y proportional to x, that is to nr, the diameter of the wire io anjr lajrer must be proportional to the linear dimension of that layer. If the thiclcncss of the insulating oovenng is constant and equal to {>, and if the wires are arraDg«d in si]uare order, Y=2(jf + i>). (15) and the condition is ^to:i) = co«rta»t (16) Tn this case the diameter of the wire increases tnth the diameter of the layer of which it ibrms part, bnt not in »o hiph a ratio. If we adopt the first of these two li^^iothcaee, which will be nearly true if the wire itwelf nearly lills up the whole epnve, tfaOD we may put n = ax. y = ffy, where a and p are constant numerical qnantities, and lt=N^ ' tsaa the sise and form of th« frcti^ where a is a constant depending upon the Aae and form of the free space left inside the coil. Honco, if we malce the Ihidtness of the wire %'at7 in the nune ratio as x, we oblain very little advantage by increa»in^ tlie ex tenuil size of the coil after the external diDieusiona have beooow a larffc multiple of tlie internal dimensiona, 720.] If inen««o of resKtanoe i> nnt roganlcd aa a defect, when the external reaiatancie i* far gri>ater than that of the gaW vanometer, or when our only object ia to produce a field of intense force, we may make >/ and Y constant. We have then = ;^1(«— ), N I ise where a ia a oonrtant depending on the vacant upace inside the^ coil. In this case the value of Q increases uniformly as th^f dimensions of the coil are incnnued, to Uial tJiere is no limit t^^ the value of G except the labour and expense of making the coiL «». Wj 0ALTAN0METRE9. Oa Suspended CoiU. 721.] In ih« ordinary gnlvannmeter a suspended magnet v» Mted on by a ftsed coil. Hut tithe ooil can be tuupended with sufilcieDt delicacj, we may determine the action of the magnet, or of another coil OD the suspended coil, by its deflexion from the position of eqailibrium. We cannot, however, introduce the electric current into the coil unless there is metallic connexion between the electrodes of the battery and those of the wire of the coil. This connexion may he made in two different ways, by the Bidlar Suspension, and by wires in q>posite directions. T!Tie bifilar saspension has alresdy been described in Art. 459 as applied to magnets. The arrangement of the upper part of the snspcnsiott is shewn in Fig. 54. When applied to coils, the two fibreti are no longer of silk hut of metal, and since the torsion of a metal wire capable of supporting the coil and tmnsmittioi; the current is much greater than that of a silk fibre, it must be taken Bpecially into iiccouat. This suspension has been brought to great perfection in the iiiHtrumL'nts constnieted by M. Weber. The other method of suspension is by means of a single wire which is coiim'cttil to ouc extremity of the coil. The other cs- treniity of the coil is Cfinnectcd to another win: which is made to bang down, in the sanie vi^rtical straight line with tbc first wire, into a cup of mercury, as is ^hewn in ¥\g. 36, Art. 720. In certain cases it. is convenient to fasten the extremities of the two win.'S to piece* by which they may he tightly strtitchcd, care being taken tliat the line ()f these tvires passes Hitough the centre of gravify of the coil. The apparatuii in this forcn may be use<l when the axis is not vertical ; see Pig. 52, 72^.] The suspended coil may be used as an exceedingly sensitive gal- vanometer, for, by increasing the in- tensity of the miiirnetic force in the field in which it hangs, the force due to a feeble current in the coil may be greatly increased without adding [ to the mau of the coil. The mag^ nctio foroe for tliis purpose may be produced by means of permanent magnets, or by electromagneta Fy. &2. S8( ELECTROWAOITETTC IltSTBlTMESTO. [7»; excited by an auxiluMT cuircat, and it may be powerfully^ coDcen- trated on the stispeiulod coil bj meAiM of soft iion ansatures. Thus, in Sir W. Thonnfon's ntcordiof^ nppamtuii, Fi^, B2, th« cnil is sus- pended betwot-n the opposite polos of the cloctromagnvte A' mi S, and in order to conccntr»t« the lineH of mngnctie fon-o on the ver- tioal sides of thi> coil, n piece of Bofl iron, D, t» IiximI brtnreen the poles of the magnets. This iron becoming inii<^etizc<(l by iodao- tion, prodnccs » verj- powerfiil fiold of force, in the inlerrals between it and the two magnettt, throug^i which the vertical sides of tJie coil are free to move, so tliat the coil, even when the current through it is vety feeble, is acted on by n eonsidenible force tending to turn it about its rertioal aicia. 723,] Another application of the suspended coil is to dctcnnine, by comparison with a tangent galvanometer, (he horizontal coi poni>nt of terrestrial magnetism. The eoil is fiuepcaded eo that it is in stable eqailihriiim whi its plane is parallel to the magnetic meridiaD. A current y is posted through the coil and causes it to he deflected into a neir porition of equilibrium, making an angle with the magnetic meridian. If the suspension is bifitar, the moment of tho couple which produces this deflexion is FeinO, and this must lie vqual to HygKOSiO, where //is the horizontal component of terreKtrial ma^ Detism, y is the current in the coil, and g is the sum of the areas of ati the windings of the coil. Hence /7y = ^tantf. 3 If A is the moment of inertia of the coil about ita axis of sua- pension, and T the time of a single vibration. J and we obtain ^y = 5^tantf. If the same current passes through the coil of a tangent gal' meter, and deflects the magnet through an angle 4^ where CistheprincijMdcoDstantof the tangentgalTanom6tcr,JVrt.710, From these two equations we obtain as of 4 I B^ tV AGtaii0 V /a tan tan ^ =tV — 0^ — f tan^ This method wbs given by P. Kobbauscb *. E1.ECTR0DTNAM0METER. ddt t,] Sir William Thomiton has conrtmclixl a sing-It instrnm^nt hy meaQS wf wliieh (In; oliKt-rviiUon* roqiiiretl t* (kh-rmiiie // and y he taadc ftiinulhtni-oimly hy th« wm# observer, ooil is jiuii]H;iKlo(t BO ni! to W in <^iiilibrium with its [>iiin« the ntagnetjc mcriiliMn, and is dcfl«ct«d from this position the current IIowb tliroug-li it. A wry nmttll mngiR-t is sup* at the c«ntrv of the wil, and in deflated by the cnrrent in the direction ftppoaite t« tliat of the dcHexion of thu coil. Let th« deflt^xion or tho eoil be 0, and that of the toni^net 0, then tlie jnnablt! jiiirt of the energy of tho system is //yysinfl + my GBin(tf— ^)— i/mco8i^— /'costf, Biflereutiating with respect to and ip, we obtain the equa- Wona of eqniUbnum of the ooil and of the muf^et respectively, Uy$eoit$ + myGco8(0—<f)+FanO = 0, —myG <:oe($—4>) + Ilm sia^ = 0. From th(«e equations we find, by eliminating ZT or y. a quadretie ^uatioD &om which y or II mny be found. If m, the ma^ctie noment of the suspendod magnet, is very unall, we obtain the followinff appTOsimate values. y = YW-t ^eosUsin^ — .ifeiu^ein^ ,-i 9 M Rin coa0 I" (i^ COB fl cos (9—0) ' g ciwtf In these expressions G and g are thi> principal elcctiic contitJintM of the coil. A its moment of inertia, T its time of vibration, m the mBgnetiG moment of the magnet, // the intensity of the horizontal ma^etic force, y the streugftli of the etirrcnt, B the dellcxion of ihe coil, and ^ that of the mn^net. Since the deflexion of the coil is in the opposite direction Ui the deflesion of the magnet, thcBc values of II and y will uUvays be I. Weber's SieefroitytaiKometfr. 7S5 ] In this instmroent a small coil is anspended by two wires within a krger coil which i» fixed. ^Vhen a cnrrent is made to flow ihrmigh both coils, the suspended coil tends to place itwlf parallel to tlie fixed coil. This tendency ia counteracted by the mumcnt of the forces itrittin); from the bifilar suspension, and it, is also atfocted by the nelion of terreBtriul ma^etism on the «u«< ^iwnded coil. VOL. II. Z BLECTSOMAOSETIC I^reTRtTMKSTS. [73S. In the ordiasry uso of the invtrarnvnt tiic i>Ianos of the two oolh aro nrarl)' nt right angles to vnch other, so tliat tlic madial action of the currents in the voiU may bo ns grvat ii< puwfibU-, aod thr pldoe of th« (UEpendod coil is ncnrl)* »t right nnglm to tlic nui<picUc Di«ridiui, so ttiut the action of tcrroxlriid tua^ctinn may be t» small as jKJBsible. Lb-t the magnetic oximutli of the piano of the fixed coil ba a, and let the angle nhicli the axie of tfao snspoDcled coil makes with the plane of the fixcul cuil be 0+fi, where is the value of thi anijiio whon the coil is in ivjuililirium and no current ie flonnnj and is the deflexion due to the current. Tlic <^()^4ltioa of «q Ubrium is ''yyty!<'oe(tf+^)-//?Vii«in{tf+j9+ft)-/'*intf = 0. Let us suppose that the ioHtrumcnt is adjusted so that a uui are both very small, and that Jf^fi is utaail compared with Wn have in this case, approximately, If the deflexions when the >«igns of y, and y, are changvd as follows, 0^ when y, is + and y, + , "i " ^ It ^1 ^» .. + » — » *4 " — t» +• then we find f If it is the same current which flows through both coita we may put y, y, = y'. »nd thus obtain the value of y. When the currents are not very constant it is best to adopt thb method, which is called the Method of Tangents. If the eurrmt* arc so constant that we can adjust ff, the utg] of the torsioD-ticad of the instroment, we may get rid of thi correction for terrestrial magnetism at once by the method of siimb. In thiit method /3 is adjusted till the deflexion is zero, so tlist If the signs of y, and y^ are indicated by the suffixes of fi be fort!, /"sin^, m ~F»ia^ = — Gffy^y^^ 11 gy^ sin «, i*sinj3, = — /'«n^4» — Gy)r\yj,— //jry^sino. Jf and ri/t =- j(i^(Binj3,+Binft-Bin ft-sin ft). 1 ■i*ias^' Fk. f.8- 7. a 340 KSCVBOMAOyETIO IHSTRtTUBSTS. Thia is th<; method nduptcd by Mr. Ij»tiinrr Ckrk in his nw of the iustniiiiisnt constructed by the El«»trioal Committal! tif British As»<M.-ititi»n. We nre )D<lebLed to Mr. Clark for the drawin of th« rlecti'odynamftmottr in Figure 53, ia which II«Imbultx nrrxngvmcDt uf two coiU is a(lo[)te(l both for the fixed and for U: Kitspcndt'd i-nil* The torsion^hnad of the iuBtrumetit, by which thu bifilur «u»ik'iision is atljueted, ia represented in Fi^. S4. T1i» Fig.6». iHfiiality of thfi tension of the tiispension wires ts ensured by their beang atlndied to the extremities of n sitk thread which fwmx ov«r a wheel, and tht'ir distitnco is reguliiti4 by two guide>whecls. which con be sot At the proper distsnco. The sti^x^nded coil can Ix! oiovitl Tertically by means of n screw iieting on the suspension-wht-cl, and horizontiklly in two directiona by the sliding pic<vs sliewn the bottom of t'ig. S4. It is adjusted in azimuth by menns of III torsion -screw, which t»ma the |oniion-h»id round a vertical nxi (see Art. -159). Tlie aximuth of the suspended coil i* aseertaimil by oboervinj^ the reflexion of a scale in Uie mirror, Hlietvn just beneath the axis of the siiiipi-ndcd coll. ■ Ib ttw K^tnl inalniinmt, th« virM Aumr'ng Ihr carnal to ■nil fhini tka i an not ^nad oat t* di>[<l»t4 in the Hfure. bol arr kipi n* cIme logniiet «■ ; liUe. ■> M to utatntloo «m1i uUior'* ckulroBupictH: kctina. CI, i I CITBBBNT-WEIOHIE. 341 The intrtninwnt originntly cinwtTOctfd by Weber is described in his Eteitrodjfmimivri'' M-tiftheithnmuufW'- It wne intended for the mcMurcnwnt of smaU oiirrent«, and Uien-forv both the fixi<d and the suspended ooiIk i:'oneixt4Hl of many windings, and the enspended coil occupid] II UrgiT juirt of the ^ystv*: within the fiscd coil than in xXk intttninii-nt nt \\\f IlritiNh Assoeiut.i(>D, w)iich wiis primarily in- tcndt^l a* a Mtandiird inslnimeiit, with which mnrv ecinisilive inxtru- metitM might be coniparE>d. 'Ilie exjierim«nt8 which he mode with it rumixh the most coniplcf* ex|xirimpntal proof of the acoiirafy of Ainp^re'n formulu ks^ a|>{>It<-d to cltiacd currents, and form an im- portant part of th« rettearehea by which We)>er has raiitixl the iiimit?riciJ determination of electrical quantities to a very high rank as regards precision. Weber's form of the electrodynamometer, in whioh one coil i* suspended within another, and is acted on by a couple tending to turn it almut a vertical axis, is probably the best tttted for absolute measurements. A method of calculating the constants of sach an arrangement ia giien in Art. 6'J7, 726.] If, however, we wish, by means of ■ feeble current, to produce a ooosideiable electromagnetic force, it is better to placo the snsjiended coil parallel to the fixed ooil, and to make it cnpabtc of motion to or from it. The siis|iended coil in I>r. Joule's current- weigher. Fig. S.'S, is horizontal, and eapabto of vertical motion, »nd the force between it and the fixed coil is estimated by the weight which must be added to or removed from the coil in order to bring it to ibe snmc ri'lative position nith rBs|M>ct to the fixed coil that it has when no current jiiuescs. The snspendod coil may alxo Im to tJie extremity of the hori- Sontol arm of a torsion-balance, and nuiy W pUoed Wtwcin two fixed coils, one of which nttiacls it, while the other re[ieU it, as in Fig. 3tf. By nrraiiging the wnla as described in Art. 729, the force acting on tlie K\ii>t>ended coil may be made nearly uniform within a small distance of the position of equilihrium. Another coil may he fixed to the other extremity of the arm of the toTsioD-bslancG and placeil between two fixed coils. If tbtt Fl|t. S6, ELECTSOVAQirBTtC ISSTItl'MBSW. two fa«peiM)e<l coils are BuniUr, but wilU the oorrent flovriti^ in oppofite directioiu, the effect of temxtrial raa^etum on the FifrM. pocition of tlw ami of th« lomion-balance will be coin]>letelj' sliminaliHl, 727.] If tho »u»pcnd«d cml b in tlie shape of a lonff solenoid, and IK uipiilile vf nioring jnrallf^l to its axia, so as to jiniis into tlie interior of a lai^r lixed solenoid having: tl>^ same asiit, then, if the cun*nt is in tiie same direction in l)oth eolenoid*, Uie ww. [tended solenoid will be siickeil into the Bxrd one by a force nhicli will be nearly uniform as long as none of the extremititw of the Bolenoide are near one another. 728.] To produce a uniform lor^ritwlinal force on a Hmall coil placed between two equal ooiU of much lurgor dimcnsioua, we should mnko the mtio of the diameter of the 1ar|^ coil» to Oie dis- tauuG between their plantti that of 2 to V^3. If no Kcud the eame currmt througih these coJU in oppoeito directions, then, in the ex- prcwion for m, the terms involving odd powers of r dis^pear, and «incD kin' a = 4 and eon' a = i, the temi involving r* disappnrt nlw, and we have n* the variable ]>art of u which indicateH n nearly uniform force on a email siifpeuded coil- The armnf^-Rient of the coils in this case is that uf Ihtt two outer coils in the gulviinometor with three coils, described at Art. Tlfi. See l'1{f. at}. 729-] SUSPENDED COILS. 343 7S9.] If we wish to BUspend a coil between two coiU pla<^ 6o near it tlmt llie distance between the mutuatly xcting u ires U » email ooTnjiiired with the radius of the ooils, the moat unirorru ion-n is obtained by tnukiii^ the radius of either of the outer coil.i exceed ;hat of the middle one by --= of the distance between the planes v3 of the middle and oot«r coils. This followg from the expression (proved in Art. 704 for the mutual induction bctwvcn two circular I currents. So mnny of the m«i6nromont« of ol«!tri«iI qtuntitin ' ilepend on obeiTvationi) of t)ii! motiou of a vibrntin^ body IJiat we , shall devote some atk'Dtion to the nature of this tDOtion, and tb*^ j best nn'tliods of obeorviny it. The small oscillationH of a body iibout a position of irtable «qm- Hbriiiiu are, iu- gcnrml, Himilar to those of a jioint act«d on by II forc^e varying din-cily lu tb<^ ditttaooe from a fixed point. In the «iu« of tbo ribrating' Iiodies id our experiments tli<-re iR alao a reitiHtnoce to tlie motion, depending on a variety of causes, suoh a» the viccotiity of the air, and that of the sufpcnaion fibre. In many electriejil intitriimeutH there is another eause of reatstance, namely, the refiex action of eurrents induced in condueting oircuita placed near vibrating magnets. These currents are indoced by ttie motion of the magnet, and their action on the magnet is, by the law of Lenz, invariably opposed to its motioD. This is in many eases tho principal part of the resistance. A metallic circuit, called a Damper, is sometimes phiecd dc a magnet for the eipress purpose of damping or deadening it vibrations. Wo tthall theroforu speak of this kind of resis as Damping, In the COM of alow ribrnttOD*, such as can bo ca»ily obMrvedJ the whole reiiistaace, from whatever causes it may nriec, appears] to be proportional to the velocity. It is only when the velocity] is much greater than in the ordinary vibrations of clectronuigneti' instruments that wo have evidence of a rcsiHtance pro[>ortional U the square of the velocity, W« have therefore to investigate the motion of a body tmhjr to an attraction varjing as thi! dixtanee, and to a tvuslance nryiiij as the velocity. I MOnOS IK A LOOAniTHMlC SPIBAU 845 731.] Tbe following application, by IVifes^or Tnit •, of th« principle of the Ilodojrrapii, eoalilps us to invcati^t« thi* Iciud of motion in a very simple manner by means of the e<juiangular spiral. Let it be required to End tbe acceleration of a particle which ilcEcribce » logarithmic or equiangular spiral with uniform angular Telocity u about tJie jKile. The property of this spiral is, that the tangent PT makes with the mdiug vector PS a constant angle a. If V IS the Teloci^ at the point P, then t> . sin a = 09 . SP. Henoe, if we draw SP' parnllcl to PT ani oqual to SP, Hie velocity at P will be given both in magnitude and direction by v=- aina SP'. ^Hence P* will be a point in the bodograph. But Sf is SP turno) tbrongh a constant angle n — a, so Unit the hodograph described [Ijf P" is the same as the original spiral turned about its pole through [an angle t— a. The acceleration of P is represented in magnitude and direction by the v«loeity of P" multiplied by the name fun-ior, -. — • UcDCC, if we porfonn on SP" the Bome <^cration of turning it • Pfoc S. 8. ^tiL, Da«. Ifl, 1SB7. 846 ELECTKOJIAOSFTIO OBSERTATIOSS. [733^ \ Uirou^li an vmgh «— a into tbc poffltmn SP^, the aoceleraiioo otP will hn c<|uul ID inuj;nitud« itii<l (HrvctiuQ to Bin* a SP^. where SP" IB etjuttl to SP turned tbroof^h nn wngle 2ir— 2(1. If wo dnw PF cqtui nnd parallel to SP", the occ-vlerntion will W w i'/| whi«h we may n^olve into Sin' a PS and —.- PK. siu'a The Tint of tbcM coinpon«nt8 is a central force towards 8 ptv- portionnl to t)ic iliitlanoo. The KCvoiii] 16 in a diroction opponte to the velocity, and einoe sin a GOB a u PA" = 2oOBaP'S=— 2 this force may \tv wriH«n Hina The acceleration of tlit- particle is lliorefote compounded of tw part-f, the tirnl uf ivhicli J* iin uttniclive force (ir, directed towards ^ and pro|H>rtionul Iri the distance, and the cceond is —2kv, a resist* anee to the motion proportionul to the velocity, where ui* , , co*« 11 = -. -. , and * = to -r^ • Bm*a BIO a If in these cxjiressionB we make = 41 the othit heoomes a ciicls^ and we have ^ = u^, and k = <i. fienoe, if the law of attraction renuuns the same, ft e ja,, and U ^ Wq SID o, or the angular velocity in diflerent spiraU with the anme law of nllntction ie proportional to the sine of the angle of the spiral. 732.] If wc now consider the mutioa of a point which la Uw projection of the moving point P on the horiaonia) line XY, we] itliiill find that its distance from f and its velocity are the hnriKODtel] compunentB of those of P. Hence the acceleration of this point is ttlxo an attraction tonnrds S, equal to n timee its distance ttma $A lo^tboT with a retardation equal to 2k times iti< vdiicity. We have therefore a complete construction for the recttline motion of n point, snhject to an attrnetion praportjona] to dit-tance from a hiced point, and to a re«i«taneo proportional l>i the velocity. The motion of such a point is tnmply thu horixont 755-1 SCALE BEAniKCS. 347 ]inrt of the ]noti<Mi of another point which moves with unifonn angular velocity in a logarithmic 8]>iral. »733.] The filiation of tbe spiral ia To <I«t«r[nin<; the horizontal motion, wc ]>nt ft ^ ss wf, a: = a + rem^, where a is the value of a? for the point of equilibrium. If we draw BSJ) making an nnfjlc a with the Vertical, then the tangents SX, 3)T, GZ, &c. will be vertical, and X, Y, Z, &c. will \>e the extremities of suceet»ive oEcillationfi, 1734.] The observations which are made on vibrutin;; bodicM are — (I) Tlie neat e- reading at tlio stationary points. Tlieee arc culled Elongations. (2) The time of passinf* a dcRnito division of the ecnli! in the positive or ncgjitive dircetion. (3j The scaie-reitding at certain definite timeB. Observationa of this kind are not often made except in-the case of vibratious of long period *. The quantities which we have to determine are — (1) The scale-reading at the position of erpiilihrium. (2) The logarithmic decrement of the ^-ibrations. (3) The time of vibration. »To deiemine tie Seadiiig af Ike Potillon of E^itiiiirium frvm Three Comecuthe Elongal'wnK. 735.] Let it,,Z2, *■, bo the observed scale -readings, eonvqiODdin^ to the elongations X. }', Z, iind let a be the reading at the ponition ,gf equilibrium, S, and let r, be the raluc of S£, ^k jTj— IT = r, sina, HProm tl "when . thcce values nc find whenee a = J'l'^II^A 'When X, does not ditTer ranch from f, we may ase as an ap- Bioximatv formula • Se« Gmh, SemlUUt du MagnetltehtM Ytrttai, 1830. II. ELECTBOMAOITETie OBSBltVATIOSS. 7'o ileferniiu lAe Logtirilkmic SttrtmviU. 736.] Tlic lognritlini of the rgtio of the amplitude of .a vibiatiwi ] to tliut of tli« ni-xt follott-in|> is calloJ th« Logarithmic DecremeDt. If we write p for tliis rutio, P = *i-«t ^ = %»P. A = log, p. Z ie called the common lo^rithniic docremeat, And A the Napienu lo^ritbmic decrement. It is innnifcal that \ = L log, ] = n cot a. Hencfl a = oot~' - > wliicli <lc1ermin»i the angle of the lo^rtthmic spiral. In mnkin^; a spodiil dote rmi nation of A wc allow the body to perform a consideralilc niimlHT of vibrations. If c, is the amplitodp of the first, and '■„ that of the «"■ vibration, ^'.-i^-oy If vre suppose the accuracy of observation to be tlic sunic li^ ' small vibrations as for large ones, then, to obtain the best value of A, we should allow the vibrations to subside till lh« ratio of c^toj e, becomes tnost Dearly equal to i, the base of ibc Napi« logarithms. Tlii.i gives n the ncarent whole number to - -t- I . Since, however, in most csks time ik vatoabV, it is bi^irt. to take the second Hit of obncrvalions bufore the iliiiiinutiou of amplitud^j bo* proceeded so far. H 737.] In ocrluiii ni^es wo may havo to determine the poettion' of eqnilibriiim from two eoii«evutive elotigatioiis, the logarithmic decrement being known from « special e>Li>criment. We have tJicn l^mf of FU/ralion, 738l] Having detcrminMl the ecalo-readiog of the point of cqui librium, n coni^iicitous mark ix pbiec<l at that point of the suli or as near it as powiblc, and thv limrN of the piuuigt* of llua are noted for several successive vibrations. Let na suppoae that the mark in at an iitikuown bat very distance f on the tiaiitive n itli- of tJie point of equilibrium, aai Of TrBBATIOM; 1^ JN the obwrvcd tJin« or the first tnneit of the mark in tb« positive I direction, ami f., t^. Sic. the time;; of the followiufr tniDsita. If T be the time "f ribrotton, iin<l /'(, P.^, P^, &c. the times of I transit of the true point of rquilibrinm. ai »here r,, t',, &o, are the successive velocities of transit, which we ay suppose uniform for tho very small distance x. If p is the ratio of th« amplitude of a vibration to th« next in sticccseion. »u I ^— r, ]• and ^B Tlie liiue of thv eccund ])aes3^> of the true point of equilibTium ia ^^ Three transits are suSicient to determine these three qimnlitieei, b«t •ny preat«r niiniber may bo combined by the mcthoil of least (Kjuares. Tlius, for five transits, Tlie lime of the third transit is, 739.] The anme method may be extended to a series of any number of vibrations. If the vibrations are so rapid that the time of every transit cannot be recorded, we may record the time of every third or every fifth transit, taking care that the directions of suceetisiye trantdte are opposite. If tho vibrations continue re;tutar for a lougr time, we need not observe during the whole time. We may beofin by observing a snllicicnt number of tran»it.t to determine approximately the period of vibntion. T, and the time of llie middle transit, /', noting whether this traiinit ih iu tlie positive or the n^^attve direction. Wo may then either go on countinir the vibrations without recording the 1itn<'* of transit, or we may leave the ap]>arat«s uiinittelicd. \\c iheii oliser^'e a If three transitd are observed at tiroes fj, ^, /,, we find The period of vibration is therefoie je oft ^ = lV2'» + '4-'.-2M-A('i-2^ + 2',-2'.+'J~|(2--fjTs)- ^' S30 UCROHJlGSTnC OBBETlTTtmL [74a mtnaA tarn* at ttamti, sad dedoee the Hate of ribTatioB T ai Uw ttBe gf anddk tna«t l". Botu^ the iinetion of this tnneil. If T sad V, the pgriodi of vibtatioa aa dedncid fnim (lie l«i> ■eU of obamBtinfM, are nckriy e^oal, «e m^ proceed to n nun acmralc drlcrmtBadoD of the petiod b^ cMabining' the two sebn of obMmtioiuL Piriding P'—P bjr f . the qnotint oc^t to be vetj ataiij ut iDt«ger, even or odd according as the tnuxita P and P' are in tb«! tuae or in opponte dirrctMAS. If thi« i* not tbe case, tbe iieneii of olwervationa is wortUesi, bat if tbe niatilt ia very aearij a whole narober «, we divide f—P h-j a, and thai Gnd the wko, \-alae of T for lh« nholt? time of swinging. 740.] Th« time of vibration T thna fimnd is the actual m«aB time of vibration, and is sabject to oorreetionii if Wf wish to dMlore from it the time of vibration id infinitet; amaU arcs and withnit dam|iin;jf. To reduce the obaerred time to the time in infinitclj email am, we observe that the time of a vibration of lunplitude c is in gti>cnl of the form r=y,(l+«c*), where « is a oooffic-ient, which, in tbe ciseof the <»n]inaiT pendolu is ^i- Now the amplitudes of the Ruocessive vibimtioDS are rJ Sfi ~\ cp-;. .cf^-', so that the whole time t^jn vibratioDs is where Ti* the Umt* dedaoisl from the obMrvations. Hf^nvc, to (ind the time Ti io infinitely small ores, we approximately, To lind the time T„ when there is no damping, we have 7; mT,ana 741.] TliP pc|iiation of the rectilinear motion of a body, attrncle^H to a fixed jioint and ri-siftted by a force varying u tho velocity, ^^ wbrrp T is till- coordinatf of tUp body at tlwj time /, and a u thq eoonliuute ul' the point of et|iiilibriuni. 74»J nBTLEXIOir OP THE OALVANOMETini. I I To solve this i<quation, leb w—a = tf-*'y; the solution of wbich is ^then (2) (3) y = CcfM{ •/at' ~ifl+a), wheni is less than w; (i) y = A-^ £/, wlicn i is ecjunl lo w ; (5) and J- = C'ci)s^(s/^— fci*/ + o'), when £ is greater than u. (6) The vmluc of « nwy lie ohtoinetl from that of ^ by equation (2). Whcji k i$ li'!W than to, the motion consists of an infinite series of o^Ilutionit, of oonstunt [H^riodic time, hut of continually dwreasin^ am[ilitudit. As A increases, the iwriodic time tK'comes longer, and the diminution of amplitude becom«it more rapid. When i (half the coefficient of resiNtaiiie) hwomes equal to or jireater than u, (the square root of the acoek-nttiou at unit ilistftnco from the point of equilibrium,) the motion ceases to be oscillatory, and during- the whole motion the body can only once pass through the point of equilibrium, after >vhi<;h it n-achetf apoxitioii of greut^wt donation, and then returns towards the point of c(|uilibntim, con> tinoally »pproacbin^, but never rea^^hin^ it. Galvanometers in vhieh the resistance it) so groat that the motion IB of this kind arc called deatt deaf- gatvanometerH. They are useful tin many experiments, but especially in telegraphic signalling, in which the exiatcnco of free vibrations would quite disguij^e the movements which are meant to be observed. H Whatever be the values of i and w, the value of a, the scale- ' reading at the point of equilibrium, may be deduced from five scale- readings, j>, q, r, t, t, taken at equal intervals of time, by the formula fc q(r'i—qf) + ripl—r')+»(qr~fi>) ■ net> ■ to t H eurr Port Ok tie OiservalioH of the Galntnomelir. 742.] To mi°a»ure a constant current with the tangent galvano- meter, the instrument is adjusted with the plane of ita coils parallel to the magii.tic meridian, and the zero reading is taken. The eurreot is then made to puss through the coils, and the deflexion of the magnet corresponding to it« new portion of equilibrium ib observed. Let this be denoted by ^. llien, if y/ is the hoiizontal mii^nctic foreo, G the coclficieut ol Iht! fpilvjDometer, and y the stronffth of the current, // tan f , (I) jn-RcmoMAosino ob8bhvatioit». p'45. If tbe coofficient of torsion of the suspeasion fibre is TMJI{tee Art^ 452), w<! must use the corrected formala y = ^(tan* + r^«OC*). W Setf- falve of the Ikfit^on. 743.] In some gulvniiorneteri the number of windings of Uie coil tlirouf^h which the current flows can be alterwl at plearank In others u known fraction of the current can he dirertecl front thr galvanoRietvr by a conductor called a Shunts. Id either case t)ie value of O, the eifect of a onit>outTent on the magnet, '\» maik to vary. Let UK determine the value of Q, for which a given error in the otwervation of the deflexion correapond^ to Uie smaUeei error of the deduced value of the Rtreiigth of the currenL Diirerviiliating equation (1), we Bad I A Eliminating G, J~ ~ n~ **" ■*^' (*^ This ia a maximum for a given value of y when the deflexion it 45". The value of should thprofore he ndjtwtwl till (Jy is a* nearly e<|Uid to ^ bk in pottifihlo ; wi Ihat for strong currents it ii ] better not to ukc too sensitive a gatvaiiomi'ter. On tie Bat Method of applying the Current. 744.] When the ohMtver ig able, by ntoatis of a key. to make or| break the conncxioss of the circuit at any instant, it is advisable io\ crate with the key in snch a nnr as to make the magnet arrive ~ftt itK i>OKition of r<iailihrittm with the least possible vcloci^. Thfta lulluuiug metJiod was devised hy Gauss for this purpose. f SuppuM- that the nutf^et is in its position of rqailibrium, and thai there is no current. The observer now makes contact for a sliorbj time, so that tliv mugnct is set in motion towards its new posit of ecjuilibrium. Ho th«'n breaks outjict. Tho force is now towa the origitml position of i-cjiiitibrium, and tho motion \* retarded, this is so managed that the magnet comi-H to rei>t exactly at thi> new position of equilibrium, and if tJie observer again makes* con- tact at thai inslant nnd maintains the contact, tlie magnet wilt remain at ivst in its uew jio^ition. anatnai a sboft^ positiofM toward^l ded. ifl ^ 745.J MEASUMKBUT 0? A CTTBREITL 808 B If we neglect tb« effect of the resistance* nnd also the iD«qn»lity ^bT the total force acting in the new and the old positions, thon, ^^nnce we wish the new force to generate as much kiiit^tic enei^ daripg the time of ita first action as the original forci^ dcBtroys while the circuit is broken, we must prolong tlie Sr^t action of the ^rfnrrent till the magnet has moved over half tJte digtanoe from the ^prst position to the aeoond. Then if the original force acts while the magnet moves over the other half of its course, it will exactly Btop it. Now the time required to pass from a point of greatest elongation to a point half way to the position of equilibrium is one-sixth of a complete period, or one-third of a single vibiatioo. The operator, therefore, having previously ascertained the time of a single vibration, makes contact for one-third of that time, breaks contact for another third of the same time, and then makes contact again during the continuance of the experiment.. The magnet is then either at rest, or its vibrationB ar« eo small that observations may be taken at oni?e, without waiting for the motioD to die awny. For this purpose a metronome may be adjusted so a* to beat three times for each single vibration of the magnet, ^1 ^e rule is somewhat more complicated when the resistance is of sufficient magnitude to be token into account, but in thitt case the Tibrations die away so fast that it is unnecessary to apply any corrections to the rule. When the magnet is to be restored to its original position, the jit is broken for one-third of a vibration, made again for an time, and finally broken. This leaves the magnet at rest 10 former position. If the reversed reading is tobetuken imme<liatelyaiUr the direct le, the circuit is broken for the time of a single vibration and then reversed. This brings the magnet to rest in the reversed rrition. Meatnrement by tie Firtt Swing. 745.] TVhcn there i« no time to make more than one obscrvatioa, the ciirrcnt may he measured by the extreme elongation observed in the first swing of the magnet. If there is no resistance, the permanent deflexion ^ is half the extreme elongation. If the re- nstonce is such that the ratio of one vibration to the next is p, and if fly is the zero reading, and ff, the extreme elongation in the first swing, the deflexion, 0, corresponding to the point of equilibrium is ^ = Aa 354 ELBCTBOMAOSSTIC OBSERTATIOITS. la this way tbe deQcsion may be calculated wiUiont wuting f«r the magnet tn come to rest in its positioa of cquilibriom. Ih mah a Strie* pf Oiaervathus. 746.] The best way of making a coDeidemble namber of meaniitf of a conaUint eurrent is by obaenring three elongations while tJuj current is in the positive direction, then breaking contact for about the time of a Ginglo vibration, so as to let the magnet swing into tbe position of Degutive <l<.>flcxion, then revcr«in|> the coTreat and observing three succesEivv elongations on the negntive oAv, thou breuking conUct for the time of a nnglo vibnition and iv- pcflting the observations on the pottitivA Hide, and m on till a soffi- cient number of observutionB have been obtained. In this way tbe errors which may arise from a change in the direction of the earth'* magnetic force during tbe time of observation are eliminAtod. The operator, by carefolly timing the making and breaking of contact, can easily regulate tho extent of the vibrations, .w a^ to make Uiem snfBviently small without being indistinct^ Tho motion of the magnet is graphically represented in Fig. 59, where the nliirii rpproscnts the time, and the ordJnato the deflexion of the maglMt. If flj ... (1( be the observed elongations, the deflexion is given by the equation 8^ = tf, + 2tfg+fl,-tf4-2tfj— tf,. I Kg. ss. Melkoil 0/ MnUipticatiwt. 747.] In certain cases, in which the deflexion of the galvanometer magnet is very small, it may bo advisable to increase Uio visible efi^ by reversing the eurrent at proper intervals, so as to set np a Ewinging motion of the magnet. For tliia purpose, ascertaining tbe time, T, of a single vibration of tlie magntt, tb corrent is sent in tbe poative direction for a time T, then in tfaj reversed direction for an equal time, and so on. When the motiod of tlio magnet has become visible, we may make the reversal <ii \ current at the observed times of greatest elongation. Tjct the magnet Ixt at the positive elongation fl,,, and let current hn wnt through tbe coil in tbe negative direction. 1 748.] TfiANSIEKT CCBBENTS. 355 Ppo I r lint of equilibrium ib then — ^, and the magnet tvill swing to a ne^tive doagmtion 0^, such that Similarly, if the current is now made positive while the magnet swings to 0j, p^i =— l'i + 0>+ 1}^. or />*fl, = fl« + (p+l)'*; >nd if the cnrrent is reversed n times io sucoceeion, we Gnd rhenoe we may find ^ in the form * = («--/>""^o) p-1 p+l 1-p- If M is a number so great that p'" may be iK^lectcd, tlie ox- ^reeaion becomes p^i = ff„ I P+l The application of this method to exact measurement requires an accamtc knowlrdgo of p, the ratio of one vibration of the magnet to the next under the influeticc of the resistances which it expe- riencex. The unceTtainties ariEiiig from thu iliniculty of avoiding irregiiUritien in the value of p generally outweigh (he advantagoa of the large angular elongation. It is only whitre ^ve witili to establish the existence of a very omall eiirreut by oauKing it to (produce a visible movement of the needle that this method in really valoable. On the Meaturement of Tramienl CurrenU. 748.] When a cnrrent tasta only during a very small fraction of the time of vibration of the galvanometer-magnet, the whole <]uaa- tity of eJeotticity transmitted by the current may be measured by the angalar velocity communicated to the magnet during the pas»ige of the current, and this may be determined from the elongation of the first vibration of the magnet. ^H If we neglect the resistance which damps the vibrations of tbo ^■magnet, the investigation bfcomos very simplo. Let y bo the int«nsity of the current nt any instant) and Q tliw quantity of electricity which it tranvniitv, then q^jydt. 0) A a z ss» BLECTBOIUOKBTIO 0ISBBTATT0K8. [74»-| Let 3i be the magnetic momettt, and A the moment of inerta at the uu^et and suspended sppamtos, A^ + MII»ae = ,Vffyoo«*. (J) If the time of the passage of the current ia very small, wc na; JDtegnite with respect to / during this short time withoat r^ionlii^ the change of S, and we find A^=MG<ioa0oJyJl + C = 3fGqoM9^ + C. (3)' Thig sheirs that the passage of the quantity Q prodnoee an aagolar momentum StGQ cios 0^ in the magnet, where $„ is the valoe of t at tti« intitant of lujsago of the current. If the magnet is initiallj io ei]uilibrium, we mnj make 0^ = 0. The magnet then swings freely and reaches an elongation tf, . If there is no reoistance, the work done against the magnetic Ebrce , during this swing is Jfi7'(l— costf,). The energy otanmttnicated to the magnet by the current is iA Equating these qnantides, we 6ad d$ = 2-^(l-C0Bfl,), whence i'-V HE sin \0^ = ^Qby(8). A Bat if 7 bo the time of a single ribration of the magnet, and wc find J_ It T m in where B is the horizontal magnetic force, G the coefficient of the galvanometer, 7* the time of a single nbmtion, and 0, the firrt elongation of the magnet. 749.] In many actual experimente the elongation is a Ka^V; and it \s then ensy to take into acoonnt the effect of restst- anoe, for wc may treat the equation of motion as a linear equation. Let the magnet be at rest at its position of equilibnnm, let angular velocity c be communicated to it instantaneooaly, and its first elongation be $x. \ firrt J amsnl ■Mist-" mm MBTHOD OF KBCOIU 357 !I1ie equtttioD of moUon U ^ = C»,Bec^fl--t'»"flco8(«,(;+^>. (9) ■When t = (i,$ = 0,MA.~ = Cu^ = p. When «,(+/9=|, tf = CrfH'"''cos^ = (l,. (10) Now -^— = <u' = co,'sec*A (12) tanja = -. w,= ^, (13) •=-^«. (14) H«oo .. = f:^.->-^^ (15) [whkh giYo the first elongation in terms of the quantity of elec- pcatj in tho trnnEieot current, nod convtrrsely, wliero jT, is the . tioM! of a eingle vibration as affecteil by the actual resigt- BDM of damping. When K is sniiiU we may use the approximate fonnula rr t Q = ^lil + h>^)9,. (17) Method of RteoiL 750.] The meUiod gntfa above EnpposeH the magnet to be at TMt in its position of Cfiuilibrium when tlio transient cuircnt it paaaed through Uie coil. If we wish to repeat the experiment we must wait fill the magnet is again at reat. In certain caeea, however, in which we are able to produce transient currcnte of equal intensity, and to do so at any desired instant, the following metbod, described by Weber *, is the most convenient for making .tinned aeries of observatjons. * JlavUtUt iri MajiuiUAm Vm{Ht, ISS8, p. W. 35? ELBCTROMAOinrnC OBSBRTATtOSS. Suppose that wc set the mnfpiet Biviagin^ hy means of a trmsiait current whose valao is Q„. If, for brovity, wii writ« 6 ./¥+l^ _.i,^'. -e • = jc; (18) M r, then the first elongstion fl, = JfQ„ = fli(My). (13) The Totocity instantaneously coinnimiicat«d to the mugoei il Btartinff is MG . ,^„, When it returns through the point of c<]uilibri(im in a negaiin direction its velocity will be », = — or-*. (21)] The next negative elongation will be 0, =-«,*-* = 5,. (SJ) Wlien the magnet returns to tJie point of eqnilibriom, its veloci^ will be „^-j,^^it\ (23J Now let an instantaneous current, whose total quantity ia — Q, be transmitted through the coil at the instant when the magnet is at the zero point. It will change the velocity v^ into r,— c, wbeie MO V = Q- m< I ir Q is gwtteT than Qo^~^^> ^^^ '■^w velocity will be negative and ' CHual to iffi The motion of the magnet will thus be reversed, and the next elongation will bo negative, <?,=-ff(Q-(2„r») = e, = -JS:c+^r^\ (25; The magnet is tlien allowed to come to ita poaitive elongatioa e^ = -«,*-* = <*, = *-* {A'Q-«,«-**). (26) and when it again reaches the point of eiinilibnum a poativ«S current whose qoantity is Q is transmitted. Tbia throws ihg' magnet back in the positive direction to the positive dongation $,=KQ-e^e-»'-; (27) or, calling this ihe first elongation of a second scries of four, n,= A'efl-e-") + fl,(-'\ (28)' Proceedbg in this way, by obecrving two elongations + and then sending a pontive current and ohscrting two alonga i SERIES OP OBSERVATIONS. 8C9 — and +,tben Bcuding » positive current, and bo on, we obtain K aeries consisting of sets of four elongations, in ea«h of wliieli a—c = «-*. and (30) If n series of clongntione have been observed, then w« find tbe logarithmic di-orement from tlie equation and Q (Vom tbe equation = S.(a-i_c + d)(l +B-**)_(^.-a,)-('/,-«,)*-«\ (32) The motion of the magnet in tbe metbod of recoil is graphicaily represented in Fig. 50, where tbe aliscissa rcpreacnU tbe time, and the ordinate the deflexion of the magnet at that time. See Art. 760. Jlfg/ioil of Miilliplication. 751.] If wo make the transient current pass ercry time that the magnet pacvcw through the zero point, and always so bjs to increase the velocity of the unngnct, then, if tf,, tf^, &c, ore the successive elongations, e^=—KQ-e-%, (33) 0s=—KQ-e-'-$.. (34) The ultimate value to wiiicb tbe elongntion lends aft^r a great vibrations is found by putting $^ = ~^ii-ii whence we find If A is small, Uic value of the ultimate elongation may be large, iMit «iue* Uiis involves a long continuwl cxi-n>riinent, and a careful determination of A, and since a small error in A introduces a large enxpr in tlie determioatioD of Q, this method is rarely useful for I enxpi tJbs tiiae of th« p«ss^ of the g nrrat vick sBttv. W CHAPTER XVn. COVPASISOK OF COILS. Erperimmfal Dttfrmina/ion of the EUdrkal ComtanU ^a Onl. 752.] Wb hnve eeen in Art. 71 7 that in * sensitiTe walvanomeUr tlie coils ebould \k of small radtus, and should contain many windings of the wire. It would b« extremely diflScult to determine the electrical constants of such a coil by direct measurement of iti* form and dimensions, even if we could obtain access to every winding of the wire in order to measure it. But >n fiict th« grcntvr number of the windinf^ arc not only completely hidden by the outer windings, but we are uncertain whether the prvscnre of the outvr windings may not have altered the form of the inner I ones after the coilinpr of the wire. It 18 bettor therefore to determine the electrical constants of the ^Coil by direct dectrieal comparison with a standard coil whose con- , stunts are known. Since the dimensions of the standard coil must be determined by I actual mea«ureinent, it must be made of t'oosiilerable size, so that Uie unavoidable error of measurement of its diameter or circum- ference may be as small as jioBsiblc compared with the quantity measured. 'Hie channel in which the coil is wound should be of rectangular section, and the dimensions of the section sboald be •mall compared with the radius of the coil. This is necessary, not I M mnch in order to diminish the correction for the size of the I ae«tion, as to prevent any uncertainty about the position of those [windings of the coil which are hidden by the external windings*. Large Uupat nhMKcnoUn w* tniiutlniM made with * aUigl* oirculkr Mo. dnctliif ring of ecmuhnble IhicknEw. whioh i« fufnciutUr Uitt t« luiiiUuD itii fonn sithnit KliT *ii|>pi)rl, Thii ii not > good plsn for * ttiuiiiiuil tontnlDettt. The cU»> tribulMti u( tlia eiURSt niUiiii Um conductor ilqiwiib du lb* r « h i ti ?o coiidaotMtj ^ COKPAfilSOK OP COILS. Tlic principal poDKtante which we trah to det<>nnin« u«— (1) Tho magnetic force at the ci-ntrc of the coil due to ■ imtt- current. This is the quantily douotod hj G, in Art. 700. (2) The magnetic moment of the coil due to a anit-rormit. Tliis is the quantity (f^. ^ 753.3 ^** dfiermitu O,. Since the cofla of tlw worlciog' gn]«-' nometer are much smaller than the ittandard coil, we place Um galvanometer within the standard coil, so that their centres coincide, the planes of both coils being vertical and parallel to the earlb's magnetic force. We have thus obtained a diflerential galvanometer one of whose coils is the standard ooil, for which the value of 6', is known, while that of the other coil is C/, the value of which w* h have to determine. H The magnet suspended in Uie centre of the galvanometer coil is acted o» hy the eurrcntri in both coils. If the strength of thr current in tlie titandard coil is y, and that in the galvknomctcr coil y, then, if these currents flowing in opposite directions prodooe a defleiion t of the magnet, J/tani = G,V-<?iy, (I where J7 is the horizontal magnetic force of the euih. If the currents are so arranged as to produce no deflexion, w« may find G^' by the equation ( <?.'=^C,. a I I We may determine the ratio of y to / in several ways. Since the value of f?! is in general greater for the galvanometer than for the standard coil, we may arrange the circait so that the whole current y Bows through the standard coil, and is tJieit divided so that }^| flows throug)) the galvanometer and resistance coils, the combined resistance of which is It,, while the remainder y— y' flows through^ uiotlier set of resistance coils whoee combined re«istanc« is S,. of ba vsrknM'pvlj. UniM »aj wniMabd flkv fai iIm CODtioatty at tb« MStol a«M th* nuia itnun o( el«<rtnrit7 to Sow nititvr tlem» lo lb* «M4Ua «* doM to ImUt of thn circDljLr ring. Tbu tho tno paik of tbe oumM Wmm* ttnoarula. BmUn thK wbea tbe curmM flow* oolj taiM rooBd the tinU, <i|i«oiKl taiv h ■CMBHij (« ftraU MDj AcUan oc the nutfuAei nrngnct daa ia Oi» cmtmit o« Im way lo or tKtu Uw tinit. If ■iiim th* cuR«)tl In U« elecUndM U oqiud t« tLat to tba dido. In Iji* otautiwiUaii of maay hntnuntBU Uio anloB ol tlui put of Ua comnt •••nw (n ki>r« Bm •ItogKlHr tort a/fla i>t. Tba iDo«t ptifpct molliod U to nwik« mw of tbo elwtnxl** in t^ (ami <f • nwi tnba, and lb* odm » win oortrod whli bniklinc nuttrial, urf |UMtd InM* (ab« ^id eooMMfie frith II, TIiaoxl«nuJ kM)an«f theelBeUoilaiwhimtliua b UN^ bjr An M3. COEFFICIENTS OF ISDCCTIOS. S68 We hare then, by Ait 276, or and y y (8) is Mny unceHainly nboiit the nctiial resistance of the iet«r coil (on account, Kny, nf itn uncertainty as to its tem- peratore) we may add resistance coils to it, so tliat the resistance of e galvanoinet«r itself forms but a small part of 7?i, and tUos introduces but little uncertainty into the final result. 754.] To determine ^, , the magnetic moment of a small coil due a unit •current flowing through it, the magnet is still suspended il tliv centre of the etnndard eoil, but the small coil is movod lid to itself along the common axis of both euils, till the same current, flon-ing in o)i|iosit« directions round the coils, no longer deflects the magnet. If the distance between tbe centres of the coils is r, we hiivc now By repeating the experiment with the small coil on the opposite of tbe standard coil, and measuring the distance between tbe ptions of the small coil, we eliminate the uncertain error in tbe lination of the position of the centres of the magnet and ' the email coil, and we get rid of the terms in g^, ;«, &c. If the standard coil is so arranged that we can send tbe current [.tbrough half the number of windin<^, so ae to give a dilTerent rnltic &,, we may determine a ncvr vnlue of r, and thus, as in Art. 431, we may eliminate the term involving y,. Il \» often poEKible, however, to determine j^ '^y direct meanirc- it of the small coil witli sufltcient accuracy to make it available caloalating tbe value of the correction to be applied to ;, in lie equation 9^=\0^r--^^ (?) vhcre y,= — -ira»(6a' + 3f'-27'), by Art. 700. ComparitOH i^Coej^eitnU oflndnetion, 75S.] It is only in a small number of eases that the direct GalculatioD of tbe coefBcieDts of iDdnction from tbe fonn and poKitton of the circuito can bo easily porfoniUMl. In order to mttaia a sufficient decree of aoRuracy, it is neoesMry that tbe dtstaimJ botwwn the oircuita should be capable of exact measurement.' But whiii the distance between the circuits is eufficient to present errors of measurement from introdacing Ibi^ errors into tho result, the coefficient of induction itsdf i» uecowarily very much reduced in rouf^nitudc. Now for many eipcrimeots it in ncocsHiry to make the ciwfficient of induction larg«, and we can only do bo by bringing the eirouitit close togitther, lo that tlie method of dir»ct mcAsufv- mcnt becomeo impossiblo, and, in order to dctftmune the cocfScient of induction, we must compare it with that of a pair of ooiU ar* rang«d so that their coefficient may be obtained by direct meuure- ment and calculation. This may be done as followt: Let .-( and a be the standard l pair of coils, B and b the ooila to 1 be compared with them. Con- nect .-f and B in one circuit, and { place the clcetrodc« of tlic gal- 1 vanomcter, G, tA P and Q, eo I that the resistance of PJQ is B, and that of QBP is S, S }imag the renKtanoc of the ga^ Tanometer. Connect a and i in one circuit with the battery. Let the current in ^ be ir, 1 that in B, /, and that in the gdnaemlbn, nfr— jr, that in the battery circuit being y. Then, if J/, is the coefficient of induction between 4 and a, and iff that between B and 6, the integiitl induction current tfarongb the galTan<nnet«r at breaking tlie lottery circuit is By adjusting the resistances S and S till there is no cnimi' through tho gnlvanometi.^r at making or breaking tJie batte: oircnit, tho ratJo of i/^ to if, may be determined by measuring ihsA otStoS. Fig. do. SEU-IITODCTIOJI. 860 H * [Tho expTMeion (8) may be proved aa follows : Let i^,j^, A'&nd BT be th« coeffiviente of self-induction of the ooils A, S, ai and the Kgalvanomctcr respeotivelf. The kinetic energy T of th« Hyst«m i« HthCQ up proximately, H The dissipation function F, i.e. half the r]tt« at which the energy Hof the currents is wasted in beating the coils, ie (see Lord Rayleigh't BlW^ o/S<fUnd, i. p. 7S) Vwbere Q is the resistance of the battery and battery coilf- H The equation of currents corresponding to &iiy varisble * ifl then B~of tlie form j 4T ^T dP wkece f w the oorreiiponding electromotive force. IHenoa we bavA i,* + r{«-jr)+ jr,y + .ff* + A-(i-» = 0, iJI-r(i-sf) + af,y+5^-A'(*-^) = 0. These equations can be at oncti integ:rat«d in regard to t. Obfltmng: that T., x,y,y, y are zero initially, if we write x—y = z we find, OD eliminating y, an equation of the form ■ AS+Si+Ci=D-y + EY. (8') H A short time after battery contact the current v ^1 have become ^Ll^^y and the current i will have died away. Hence H This gives the expression (8) above, and it shews that when the ^ total quantity of electricity passing through the galvanometer is zero we must have jE = 0, or i/,ff— JJ/,5 = 0. The equation (8') further shews that if there is no current whatever in the galvano- meter we must also have i> = 0, or if^.^)— JfiX^ = 0.] Com/>an40* of a CofJ^eni <jf Srff-iiuluiTlioK teith a Co^jKurBtaf Mutnal jHilnction. 756.] In the branch AF of 'Wbcaitstone'a Bridge let a coil he inserted, the coefficient of self-induction of which we wish to find. Let OS call it L. • [Tlie iarMttekUan b KjnMebrMkets, IjiIw-h ttoia Mr. Fltaihio'iiiotwoTrD/iwat Cteffc M&ivcU'i XMtuN^ pOMiMM M Diotuiii.'hi'ly iiiiumt m Muz pMt nf tb« Ual iMturs dclir<T«l bv thn Profcaaca. Id Mr. FlMnlag'* notw tha (Jiu «f tiw coEpari- BMot dlffcn ftoat thai ^voa In lh« Uit in hiring th« batuny and gslvanorndter IaUnihM>c«d.] 366 OOMPABISOH' OP COTLS. I Id the oonnectinff wiru betwceo A and tlic bstii^ry another Mil is iDfiert«d. The covllicicnLt of nmtuiit iiMluotioo between Uiix coil and the coil in AF in M. It may be meaaared hj tlie metliod ^ described in Art. 753. fl If the cuiTCnt from J.io Fiax, and that from J U> JliBjff th«t ~ from 2 to A, through S, will he t+j. The cxtcrnul clcctromutivo force fron ^itoi'w \ A-F=P.^lpM(^.±).p) The external eleetromotive force alonff All is J-Jr= Qy. (10) If the galvanometer placed betw«w F and U indicates no currvot, either truH«ii.-Dt or iwrmanent, then by (9) and (10), siBCc U—F= 0, Pr=.Q.y: (11) M Fis- 61. and whence (U) (13) I « Since/' i» always posilive, jV muet be negatircnnd tbereforo the current must flow in ojipoKito directions through the coils plnved in P and in B. In making the experiment we mfty either begin by adjusting the resistances so that PS=qR. (H) which is the condition that there may bo no permoiuint curreat, and then adjuxt tliii di»t«iice between tlio coil* till the galvanometer, ceetca to indiciite a. transient current on making snd breaking the batteiy connexion ; or, if thin distance is not capable of adjuitmeot, we may get rid of tlie transient current by altering the restslAooes Q and S in such a ^ay that the ratio of Q to 5 remaina constant. If ihis double adjuntment is found too troublesome, we may adopt a third method. Beginning with an arrangement in which thel tnuuieiit current duo to sclf-inductton is slightly in exoees of that due to mutual induction, we may get rid of the inequality by in- Betting a conductor whose resistanoe is T between A and Z. The condition of no pi-nnanent current through lhi> tn>lv.tnomebor is not aficct«d by the introductioa of ff' Wo may therefore get rid c^ ^i^^ a ci 1 757-] SBLF-IKDrCTION. 367 the traniiiciit current by ndjaetiDg the ro-Ul^tce of ft' ftlooe. Whoa liii! w douc the vnluo of L is (15) CoMpariaon of the CofJJieiaili of Seff-mlnclion of Two Coil*. 757.] Insert the coils in two adjacent branches of WheatBtone'fl Bridge. Let L and N be the coefflcieots of self-induction of the o(»1b inserted in P and in R respectirely, then the condition of no g»!T8nomet«r current is and PS = QS, for no permanent current, -^ = -=■, for DO transient current. r it (16) (17) (18) I Hencf^j by a proper adjustment of the resistances, both the per- nuuient and the transient current can be got rid of, and then the ratio of £ to iV can be deti^rmined by a oomparisou of the kiesistuioc*, 366 OOMTABISOK OP C0IL9. In the connectiDg wire bctwoen A nnd tlu> Hstteiy a inserted. The ooefficiont of miitmil induction br and tho coil in AF is JA. It ma; be mtiiutuTei* described in Art. 756. If the current from A\o FUx, and tUat fr from SSUt At thr The external c' AU>F\» M V A-F=Px The . t« along / -STASCX w . Fi7.«l. and wheDoe Since L \» aI\Tay current must t!i ^ in P and in B. by adjosting tf which IK tl and then n' ceases to i batterr c we nius tC antl Ift^ a thit tra"- *^ «rtor is defined as tlie ratio of thr , live force to that of the cuitenl ,.jotor. The determination of tit tf- -.ruiiffociie nieaHure can be made by vMr, when we kooiv the value of the .. determination of the vnltic of tht □lEciilt, as the only cniw in wliich ire . J,' is when it arise* from tlic relative ..-•■^■i-ct to a knoTvn magnetic Hj-vtem, ^^Btion of the ri}«i»1an«c of a wire in «ai made by Kir«hhoir*. He employed •^ J.and Af, and calculate thoir coefTiciest of mutual induGlion from the geo- ^ metric*] datA of their form and apoation. "Hieee coils were jiUced a in circuit with a (^vanonietor, G, = and a battery, B, and two potntc \ oftho circuit, /*, between the coils, .^■' and Q, between tlie battery and [galvanometer, were joined by the ^ |C^ was to be measured. ^''wly it i§ divided between the wire and . imd produces a certain permanent de- r. If the coil ji, is now removed quitikty iinbbM- •1i« iDlnulUit Mocirtor tlrlcniailiw CHAPTER XVin. ELECTROMAGNETIC CKIT OP RESISTANCE. (h lie J)«iefmiiuition of fie HetUUtnee of a CoU in Sleetn- magneik iteature. 768.] TiiK reRiHtance of a conductor in deGned ae the ratio of I numerical value of the clcetromotivi' force to that of the which it produces in the conductor. Tlie dGtenninatJOD of value of the current in elcdromagnetio measure can be matle means of a standard galvanometer, when we know the ralue of eartli'e magnetic foroe. The determination of the vslne of electromotive force is more difficult, as the only case in whidi wf^ can directly calculate its value is when it arises from the relative motion of the circuit with respect to a known magnetic tystcin. ^ 759.] The first determination of the reastanoo of a wire ifl^ electro maniple tic measure was made by Kirchhoff*. He employed two coils of known Ibnn, ^j and A^, and calculated their coefficient of mntual induction from the geo- metrical data of their fonn and position. These coils were placed in circuit with a galraDompter, G, and a b«ttery, S, and two point« of the ciicuitj P, between tho coils, and Q, between the butU^ry and galvanomDlcr, were joined by the wire whose resistance, R, mu to be measured. When the current is rtoady it is divided between the wire the galvanometer circuit, and prt^uces » certain iwrmanent lloxion of the galvanometer. If the coil Ai \a now remorcd quickl^ F1k.CS. [ckl^ ■ • BaitimnaiB dtr OctutentM tod wslohar din IbMuUH bdadtttf •UtirliB SbNlineaUiiMfl.' i'oyjp. daik. )»ri (April 1M»>. KIBCiniOPF*S METHOD. Ota A^ Aiul placed in a position in which tlic coefilciont of mutuftl [induction between A^ and A^ is zero (Art. 53S}, a current of induc- bioa i* produced in both oircuita, and the galvanometer needU ivw an impulse which produce* a certain tmneiient deflexion. The rcsiiitnDce of the wire, H, ia deduced from a comparison etween the permanent deitexlon, dae to the steady current, and th^ Dsti^nt deflexion, due to the current of induction. Let the resistance of QG-i,/" be A", of PA^BQ, B, and of PQ, R. Let X, M and A' be the coefGoients of induction of A^ and Aj . Let i be the current in G, and y that in £, then the current from P to Q is i—^. ^Lei E be the electromotive force of the batter/, tlien (^K+B)x-By+~(U+My) = 0, (I) Wl ~Bi+iB+Ii)y + ^(Mx+Ny)=E. dt (2) When the currents are constant, and everything at rest, (ir+i?)A-fly = 0. (S) If M now snddenty 1>ei?oraes z^ro on account of th« separation of Ai from Ja« t''*'*') intt'gratiug with respect to t, (K+Il)x~Ry-My = G. (4) -Rx+(B+S)y^3fi =JEdl = 0; (5) Substituting Uie valoe of^ in terms of « Irom (S), we find x_M (B^Il)(K+Ii) + B'> ,. i~ B {B+B}{K+B)-B' "' h 'When, ns in KirchhofT's experiment, both B and K are large punxl will) B, this (-quation it reduced to i R ^*' thiM quantiticd, x is found from the throw of tlic gah-aoomctcr ) to the induction current. Svc Art. 718. The permanent cur- rent, A, i* found from the ])vrmAncnt dcHcxion due to the eteiuljr cumeDl; mw Art. 746. M w found cither by direct caliiilntion from tiro geometrical data, or by a comparifion with n pair of coiU, for which this cadculation bae boon made; see Art. 75S. From ■VOU 11. B b wa mflT OF BEeiSTANCI [760. tbwe three quantities R can b« dctorminod in eloctronuigo«tic mcaeare. TI10BC mcthoda involve the determmatioD of the period of ribra- tion of the gah-anometei ma^aet, and of the logaritlimic dvcranu-nt vritM oscilUtions. freier's Method iy Traiu!fHt Currentt*. 760.] A coil of considerable etze is mounted on an axle, so as to be capable of revolving about a vortical diameter. The wire of tbu coil 18 connected with that of a tan^nt ^Iv&nometer so as to fonn ■ nng'le circuit. Let the reeistance of this circuit be It. Let the large coil be placed nith ita positive face perpendicular (o the mogDctic meridian, and let it be quickly turned round half a tulton. There will bo an induced current due to the earth's Di^tic force, and the total iiuimtily of ck'ctrieity in this curre nt il electromagnetic miusure will be where ,7, is the magnetic moment of tiie coil for unit current, whi* in the catso of A large ooil may be dctcnnined dirMtly, by nten-^ euring the dimeaaioas of the coil, and calculating the sum of the areaa of it« winding*. // is tlie horixoutal oomponoDt of borrcstria) ma^ctiem. nnd R is (he rpHistance of the vireuit formed by the ooil and giilviinomcter together. This current sets the magnet of the galvanometer in motion. jH If the magnet ia originally at rest, and if the motion of the coil^ occupies but a Kmall fraction of the time of u vibration of the magnet, then, if we neglect the resistaaoc to the motion of the magnet, wo have, by Art. 740, // T Q = -^-2mn\0, (25 where is the constant of the galroDometer, T » the time of vibration of tlie magnet, artd $ is tbe obMTved eloogaljon. Prom these eqaationH we obtain The value of U docs not appear in tlus remit, provided it ia same at the poeitien of the coil and at that of tJie galTanotnetcf This ehould not be asauuied to be the case, but abonld be tested comp&ring the time of vibration of the asame magnet, flr«t at one of tfacco places and then at tbe other. * £Ml. lUaa.: at Fogs., •^"- ^'^='^- >>' (1851). f (3 N I I > 871 701.] To mnlw n »crie« of observation)) Wcbcr began with the vriil jKintllel to the magm-tic miTiiHan. Hu tin-n turned it with its |Kj(iitive face nortli, Hiul observed llic (iret eloni^tion due to the negative current. He ttien observed the second olongution of (Iio fn>eiy g^^ini^ing magiict, and on tLe return of the magnet through the point of equilibrium he turned the coil with its positive fnou Kouth. This eauned the magnet to recoil to the positive side. Tho eeries was contiuiicd as iu Art. 750, and the result corrcuted for feAistauce. In this way the value of the resi^tunce of the conibinccl circuit of the coil and galvanometer was ascertained. [n all Biich exporiments it is neceaaary, in order to obtain «n0i- tiently large deflexions, to make the wire of copper, a metal which, though it is the best conductor, has the disadvantage of altering conaiderahly in wsistaui'e with alterations of temperature. It is also very difficult to aeccrlain the temperature of every part of the apparatus. Hence, in ortlcr to obtain a result of permanent value from such iin experiment, the rcKistuncc of the experimental circuit ebould he compared witii that of n cttrefully constructed reeistance- ooil, both before and aftc^r eJich vxperimi.'nt. fFeber^t Method b^ ohervhiif the Decrement of the Otcillations of a MaifHtt. 762.] A magnet of considerable magnetic moment ie suspended at the centre of a galvuuomuttr eoil. The period of vibration and the logarithmic decrement of the oscillations is ebserveil, firnt with the circuit of the giilvanonmUT open, and then with the circuit cloaed, and the conduL-tivity of the galvanometer coil is deduced ftom the effect which the currents induced iu it by the motion of tiie nttgnet have in resisting that motion. If 7' is the obiterved time of a single vibration, and A the Na- pierian logarithmic decrement for each single vibration, then, If we write ff ,,. and a = y (8) I the equation of motion of the magnet is of the form ^ = Ce-*' eos {i»l+ 0). (3), [Thi* expresses the nature of the motion as determined by obser\';i- ftioa. We mu»t compare this with Uic dynamical equation of kmottOD. D b 3 872 [76J. Let M be tbe coiifficit.'iit of induction between the galvanometer coil and tlic Ku${>codud mngnct. It is of the form where (?,, &,, &c. are coefficients belonging to the coil, ^,,^, ft«. to the magnet, and P\{6), 1*,{fi), &e. uv zonal harmonics of the angle betweea the ax<>s of the coil and the magnet. Si>e Art. 700, By a proper arrangement of the coiU of the gulvanometcr, and by building ap the suspended magnet of several magnet* placed side by side at proper distances, wc may canso all the ternu of M »n«r t' lirst t(j become insensible comxHirod with the first. If we uW |i ^ =: 9~^> ^^ ^^3 writ« 11= Gm»m^, (5) whore is the principal coeffiirient of the galvanometer, m \t taagnctic moment of the magnet, and lit the angle between axia of the mugnet and tlic plane of the coil, whicli, tu this periment, ik always a small angle. If J, in ilie coetflcieiit of seir-inductioii of tli« coil, and R resistance, and y the current in tlie coil, ^(£7 + ilf) + fi). = 0, (I or 2-^+J?y+(?jfflC08<6-:^ = 0. t at at * The moment nf the force with which the currrent y acts on 11 mag^Dct is y -^ , or Gm y cos ^. The angle ^ ia in this experime Ki small, that we may Mippose cos ^ = I. Let us BuppoM titat tJio equation of motion of th« mag^nefc ' the vireuit is broken is I ^^-^'^?^^* = 0' (8) where A is tlte moment of inertia of the suspended apparatus, B expresses the resistance arising from the viscosity of the air an of the suspension fibre, &c., and C^ oxprwnMs the moment of the fon-e arising from the nuth's magnetism, the torsion of the sua- pcnsion nppnraius, Stc. tcniling to bring the magnet to its poaiti< of equilibrium. The equation or motion, u affected by the tnirrcut, will be infl Ihe 1 ia-l WRDER 8 IfKTHOn. 873 I To d«tennuie the motion of the magiiei, we have to combine ttiU oijiuition with (7) and eliminate y. The resnit is linear difiercotial equation of the third order. Vie hnve no occnsion, however, to solve this equation, liecauM the (Iiita of the problem are the observed elements of the motion of the magnet, and from these we have to determine tlie valae ofS. Lot 00 and wg he the values of a and w in equation (3) when the ircuit i» broken. In this cnso S is intinito, and the eqnation is doccd to the form (8). Wo Ihtu find Solving^ equation (10) for ^, and writing j7=— (a+iw), where i = •/—!, at (12) find a-|-I« Since tbc value of w is in general much greater than that of a, the beet value of R in found by equating the terms in f m, & = 2'<(«-''o) ■Oo W« maj also obtain a value of £ by equating tbfi termH not Iving I, but as theee terms are small, the equation is useful Hjr M a means of testing the accuracy of the observations. From tfaeae equations we find the following testing equation. Since LAn^ is very small compared with G^m*, this equation id equation (t4) may be written In this expression G may be determined either from the linear messnremcnt of the BJilvnnometcr coil, or better, by comparison with a standard coil, according to the method of Art^ "53. /( i« the moment of inertia of tbc magnet and its suspended apjuratu^ which is to be found by the proper dynamical method. <u, o,,, a ad Og, are given by obocrvntion. TTKIT OP RESISTAKCB. The detenninntion of the valne of m, the magnetic momcat ef tlip euspcnded magnet, is the moit difficult part of the invcstigatioii, becAUfre it ib iiffect«d by temperature, by the dtrtLV ma^tivtJc force, and by mL-chantcal violence, so that ^n?at cnro must be taken br measure this quantity when the mugnet u in the vcty same ctreuiQ- stances as when it ig vibratiug. . The second term of H, that which inToIvw L, in of teat import- I ance, as it te generally xmall compared with the first term. Tbt va\u6 of L may be determined either by ejilculation from the koown form of the ooil, or by an esperiaient oq the extra-current of to- ductioD. See Art. 756. Tiomion'a Metkod f^ a Sevolviiig CoiL 768.'] This method was suggetited by Thomson to the Committee of the British Aiisociation on Electrical Standards, and the ex- periment was made by MM. Balfour St«wart, Fleeming' JeDldo, and the author in 1863* A circular coil is made to revolve with uniform velocity about a , vertical axis. A Bmall magnet is eosponded by a silk fibre at the centre of the coil. An electric current is induced in tiic coQ by tho earth's magnetiEm, and also by the suspended magnet. This current is periodic, flonnng in opposite dtn.ictioR8 through the wir« of the coil during dif^rent parts of each revolution, but the effect of I the current on the Kuwjiendcd magnet is to produce a deflexion from] the magnetic meridian in the direction of the rotation of the ooil. 764.3 Let Jl be the horizontal component of the earth's nu^J netiem. Let y be the strength of the current in the coil. ff the total area inclosed by all the modiDgs of the wire. G the magnetic fotce at the centre of the ooil due to uoitr^ current. Z the coefficient of self-induction of the coil, il/thc magnetic moment of the suspended magnet. the angle between the plane of the coil and tho magnetio meridian . ^ the angle between the axis of the «a«peadcd magnet andJ the nu^nctio meridian. A the moment of inertia of the suspended magnet. ifffr the coefficient of torsion of the suspension fibre. a the azimuth of the magnet when there is no torsion. S tho resi«t«nc« of the coiL • 8m Jbpori (/ At nMA AmtUOon/wr IMS. 765] THOUSONS UETHOD. 376 I The kinetic f acr<;^ of the system la The firet term, \Ly^, expresses the enei^y of the current lui dependiD^ on the coil itself. The second term depends on the mutual nction of the current itnd terrestrial tDagnetism, the third on thftt of the current und the magnetism of the saepended magnet, tJie fourth oil that of the magnetism of the suspended magnet and ifrrestrial mngrictijim, and the Inst expresses the kinetic energy of tbc matter eomposing the magnet and the suspended apparatus which moves with it. The }»otential energy of the suspended ftpporatus arising from tlie torsion of the fibre is Mil r= »-2*o). (a) The electromagnetic momentum of the current is jj = .p = Ly-IIg sin 6-MG sin («-*), («) Pud if S is the resistance of the coil, the equation of the current i« (5) (8) dtdy , Mnce $ = iot, (B+JL-^) y~ nso)C09$ + 3fG{u—i))cos{6—it>) 765.] It is the result alike of theory and obeerratton that 4>- ^hc azimuth of the magnet, is subject to two kindu of periodic variations. One of these is a free oscillation, whose periodic time depends on e intensity of terrestrial magnetism, and is, in the experiment, seconds. The other is a forced vibration whose period is that of the revolving coil, and whose amplitude is, as wu shall insensible. Henoe, in determioiDg y, vtv may treat ^ tut eennbly constant. We Uius 6nd Hff* S^ + L'^* {Scos0 + LoiBin0) MG<. WTL*^ {RcQa{6—^)-^lua.ix.{$—^)), + C/ {9) The last term of thix exprcsuoa soon dies away when the rot«- ioQ is continued uniibrm. 376 UHIT OF BESiSTAVCK. [766. The equation of motion of the Huapended magaet is d^ T dT dld^ dV _ (10) wbeooe A^—MQyQOi{0~^)+MII{faa4t-^r(^—a)) = 0. (11) Substitutiog: the value of y, and arranging the tcrmti ncconling to the functions of multiples of 0, then we know from ol>*erTati' tbat ^ = ^„ Jrbe-" cos 11/+C cos 2 (tf— (3), {\i where ^^ ifi the mean value of ^, and the second term expnwi the free vibrations gradnally decaying, and the third the forced vthrationa arising irom the variation of the deflecting current. Soginning with the terms in (ll) whiob do not involve 0, which must collectively vanish, we find approximately -g^^ J77^ (^ cos *o + /,» sin *,) + 0J«| = 2J/Zr(8in4o + r{^-o)). (1$) Since L tan </><, is generally small compared to 6g, th« »olati(m rf tlic quadratic (13) gives approximately 3 ced 1 A = .^^.ec,,-|^(l^-,)Un.*. 9U Og^Gi I -(^Vdf-')'--*.}- (M) If we now employ the leading term in this expreesion in cqua^ tionR(7), (8), and(l]), weshall find t at the valitoof « in cciuatioD (12) is A/ ~7~ scc^o- That of c, the amplitade of the force^^ vibrations, is i-^sin^g. Hence, when the coil makes m«Qr revolutions during one'frcc vibration of the magnet, Ihc amplitnde of the forced vibrations of the magnet ie very email, and we ma; neglect tJie tenns in (1 1) which involve c. 766.] The roBistanco is thuK determined in electromagnetic m< sure in terms of the velocity w and tb« deviation ^. It is n< neceeeary to determine H, the horizontal tcrroitrial magnetic foreej provided it remains constant during tbc viperiment. J noi^ M To determine js we must make use of the suspended magnet to deflect the magnet of the magnetotneter, as de-scrilx-d in Art, *54. lu Ibis experiment M should be aaaall, 00 Uut thia correcUou, comes of Bcoondaiy importance. JOI'LB's METUOfi. S7T For the other corrections required in this experimeDt see th« 1 ^the BnVuh AteoeiaHon/or 1863, p. 1B8. ■Joale'a Calor'meiric Method, 767.] The beat geDeratod by a current y in possinj; through ft leoDductor whose resistance is £ is, by Joule's law, Art. 242. i = j-fRfdt. (I) where J is the equivalent in dynamical measure of the unit of heat employed. Hence, if It is constant during tlie experiment, its valae ia ff.lf- This method of determining R involves the determination of i, [the heat generated by the current in a given time, and of /, the square of the stren^rth of the current. In Joule's experiments *, A was determined by (he riae of tem- eniture of the water in a vessel in which the conducting wire was It WHS corrected for the effects of radiation, &c, by ltd experiments in which no current was passed through the wire. The etrength of the current was measured by means of a tangent IvBOomotcr. This method Involves the determinatioa of the intensity of terrestrial magnetism, which was done by the method in Art. 457. These meiisurt^ments were also tested by the . weigher, described in Art. 720, which measures y* directly. be most direct method of measuring i y^ dty however, is to pass the current through a self-acting eleotrodynamometer (Art. 725) vitJi s scale which gives readings proportional to y, and to make Itiie observations at equal intervals of time, which may be done 'approximately by tatinjj the reading at tlie extremities of every vibration of the instrument during the whole course of the esperi- aent. • Report of ih* JlritUb JMtortatien far IS67- CHAPTER XTX. ooiErXKBON of the electrostatic with tub electro* magnetic ukit8. Determination of the Nnmher of EUdroitaiic Units of Etectrieitg in one Eleelromagnetic Uitit. 768.] Tub absolute magnitudes of the electrical aaits id Wtb ^steins depend on th« units of length, time, nod maM whicit we adoptj and the mode in wliich thoy tk-pcnd oh these units i« different in the two systems, so that the ratio uf the (tiectrical nnitt will be expressed by a diSeront number, ncoording to the different units of length and time. It appears from the table of (limenKions, Art. 628, that the mimlK'r nf electros t^itic unite of electricity in one electromaignetic unit variett inversely as the magnitude of the unit of length, and directly as the ma^itiide of the unit of time which we adopt. If, therefore, we determine a velocity which is ropreeentcd nn* merically by this number, then, even if wo adopt new unit* of length and of time, the nomher representing thiw velocity will still bo the number of electrostatic units of electricity in one <;Iectro> magnetic unit, iMcording to the new system of measurement. This velocity, therefore, which indicate* the relation betw< electrostatic and electromagnetic phenomena, is a nataral quaoti of definite magnitude, and the measurement of this quantity is one of the most important resoarcbes in electricity. To shew that the quantity we are in search of is really a velocity, we may observe that in the case of two parallel currents the attzac- tion experienced by a length a of one of them is, by Art. G86, F=2CC'%, ' where C,C *n tine numerioal values of the eorrenta tu tro- I ?69.] RATIO EXP8E88BD BY A VBMICITT. 379 I magnetic mensiire, itnd b the distance between them. If we make is=2a, then jp_ CC'_ Now the quantity of electricity transmitted by (hi' current C 'm the time / is Ct in olectroma^Detio measure, or nCl in electrostatic meiisure, if n is the number of electrostatic units in one clectro- uagiietic unit. Let two smail conductors be charged with the ^uantitaec of electricity tmngniittcd by the two currents in the time t, KoA placed at n distance r from each other. The rejiolsion between thetn will be CC'k*t' F' = Bib. H Let the distance r be so chosen that this repulsion is e^ual to the H&ttractioD of the currents, then ■Hence r=«(', Bor the distance r must increase witli the time i at the rate n. Hence ft !b a velocitj*, the absolute magnitude of which is the —^siunc, what«vcr units we asaiime- H 769.] To obtain a phytrical conception of this velocity, lot uh inuf ■^ne a piano surface charged with electricity to the clcctrosUitic snr* ■ faoc-dcnsity (t, and moving in its own plane with a velocity r. Thia moving elwitrificd surface will be equivalent to an electric current- Kvhcct, the Rtrcngth of the current flowing through unit of breadth ll ■tro lof the eurJace being av in electrostatic measure, or - trv in elec- magnelic measure, if n ia the number of electrostatic units in one eIet:troma^netie unit. If another plane surface parallel to the first is eleotrilied to the surface-density a, and moves in the same direction with the velocity r', it will be equivalent to a second oarrent-sheet. He electrostatic repulsion between the two electrified surfaces is, Hby Art. 12i, 2ir<ro' for every unit of area of the opposed surfaces. The electromagnetic attraction between the trto curreut-shceta is, by Art. U53, 2irMK' for every unit of urea, w and a' being the surface-densities of the currents in eleelromagitctic meuKure. But 1 and w'= -irV, so that the attraction ia The ratio of the attraction to the itrpuldion U ocioal to th»i of vv' to n'. Il^noe. tiinoc tliu attraction nod tho ropulaiuii are qoan* titles of the same kind, n muNt W a ()tinDtitj' of ttie same Icitit] mi r, tliat is, a velocity. If vif. Dow HupjiuKe tlic velocity of esch of Ibe niDvlQ^ planes to be e^uai to », tlic attraotion nill l>e erjtml to the repulsion, and there will bo no RMtclianical action between tltem. Hence we may define the ratio of the electric unita to be a velocity, such that two electrified surfaoes, moving- in the samo direction with this velocity, have no mutual action. Since this velocity ii about 288000 kilometrcB per second, it is impossible to nuike ihit experimeut above described, 770.] If the electric surface-density and the \Trlocity cao bo made so great that the magnetic force is a measnrablo iinantity, we may at least verify our supposition that a moving I'lcGtrifivd body it equivalent to an ciccti'ic current. We may assume * that an electrified sarihec in air n'ottld begin to discbarge itself by sparlcs when tho electric fonoc S«« reaches the value 1 30. The magnetic force due to the currenL-aheet is 2jr(r -■ The horizontal magnetic force in Britain is about 0.1 T5. Hence a surface tloctrified to the highest decree, and moving with a velocity of 1 00 metres per second, would act on » magnet with a force equal to about one-four-thousandth part of the enrth'e hori- zontal force, a quantity which can be measured. Tlic eloetrified surface may be that of a non>conductiDg disk revolving iu the plam of the magnetic meridian, and the magnet may be placed «Iom to the ascending or descending portion of the disk, and protectwl frnm its eloctroatatic action by a screen of metal. I am not aware Uiat ^^ tlii:< experiment has been hitherto attempted. ^^P I. Comparison <^ Uniii of EfedrieUjf. W 771.] Since the ratio of the electromagnetic to the electrostatic I unit of electricity is represented by a velocity, we sliall in future I denote it by the symbol o. Tlie first numerical determination of I this velocity was made by Weber and Kohlraosch t. ■ Their method was founded on the measurement of the same ■ quantity of electricity, first in electrostatic and then in electro- I magnetic measure. I The quantity of electricity measured was the charge of a Lcydei ^^^ jar. It WM nieuured in electrostatic measure as the product of i I i * Sir W. TlwmBca, & 8. Pme. ovB«pcfat, Atf. alx. t f MArMfyMMMe MaaA^ntmmtm i and Pogg. Aim. Mix (Avf . 10, H 77I-] METHOD or TTEBES AKD EOHLBACSCn. 381 » capacity of tho jar into ttic difforenoc of potential of its coatiogs. The. cjijuwjity of thejiir was di-termiiicii by compiirisDi) with that of a sphtire BUBjwiidrtl in an o|icu s]iace lit n ilistuncc from other Iwdios. The capacity of hiioIi a sphere is cxjin-ased in electrostatic m>>aEiirc hy it« radius. Thus the capacity of tlic jar lany be found Knd cxpn^^ed a^ a certain leu^.h. See Art. 2S7. TIic dilTerenci' of the potj^ntials of tJte coatings of the jar was mca- cured by connecting the coatings with the plectrodcs of un electro- meter, the constants of which were carefully determined, so that the differenee of tJie potentials, E, became known iu electrostatic measure. By multiplying this by c, the capacity of the jar, the chu^ of tlie jar was expressed in electrostatic meosarc. To determine the value of the charge in electromagnetic measure, the jar was discharged through the ooil of a galvanometer. The ; of the tranfiient current on the magnet of the galvanometer unicated to the magnet a certain angular velocity. The maginet then swung round to a certain deviation, at which its Telocity was entirely destroyed by the opposing action of the earth's magnetism. By observing the extr«me deviation of the magnet the quantity lof electricity in the current may be determined in electromagnetio easare, aa in Art. 748, by the formula rhere Q is tJie quantity of electricity in electromagnetic mMson. ^e have therefore to determine the following quantities: — //, the intensity of the horizonljil oomponeul of terrestrial mag- aetism; see Art. 4 S6. 0, th« principal constant of the galvanometer; see Art. 700. T, tbc time of a tingle vibration of the magnet ; and $t the deviation due to the transient carrent. The value of v obtiuned hy MM. Weljcr and Kohlrausch was T = 310740000 metres per second. The property of solid dieleetrios, to which the name of Electric Absorption has been given, renders it difficult to estimate correctly 'the c«pacity of a Leyden jar. The apparent capacity varies ao- ooidtng to the time which clap^^cs between the charging or dis- charging of the jar and the miAsiircmont of the potential, and tJte longer the time the greater is the valno obtained for the capacity of lie jar. COMPARISON OT irarira. I HoDCO, eiuce the tim« occnpiod in o1>Uiniag a reAdiog- of tin elcctroiueter is large in comparison witti l.lie time during- which the diidiargv tUroup^h the galvanometer takes ptace, it is probable that the estimut« of the dicchargo in electrostatio measure is too h^,l and the vuluc of v, derived rrom it, ih probably alao too high. IL » txpretsed as a SeaU(anM. 772.] Two other methods for the detenaination of v load to an cxprcKsion of its value in terms of the reaistance of a g:iven con- ductor, whieb, in the electromagnetic sjrstem, is also expressed ae a velocity. In Sir Willium Tliomson's form of the experimeoit, a ooDstaot current is made to flow through a wire of great reeistonoe. VIm electromotive force which urgeit the current through the wire is mea- sured electrostativally by connecting the extremities of the wire with the eleetrodcH of an absolute electrometer, iVrts. 217, 218. Ute , strength of the current in the wire is measured in eloctromagnctic M measure by the deflexion of the suspended coil of an elcctrodyna- " ntomotcr through which it passes, Art. 72S, The resistance of tJi* circuit is known in electromagnetic measure by comparison with a standard coil or Ohm. By multiplying the strength of the current by thin resistance we obtain the electromotive force in clcctro- niftgnetic measure, and from a comparison of this with the elcctio- stutic mcasun^ the vuluc of v ts obtained. ThU method rdiuircs the eimoltaneous determination of two forces, by means of the electrometer and clcctrodyoamometer re- spectively, and it is only the nitio of these forces which appears in ^ the ivsult, ^^P 77S.] Another method, in which these forces, instead of being ^^^ separately mciistirt^'d, are directly opposed to each other, was em- ■ ployed by the present writer. The ends of the great rcsistjince coil I are connected witli two i>arallel disks, one of which is moveable. B The same difference of potentials which sends the current through H the groat resistance, also causes an attraction between these diaka.! ■ At the nme time, an electric current which, in the actual experi- H ment, was distinct from the primary currents <>■ tent through two B coils, fastened, one to the Ijnck of the fixed disk, and Uio other to B the l>sck of the moveable didc. Tlie current Bows in oppgsit^ B directions through these coils, to that they repel one another. By B adjusting the distauoe of the two disks the attraction Is exactly ^^^ balanced by the rcpolsion, while at the same time another ohBenrer, ^^^^^^■B^HiM^^M^^i^rt^^H » SyiT METHODS OP THOMSOK ASD MAXWELL. 883 ipy BMftDS of a difTtTODtial f^Ivanomef«r with shunts, tlet^rmines the Tttlio of the primary to tho secondary cumnt. In this experiment tlio onlymcasiiromeut which most ho roferrod to a material irtandurcl in that of the g^reat resistance, which must be detcTmiDed in ahsoluto meiisnre hy comiiarison with tlic Olim. The other meaaurGmcnts are required only for the deti;rmiiiution of ratios, and may therefore be determined in tcmis of any arLiitrary unit. Thus tile ratio of the two forces is s ratio of eqnnlity. The ratio of the two currents is found by a comparison of r«i(iHt> ancea when there is no deflexion of the dtiferential galvanometer. The attractive foroe depends on tlie stguare of the ratio of the diameter of the disks to their distanee. The repulsive force depends on the ratio of the diameter of the coiU to their distance. The value of v is therefore expressed directly in terms of the rttsistance of the great coil, which is itself compai'ed with the Ohm. The value of c, as found by Tliomson's method, was 28.2 Ohms * ; by MaxwtU'g, 28.8 Ohmsf. III. Elecfrotlalic CujiacUy in EUetromagnelio Measure. 774.] The capacity of a condenser may he nsct-rtaiued in electro* aetic meaoure hy a compariiiOD of the electromotive force which liKes the charge, and the quantity of electricity in the current ' discharge. By means of a voltaic battery a current is maintained 'through a circuit containing a coil of gnuit resistance. The con- denser is oharged by putting its electrodes in contact with those of the resistance coil. The current through the coil is measured by tlie deflexion which it produces in a galvanometer. Let ^ bo this deflexioB, then the current is, by Art. 742, ys=-g-tiui0, 'here If is the horizontal eompooont of terrestrial magnetism, and I'C ia the principal constant of the gnlvanometer. If A is the resistimoe of the coil through which this current is to flow, the ditTereace of the potentials at the ends of the ■ Btjiori t4 Hn'tUh Anwialimi, 1 fiflO, p. 4S1. i PhU. TtaM., ISeS, p. 61S ; Mxl £qwn of BriMi Aitoeialion. 18S9, p. tS«. coupARisoH or oyiTS. and tlie charge of electricity produced in the condeaser, wluw cApacitj in electromagnetic m«iieiirD is C, will be Now let the electrodes of the condenser, und then those of Hw ^Iranometcr, be disL-ooDocU'd from the circutU, and let the magnet of the galvanomotcr he hroiight to rest at its position of equili- brium. Then let the electrodes of the condenser be connected witb those of the gnlvanomeb.>r. A triinsicnt current will flow tbron^fa the galvanometer, and will oause the ma|*nct to swinff to an ex- , trcmc deflexion 0. Tlien, by Art. 743, if the dtMhargv \a equal to I the chm-jje, }£ f We thus obtain as the value of the capacity of the coodi electromagnetic measure _ T 1 2m\\& ^ ^S ^~- ^^ ^ ■ • T It tan <t> The capacity of the condenser is thus dctennined in terms of th« following quantities '— T, the time of vibration of the magnet of the fralvanometer &om rest to rest. J?, the resistance of tho coil. 6, the (Txlreme limit of the ifwing produced by the discharge. 0, the conKtant deflexion due to the current through the coil R. This methoil was employed by Profeinor Flecmiufir Jcnhin in dflt«r- minin^ the ca[)acily of condenaent in electroma^ctic meaaorc*. I If c be the capacity of the same condenser in electroetatic meo- ^ sure, as determined by comparison with a condeu^er whose capaci^ can be calculated from \U geometrical data, T ism\0 T!ie quantity v may therefore be foond io this way. It depends I on the determination of R in electromagnetic measure, but as it " involves only the square root of i?, an error in this determination will not alTect the value of f so mnch as in the method of Arts. 772, 773. Inlrmiilcnt Current. 776.] If the wire of a battery-circait be broken at any point, and • fi9M ofStkiA Jbtotiatiim. ie«7. I WtPPH. S85 tli« broken eii^a connected with tti« oU'c^trodot of it coiiKlpn«cr, tin- I current will flow into the condonwer with a «treng:th which dimin- isliee as the diSerenoe of the pot«ntiuU of tbv vondcneer incrciucs, so t]iat when the condvnser luw roceivcd the fiiU chikrj>o com- spondin^ to the electromolivv force acting on the wire the current I oeBMfl ontirelr. If the electrodea of the condeiuvr btc now disconnected from the ends of the wire, and then again connc^et^Hl with them in the ■i reverse order, the condenifer will <Iiii«tiar^ itself thiwigh the wire, and will then become roe)iurg«d in tho opposite wnj-, eo that a traosietit current will flow through the wire, tlie total quantity of * which is equal to two chitrgCH of the condenser. By menns of a piece of inechnnijim (commonly called a Commu- tator, or KJppe) the operation of revcning the connexione of tb« condenser can he repeated at regular int«rvaU of time, each interval being equal to T. If thia interval is sufTiciently long to allow of the complete diw^hargo of the conden^r, the quantity of electricity tnin«nitt«d by the wire in each interval will Ik* 2 KC, where E k iho electromotive force, and C is the ciijnwity of the condenser. ■ If the magnet of a gj1vunometerinclud«<l in tlic circuit is loaded. BO as to swing eo slowly that a great many discharges of the eon- I denser occur in the time of one free vihrntion of the magnet, the Bnccession of discharges will act on the magnet like a steady current whoee strength is 2/.'C If the condenser is now removed, and a resistance coil substituted for it, and adjusted till the §teady current through the galvano- meter produces the same deflexion as the succession of discharges, and if i? is the resistance of the whole circuit when this is the case, m. ^^1^. Ml ■ yUe mt B motion to R R = T T id- P") We may thne compare the oondenser with its commutator in * motion to a wire of a certain electrical rcBistanec, and we may make use of the different methods of measuring rcBistance described iu Arte. 315 to 357 in order to determine this rcsisfanoo. 776.] For this purpose we may suhstitutv for any one of the wires in the method of the Differential Galvanometer, Art, 340, or in that of Wlieatstone's Bridge, Art. 347, a condenser with it* com- mutator. Let na suppose that in either case a sero deflexion of the VOL tt. C C COWPABISOS OP TTNnS. [777- 'a galvanomoter lias bcvn ol>t«tn(i], firsl with the cotvleoaer and com- tDutfttor, and then wiUi a ooil of rvtirtancc R^ io its place, tl tli« qtiautity —^ will bo measured by the renstaooe of tlio circui which thf coil /?, forma part, aiid which in oom))l(rt«d by the n- tnaiodcr nf thfi oonducting system including tho battery. Uma tli<! rcBistanw, If, which we have to ealcuinte, i* wjuaJ to 7?,, tint of the n-KiittAnce coil, together with H^, the re^isltuice of th« mainder of the system (including the battery), the extroinitjef tliv ri'»istiiDCe coil being taken as the electrode* of the oyKtvtn. In the cases of the diflercntinl gah-anumet«r ii»d Wbentcl Bridge it is not neoeaaary to make a second experiment by su tuting a resislanoe coil for the coudcnscr. The value of tlic Tv»i auoe required for this purpose may be found by calculation (nta the other known resistances in the system. Using the notation of Art. 347, and supposing the condenBTf Aod oommutntor substituted for the coaductor JC in WheatMt<me'ii Bridge, and the g»1 vniKuncti-r inserted in 0.^, and tltal the dtiflt'xion of the giitvuuomctcr is xcro, (lien we know that the resistance of coil, which placed in AC would give s zero deflexion, is The other part of the resistance, R^, is tliat of the sj'^tem of co' ductors AO, OCf A£, BC and OB. the points A and C being- con- sidered as the electrodes. Hence g __ ;i(o + a)(y4a) + w(y + a) + ya(g+a) * (c + a)(y+o)+j9(fl + a + y+a) $ In this exprettsiion a denot«s the internal rvviKtitnee of the bftttery and its connexions, the valne of whieh cannot be detormined with crrtuinly; but by m;)king it small comiwrcd with the other re«i»t> ances, tliis uncertainty will only uligbtly afTiwl the value of &^, The value of the capacity of the condeniier in eluclromagn measure is "p 777.] If the condenser Itas a lai^ capacity, nod tlie commutator is veiy rapid in its aotioo, tlie conden«er may not bo fully discbai;g^U The equation of the electric current dorii^ t^lfll 4 at each reversal, discharge is « + ^C^?+.jSC«0, (6i where Q b the charge, V the capacity of the cotulenBcr, It. tl WITH COIL. 387 reoistaDoe of the rest uf the system betn-e«n the vlt^ctrodos of the ooDdoiiser, and E the eleetrouaotivo force due to the cuniivxiouH ' with the battery. t Hence q = {Q„-i- BC)e'^-EC, (7) whore <?„ is the initial value of Q. If T i& the time during which eontaet is maintained during each dtBchorge, the quantity in each discharge is q = 2Ec i-t 1^ («) 1 + ^ «.c- By makings e and y in equation (4) large compared with ^, a, or tt, the time represented by Rfi may be made so email compared with r, that in calculating the value of the exponential expreseion we may use the volne of C in equation (5). We thus find where B, is the reeistance which must be substituted for the con- deoficr to produce an equivalent effect. R^ ia the resistsnce of the reat of the system. T is the interval between the beginning of n dischargu and the beginning of the next discharge, and r is the duration of contact for each discharge. We thus oUtaiu for the ] corrected \-aIue of (7 in electromagnetic measaro 1— fl *» '^ IV. Gmparitm of t/ie Efeetroatalie Capaeiiy <f a Cond<m<-r with lie EUetromagnftk Capacity of Sflf-inductitm of a Coil. |_ .J76.] If two points of a conducting circuit, between which the rc«iHtnace is S, are connected with the electrodw of a condenser whose capacity is C, then, when an electromotive force act« on the cironit, part of the current, instead of passing through the resisluncc R, will be employed in charging the condenwr. Tie carrent through R will tlierefore rise to its final value from iMtro in a I gradual manner. It appears from the athematical theory that the manner in which the current tJirough cea u Tig.t9. 3S8 cosTPABisos OP fjwra. It rises from ziyro to its fina] value is czprcMcd by a fononla U exactly tbe sunv IcinO u« tbat which vxpri-^se* tbo value of a cur* rvnt urfreU by a uoQKtunt electrroniutive (otou tbrougli the doU of aa el«ctromii{^nct. Henoe vie may plncc a condenser uad nn electn>- tnftgnetr oti two opposite mtMnbers of Wheatslone'ii Hrid^ in exh a wuy tUal. the current thraogh the galvanometer in alMniys xen, even lit Uu- iuptmit of making or Itrealcing tlic buttery circuit. Id the 6giire, let P, Q, R, $ be the resistaDces of the four mcin- bere of Wbeatetone'e Bridge respectively. Let a coil, wIiom- &» Hi- cieut of 8clf-induetion is L, be made part of the member AI/, wImv resistance is Q, and let the electrodes of a condeneer, whose capacity is C, be eonuected by pieces of small resistaoce witb tbe iMiinU F and Z. For tlio sake of simplicity, we shall assume that there in on current in the ^galvanometer G, the elec-trodee of which are con- nected to F and //, We have therefore to determine tbe condilioo that the potential at /' may be equal to tbat at II. It is only when we wish to estimate the degree of accuracy of the method that we require to calculate the current through tbe galvanometer wh^a this condition is not fulfilled. Let « bo the total quantity of electricity which haa pancd through the mcmlicr AF, and e that which has passed through FX at the time /, then x—: will be the charge of the condenser. Tlie electromotive force acting between the electrodes of tbe eonden«ci is, by Ohm's taw, R jr , so that if the capacity of the oondenscr at 0) Let y be the total quantity of electricity which has passed throng the member A/l, tbe eicctfomotive force from A toll must be wjtial to that from A to /', or (2) ^tie^^dt' ^ dt Since there is no current through the galvanometer, tbe quantify vhioh has patu.ed through ///must be abo^, and we find (3) Substituting in (2) the value of r, derived from (t), and com- paring with (3), we find as the condition of no current through Ibu galvanometer m CONDESSER COMBINED WITH COIL. 889 r The conilition of no 6nni cumeiit is, as in tlie ordinary form of I Vh^aUtone's Bridge, Qg _ $p_ /^\ The condition of no current at making and breaking the battory oimexioD ia /: •^=XC. (6) Here V. and AC arc th« timc-coiisUnts of the merabem Q and R respectively, and if, by varying Q or H, we can adjust the members of Wheatstone's Bridge till the galvanometer indiciit«M no iiirrent, cither at making and breaking the circuit, or when tliB current ia steady, then no know tliat the time-constant of the coil is equal to that of the condenser. The coefficient of wif- induction, /, ojin be determined in electro- magnetic mensiire front a compiirison with the coefficient of mutual induction of tiro drcait«, whose geometrical data are knowD (Art. 7oG). Tt is » (juantity of the dimensions of n line. The eajmcity of Ihtj eoi)den:«er can be determined in i-lcctro^latic mcftKUrc by comiiarison with a condenser whose geometrical duia are known (Art. 229), This quantity is also n length, e. The vioc- troinagnctic mcaoure of tiie capacity is c=4. (') I ■ Subatitnting this value in equation (C), we obtain for the value where e is the capacity of the condenser in electrostatic measure, »/. the coefficient of self-induction of the coil in electromagnetic meaiiure, and Q and II the resistances in electromagnetic mca«uru. ITie value of r, as determined by this method, depends on tho determination of the unit of resistance, as in the second method. Arts. 772, 773. K V. QmliinttfJoH of tie BUcfro^attc Capacity (f a C<md<nur teitk ike EUeffomagK^ie GipaciSy of Srlf-indtietitni of a CoU, 779.] Let C he the capacity of the condenser, the *urfaces of which are connected by a wire of resistance R, In thii: wire let the coila L and L' be inserted, and let L denote the xum of their ca- pacities of self-induction. The coil L' is bung by a bitiUr snspen- sioa, and consiste of two coils in vertical planes, between which 390 OOWPARlSOlf Of UKITS, [77^ pn^cfi a vi-rticnl axis which carries the tnagnet 3f, the axis of wbiek roTolvog in a horizootal [ilaae between the coils L'//. The coil I hoe a large coefficient of yclf-inductian, and is fisetl. The eik- pended coil L' is protected from th* currents of air caused by fbe rota- tion of thv nuLgitct by encloeing tb rotating yaii» in a hollow esse. Till,' motion of the mog-uet ctirrent« of induction in the ^-oil, and tilt-He are acted on by the so that the plane of ike si coil is deflected in the direction of the rotulion of the magnet, lirt us determine the Klrcngtli of tic induced currents, and the mitgnit of the detk'xioD of the 8wpei>«]< coil. Let X he the charge of electricity on the npper Btirface of the condenser C, then, if £ is the electro- motive force which produces this charge, we have, by the theory of the condenser, g — crjf. (j Hg. •}*. tic 51 We have also, by the theoiy of electric currentR, 1 («) where 1/ is the electromagoetic momttntum of the circnit I/, when the axis of tlie magnet is normal lo the plane of the coil, and in the angle between the axiei of the magnet and this normal. The equation to determine x is tlierefore aP at at If the coil is in a position of cquilihrinm, and if tlic rotation the magnet is uniform, the angular velocity being a, e = n(. (4) The exprowion for the current consists of two parte, one of wbidi is in<te|>cndei)t of the term on the right-hand of the equation, iiod diminislieM acoording to an exponcutial function of the time. The other, which may be called Hiv forced curroot, depends entirely on the term in 0, uud may he written COSDESSEE COMBINED WITH COIL. Sdi Finding llie values of A ami £ by »uWtilutiou in the wiUJitton (3), pe obtain JtCnco9d-(l-Cf.n^«a0 Tb« momtrnt of thv force with which the magnet acts on (he coil V, in wliicli the currvnt m it flowing, heinf* the reverse "f that KDg on the mu^oet Ibe coil bcini* by supposition fixt-d, i« 1-nahy Intpgrating this expression witli respect to ( for one revolution, kod dividing by the time, we find, for the mean value of 0, = -l («) If the coil has n considerable moment of inertia, its forofd vibra- tioaa will be viiyemiiU, and its mean deflexion will be proportionui 0. Lut i)|. i>j, D^ be the observitl deflexions corrcfrponding to nn- iikr velooitius n, , n^, Wj, of the mngTiet, then in gcnenil P~ = (^-C£ny + R'C\ (9) [where Pisa constant. Eliminating P and R from three equations of this form, wo find IC'L* |^K'-V)+J^(V-«.')+J^{V-VJ «»' ">*«.* »i (10) If «ig is «ueh thnt CLti^' = 1, the valne of -^ will be a minimum Ifor this vnliic of ii. The other vnlucs of a should be tnken, one Igrvater. und the other Ivss, than i«^. value «f Cf/, determined from this c^tmtion. is of the dimca- I of the Kqitare of a time. Let u« call it t', IF C, he tlie elect rusLitic measure of tho capaeity of the oon- r, and L„ the electromajrnetic measure of the self-induction of » coil, both C, and X„ are lines, and the product C,£. = ^C.L. = v'C^L„ = v't'; (H) tr= —s—i UHl (12) whew r* ifl the value of C*£*, determined by this experiment. Th« k 392 [7^ experiinetit here so^i^fibtd as n method of drlermmin^ r is of tlw aatntt luitur^ as odo dcecribed by Sir W. R. Grovp, Piil. H/^^ March 181)8, p. 184. S«e also rcmurks on tliul *.-xi>unmvQt, by ynteat writer, in the ntimbcr for M»j I8G8. « 'J VI. Elffctro$ia(ic MeaMrettttttl of ItetiifttHCf. (See Art. 385.) 780.] Lt't u condenser of capactly C be (lisctiarged t-hron^b a conductor of remittance R, then, if « is tlie diarge at aujr iostant, Hence » = Xf,e *-'. If, hy any method, we can make contact for a tihort time, whJvli i» accarat«ly known, so as to allow the current to flow throug'h conductor for the time I, then, if Eg and £, are tlie readings of . electrometer put in connexion with the condenaer before aod the operation, IiC(iog. S^- log. A',) = t. (3) If C is known in electrostatic meamirt! as a linear <)uantity, X may be fonnd from this equation in electrostatic meawirv an the reoiprooal of a velocity. If jK, i» the numerical value of the resiiiUnco as tlius detorniinod, aocl Ifm ^he numerical value of the resistance ia elect roma^etie measure, ff ^^ Since it is noceesary for this exporimcnt that S shoold be very great, and since H most be small in tho electroma^etic experi- ments of Arts. 7li3, Sic., the es|)criinei)t« must be made on e«|i*Tuti) conductors, and the rceistaneo of these uoodugtors compared by tho ordinary methods. CHAPTER XX. BLBCTBOMAONETIC TBEOBT 0? UOHT. ■ be at ^ftttrib 78).] Ik WvithI parts of this trentie? aa attempt lias beffn rand« explain «U>ctroniii^i'tic ph«nomeiia by means of mfichanicRl Ktion trntiMiiittAiI fr«m one tMnJy to another by meimii of n mL-tUiim oceupjing Uik ajiaw between tiwm. The UDdiilutory theory of liglit alHO aiisuincs the existence of a medium. Wo have now to Hhvw tliat the pro[ierttes of iJic clectrom noetic medium are identical with thoBe of the luminifiTOti* mt^diiim. I To fill all space with a new medium whenever any new f\te- nomenon is (o be eitplniiiiil ih by no mcitOK philosophical, but if Uie study of two diQeriMit branches cif ecienoo bax inde{>endently Buggeated the idea of a medium, and if tlie propcrtitw which nauat be attributed t-o the minliiim in order to iKVount for eleclro- etie phenomena are of the Hnmc kind ns tlioae which we ttribute to the luminiferous medium in order to account for the phenomena of liffht, the evidence for the physical existence of the medium will he considerobly stren^hened. But the properties of bodies are capable of quantitatirc meaflnro- meoti We tJmefora obtain the numerical value of oome property of the medium, sueh ae the velocity with which a disturbance is pro- pagated thron^h it, which can be catculnted from electroma^iictio expcrimentfi, and aleo observed directly in the ease of light. If it should be found that the velocity of propagation of electromagnetic dietorbancee is the same as the velocity of light, and this not only fn air, but in other transparent media, we shall have strong reasons for believing thiit light is an electromagnetic phenomeuou, and the uombinatioD of the optical \^ ith the electrical evidence will prodiic* a conviction of the reality of tJte medium similar to that which we obtain, in the case of other kinda of matter, &om the combined evidcBCtt of the Kcmn's, 894 EUCTBOUAGHETIC TUEOBY OF LIOIIT. [7 782.] When light is omiUed, a oerUin amouui of coerg; exitendod by the luminous body, and if lh« liglit im aUxorbcd another body, tliiit body heoom^a heated, ahewiD^ tiuit it hu ceived energy from without. Durio}; the interval of time afi«r light left the first body and before it reaebed the second, it have cxist«d as oner^* in tho intervening space. According to the theory of emiFsion, the tranMmieHion of on U effected by the actual tranHfercnec of ltght-cor}>nKCu1ca from luminoue to the illuminiited body, currying with tlivm tlieir ki: encrjjy, tfsctber with any other kind of energy of which they be the rt'cepfaclcs. Accoi'<lin<7 to the theory of undulation, there is a material medim which fills the space between the two bodies, and it is by the actiw of contif^oiis parts of this medium that the energy i« paaeed on. from one portion to the next, till it renches the illuminat«d body. The lumiuiferous medium is therefore, during the poMRgc of ligbt through it, a rcceiitacle of energy. In the uudulatory thtwr}', » dvveloi>ciI by Huygciis, Frcsncl, Yonng, Green, Sec., this euerg;' is suj)])OKcd to be partly potential and purtly kinetic. Tlie potential energy iK Hup[)u«ed to bo due to the distortion of the eltmcnlMj* portions of the medium. We must therefore Tiegaid the mediutn h ebisliv. The kinetic energy i$ jitippoMHl to be due to the vibratory motion of tlio nicilium. We mu«t therefore regard the medium a * having a 6nite density, ^M Id the theory of electricity and mtgnettmn adopt4>d in llii^^ treatise, two forms of energy are recognised, the e)eetruiitatie antl the electi'okinvtic (see ArU*. 630 and 636), and tlieae are suppotHul to bare their mat. not merely in the electrified or magnetized bodies, but in every part of the surrounding space, where electric or magnetic force is observed to act. llenoe oar theory agrees with the unilnlatory theory in assuming the existence of a medjiun which is capable of becoming a receptacle of two forms of energy*. 783] Let ns next determine the conditions of the -|>ro]ngMtion of ail electromagnetic disturbance through a uniform medium, which we shall suppose to be at re«t, thut is, to have no motion except th: which may bo involved in cloctromagiKtic distnrbftnce*. J * ' Far my own )>*rt. coiialdnriiii; lliK rHtilioli •>t k vicunin (o Om majpieUc fi sad lliQ KuUHral cltkmrtiir i>f iiiunistio phuianuak •xt(<rti>il to Ihv ina|MI, I un inclinad to tliv antiiiu tliitt ia tha truumiBiiMi of tho forca Ihera u asdi wi tetiaa, •xMfittt la the ma^rt. thui that cbn •ffocto a(« iuanl;r aHtxi i na sad itfmUaa ■> > dUtWNo. Stub sn action nmy bu a (nuutluD ^ Uie nlbcr ; W U U a»t M ■!! nnlikei* Ihul, IT iian ba ka Mthrr, It ilioiilil lixvn oUiw met tbu Ktiofilf ik* cuiiBjiataoe td. ndiMSuiu.'— PmmI>^'b I'^f^malal /ionircAct, 9»i6. fa jS^.] PBOPAOATION OF RLKCTROMAOSETIC DIPTl'RBANCES. 395 Lot C be the specific conductivity of the medium, K its speciGe [capRcily for cloctrostjitic induction, and n its mai^netic 'pcnne^ ibilily.' To obtain the general <>>{iiut Ioiik of cIcotromafTDctic disturbance, Ifte shall exprcM the true current G in terms of the rector potential fil and the electric potential 4*. The true current 6 is made up of the coniltiction current j^ and the variation of the electric displacement Ti, and since both of these I depend on the electromotive force @, we find, as in Art. 611, But since there is no motion of the medium, we may express the :electromotive force, as in Art. 599, I g=_a-U1'. (2) H«,o. 6 = -(c + ±;.|)(f+v*). ■ (3) I But we may determine a rehition between S and SI in a diOVrent way, as is eIicwd in Art. 616, the equations (4) of which may W twritten 4ir*iS = 7»?I+ W, (4) . . iiF dG dii Combining equations (3) and (4), we obtain ^(4«C+A'^)(2^ + V*) + V*a + V/=0, (6) rhioh we may express in the form of three equations as follows— ,.(4 (0 dt^^dl ^ ds- Ihwp arc the g<'ncral M{n»tions of elwtromagnetic di:sturbaiieoa. If we difTerentiate these equations with reiipect to jt, y, and t recpeetivcly, and aM, wn ohlain H^ If tltc medium ia a non-conduct<Mr, C = 0, and 7'^, whieh is ^'proportional to the volnmo-denjtity of free electricity, b indejiendent gf t. Itence J must be a linear function of /, or a constant, or zero, and we may therefore leave 4 and '\ out of account in concddering eiiodic disturbances. 396 BLKCTROMAGSETIO THEORT Of LIGHT. {.7h\ (»)' Propanatioit ^ Unduiatioiu in a Non'<:imdtielittg Medium. 7&4.] Id this case, (? =: 0, And the cxiujitions become The etiuatioDs in thia form ar« similar (o those of th« motiou of an elastic solid, and when the initial conditions an* {^iven, tbp, solution cao be exprtwed in a form gircn by Potsson *, uud appliedl by Stokes to the Tlieory of Diffraction t- ' (10) Let us write r= ^rz dF If tb« vnluea of f, O, ff, and of j- . .,- t T7 are giT«a at erVjl point of space at the epoch (/ = 0), then wo can determine theii] TaluM nt nny eitV>sc[|ucnt time, /, as folloWB. Lot be the point for which wo wish to detrnninc the valae of /"lit the time /. With as centre, and with radius ft, dt-wrilie a sphere. Find the initial value of /'at every point of tlie oplu-riral surfaa.', and take the Huan, F, of ail thofo values. Find aivo tbo^j initial value:* »f rff the mean of these values be at cvcrj' point of the spherical surfiico, and lebl r« Then the vnlue of /'at the point 0, at (he timo ', ix '=i(K)^'f- 4i dt Similarly = s(^')-'f- ^=^(^0^-4'- 785,] It appears, t]>eTief<>re, that the condition of thing* at tb€ point at any instant de{iends on the c<»idition of thing* at distance Vt and at an interval of time t ptevioosly, so that anj disturhance is propagated through the medium with the velot-ity Let us suppose that when t is zero the quantities ?I and 31 ■ MtM. dr TArtot.. Una. Ul, p. IM. ■t- UamhrUtgt rnmtattiotu, vol. ix, {>. 10 il830>. nmk rs?-] ■nttocnr of i-TGrrT, esce|vt tritbin a cerUin Rpace S. Then tKeir vu1uk« at at tlie time ' will be usero, unless the spherical surfiK'e duHi'tilied aliout lO as centre with Tsdius Fl lies in whole or tn part withia the I ^. If is out«ide the space S there will be no disturbance ■t until n becomes equal to the shortest distanee from O to the B|)a«e S. The disturbance at will then beg'in, and will go on till \Fl is equal to the {rreatest distance from to any part of S. The Idisttirbancc at will then coase for ever. 786,] The qtiantitr F, in Art. 784, which expresses the velocity of propa^tion of electromagnetic disturbances in a non-conducting Kxnediuui is, by equation (9), e<iual to ■/Kl If the medium b air, and if we adopt the electrostatic system it meaanrcmeat, K = I and fi = — , so that T = », or the velocity of propagation is numerically equal to tlic nninber of electrostatic ■unite of electricity in one electromagnetic unit. If we adopt the eU-ctromagnetic system, A'= -y and (4= I, so that tlie equation r= II ie still true. On the theory that light is an electromagnetic disturbance, pro- pagated in the same medium through which other electromaffnetio action* lire transmitted, F must be the velocity of light, a quantity Illio vuluv of which haa been estimated by several methods. On the other haud, v is the number uf electrostatic unita of electricity in 0D« electionuif^ctic unit, and the metliiidG of determining this quantity bave been described in the lust cha[>ter. They are quite inde- liendcnt of tJie methods ol' iindiug the velocity of light. Hence the agrocmont or disagreement of the values of Tand of v furnishes a te«t of the electromagnetic theory of light. 787.] In the following table, the principal results of direct observation of the velocity of light, either through the air or through the planetary spaces, are compared with the principal results of the comparison of the electric units :— Vslvdtj oT Li|[)it (iiilitrw par Hoomt], Kzeau 314000000 Aberration, &c., and ) Sun's Parallax J'" ^'oDcault 298S6O00O . 308000000 FUUo at Eloctric Vnitn. Weber 310740000 •Maxwell... 288000000 •Thomson... 282000000 [Tic nperimenU of (b« ComiDiUv* Of Um Dril^h AvMUtion fur thq Jcttr- (ID of tlw uiut of raiMtanM ia abtolut* UMuun ban) natndj bMO irpeatoJ 89e KIECTROMAOITETTC THEORT 07 LtOHT. It is maniPcgt tlmt the velocity of light and the ratio of (Jw* nri> <|uim(itii-« of tho samo order of magnitude. Neither of i can 1)0 »iul ti be detcrtniacd us yet with such a de^ee of ac M to CDRhle tis to assert that the one is greater or Ices than other. It is to Iw hoped that, by further ex{)criinents the relttk between the mitgDiiudcs of the two quaiititici) niuy bo more CQnt«ly deLermininl. In th<> meantime our theory, which assertit that thcso two qoaa* titles are equal, and ossig^DH a physical reason for thin cjiinh'ty, ii certainly not coutnulicled by the corapurison of these results rodi aa they are. 788.] In other media than air, the velocity f i» inversely pro- portional to the (iquarc root of the product of the diflcetric and the ma^etic tnduotive eapaciliex. According to tlie undulutor^' theoij*, the velocity of light in difTcrent media is inversely proportiooa] to their indioee of refraction. There are no transparent media for which the magnetic cn|Micilj' diSers fi'om that of air more than by a very amall fraction. Uence the principal part of the dilferenee between these media muat depend on their dielectric capacity. According to our theory, therefore, the dielectric capacity of a transparent medium should be equal to the square of its index of refraction. But the value of the index of refraction is different for light of different kinds, being greater for light of more rapid Tibi»tioi)& We must therefore select the index of refraotion which correeponda to waves of the longest periods, because theae are the only wavea whose motion can be compared with the slow processes by which we determine the capacity of the dielectric. 789.] Tlie ooly dielectric of which the capacity has been hitherto determined with sufficient accuracy is paraffin, for which in thu solid form MM. Gibson and Barclay found * K= 1.975. (12) Dr. Gladstone has found the following values of the index of refraction of melted paraffin, sp. g. 0-779, for the lines A, D and i?: — T«inpwMuM 54 '€ A 1.4306 D 1.4357 B 1. 4499 a7'C 1.4294 1.4343 1.4493 ti Dr. KchiuM' Kt til" CkVMdliA l^ihunl/nj. wMk tb* , . , , , ^t" eont, (mailer thui it *<■ iutuiiifed Uihr. Ttw afbet IBpOk tb iattt of Uie t^eeuiu nnlu ■■ fi*ca bj M«i*rdU uul Tbanuna woalil mini's tibta lij t»t pe> cnnt.] • mi. Tnuu..lS7l, p. SJi. 1 ?90.] 300 b^ I which I fiii^ ttiat iim index of ttfnwtioii for wave* of iiirmiU ig^h w<iul<l bo ubotit t.432. e square root of K is l.*05. c (linVrenot bct.wittin thoMO nunibeni in greutiiM- thttn can be ae- >uated for by errors of observation, and shew^ Hint our thoorit* of e structure of bodies must be much improved before we .can deduce their optical from their electrical properties. At the same tim«, I think that the agreemeDt of the numbers is such that if no greater discrepancy were found between the numbers derivetl from ihe optical and the electrical properties of a considerable nimiiier of lubstauces, we should be warranted in coocludinjf that the square root of K, though it may not \te the complete eipression for th« iodex of refraction, is at least the most important terra in it *. I Pittne ffaoet. 790.] Let us DOW confine our attention to plane waves, the front of which we ehatl suppose normal to the axis of s. All the quan- ties, the variation of which constitute)) such waves, arc functions of s and i only, atid aro independent of x aud y. Hence the equa- tions of ma^K-tic induction, (A), Art. 391, are reduced to dO '=-di' dF -J-. 6=jr' '^ = 0. da (13) br the mat^nctic dirturbanco is in the plane of the wave. This agrees with what wc know of that difiturbanoe which constitutes Putting' po, fi,S and ny for a, (i and e respectively, the eqnattona ;l«Gtnc currents. Art. 607, become rfS d*F ^ iv^^ = --^=- — dz da i^^p= _=__, (») Hence the electric disturbance is also in Ihe plane of the wave, and if the magnetic disturbance is confined to one direction, ray that of * pn ■ iMHr md to the Rojal Booitty on Jane li, 18TT, Dr. J. UepUnam givM i« ronlti or expcrimonla dihIo for tha purpoM of detPnniDtng tile ipoelfic ItiiiiK^tiro itir* otruioot kind* of sfi^. TticM renilM do not vurlfy tli« ibmndli-al ouii- oliMMu urliti) hi III Uio text, liiii valii* of K Mtix in •oir'h ram in "lora at Uial of Uio aqiiani of tlm r»rnul!v<> !i»li». In * mlmwiurat jnyn to thu KnykI i^-ialy, Ra4 on Jkn €. tfisl, Dr. Hopkimixi find* tiuit. if tim dmolc the Jn-jci of Tefr.iction bv «a«H of iiifinhv Unc^, [lien K • fi'w fbr hydnoarLoiiL but for uJuul and <cDvUl>la dil<K>,.<«i.) 400 tiTOHT. M r.tiie electric tlisturbanoc la confiaed to the perpcmliculnr dii or that of jr. Dut vre may citlculitte tho electric disturbaooe in auotber for ify, j/, i UTC the coiiipoDCnt« of electric displacement in a couilactin^ mt^liiun, d/ df M 1£ P,Q, R an) tliv compooentit of tli« electromotive force, A' K ^ . K •^=n^' '=4^«- ^=o^= (M and Kince there is no moUoD of the medium, equations (B), Art. 59^1 become ?=- dr O- ^^ R=- du Uenoe K d*G CMuparin^ these r»)ii«e with those ^ven in eqnatioD (11), wv Gwl d^F .. d'r dz^^^" de ' ^G .. d»G Q = K,i dfi" The Grst and fiec^ind of these equations are the equations of pagatioQ of a plane nave, aod their wlc tiou is of the welUkuown form -^ ^- (20) The solution of tlic third equation is ■ Jl^A + Bt, (21) where A aud Ji are functions of ;, // to therefore either couatant or varies directly with the lime. In neither case ean ij (die part in the proportion of waree. 791.] It appears frotn this that the directions, both of the ma^ettc und tlie cloi'tric ditAurlMWOH, lin in the plane u: lhi> wave. The malhvmaticnl fonn of diKtiirhattcc therefore, Of^rrnF wi(li thai the disturbance which concliluleii light, being transverse to the direction of propagation* Fig. GS. BNEBOT kVD STBE89 OF BADIATION. If we suppose G = 0, the diaturbauoe will correspond to a plniic- slarizi'd ray of light. The magnetic force is in thia case par&Ilel to the axia of y aiid 1 IP vaa\ to ~ -^, aiid the «lcclromotivc forco in parulU'I to the axis of ^aod equal to — -^. The magnetic force is therefore in a plane Ipcrpvitdiciilur to Lhitt which contains the electric force. The value:* of the ina^Detic force and of the electromotive force 1 itt a ^iveti inhalant at different points of the ray are represented in rig. 65, for the case of a simple harmonic disturbance in one plane. This corresponds to a ray of plane- polarized light, but whether the plane of polarization corresponds to tho plane of the magnetic Lirbance, or to the plane of the ck-ctric dieturbaoce, remaing to kteen. See Art. 797. Energy and Slreta of PaJialion. 79~2.] The electrostatic energy per unit of volume at any point of ie nave in a nou-oonducting mediuu] is a (22) lie cloctrokinclic energy ut the same point is (23) 8;r inp, ivfi dx virtue of equation (20) these two expressions are eqaal for a single wave, so that at every point of the wave the intrinsic energy ,of the medium is half electrostatic and half electrokinetic. Lct/> be the value of either of these quantities, that i«, cither the Blectrostatic or the electrokinetic energy per nnit of volunie, then, ^in virtne of the elec-tro^tatic etat« of the medium, thert- \<s a U-n«ioD whoMi magnitude is p, in a direction parallel to x, combined with a priwsnre, also equal to/), pamllcl to j/ and :. See Art. lOr. iln virtue of the electrokinetic state of the medium there is a cnKJoD equal to /i in a direction parallel to j', combined with a treffiure e(]uiil to p iu directions parallel to x and .*, See Art. 643. Henoc the combined effect of tlie electroKtalio and the clcctro- [inolic rtlrecws is a pramre equal to 2p in the diixction of tho iropAgation of the wave. Now 1p also expresses tho whole enCi^ R unit of volume. HencL- in a medium iu which waves are propugaletl tlieie ia a VOL. 11. D d lOirenC THEORT OF uonT. pressure in Uie direction nonnal to thv warcKj adiI untni equal to the enerj:)- in nnit of volume. 793.] Thas, if in strong eunligbt the energy of the light «hi«4 fftlle on one eqaare foot ie 83.4 foot pouiidit per second, the mtn ener^' in one cubic foot of sunliglit is nhoot 0.00000008^2 of a fix* pound, and the mean prcsauro on a Kiuare foot is 0.0fl0oo00d83af > ixinnd weight. A flat bodjr exposed to sunlight would cx|tenMtt this pressure on its illnminalwi side only, and would th<'refore tt lepcUcd from the side on wliit-h the light falls. It is prol>ablc thil a mnch gn-nt^-r energy of ntdiiition might W obtained by mems at the oonct-atrat^-d rays of tlw vlectric lamp. Such rays falling on a thiu oiflaliicdisk, delicately suspended in a Tacuum, might |>erhai>t produne an obstTvabU; mechanical eOect. When a diatarbane* of any kind eonniits <^ terms involving aiiies or cMioes of angtn which vary with the time, the maximum energy is double of Um! mnui energj'. Hence, if /* is the mazimnm electromotive force and ii the uaximnnt in^>n«tic force which are calkKl into pby during the propagation of light. — P^ = -— 3* = moan energj" in unit of voluniei, 8 ff Hit (RP With Pouillft's data for the energy of siiulight, as quoted by TliomsoQ, Tmat, H.S.E,, 1854, this givej in electromagnetic m' ture P = 60000000, or aboat COO Daniell's cells per rodtre ; j3 = 0.193, or rather more than a tenth of the hotizoutal netio force in Britain. ru 111 PropaffaiioH vf a PlatK Ware in a CiytfalliuJ ifediam. 7M.] To calculating, from data ruriiish<;d by ordinary electr magnetic esperimonts, the electrical phenomena nhieh would result from periodic distDrlumoes, millionii of millions of which occur io second, we have already put our theory to a vert' severe test, eve when the medium is supposed to be air or vacuum. Itut if attempt to extend our theory to the case of dense media, we U involved n<tt only in oil thi' onlinsTT difficulties of molecular tbcorii but in the deeper mystery of the relation of the molecuU-s to elcctromagnctio medium. To cviidr tlii'si* diflicidlics, wc fhall assume that in cerl^iin mtilij the specific cnpucity for elvclTostntic induclion is ditrenml in dif ferent dlrecttouB, or in otlier words, the ekotric diephiccnieat, it BOCBLE REFHACTION. ■d of bein^ in the same direction as the; electromotive force, un<l proportional to it, is related to it hy a system of linear c-(|UiitioDS limilar to those given in Art. 297. Il may be shewn, tw in Lrt. 436, that the systi'm of «oefncients must he symmetrical, so Itnt, by » projier choioe of axes, the eijuatioiis become /=T7^-^- 5 = Aa;«2, i = ~K,s, (1) pwbere ^|, K^, and Kj »re the principal inductive capacities of the medium. The equations of propagation of disturbances are therefore d^J^ ^ d^G tfiG d*G dx' dy* dxdjf dUI _ ,d^F d^*. = ^■f' ( rf^ - d^J d.'djs d'F = ^^^Kdfi~A.dV' dydx djrdy ■' ^dfi Jydi dzdx dylf: ~ "*" W/* d£dt> i (2) 795.] If /, m, n are the direction -cosines of the normal to the I wuve-front, and V the velocity of the wave, and if h^my-^-nz— Vl = w, (3) knd if w« write f", 0", //", +" lor th« second different ial ooeffi. [ei«nt« of f\ Q, II, ♦ nspcetively with resjiect to v>, and put ^iM = ::5' ^j'* = ^. A*)fi = 3' I wh«rc a, h, e are the three principal velocities of propagation, the ' equalionc become («a+„E_ ^) r'-i»icr-nm"~ r*"^ = o, n -ImF" -i (»» + ;»- ~) ff'-mnir- f*^ - = 0, -«//"'-«nG"+(/»+M»-^)fl''-r*"^ = 0. 796.] If wc write 1^ «* •« («) F"-*^ ==£'. (») (7) ' wc obtain from these ecjuatiouH rV{VF''-i^') = 0, rv(rji"-n¥') Uenee, eitb«r Fss 0, in wbicb caee Uiu w«ve is not propagated ut Dda ii 404 ELBCTBOSIAONBTIC TRRORY OF LrOHT. [79J all ; or, V= 0, which leads to ILc cciuntion for f g-ivon hy FresofJ ; or the quantiUn within briirkcU vaninh, in whioh caiw th« vector whose componcntv arc t", O", H" U nnrnial to the wavo-fronl and proportioniil (o th« electric volume-density. Since the medium is a non -conductor, the electric density at any given point ia con^toiit, and tlicn^forc the disturlmnee indicated by these equations i» not periodic, and cannot constitute a ware. We may therefore con«d«r •V'= in the investigation of the wave. 797.] The velocity of the propa^tion of th« wave is therefi complett'Iy determined from the equation P = 0, op I* M» I = 0. (8J There arc therefore two, and only two, values of f^ correspoodio, to n {fivcn direption of wave-front. If A, fi, r are the direction^oosines of the electric corrent wl component* are «, v, «p, then ;A + m;t + iif = O; (1 or the carrcnt is in the plane of the wave-front, and its dinctioi in the WKvc-front is dfrtcrmined by tlie equation These equations are identical with those given hy FresncI if we dftltne the plane of polarization as a plane tlirough the ray per- pUHJicular to the plane of the electric disturhaitce. According^ to this electromagnetic theory of double refraction the wove of normal diKturf>anoe, nhielt conctitutex one of the chief (lifficnlties of the ordinary theory, do<M not exist, and no new aaaiimption is required in order to account for tite fact that a ray polarised in a principal plane of the crystal is refracted in the ordinary manner*. Refatitm ielwcm S?«irie ConducticUf aitd Opaeif)/. 7^.] If the medium, irutead of being a perfect insulator, ia conductor whoso conductivity per unit of volame is C. the di turbonce will consist not only of electric displocemcnte but of cnrreat* orcondnclton, in which electric energy is transfoTmod into heat, to that the unduliition is abfiorb^d by the niedinm. * S«t SlolMa' ' Itcpoet on Double RrfbaUen.' Brii. J^te. Btrot*. ISOi, p. US. J ■th SOO.] COKDUCTIVITr ASD OPACITY. 405 ir the distarbaBc« is expressed bj- it circular function, we nuy write /?= «-"«.«(«(- J.-), (1) for this n-ilt Katisfy ttie ei|iuit)OD d^f „d^F , „dF Bridcd y"-/)* = p.Kn\ (3j [ud 2jt7 = 4irjxC7«. (4) The Telocity of propagation is and the coefficient of abaorptiOD is p=2ii^Cy. (G) Let R be the resistiince, in elcctromagnolic measure, of a plate whose length \b t, bresidth 6, and thickauss ;, The proportion of the incident light wliieh vrill be tniusniitted by tlii« i)Ule will be 799.] Mo»t transparent solid bodies are good insulators, and all ^ood comluc'lors are rery opaque. There are, however, many ex- cq>t!ofU( to the Uw that the o]>acity of a body is the greater, the grcat^tr ilti conductivity. Kleotrolytes allow an electric current to pass, aud yet many of them are transparent^ We may supjmse, however, that in the oase of the rapidly alternating forces which come into play during the propagation of light, the electromotive force acts for so short a time in one direction that it is unable to eOect a complete sejiaration between the combined molecules. When, during the other half of the vibration, the electromotive force acts in the oppoiite direction it simply reverHca what it did during the first half. There is thus no trne conduction through the electrolj-te, no loss of electric energy, and consequent!}' no absorption of Hgbt. SOO.] Gold, silver, and plitttinmi are good conductors, and yety hen formed into very thin ptuten, they attow light to pass through th«ni. From experiments which I have made on a pii-ce of gold leaf, the icsixtanee of which was deU^rmiiuHl by Mr. Hockin, it appeuv that its tranitpareQcy is very much greater than is cou- nrtcDt with our theory, unless we suppose that there is less loss 406 ELECTROMAGNETIC THEORY OF UaRTT. of Anergy when the dectron)otiv« forces are reversed for every seoi- vibretioQ of lig'ht than when they aot for sensible timeis, u bi osr ordinary experiments. 801.] Let ua nest consider the aise of a nie<Iium in which lh< conductivity is Inr^ in proporfion to the iniluctive capacity. In this case we may Iottv<- out tht; term involvinjj K in the eqii»- tions of Art. "83, nrnl they tlion hecoine <fF Each of these equntions is of the seme form M the equation of diffusion of heat frivcn in Fourier's Traifede Oaietir. 803.] Takini; the firet as an example, the eomponent /* of vector- potential n-ill vary according to time and podlion in the skta way na the tnnpcrahire of u homoireneoiis solid varies according to time nnd position, the initial nod tho sarface conditions beinfj^ made to corresptrnd in the two csokx, and the quantity 4xiiC bein: numerieully equul to the reciprocal of the thermomctric corulitctivit of the inihftliincc, that is to say, the numitr of unit* of r-ofnmt tht tuhitance teiUi would (tf^ ieaUd one degree iy tie heal tchu-h /Mittt through a ittiit eiihe of ike tuLifanee, Iteo ofipotitefaeet o/" which differ bjf one degree of temperature, wiUg Ihs other facet are imfiemeable to heal*. The diiferent problems in thermal conduction, of which Fou hnx given the solution, miiy be tnuififormed into problem* in diifuiiion of electromngnet ic quantities, remembering that /', O, ff are the componeiila of a vector, whcreu« the tcmpemture, in Fouricr'i problem, is a wahir (juantity. Let UK t«ke one of the case« of which Fourier has given a com- plete volution t, that of an infinito mMlium, the initial state of which is given. I It fc/o iricifl 1 [ nf Btat, p. ISS Snrt adkloD, p. 2ii tmrlli »JUkn. I An. asi. Tlin •<|UMi«) wbiMdvUnBiMa 111* (oNpontanh ' r a tima ( In t«niui Dr/(a, 0. y>, tik* isitial tsmpcmtaMafeJ • 8«a UmwiOI'* Theory «/ Btat, p. ISS Snit adkloD, p. aOA tmrlli »aUkn. 1- TroiU it. la CAdl'ur. An. 881. ~ . - *, M ■ |">fut <j-, y, i) alWr i tbc pgitti (a. «, 7), !■ wbtn i !• Uie iharawm*tria MnlocUTity. '/i'.B.yX !04.] ^ESTABLISH UKKT OF THE BrSTlSCTIOS OF rORCR. 407 The stslo of any point of the meilium at th« timo i it fonod by takiD" the average of the state of every part of tlii> mediant, ifche wci|^ht assigned to each part in taking the average being I where r is the distance of that part front thepcHnteonsidered. Tbia average, in the case of vector-quantities, ia niost ooiivenicaUy taken iby considering each component of the vector separatt^ly. 803.] We have to remark in the first plaoe. tliat in thi* problem the thermal conductivity of Fourier's medium is to be taken in- Tetsely proportional to the electric conductivity of our n»ediuitt, 00 that the time rL^plirvd in order to reach an a.iiii^'iteil vta^ in the procees of diffusion is greater the higher the electric conduct- ivi^. This statement will not appear paradoxical if we remi'mhcr the result of Art 655, that a medium of infinite condiietivity fi)rmn ^a complete barrier to the process of diffusion of ma^etic force. H In the next place, the time re<iuisit'> for the production of an Vsasigned stage in the process of diffusion is proportional to the sijuara Bof the linear dimensions of llio system. B There is no determinate velocity which can he delincd as the velocity of diffusion. If wo attempt to meusurc this velocity by ascertaining the time requisite fur the production of a given amount of disturbance at a given distance from the origin of disturbance, wc find that the Miiiillcr llie Kelcetcd value of the disturlwince the greater the velocity will apiK-ar to Ihs for however great the distance, and however small bhu limi;, the value of the diaturbance will differ I mathematically from xoro. This peenliarily of diffusion disfinpruishes it from wave-propaga- tion, which tukcx ptace with a definite velocity. No disturbance take* place at a given point till the wave reaches that point, aud wlien the wave Iiasi passed, the disturbance oeasca for over. 801.1 Let tis now investigate the process which takes place when tan electric current begins and continues to flow through a linear circuit, the medium surrounding the circuit Wiug of finite electric DOlulootix'ity. (Com|>nro with Art. OGO.) When the current begins, its timt effect i» to produce n current of induction in the parts of the mt-diuni close to the wire. The direction of this current in opposite to that of tJie original current, and in the first instant iitt total quantity is equal to that of the original current, so that the electromagnetic effect on more distant ,rU of the medium is initially aero, and only rises to its final L ELECTROIIAOSETIC THSORY OF UOHT. vilite sm the induction-current die^ away on accoant of tbe el«etra reHiiitance of the muJinin. But aa th« ii)i]»c1 lon-cnrrent close to the win dies away, a tie* induction-citrrent in ^nerated in the medium heyond, so that the •pace oocupiod by the induotion-current is continually becoming wider, while iU intensity is continually diminUhiR]*-. T\m diiliuion and decay of the induction-cnrrcot is a plirao- menon precisely anatcg^ous to the difTusion of heat from a part of the medium initially hotter or colder than the n^t. We most remember, however, that since the current is a viTtor qoaulity, and since in a circuit the current is in opjMUto directions at op- posite points of the circuit, we must, in <4tlcuUling any ^iren com* poDcnt of {he iiiduction-current, compare the problem n-itb oae in which equal quantities of Heut and of cold are diffusod bam neighbourin'^ places, in which rase the effect on diiitant points viJI be of a smaller order of ma^iitude. 805.] If the current in the linear eircnit is maintained oonstaat, the induction current*, whieli depend on the initial chanf^ of state, will g:radua]ly be diffused and die away, leaving tbe medium in its perninnent state, which ist auato^us to the permanent state of tbe flow of heat. In thiti statv we have throughout the medium, except at the part occupied by the cireoit, in which vt/-^4,^ rtO=<„r. I (3) Tlicsi! cquntions are snffieient to delOTmine the vatnes t>{ P.G.M throughout the medium. They indicatr that Ihi-rc arc no cunvots exoej)t in tJio circuit, and that the mngnctic force* nre simply those due to the current in the eircnit aecor<lin^ l« the onliniiry theory. TTic rapidity with which this permanent nUiU- i» csdihlishcd is go ffrcat tliat it oould not be measured by our ex|«erimenl«l methods, exwpt perhapii in the case of a very brge mas* of a higlily con- diicling medium nuoh as copper, NoTK — In a paper puldishcd in PogrsendorlTs Annals, June 1867. M. Lorenr. luw dedooed from Kirebhoff's equation of cli-etrir cur- TvnU (I'offg. ./«»,cii. 185G), by the addition of certain tcnu* which do not nflWt any experimental rcstilt, a new set of e(]uati<ins, tiMli- cating that the distribution of force in the electromagnetic 6cld may be Coticeirml as ari»ing from tbe mutual action of contignoua < >5] L0BKSZ8 THEORY. 409 jplements, and that waves, consietiDj* of transverse electric carnmt*. ay be propagated, mth a velocity eomparaWe to that of light, in ^non-eondtictini* mcdiii. He tbcrcforo rc^rds the digtiirbanoe wlitcb ^ COD eti tilt es It^ht ns identical with these electric oiirrents. and he [ shews that condiioting media must bo opaque to such radiations. ThcHi' conehisions are cimilar ffl those of this ohnpter. thoiif^h Fobtaincd hy an entirely diflVTeiit tncthfxi. The theory given in this etiapter wns fir«t published in tbc PMt. Trant, for 1803. CHAPTER XXI. MAOKETIC ACTIOIf OV LIOHT. 806.] The most important step in establishing a rclatioii between electric snd magnetic pb«nDmet)a unci thoBO of liffht must be th* discovery of some instance in which the one Get of phcnomeoft » afTvctfid by the other. In the search for such phi>nomenn wo must be guiJctl by any knowledge wc may bav« uliyjuly obtainetl with r«N[ioct to the mathciDaticiLl or geomi-tricul form of th« qoantitie* which wp wish to compare. Tliu», if we ftndciiTour, as Mr*. Sonutr- ville did, to magnctixe « needle by nKonit of light, wc must re- member tliuL tlie didinction 1>etween magnetic north and itoutJi it ft mere matter of direction, and would be at onee rereraed if reverse certain eonventiona about the use of matlieiuatical eigaa. There is Dothiug in magnetism annlof^us to those phenomena chK^rulysis which enable us to diatiiiguieh positive Irom negative rloctricily, by obscrvins that oxygfin appears at one pole of a cell and liydnigi-n ut the other. Hence nc must itot cxpoc-t that if wo make light fall on on« end of a needle, that end will become a pole of a ixTtain naniv, for thu two poles do not differ as lit;ht docs from dnrknecs. "Wo might expect a bctt«r result if we cauMil circubrly polariz<MLi light to fall on the needle, right-bandit light falling on om^ end and Ieft<lianded on the other, for in some reepecte these kinds of light may be said to be related to each other in the same way as the poles of a magnet. The analogy, however, is faulty even here, for the two rays when combined do not neutralize each other, but produce a plane polarized ray. Famday, who was ac^iuainted with the method of studying thit strains produced in transparent solids by means of polarized light, made many cxporimenta in hopes of detecting some action on polar- i£<-d light while passing through a medium in which electrolytic conduction or dielectric induction exists *. He was not, iiowever, ■ StftHtrntaf Stmrdim^kdlH smI SSI6-2nO. I 1 FARADAY 3 DISCOVERY. 411 (o d«tect an}- adtoD of this kind, thoug^h the experimenU were nged ia the way b«st adapted to di§oorer effects of tension, !« elei-tric force orciirrent being at right anglofl to the direction the ray, and at an angle of forty-five degrees to the plane of polarization- Farsdar varied these experiment* in many waya with- out discovering any action on light due to eWtrolytic ourrenta OP to static (.'leclrie induction. I! lie succeeded, however, in eetabliahin;* a relation between light ^od mignetisra. and the experiments by which he did so »re <1e* ■scribed in tlie nineteenth eeriee of hifi Experimentai Bc*Mr<^Jir». Wo •hall tak« I^raday'a dieooTery as onr etarting-point for further investi •Ration into the nature bf ma^etiem, and we ehatl tborcfon) I describe the phenomenon which he obeervcd. 807.] A niy of plane-imlarizod tight ie transrDitt4.'tI through a traiivjinrcnt diumiignctic medium, and thv plane of its polarizjition, wIh'H it emerge* from the medium, is ageertain^'d hy obftcrving the position of an unuljRcr when it cuts off the my. A magnetic force is thi-n made to at-t to that the direction of the force within the tranHjiar^'iit medinm coin<-id«! with the direction of the ray. The liglit at once rcapt>o(int, but if the auiilysiT i» turned round through ■ certain angle, the h'ght is ngnin cut off. This iihen^ that the eflVet of the magnetic force is to turn the plane of jwlarizal ion, round the direction of the ray as an axin, through a ei'rtain angle, im>a«ur('d hy tlte angle through which the analyser must be turned in order to cut off the light. 80H.1 The angle through which the plane of polarization is turned i« projiortional— (1) To the diftance which the ray travels within the medium. Hence the plane of polarization change* continuously from ita posi- tion at incidenne to ita poaitJon at emergence. (2) To the intensity of tbe resolved i<art of the mngnetio force in tlie direetion of the ray. (3) The amount of the rotation dejwnda on the onturo of the medium. No rotaljoa Itaa yet been observed when the medium is air or auy other gaa. These tiiree statements are included in the more general onv, that the angular rotation is nnmerically equal to the amount by which the magnetic potential increases, from the point at which the ray enters the medium to that at which it leaves it, multiplied by o eoeflieient, which, for diamagnetio media, ifl generally positive. SOd.J In diamojfnetic aubetanoea, the direction in which the plane 412 MAQKETTC ACTIOS ON UGITF. of polariution h miwle to rotate ts the same as the direction in viaA a pocitive current must circulate round the ay in order to pi a magnetic force in the same direction aa that which actually rniU in tlie medium. Verdet, however, disooTered that in oertnin fcrroma^trtio medi^ as, for instance, a strong solution of percliloridc of iroo in wm^ spirit or ether, the rotation is in the opposite direction to the eujntf wfaieb would prodaoe the ma^Dctic forc«. This shews that the diSerence between rerromn^ctic and dia- magnetic i-iibstanecs doi!« not arise merdy from the ' nm^ctic per- mcnbility ' Wioff in the first case greater, and in the twoond l«as, than that of air, hut that the pioperttn of the two claaws of bodies arc really opposite. TIr- powtrr acijuirwl hy n Kubstanco under the actioa of magnetic foroc of Totatinff the plane of ]>olariz»tion of light i^ not enutlyfl proportional to its dianiagnetic or ferromagnetic magiiiitimilnlity. ^ Indeed there are cxeet)tion!i to the rule that the rotation in ixwilive for diamagnetic and negative for ferroiaagitutie aubstanoca, for neutial chromate of potash is diamagnetic, but produces a n^ative rotatloB. 810.] There are other substances, which, independently of tbt . application of magnetic force, cause the plane of polariEation to'fl turn to the right or to the left, aa the ray travels throu;^ the sub- vtance. In some of these the property is related to an axis, as in the case of qaartz. In others, the property is independent of direction of the ray within the medium, as in tnrpentinc, soluti' of sugar, Jtc. In all these substant-cs, however, if the plane polarization of any ray is twisted within the medinm like u nt;ht> handed screw, it will still be twistwl like a rig'lit-hunded si^n-w if the ray is transmitted through the medium in the oppositft direction. The direction in which the observer has to turn his analyser in order to oxtingiiisb the ray aAer introducin<^ the medium into ita path, is the same with reference to the observer whether the ray coroe^^ to him from the north or front the south. Tlie direction of thefl rotation in space in of courw reversed when the direction of tltc ray is '' reversed. Bnt when the rotation is produced by magnetic action, its direction in spooc is the same whether the ray l>e tmveUiug north or sonth. The rotation is always in the same direction as that of the electric current which produces, or would produce, the acinal magnetic state of the field, if the medium belongs to the (lOsitJTO elaas, or in the oppoaitc direction if the medinm bvloug» to the negative cbus. IS m tio^ 3 olH I ;i2.] STATEMEST OV THE PACTS. ^B It follows from this. Unit if tlic nty of lij^ht, alter passing throug^h ^BJi« m«diiim from norlli to Month, is ri'llocti'd hy a mirror, eo ns to ^Bvtiini through the mpiliuin from eoulh io north, tho rotatiou will ^H^ doubled wh«n it re^ilbt from magtu-tic iiclion. When thi; rota- Htion dqipo'l* on tho nntiiix- nf t hu ineilium aloiu*, as in tiirpoi»tiii«, &v„ Hthc ray, whtii rcflcctitl bai.k throiif^h the mwiium, emerges in Hut ^HuM plnD« la it entered, the rotatinn during thu lintt poMOgft. ^Hhroiigl) the medium having been exaetlj' reversixl (lurio^ the ^B 811.] The phy«ioal esplanntion of the phenomenon present* con- siderable diflicultics, which can hardly be said to have been hithi;rto overcome, either for the magnetic rotation, or i'ur that which certain media exhibit of themselvefl. We may, however, prejxirt! the way for such an explanation by an analyeia of the observed lots. It is a well-known theorem in kinematics that two uniform cir- ular vihrutions, of the same amplitude, having the same periodic lime, and in the same plane, but revoh-ing in opposite directions, are equivalent, when compounded together, to a rectilinear vibra- The periodic time of this vibration is equal to that of the Pcircular vibrations, its amplitude is double, nud it« direction is in ithe line joining the pointii at which two. particles, dt^'Scribing the Bironlar vibrations in opposite directions round the same circle, rould meet. Hence if one of the circular vibrations has it« phase lecclerated, the direction of tlte rectilinear vibration will be turned, the same direction as that of the circular vibration, through an angle equal to half the acct-Iomtion of phase. It can also bo proved by direct optical csperintcnt that two ray« of light, cimilarly-polNT7zed in opjwsite directions, and of the same intensity, become, when united, a plane- polarized ruy, and that if by any means the phase of one of the cireHlarly-polarized rays is accelerated, the plane of polarization of the remiUaut ruy in turned mind half the angle of acceleration of the ph«»c. 812.] We may therefore express the phenomenon of the rotation of the pinue of polarization in the following manner: — A plano- >Inri7.cd my fallo on the medium. This is equivalent to ttvo cir- bnlarly-polarixcd ravK, one righi-handed, the other tefl-handed (as (ganin the olmerver). After juiasing through the niediuni the ray I still plane-polarized, hut the plane of polarization is turocd, eay, I the right (a.* reg;ird=i the observer). Hence, of the two circularly- rized rayo, that which is right-handed muat have had its phase Kisnnc AC [8r witL to the obDok IIS- tbroi^lki mediatn. In iOhKT wsiia, th* ri gh l-lMMJrf i^ bas perfbroMd a giwbf BHobn tf libntiMH. hw -laglk. wit^B tbc Brdioat, tkaa tfce hft l—iM nr which baa Um Tlua BHid* of (Utia^ what takes plaee im qait« iailffpeikdeat tf aay tivMrf of 1%^ for tkoagk w* am toA Ions as wsn-lea^ cnrnlar^polafiiatiaB, ke., wbidi onj bi aapoented in oar ntadi with a i«rti«dir Smm of tk« nada l atoty Xheorj, the rgaaoaiiif » iaiepadMA of thu Mwciition. aod depends unljr on fiwrts pnnd by cxpRUBvnt. SIS.] lit i1 iiiaiiidnr tiir irraligTmliT nf rrnr iiniiiiiii iiji st a gimi iartant. Anv Bntlolatian, the tnotiaD of which at mth point is areolar, mxj W rapnaeBtcd by a beGx or aemr. If Uw ■ovw is made to nrolTC about its axis witbovt anr longitnltiul aoUoii,caelt particle will deanibe a ciide; and at the mmt tunstlM ptopagatsoa of tfas aod d at io a will be fcpw aeBte J by tb« appannt lawitudiasl notion of tbe nmilariy HtaBted pBCts of tbe tbraad of the screw. It t* easy to see that if tb« screw is rigfat-iiandnl, aad th« obsenr«r is placnl at that end towards which the undoUtioB tiarel^ the motaoo of the senw will appear to him left-I that is to say, in the opposite i rectioD to that of the baods ■ wsfe^ Hsaee such a ray bas beeo called, onginally by Fmuih writei*, bat non lij the whole eeieotifie worid, a lefi-ba colarly-pobrized rsy. An^bt-hiuided nrcoUrly-i ized ray is rrpreseoted in natiDrr by a left-haodifd In t^. 66 the right- luln■l*^i K J, oa the rigbt-hiinil or Uie H^fi': represents a left-luuided ny, luul tbt^ left-handed hrlix /Aon thr li-(l' band, repraeats a right-bawled ray. 814.] Let us nun- rfiaaider two KOt-h Tvyt which bar* the same Tbey wv i^eometaually aUka in. -h— de^_l •oaitedfl ods4>f ^^ ware-leagth within the nedinm. m I fl&l 415 fall rc¥p<.-ct«. oxcc'i)! that one is t1i« pervernon of the other, like it« uma^o in x luokiiig-gIii»s. Ouu of thom, hou-ev«r, siiy A, baa a uliortcr period of rutntion thnii the other. If the motion is entirely rau0 to the forces culled iato piny hy the displacement, thJa ebetvs kbkt greater forces are cnllcd into play hy the same ilitplaeement hrfa«n the configiimlioit is liki^ A than when it is like B. Ilenco in Blii* ouc the le(V-huiided ruy will be aroclerated with rE«|>ect to the pight.'hitnded ray. and thi* will be tlio case whether the rays are hraveUiiig from S to S or from S to A'. 1 This therefore is the explanation of the phenomenon av it is pro- Muccd by turpentine, &e. In these media the displacement caused my n circularly-polanzed ruy calls tnl^ plav grcjiti^r forces of resti- Itution when the onli^uration is like A than when it Js like B, The forces thus defiend on the eoiifig-uration alone, not on the dir«c* tion of the motion. But in a diamagnetie medium acted on hy magnetism in the direction S/V, of the two screws A ami B, that one always rolales with the greatest reloeity whose motion, as seen by an ey« looking from S to y, appears like that of a watch. Hence for rays from S no X the right-handed ray B will travel quickest, but for rays from N to S the left-handed ray A will travel quickest. 816.] Confining our attention to one ray only, the helix B has [exactly the eame configuration, whether it reprcHints a ray from S isK one from N to S. But in the first instsncu tho ray travels ^ftod thervfonc the helix rotates more rapidlj-. Ilcnee greater forces are called into play when the helix is goin^ round one way tluiD when it is going round the other way. The fbrct*, tlierefore, I do not depend solely on the configuration of the ray, but al«o on the direction of the motion of its individual parta. 816.J Tlie diatiirbance which constitutes light, wliatever it» physiml nature may be, is of the nature of a ve<;tor, perpendicular lo the din«tioii of the ray. This is proved from the fact of the interference of two rays of light, which under certain conditions producea darkmiiu, combined with the fact of the ood interference of two rays polarized in pinnee perpendicular to each other. For since the interference depends on the angular position of the planes of polarization, the disturbance mnst be a directed quantity or Te<)tor, and since the interference ceases whi'n the planes of polar- ization are at right angles, the vector representing the disturlmuce must be ])erpendieular to the tine of ioWnoction of these pbnes, m tliat is, to the direction of the ny. 416 MAGNETIC ACTIOS OS LIGHT. [8.] 817.] The disturbance, hemg & rector, can bo resotved into cna pon«ats pamllel to x and jf, ihn txis of 2 bein^ |>ara)I«I to diivcttoD of tbe my. Ix.-t ^ and 7 b« tlwse componi-uU, then, lo^ case of a my of homo^neous circuUrly-poUrized ligrht, f = r cos tf. If = r Ein 0, wborc = Ht—q:i-a. (ij III tkft^e espreasions, r denotes the loa^Uudc of tfa« vector, wij 6 the angle which it Duike« with the direction of Uie axis of*. The ]H;rii>dic time, r, of tbo disturbance k snch that iar = 2ff. (X| Tho wavc-leo^tb, k, of the disturbance is such that ;A = 2ir. [4| The velocity of projMigation is - • The phiute of the didrtiirbancc nheo t and r are both zero is a. The circularly-polariziKl light i» rigrht-handcd or lvlY-handi:J uec'oiding aa q \» wgalivc or jioKitivc. Its vibrations are in tb« positive or the negative direction of_ rotation in the plane of [x, y), according as a is positive or n^aliv Tlie light \» propag-.ited in the prisitire or tho negative direction of the axiH of f, uccording as n and j are of the Game or of opposite signs. Id all media n varies when q varies, and y- it ain'ays of tlie same Ei^ with -■ H Henee, if for a given Dtimtrical value of a the valu« of - is greater when it is positive than when * is motive, it follows that for a vahiv of q. f>tven both in inx^iitiidc and sig^, the posittva value of a will be gr(«(«r than the nt-gative value. Now this is what is observed in a diamaguetic medium, aeted on by a magnetic forcff, y, in the ilircction of ^. Of the two circularly. p»lariz«d mys of a given periud, that is avcelfmti>d of which tli^j direction of rotation in the plane of (•b, ^) is positive. Hence^ o( two circiilurly-iiolartiti-d rays, both left-handed, whose wave lengtl within the medium iK the mme, that has the ahortsst period wl direction of rotation in the phtne of ty is positive, that is, the raji which is pro|)Mguted in the pokitive direction of e from south till north. Wc have therefore to account for the fact, ttiat when in thaj equations of the syntcm j and r are given, two values of m f8i9.] ENEROT OF THE MKDKTU. 417 ^^w SBtisff the equatioDs, one poiitire ami the other oe^tive, the positive raluc beinjS' oumerically g^reater than tho nej^ative. 818.J We may obtain the et^iialioRH of motinn from a conaidera- ioD of tbo potentifti and kinetio ener^es of the medium. The ]>ot«ntiul energy, r. of the system depends on iU confii»uration, tlint is, on thv relative poeition of its parts. In so far as it d^peods on the dii^litrhanoe due to circularly- polarized lig:ht, it must be a fuDctioii of r, the amplitude, and ^, tbo coclficii'Dt of tonion, only. It muy h(- diirtTont for positive and ne^^tivo values of f of oqnal Dumerieal value, and it probiihly is so in the catw of mudi« which of themselve* rotate the plant- of polarization. "Hie kinclie energy, T, of the system \» a homogeneous function of the second degree of the velocities of the systflm, the eoelficienta of the different terms heing fiinctioiis of the coordinates. 8IO.3 Let us voiLsider the dynamiuil condition that the ray majr Ite of constant intensity, tJiat if, that r may b^ constant. Itfgrange's et^uatJon for the fort^e in r become* 41 dr df^4lr~^' ISioee r is conrtant, the 6rst term vanishes. We have thrrt^fore the 'in which q is supposed to be j^iven, and we are to determine the value of the an^Ur velocity 6, which we may denot* by its actual value, n. The kinetic enert;y, T, contains one term involving n' ; other terms may contain prodacts of n with other velocities, and tho re«t of the terms are independent of n. The potentiid energy, /'. is entirely iDdcpcndcnt of «, The equation is therefore of the form AKfl + B»^C=(i. (7) This Wing a quadratic equation, gives two valnos of n. It 8p{>E-nr(t from exjieriment that both values are i^al. that .ono is positive mid I the other negative, and that the positive value is aumericidly the greater. Hence, \^ A w positive, botli B and C are aegativc, for, if K, and «, are the roots of the equation, J(a, + >4)+/t = 0. (8) The coeflicient, ?, tJiorHorc, is not zero, at least when magnetic force a«tK on the medium, We have therefore to consider the ex- [presaion Bn, which is the part of the kinetic energy involving the [finA power of n, the angular velocity of the disturhanoo. VOL. It. £ e meiTETic Acnov or uobt. [83C 820.] Every term of 7 ia of two dineiuiioQe u rOf^rds Teloctl]| ileacv tbe t«nn> iDvolrmg^ n must iaTolve Mino other vdt TTiis Telocity cuutot be »■ or j, becauie, io the cmtK we consider, ' r and ; are coiutaDt. Uenoe it u m velocity wbivh exists in tbt medium independentlT of (hat motion which cotutitattv light. It roust also be a velocity related to x in such a way tluit when it ii multiplied by • tbe resnlt i» a scalar (|aanlity, for only mxlar ^van- titics can occur as temu in tbe valoe of T, which i« itMlf ecsltr. Hence this veloci^ must be in the same direction as «, or in tb» f^poeite direction, that is, it mt»t be an anyutar veiocity nboot tlit axis of t. I Again, thif velocity cannot be independent of tbe ma^otic force, for ir it were related to a direction fixed in the medium, the phe- nomenon would be different if we turned tbe medium end for end. which is not the caae. We are therefore led to the eonclusioa that thin velocity is an invariable accompaniment of tbe magnetic force in those media which exhibit the ma^^netic rotation of tbe plane of polarization. 831.] We have Ik«q hitherto obliged to ose language which \t perhape too euggeetive of the ordinary hypotheeis of motion in the nndnlatorii- theory. It is easy, however, to state our reenit in a form free from this hypothesis. Whatever tight is, at each point of space there ir something going on, whether displacement, or rotation, or sometbin^^ not yet imagined, but which io certainly of the nature of a vector or di-^ reoted quanti^, the direction of which is normal to the dirccliooH nf the ray. This is completely proved by the phenomena of inter- fercDOe. In the cnM of drenlarly-polanzed light, the magnitude of this Ti-ctor remains always the same, but its direction rotates round tbe direction of tbe ray so as to complete a revolution in the periodic time of the wave. Tbe uncertainty which exists as to wbethf r this vector is in tlie plane of polarisation or perpendicular to it, docs not extend lo our knon'Iedge oflhe direction in wbiob it rotates in rif>ht- faandcd nnd in Icft-hnmled circularly-pobrixed light respectively. The direction and tbe angtilar velocity of this vector are perfectly known, though llie physical nnluro of the vector and its absolute direction at a given iniilant are uncertain. When a my of oircularU-'poiarized light falls on a medium under the action of magnetic force, its propagation within the medium is affected by the ralation of t> '' *>od of rotation of the light to MOLECirLAR V0KTICE3. 419 lie direction of Lbe mag^tiv^tic force. From thi> we conclude, by tlic >ning of Art. 817, Uiat in the medium, when under th« action '^of magnetic force, some roUtory motion ia going i>n, tlie axis of ro- tation heing in the direction of the magnetic foroea ; and tliat the rate of propagation of circnlarly-polarized tight, when the direction of its vibratory rotation and the direction of the magnetic rotation of the medium are {he same, is dificrent from the rate of jtropaga- Uon when these directions are opposite. The only reseiublance which we can trace between a medium iroogb which circularly-polttnzcd light is propagated, and a me- linm through which lines of magnetic force pass, is that in both Ifre is a motion of rotution about an axis. But here the rct>cm- cc HtojiH, for the rolut.ioQ in the opticjil phenomenon is that of be vector which represents the disturbance. This vector is »1wny« erpcndicular in the direction of the ray, and Totate^ abotit it a 'known number of times in a sicoud. lu the magnetic phenomenon, that which r(>t»tes has no properties by which its eides can he iia- tingut«he(i, (to that we euiinot determine how many times it rotates ^_ in a iteeond. ^K Thera is nothing, therefore, in the magnetic phenomenon which ^Hcorresponda to the wave-length and the wave-propagation in the ^■optical phenomenon. A medium in which a constant magnetic foroe ^iis acting is not^ in consequence of tliat force, filled with wnvc« travelling in ojie direction, 8s wlien light is propagated through it. The only resemblance between the optical and the magnetic pheno- menon is, that at each point of the medium something exists of I the nature of an angular velocity about an axis in the direction of itbe magnetic force. On lie Ifypo/Jifti* <jf Moleeuiar Fortiea. 822.] The contti deration of the action of magnetism on polarized light hsds, iu> we have set-n, to the conclusion that in a medium iiuder the action of mngnctic force Kometbing Iwlonging to the nme mathemntical clans as an angular velocity, whose axis is in the ■direction of the magnetic force, forma a part of the phenomenon. Thia ang:ular velocity cannot he that of any portion of the me- dium of RCHHible dimeuMona rotating us a whole. We must there- fore conceive the rotation to 1>e that of very small portions of the Itncdinm, each rotating on ita own axis. This ie the hypothesis of molecular vortices. The motion of these vortices, though, as we have shewn (Art. 576), Be z 4S0 KlQKfinC ACTI05 OUT LTQBT. [M it doBB not aseiblT kSect tbn vinbli- motinna of large bodies, bs; W swh as to afcot that vibntory motion on wbJdi tbe propagatmi of fiflrtt a eetr Sia g to the nsdalatory tbeoiy, dopends. The Aa- flMCBteato of the nxdiam, diinng tbe propoi^tioti of light, trill prafaw a £Btoriaii« of the Torti«c% and the vorlioes when m distanlicd wmj raaei on the medium m u to affect the mode o( fnfu e a tix ta of tlie ny. 8StS.] It is inpoMable, in oor pnwDt ttato of t^oranoe aa to the natitre of the Tortices, to awign the form of the law which oonnectt tbe dufilaoeakent of tli« mediom with Uic viuiatioii of the vortices- We shall tbeieTore assome that tbe Tariution of the rorticea caoud bv tbe disf U<«nKat of the medium U subject to the same conditioiui whM-h Ilelmbottz, ia his great memoir on Voitex-motion *, hu efaewQ to regulate the Tariation of the vartioca of a |M>Tfcct liqoid. Ilelmholtz's law may W stated as foUows : — Let P and Q be t«o Deighboaring particles in the axis of a vortex, tlicu, if in oona^ ^Beaee of tbe motion of the fluid tlwae parttcles arrive at the poiat* ^^, the line P^Q" will represent the oew direction of the az» of the Toitex, aad its strength will be altered ia th« ratio of /^^-toPQ. Ilence if a, p, y denote the components of the strength of a Tor- lex, and if (, ih C denote the displacements of the medium, tlic valiu of a will become 1 o'=«+BTi+fl^+y3i, We now aasome that the same condition ia satisfied during thoj amall displacements of a medium in which a, ^, y lepreaeut, not the component* of the strength of au ordinary vortex, but tb« components of magnetic force. 824.] The com|ionen(« of the angular velocity of an element o( t}iQ medium are , il /4C dn>. (%y • CnlU't Ja-ntml, ToL 1*. (IS^S). Timiuktwl bj 1UI. PIUL Mag . Jul; , 11167. WOLFCUUR VOBTICES. 421 The next ntcp in our Iiypollicsis is tbe assumption that tliv kinetic enei^ of tbe medium contuins n term of Ok' form l!nitt is eqniTalciit to supposing that the anf*u1ar velocity acquired by the element of Ihc medium during tho proi>a<fation of li^hl ia a [qoMttity whieh mAv enter into combination with that motion by which nugnetic pbcnomona are rxplninod. Id order to I'nrm thu ei|iiat.ions of motion of the medium, we must [expreBs its kinetio energy in tLTm» of the velocity of its parts, the components of wliich are f, i], C We therefore integral* by parts, and find [2C J Ij {a»j + fin^+yoif) dx dy d» = cff{r,-H)di,dz^cff{aC-yt}dzdz-^cff{pi-<in)ds<}y The double intt^rals refer to the bounding surface, which may tie mippOHMl at an infinite distance. We may therefore, while in- vextigatin^ what takes place in the interior of the medium, confine our attention to the triple integral. 825.] The part of the kinetic energy in unit of volume, expressed by tliis triple iatej^nil, may bo written I iilC{imi-rtv + («y, (5) where «, r, w are the components of the electric current as given in equations (E), Art. G07. It appears from this that our bypotlieais is e([uiTalent (o the imption that the velocity of a particle of the medium whose poncots arc ^, ^, f, is a quantity which may enter into com- bination with the elcctrie current whose components are », w, w. 826.] Returning to the expression uuder the sign of triple inte- gration in (4), substituting for tbe values of a, )3, y, those of w'ljS', /, as given by equations (l), and writing If Tk ^" ":&+' dy-^^di' ^■tbe expression under the sign of iutegratiou become* (6) fn the cau of waves in planes normal to the axis of < the displace- 422 MAOSETIC ACTIOK OS U6I1T. [Sa; tatfata sre fiuictions of * aod t otiljr, so that ~ = y — ^ ami Uw expieesioB is reduced to Cy («J The kinetic energjr per unit of volome, so &r as it d«p«nda « the vrtocitiei of dtsplaoement, may now be written wlwre f> is tb« density of the mediom. 827.] The <y)mpoii4'ntd, X and }', of the inprfntcd force, refemd to unit of Toliune, may be deduced from this by LdgniQ^'s ecjos- tioDa, Art. 561. We observe tbat by two suecewiTe int^catiaw by partd tti regard to ;, and the amission of the double int«j^k at tli« boundiog surface, it may be shewn that H«ice Tlie cxprcwioD for Ute forces are therefore given by ^icse forces arise from the artion of the remainder of (lie mcdiam on the element nnitcr consideration, bik] muat in the ease of isotropic medium be of the form indicated by Oauchy, J = J.^+^,J|+*C ,^+.*.^ + &C (13) 828.] If we now take the owe of » circularly-polarized ray for which f = r oo« (»/— yr), 7 = rsin(i»(-yr), ()^H we Bod for the kinetic enei^y in noit of Tolume ^^ r= ipr'H'—CYf'fM; and for tlie potential energy Jn unit of volume vrbeie Q ia a funotioo of ^ '. 829.] RXPBESSIOK POE THE ROTATION. 423 The oondilion of frw j»roj)iigalion of the my givuti in Art. 820, equation (6), in df _dT . . which gives p»*-2Cyy''» = Q, , (XS) whrace tlie value ofn may be found lu terms of j. But in the cnse of a ray of ^iven wave-[ieriod, «ct«d on by I maeiiclic forci^ what we want to determine ia the value of -^ , when n da "* ■e constant, in t«rms of j^> when y ie conatant. DiiTerontiatinff (18) {%^*—2Cyq^)dn—{J^-^'iCyqH)dq-2Cq^Hdy = ^. (13) We thus find J- = - ^'i'; , % . (20) 829.] If A is (he wave-length in air, and t the corresponding index of rcfmclion in the medium, q\ = 2-!ti. H\ = 2ffF. (21) The ohaijge in the vahie of q, due to magnetie action, is in every I an exceedingly email fraction of its own value, so that we may it*! dq ? = ?o + /j,y. (22) where fo U the value of q when the magnetic force is sero. The LIMgI<-', 0, through which the jihme of jiolarizntion jg turned in sing through a thii-kness c of the medium, is half the sum of the positive and negative values of jc, the nign of the retmlt heing changed, becauiie the sign of q ia negative in equations (14). We thus obtain ' dy {l-X^\ 1 ' FpA The second term of the denominator of thia fraction is approx- imately equal to the angle of rotation of the plane of polarizntion during its passngu through a thickoesv of the medium equal to half a wuvo-len^h. It is therefore in all actual ciuen a quantity which wo may nirglcvt Jii com]>unson with unity. Writing 1^=01, (25) vp we may call m the coefliuiunt of roftgnctie roUtion for the medium, a quantity who»ie value muxt he detenu iuMl by observation. It is found to be positive for most <riuDiugnetie, and nt^tire for some 424 UAGNETIC ACTIOS OS LIOBT. [Sja panunagnetic luedia. th«orv Viv hare thervforc u tlie Goal result of xm 9 = MCy'l{i-X^), (26) where is Uie aiif[tilAr rotation of the plane of polarization, » ■ oonsUnt determioed by obaerration of the metiinm, y the intensi of the iBag:nebc foree resolved in the direction nf the ray, c length of the ray within the raediam, A th» wavc-lcngUi of light in air, and > ite index of refraction in the medium. 830.1 The only test to which this theory has hitherto heen ffl jectcd, i« that of comparing the valnes of for difTvn-nt kind« light passing through the same medinm and acted on by the eame magnetic force. This has been done for a oonnderable nnmher of media by Verdet*, who has arrivc^l at the following results : — (1) TIk' magnetic rotations of the planes of polarization of TKyt of diflVtrent cuK>un> follow approximately the law of the invi squan' of thv wnre^lejigth. (2) The exact taw of the phenomena \» alwa}-e such that ihti pro- duct of the TotatJon by the leqnare of the wuvc-loiigth increaacs from the le«»t refrangible to the most refrangible end of the spectrum. (3) The aubstances for which thia increase is most seoaiUe are also thuRc which have tlie greatest dispervive power. He also found that in the eolation of tartaric ncid, which of itself produces a rotation of the plane of polarization, the mi^netic rotation is by no mt-ans proportional to the natural rotatiou. In an addition to the same memoirf Verdet has given the results of very careful experiments <m bisnlphide of carbon and on cn»fiote, two snbstaoces in whioh the departure from the law of the inrervG square of the wave-length was very apparent. H« has also cmn- p«rc<i tJiese results with the numbers given by Ihne difiereut foi^ (I) «-«.y^(,-i^); (II) '' = -yjJr('-^^)i {III) $ = mcy (.<-4y The first of these formula^ (I), is that which we have already ob- tained in Art. 829, e<iuation (26). The seooad, (II), is that which • B*cti«rdMt lur Im pNtiritftA optlqun il><Telopf^(B 6aiu Iw ooorp* tn4u|iM«Ui par l^uUoii <1u nuffndiJim^ «** putJaL Cvnptn lUndui. X. t>L p. «M (4 Apifi, IBM). -t Cempla OhmNw, IrU. p. «» (19 Oc*.. 18«>. 830.] PORMULA FOB TH8 ROTATION. 425 ' Tcralte from substituting in tho cquuUoiui of motion. Art. 826, cqua- 4»( fPn tions (10), (U), tenns of the form ~^and — '^~, mst«ad of ^-rj. I am not aware tlint this form of th« equalioti bus suggested by any pbysical theory. The third fortnuta, (III), reeulU from the pbysicul thi'ory of M. C. Noumaon'*, in which the equatioDs of motion contain terms of the form -p- aud — jrt. ' lii III U IB evident that the values of given by the forniuU (III) nrp 'not even nppr^ixiniati-ly proportional to tho invert gquure of the wavf-lcnglh. Those ^iven hy the f'lnnulti- (I) and {II) satisfy this condition, mid give vahies of fl which agree tok-nibly well with the < obccrrwl vidut-s for media of moderate disiiersive power. For hi«nil> ' |>hide of OJirbon and creosote, however, the vnlues given by (II) differ very much from those observed. Those given by (I) agree better ' viHh observation, bnt, though tho agreement ia somewhat close for bisulphide of carbon, the numhera for creosote still differ by quan- tities much greater than con be accoauted for by any errors of observation. Mojfneiie liolafion "/ the Plane cf 'Polarization (from Frrdef^. Buulphido or Cnrboa at Sl'.UC, Llnca of tti* •{HK-tmtn C D B F Obaorvod rnlatiun 692 768 10(10 1234 1704 CUauUtod by 1. S8B 780 1000 I'M* 17IS 11. OUS 772 1000 I-.'I8 1A40 III. 043 0(7 1000 1031 lOSl 1 Kotation of ths i*j . K - as'.ss". CnonMt at !t ".ac. -Linca of tho nptctrum C D s F a ^DbaWTod raUtiao 573 758 1000 IMl 1721 CUcutalolby 1. tir 7S0 1000 ISIO 1608 u. 6SS 7H9 1000 1-.iOO 15U5 in. »7« 993 lOOO 1017 1011 I RotKtdon ol th* n>; E •• SIDES'. * 'Enluwre tontiilur quomcKlo lUt at lucii pluium pol&rilatiotiii par vlret oloo- triow *cl DU^eticM diplinttur.' Ilali* Sajri'iiini. i^iS. t Tb«« ihrot form* of tho uqi)»tiniiii iif million vtrn- finl iniitB"ttfit liy Sir O. H. Alij {Pim. Mail., June I<i40) u x uiiWiiH of muljiin); tlio iilioSdiiiuiion Ihi-ii r*n-RtIy diMDvamI by Fanulny. Hsu Cull*^ b.id pnrioiiiilj' tu^gwiwl vquMiona ooutaiuing of tlw fbnn is ordsr M rtpraunl niBtlininftlicaUy the pheaomnis of nnutK. ajr Him* •quktiunii mm oiTarwl hy Mm Cullu^h utA Airj'. *iinl m |,'i<^iig s mecibMiiaal nplMMlim of lh« pbvnoiiieru. but •■ ■h«<rui)' tbM tba phenamcnii mity b* njilftinvd bj aqusliaiw. trhioa equuiotw Sppoar to be *uch m ralgbt poailily be deduiwil (ivm t plsiiMbJo mtcluidGBl wBumplloiL ftltliauirlt no iiioh BMiuiipticui Iua i4t boon 426 MAOSETIC ACnOK OS UGWC. [83'. [II* 1 Wi) lire 80 little acquainted n-ith the details of the molecukr coDutitutioD of bodies, that it is not probable that any satislaeton theory can be formed relating to a particular phenomenon, wich u | that or the magnetic action on lif^ht^ until, by an induction foumlftl on a number of difTerent cases in which visible phenomena are fonod ' 1o dcpcn<l upon actions in wbtch the molecules are concerned, wt leant something more definite about the pro{)crties wbicli must bt attributed to a molecule in order to satisf^' the conditions of eb- «rved fiwts, Tiic tli<.>ory proposed in the procedinf; V^f!^ >" evidently of a i proviKioniil kind, re^in;; as it does on unproved hypotheaee relatio^fl to the nature of molecular vortices, and the mode in which they are afFccU'd by the displace m<.>iit of the medium. We must therefure regard any coincidence with observed facts as of much lees adeiitific value in the theory of the magnetic rotation of the plane of polari- zation than in the electromagnetic theorj'of light, which, though it, involves hj-potbeses about the electric properties of media, does do t-p<rulute ax to the constitution of their molecules. 831.] N'oTB.— The whole of this chapter may be rcigarded as an' expimMou of the exceedingly important remark of Sir William Thom«m in the ProeeediNSi o/the Reyal Socirty. June 1856 : — 'The magnv'tic influence on light difcovered hy Faraday depends on the direelion of motion of muviug particles. For instance, in n medium possessing it, parf.iclfw in n utraight line {nrallcl lo the lines of mn^uetic force, displaced to a helix rviund this line us axis, and then^ projected tangentially witli siicb velocities as to di^scribc eircles,fl| will have ditrt-rent velocitien according an their motions are ronnd in one direction (the aime u» the nominal direction of the gidvanic current in thi- magnetizing coil], or in the contrary direction. But the elastic reaction of the medium muat be the Ksme for the same displacements, wltatevor be the velocitieH and directions of the irttt- tides 1 that is to say. the forces which are balanced b; centrifugal foroe of the circular motions are equal, while the luminiferouafl motions are unequal. The ab§olute circular motiooB being there- fore either equal or such as to transmit eq<uil centTifugul forces to the pnrtielee initially considered, it follows that the luniiniferous motiims are only components of tJio whole motion ; and that a less luminifirrouji component in one direction, Compounded with a mo- tion existing in the medium \vlw(i franitmitting no light, gii'cs aa cqiuil rt-Kultunt to that of a greater himiniferoun motion in the oon-J tnry dinxtiou compounded with the same non-luminr>u.> i>iiili.iti.f AEGUMENT OF THOMSON. 427 I I I I think it is not only impoHiibIc to coniseiro any other than this ; dyniitnicnl tixjiliinntiun of thv IwH: that circularly-polarizml Hg^hi .^ransinittt'd Uinnigh mftgnctizeil glass pantile! to the linos of maff- netixiiig lori,'!', with the aame qnality. right-huiiOcd iilways, or h-fk- handed alwnya, ie propiigate<) at dilferent mios ucconlinf* as its coumc is in the direction or is contrnry to the direction in which a north magnetic pole is drawn ; but I heli^ve it can be demon«l ruled that no other explanation of that fact ie possible. Henee it appears that P'araday's opUeoI discovery affords a demonstration of th« rfr- altly of Amp^r«>'8 explanation of the ultimate nature of magnetism ; and gives a definition of msgnetization in the dynamical theory of heat. The introduction of the principle of moments of momenta ("the conservation of areas") into the mechanical treatment of Mt. Bankine's hypothesis of*' molecular vortioeti," appears to indi- cate a line perpendicular to the plane of reatiltant rotatory nio- mentam ("the invariable plane") of the thermal motions as the magnetic axis of a magnetized body, and suggests the resultant motneot of momenta of these motions as the definite measure of the "magnetic moment." The exphination of all phenomena of viectrotnagDetio attmction or repulsion, and of electromagnetic in- dnotion, is to be looked for simply in the inertia and pressure of the matter of which the motions constitute heat. Whether this matter is or is not electricity, whether it is a continuous fluid inter- prrmeating the spaces between molocnUr nuclei, or is itself mole- cularl}' grouped ; or whether all matter is continuous, and molecular hcterogcucoutinesH consists in finite vortical or other relative mo- tions of contiguous parts of a liody ; it is impossible to decide, and porfaapn in vain ti> spi'eulate, in the present state of (icicncc' A theory of moiccuhir vorti(«8, which 1 worked out at consider- able length, was published in the PAH. Mag. for March, April, and May, leci, Jan. and Feb. 1802. I Utink we tiavc good evidence for the opinion that some pheno- menon of rotation is going on in the magnetic field, that this rota- tion is ]ierformed by a great number of very nniull portions of maU«r, each rotating on its own axis, this axis being parallel to the direction of the magnetic force, and that tlie rotations of these dif- ferent vortices are made to depend on one another by means of some kind of mechanism connecting tliem. The attempt which I then made to imagine a working mode] of this mechanism mnet be t«kea for no more than it really is, a de- munetralion that mechanism may be imagined capable of producing MAGNETIC ACTION ON LIGHT. m^ B connt^sion mechanicallj eqcivalent to tlie actual conaexion of tbe jiarte of the elcctroma^Gtic fiold. The problem of determining' the mecbanism required to establish a given species of connexioo be' tween the motions of the parts of a system always admits of an inGnite namber of fioIatioQE. Of these, some may be more clniDiy or more complex than others, but all most satisfy tbo eonditions of mechanism in general. The following results of the theory, however, are of higher valne : — (1) JIagnetic force is the effect of tbe centrifugal force of the vortioce, (2) Electromagnetic induction of currents is the effect of the forces called into play when the velocity of the vortices ie changing. (3) Electromotive force arises from the stress on the connecting mechanism. (4) Electric displacement arises from the elastic yielding of the connecting- mecbaniEm. CHAPTER XXII. FBRROUAQKETfSM AKD DUHADNBTIRM BXPLAIKED BY MOLECULAR Cl'nRENTS. 0» EUelromagnetic Theoriet of MaffaetUm. 832.] Wb have seen (Art. 380) that the action of magnets on one another can be accnratt-Iy reprcsfntt'il W the attractions and repulsions of an imiiginary sitbstuiicu called ' mti^nctlc inatt«r,' We have shewn the reasons wliy wo mii«t not suppose this ma^etio matter to move from one part of a inng^iiet to another through a wnsikk) dirtancc, as at fintt et^ht it a[)]«ars to do when wo msg- DVtue a bar, and we were led t" Poissou'a liypoUieHis that the SiagDCtie matter is strictljr confined to single muleoulfts of the mag- imUo iubstAnci-, so that a raa^uetized molecule is one iu wliich the kinds of mugnvtie matter are more or less seipnratod to- wards opposite ))olcs of the molecule, but so that no part of cither can ever bo actually separated from the moU-c-ule (Art, 430). These arguments complctt^ly establish the fact, that magnctiza- tton is H phenomenon, not of large masses of iron, but of molcculea, that is to say, of portion* of the subsfancc so small that we cannot by any mcebanical method cut one of them in two, so as to obtain a north pole separate from a south pole. But the naturo of a mag- Detic molecule i» by no means determined without further invcsti- g*tioD We have seen (Art. 142) that there are strong reasons for bdieving that the net of miignetixiug iron or sUx) does not consist in imparting magnetization to the molecules of which it in eom- pose<l, but that these molteules are already magnetic, even in un- magnetised iron, but with their axcx placed indifferently in all directions, and tliat the act of magiK-tixation consi^a in turning the moleeule« so that their axo* are cither rendered all parallel to one direction, or at Uast are deflected towards that direction. Bt.ECrmC THBOBY OF MAOSOTTSlt 8S3.J Still, bowwur, wo hkvo arrived mt do cxpUnatioQ nf U>v nature of a magavtiit molociil*?, limt i«, wo have not rocognized tu likeness to nny other Uiing of which we know more. Wo \tan thei-efore lo oonsider the li}'j>(>tlie«is of Aini)^rA, lliat the m»gnetim of the molecule is due to an electric current constantly cinmlattng in some closed path within it. It is possible to produce an exact imitation of the action of nay magnet on points external to it, by means of a aiiecX of clwtric currents properly distributed on its outer eurfooe. But iJie action of the magnet on points in the interior ia quite dilTercnt froin the action of the eltctrio currents on corresponding points. Henco Am* pdre conolnded that if magnetism is to be cxplaiiu'd by mvniM nf electric currents, these currents must circuluttt within the molvculc» of the ma^et, and must not flow from one molecule to anotJief. Ah we cannot experimentally measure the magnetic action at a point in the interior of a molecule, this hypothea> cannot be di»> proved in the same way that we can disprove the hypothesis nf currents of sensible extent within the ma^et. Desidea this, we know that an electric current, in passing (Vom one l>nrt of a conductor to another, meets with resistance and gene- nit<* bwit ; so that if there were currents of the ordinary kind round pnrti'iiis <if the magnet of sensible size, there wouM be a constant expenditure of energy required to maintain tliem, and a magnet would be a perpetual source of heat. Dy confining the circnita to gthe molecules, within which nothing is known about renstanoe, we Day assert, without fear of contradiction, that the current, in cir- culating within the molecnie, meets with no resistance. According to Ampere's tbeorj', therefore, all the phenomena of magnetism are due to electric currents, and if we could make ol>- siTvaliona of the magnetic force in the interior of a magnetic mole cule, we should 6nd that it obeyed exactly the same laws as the force in a region surrounded by any other electric circuit 831.] In treating of the force ia the interior of magnets, we bavi Bupposed the measurements to be made in a small crevasse hollowed out of the substance of the magnet, Art. 395. We were thus led to oOQsider two different quantities, ihe magnetic force and the. magiuitic induction, both of which are snpposod to be obeerved in ft space from which the magnetic matter it removed. We were' not imppoaed to be able to penetrate into the interior of a wHic molecule and to obeenre the force within it. If we adopt Ampire^s theory, wc consider a magnet, not as '36.1 HOtECtTLAS CBKRESTS. I ntinnoiw sabetonee, tlio mn^ctization of which varies rrom point point acoonliii^ t<i in>an- msily c(>ni.'cive<l law, but ns n tniiliittide of inolcciilui, within ciich of which circulutcs ii Kjirtoni of electric Current)!, tjivin^ rise to a dielributioH of mBgoetii- force of CKtrcmo complexity, the direetioTi of Iho fori-e in the interior of a inoleetilc \mag generally' the rever»e of that of the average ioroe in ita neigh- bourliood, and the magnetic potential, where it ex'iaU at all, Imiiig a function of as many degrees of multiplicity as there are molecules in the miiKm't. 835.] But we shall find, that, in spite of this apparent complexity, which, however, arises merely fiom the coexistence of a multitude of simpler parts, the mathematical theory of ma^etism is greatly Bimplilied by the adoption of Anip&re's theory, and by extending our mathematical vision into the interior of the molecules. In the firet place, the two deGnitions of magnetic force are re- duced to one, both hecomlDg the samo as that for the ^ncc outside be magnet. In the next place, the components of the magnctio everywhere satisfy the condition to which those of induction luhject, namely, da d/i _ dy _ ^ dx ^y 'di = '- by In other words, the distribution of magnctio force is of the same nature as that of the velocity of an ineompre.i.tihle fluid, or, as we have expressed it in Art. 23, the magnetic force has no convergence. I Finally, the three vector functions — the electromagnetic momen- torn, the magnetic force, and the electric current — become more simply related to each other. They are all vector functions of no nvergcnce, and they are derived one from the other in order, by thi^ Niinic prociiits of taliin<f the space- variation, which is denoted by Hamilton by the symbol V. 886.] But we are now considering magnetism from a physical lot of view, and we must enquire into the physicul proportic* of tlte molecular currents. We assume that a currunfc is ciiculatiog io a molecule, and that it meebt with no resistance. If £ is the coefficient of aelf-induction of the muleeuhtr circuit, and ^the co- efficient of mutual induction between this circuit and some otlicr cirvait, then if y is the current in the raolwul«, and y' that in the other circuit, the e<|aatiou of the current y i« ^iLy + 3ty') = ~Ry: (2) 432 [8.^" and sinoG by the hypotliesis there is do reaistAiicv, i? ^ 0. Bod «« get hy int^ratioa IiyTify'=: constant, = Ly^, say. ( l>ct us Giifipose tlint the area of the projeclioa of the tnolscnlu circuit oil a place perpendicalar to the axis of th« molcculo is J, this axis heingf defined as the normal to the plane on wbldi iht projection ia greatest. If the action of other ciimnt« prodocec ■ magnetic force, X, in a direction n-hose inclination to th« uxia of the molecule i§ $, the qaantity My' becomes XA cos 8, and we haw* as the equation of the current Ijyi-XAe<M0 = Ly„, (' whore ya is the value of y when J* = 0. It appc^'nrs, therefore, Uiat the strvn^b of tbc molecalar cnrrent depends entirely on its primitive value yg, and on the intensity of the magnetic force due to other current*. 837.] If we supimse that there i* no primitive currunt, but tiiat the current is entirely due to induction, then y = — -y- cos tf . (5 Tha se^tivo sign shews that the direction of the induced cur- rent is oppoiiitu to that of thi- indueicj; current, and its magnetic action is such that in the interior of the circuit it acts in the up* posite direction to the ma^rnctic force. In other words, the mole* cular current nnU lilic a Kmall ma^et whose poles are turned towards the poles of the iianie nume of the inducing magnet. Now this is an action the reverse of that of the molecules of iron under msj^uetic action. Tho molecular current* in iron, therefore, arc not excited by indtietion. But in diamu^^etic suhstances an action of this kind is observed, and in fact this is the explanation of diamagnetic polarity which was lirst given by Weber. I 4 Weiert Tieorjf of Liamagnetitm. 83B.] According to Weber's theory, there exist in the moleeules of dianiaguctie substancea certain channels round which an elertri' cumnt can circulate without reaistance. It is manifest that if snpiKwc tlicse channels to traverse the molecule in every directi< this amounts to making the molecule a perfect conductor, Be{>inning with the assumption of a linear circuit within the m Iccule, we have the strength of the eurruat given by c<[uutioD PERPECTLT COSDUCTIXO MOLECrLES. 433 The mn^Detio momcut of tke current ie tJie product of iU strength by the aroa of the circuit, or yA, aud tlie resolved part of this in the direction of tlw magnetizing force is yd cos 0, or, by (5), _^co8«tf. (e) If there arc i such molecules in unit of volume, and if their axes are distributed inditlercutly in all directions, then the averat^ vnlac of txm'0 will be i,nDd the intensity of magnotiscation of the substance will be ^ nXA' .. Neumann's coefficient of magnetization is therefore The magnetization of the substance is therefore in the opposite directioQ to the magnetizing force, or, in other words, the substance is diamafpietic. It is ftlso exactly proportional to the ma^etizmg foroc, and does not tend to a finite limit, as in the case of ordinary ma^etic induction. See Arts. 442, &c. 8S9.] If the directions of the axes of the molecular channels are arr«nj^>d, not indifiertrntly in all directions, but with a preponder- ating uumber in certain directions, then the sum oxtmded to all th» molecules will have different values according to the direction of the line from which is measured, and the dis- tribution of these valueti in different directions will be similar to the ditttributioQ of the values of momeuts of inertia about axes in dif- ferent directions through the same point. Such a (tistributioa will explain the m.-ignelic phenomena related to axes in the body, described by Pliicker, which Faraday has called Magne-cry stall ic phenomena. See Art. 435. 840.] Let us now consider what would be the effect, if, in^trad tile electric current beiiit; confined to a certain chanmd nithia ! molecule, the whole molecule wertt stippimod a poriV-cl conductor. Let us begin with the case of n b(tdy the form of which is acyclic, that id to say, which is not in the fonn of a ring or |ierfor.it«i body, and let us suppoM that this Uxiy i* everywhere surrouuded by II tliin ahcll of perfectly conducting matter. We have pwvcd in .^rt,654, that a ilused sheet of perfectly coq- ^—^ '"tter of any form, originally free ^m ounente, becomes. 434 BLEOTBIC TUKOBT OF UAONETISJL [S4I. when exiwsec] to external magnetic torve, a cnrrest-sheet, the artion of which on every point of the interior is Euch as to nuke thr magnetic foroe Tero. It may ai«ist as in understanding tJiis case if we obeerre that the distribution of magaeiac force io the neighboorhood of aacfa ■ body 18 similar to the distribution of velocity in an iDcompresdUe fluid in the neighbourhood of an impernuus body of tb« same farm. It ia obvious that if other coiiductia}> shelU are placed within the first, since they are not exposed to mu^^netic force, 00 cnrrentf will be excited in thenL Hence, in a solid of purfoctly conducting materia), the effect of magnetic force is to fp'ncratc a system of currents which are entirely conBncd to the siirfncc of the body. Sll.] If the conducting body i* in the forui of a inhere of nuiita r, its magnetic momeot may he shewn to be and if a number of such spherce are distributed in a mediam, «> that in unit of volume the volume of the conducting matter is V, then, by putting *, = I, i, = 0, and p = f m equation (l 7), Art 314, we find the coefficient of magnetic permeability, taking it as the reciprocal of the rcsistunce in that article, viz. 2-2 i' f = 2+i' "M (10) <") whence we obtain for Poisson'a magnetic coefficient 4 =-♦*-, and for Ncnmann's coefficient of magnetizatioiHjy induetion 3 ^ Sinee the mathematical conoeptioD of perfectly condttcting bodies leiuls to nwulta exe«cdingly different from any phenomeoa which we ran observe in ordinary conductora, let ua pursue tlie subject , BOmowhat further. M 842.] Returning to the case of the conducting channel in th^^ form of a closed curve of area A, as in Art. 83f>, we have, for tha moment of the electromagnetic force tending to increase the angle 9, m ~~p- siaOcoa0. /13) Tliifl force is positive or negative according as ^ is lest or great than a right angle. Hence the cffcet of magnetic force on a per leotly Donducting chanr.L-l tr-i:!-i tn lura it with its axis at ligbi (12) 843-] HODiPIEB THBOST OF TITOUCBR MAflJfBTISM. 435 I aaglM to thv lin« of magrwtic forc«, that is, so tlmt tlie plane of the ohannel bocomes punillcl to the lines of foKtf. An efr«ct of n simikr kind may be ohserved hy pWiog a pt-nny or a copier ring bcttvoen the pnleti of an elect roinig^ot. At the instant that Uie inignct in ezoil«d tJio ring turns its plane towards ihe uxi.il diniction, \nil thiti tbrce vnniiithfg ax soon as tho ourrtrotK are deadened by the resistance of the copper *. 843.] ^Ve liave hitherto conHidered only the owe in which the molecular currents are entirely excited by the oxterna) magbetie force. Let u» nt'Xt examine the bearing of Weber's theory of the magneto-electric induction of molecular currenta on Ampere's theory of ordinary magnetism. According to Ampere and W^iber, the moleuular currnnttt in magnetic substAnces arc not excitM by the external magnetic force, but are already there, and the molecule itself is acted on and deflected by the elect roma^ietio action of the magoetie force on the conducting circuit in which the cmrent flows. When Amjiirw devised this hypothesis, the induction of electric cur- n-Dtx was iiol known, and ho made no hypothceis to account for the existence, or to determine the strength, of the molecular currents. We are now, however, bound to apply to these currents the same laws that Weber applied to big currents in diamagnetic molecules. We have only to Biippose that the primitive value of tbe current y, when no magnetic force acts, is not zero but Vq. The strength of the current' when a magnetic force, .V, acta on a molecular current of area A, whose axis is inclined to the line of magnetic forcei, is XA „ (u) and the moment of the couple tending to tnm the molecule so as to iucreosc fl is „ X^A* . , . ~y^XA aintf + -—^ sin 2 0. (16) Hence, putting . A i*y„= m. 'Yo = S. (16) inibc investigation in Art. 443, the equation of equilibrium becomes Xtiia0-BX*mn0C960 = Ds[a{<t-9). (17) Tbe resolved part of the magnetic moment of thv current in the direction of A' is yAco»9 = YtfAcosff j-~coif9, (18) = tHCm&(l—SXoot$). • 8oe Fkndjiy, Krp. Ha.. 2310, Jtc (19) 438 StStnftlC TltKOBT OP 3UG!7EnS3r. [84^ 844.] Th«e comlitionii difier from those in Weber's tiiooij of nugnoUe indnctioD hj the terms involving- the confficifnt S. IT BX u nnall oumjiared with unitj*, the results will approxioulv U 1ho«c of Weber's theoiy of ma^etism. If £X is large oompORd with unit}-, the results will approxinwte to those of Weber's tbooiy iff iiaBiigoistJxm. Now the greater y^, the primitive vslae of the moIeeuUr current, the HmiilliT will B become, ntxd if £ is also larffe, this will aUo dimituMh B. Now if the cnrrent flows in a ring ohaanel, the vahie of £ depends on log ^ , where S is the ndius of the mean line of the channel, and r that of its section. The emalter therefore ihf section of the channel compared with it« an-a. the greater will be I, the coefficient of self-induction, ftnd the more nosrly will the plie- nomcnn agree with Weber's origiiiul <licr>rj'. There will be tbu ditferencc, however, that us X, the magn*aixtiig force, increases, the temporary magnetie moment will not only reach a maximnm, but will afterwards diminish as X increase*. If it should ever be experimentally proved that the t«tnporaiy ma^etizatiun of any snWtanoe Btst inereaseii, and then diminii)h«a as the ma^etizing force is continually increased, the evidence of the existcncu of these molecular currents would, X think, be mised almost to the rank of a demanstration. 845.] If the molecular currents in diama^etic mbetODon are conGnod to definite channels, and if the molecules an capable of being deflected like those of ma^etie subetanoes, Uiuii, as the mag- netising force increnscH, tlie diamagnetic polarity will alwavs increase, bat, when the force ti« great, not quite so fast oa the ma^^etiziojf force. The small abwolute value of the diamagneticcoeflidentBbewB, however, that the di^Recling force on each molecule must be small compared with tJiat exerted on a magnetic molecule, so that any result duo to ihia deflexion is not likely to be perceptible. If, on the other band, the moleoular currents in dinm^netic bodies are fre« to flow through the whole sabstance of tlie molecules, the diamagtivtic polarity will be strictly progwrtional to the mag^^ nctizing force, ant) its amount will lead to a determination of th^| whole space ocoipie*! by the perfectly conducting maates, and, if we know th« number of the taolecnlai^ to the determination of ihi of each. CHAPTER XXIII. THEORIES OF ACTION AT A DISTANCE. On tie Explanation of Ampir^a Formu/a ykn Ij/ Oaiui and Wehtr. 846.] The attraction between the element* dt and d/ of two circuite, carryiog- electric curreut^ of intensity i and i', is, by Ampdre's formula, th. itdsd^ . d'r dr dr-^ the curreatfl being estimated in electromag'netXc unit*. See Art. 526. Hw quantittea, whose meaning as f hey appear in theee expreaeiona we have now to interpret, ace dr dr , d'r ' *~'' .UT/' ■""• d^' lud the most obvious phenomenon in which to seek for an inter- pretation founded on a direct relation between the curronbs is the relative velocity of tlic electricity in the two elements. I 847,] Let 03 therefore consider the relative motion of two juir- ticlea, moving with constant vclociticE r and v' along the ek-menta ds and d^ respectively. The square of the roktivo velocity of iheiie particles IB tfi= i>*-2vtfco9t+i^i [ ftnd if we denote by r the distance between the particles, <*f dr . dr u'^'di-^'^di" ^y^-^2r^ d'r dt'd^ ^x^- d»r di^' (3) 438 ACTION AT A DinAKCE. im vrbere tlie symbol d indicates that, in the qtuatity OifTor^ntiatei the cootdinates of the particles are to be expressed in t«nne of the lime. It appears, therefore, that the terms involvioi; tKe product rf' i^j the eqoationa (3), (5). and (6) contain the (^nantities occurring i^H Wu thcrcrore cnd^avoaT t« But in order to (l) and (3) which we have to intor]>ret. exprow(1)nnd (2) in termaof a", jr\ , and ^. do so we must gtrt rid of the first and third tcrm» of co^ili of thcM cxpreaaioDf^ for Ihcy involve quiintitit-s which do not ap{>cAr m tli« rormiifai of Aiin)Jre. Ht^nce we cannot esjilain the electric currenl ns a tjnn»rer of electricity in one direction only, btit we must coi hiae two opposit« streams in each current, so that the oombi etTecl of the tcnns involving e* and r^ may be zero, 848.] Let us thcrof<»re rappo»c tltat in llic first clement, tit, we have one electric purticle, f, moving with velocity r, and another, r, , moving with vctoeity t',, and in the same way two particW, ^ ani f'l, in da', moving with velocities v' and u', respectively. The tcnn involving r* for the combined actJon of tJiece partici Similarly 2 (v'W) = {v'^/ , r'.V,) (* + e,) ; ■nd 2(cp'</) = (w + Pie,){»'«'+f',/,). Id order that £(r'tt'] may be zero, we must have either rvnt 1 I (r (9) (9) e'+/, = 0. or '< + r, V, = 0. According to Feehncr'a hypothesis, the electric current oonsis of a current of positive electricity in the positive direction, com- bined with a current of ne^^tivo electricity in the nefr-itive direc- tion, the two current* being exactly equal in numerical magnitude, both as respects the quantity of electricity in motion and the velo- city with which it i* moving. Henee both tlie oonditioos of (10) arc sstiKfiecl by Kechner's hypothesis. ^^ But it is sufticicnt for our purpose to assume, either — ^| That tlie quantity of positive electricity in each element la on- merically equal to the quantity of negative clectricitj-; or — That the quantities) of tlie (wo kia<ls of elcctncity are invcrecl;^ a» the wjiiarce of their velocities. Now we know that by charging tli« second conducting wire at whole, wi- can make tf'+«'| either iKKtitivo or negalirc. Such a cbari^ wire, even without a cun«at« according to this formal^ would act on the first wire carrying a carrcnt in whicfa ^t-i-t^ FOHlIPtAE OF GAU83 AND WEDER. 43d I haB ft valiw difieriog from zero. Such an action has never been observed. Therefore, since the cjuantitye'-ftf'i maybe shewn experimentally not to be always zero, an<] since tb« quantity i^e + v'^e, is rwt capable of betnf^ experimentally tcslwl, it is better for these specu- lations to aasume that it is the latter (juantity which invariahly vanishes. 840.] \Vbat«ver bypothivis we adopt, ther« can be no doubt that the total translVr of electricity, ruckooed algebraically, along the first circuit, is rvprctii.'nt«d by p<+r,e, = «'(/*; where i; is the numlx-r of units of elatical electricity which are transroitted by the \init electric current in the unit of time, so that we may write equation (9) X(r^r^) = cHi'^ift»'. (11) HcDfie the earns of the four valui.'g of (3), (5), and (6) become 2(^b2) =_2c*HV*.i«'co««; (12) Jrdr ^' (13) Hi) and we may write the two expressions (t) and (2) for the attraction Wtween d^ and (// -■-[?(:'i':«(l^)')]- (■•) 650.] The ordinary* expression, ia t^e theory of statical elcctri- city, for the repul^ion of two electrical particles e and / is -^, and (IT) h gives the electrostatic repulsion between the two elements if they are charged as wholes. Hence, if we assume for the repulsion of the two particles either of the modified expressions we may dedocc from them both the ordinary cIrctro»lnttc force*, and the forces acting between cnrrents as determined by Ampere. 440 ACTION AT A DBTANCB. [851- 851.] Th« first of these expremoita, (18), wiw dUoovered \ff' Gauss* in Ja)y 1835, and iDt«Tpret«d by him as a runilameiital b* of electrical action, that ' Two elementit of electricity in a state «f j relative motion attract or repel one another, but not in the way as if they are in a state of relative rest.' "niiit discorcry «» I not, »o far as I know, puhliiibed in the lifetime of Uaaa», m that tlttj aecond espreesion, which was discovered independently by W. Weber.! and published in the first part of his celebrated E/eJtltvJfaamutie' .ifa/islpfi/immiiu^eti t, was the first resolt of the kind made known j to the scientific world. S52.] Tlte two expressions lead to precisely the same rvsult whai < they are applied to th« detemiination of the mecfaaoica) force be- tween two electric carrents, and this reeolt is identical with that ; of ArapSre. But when they arc considered as exprewnonM of tlie , physical law of the action betw(«n two electrical particle*, we ate . led to enquire whether they are consistent with other known bcU ' of nature. Both of tbeee expressions involve the relative velocity of the! particles. Now, in establishing by mathematical reasouing- 1h« well-known principle of the conservation of energy, it is generally asMnmed that the force acting between two particles is a function of I tlie distance only, and it is commonly stated that if it is a rnnctiOB] of anything else, such as the time, or the veloci^ of the particli the proof would not hold. Hence a law of elcrtricid action, involving the velocity of particles, has sometimes been supposed to be incoosisteDt with, principle of the conservation of onerpy. (i53.] The fonnula of Gau«8 is inconsistent with this principle, and muit therefore be abandoned, as it leads to the conclusion thai energy might be indefinitely gmeratevl in a finite system by phystoal means. This objection docs not apply to the formula of Weber, for he has shewn ] that if we asHume as the potential energy of a system coosistiDg of two electric jttrUeles, (80) tthe repulsion between them, wbioh is found by difiVrontiating this j i|nantity with respect to r, and cfaanging the sign, is tltat given hyj the formula (19). *=^[-^(^«)']' • HM»(0llttlMM«dWin.inT),*nLr.B.«18. + JU. ZfOafM. Oa, Lriprif (1M«>. : A«y. Amn^ luUL |>. Sir(l«8). 854-] HBLH1IOLT2i) CRITICISM. 441 Ilence the worh done on a moving pitrticle hy tlie repnUioa of A partide is ■f'o— Vi> vhcre ^„ imd ^, arc tiic vuIdck of ^ at the Dning aad at the end of its ]jat,h. Now ^jl il(-[)cnd« ouly on th« nee, r, and on the velocity re«rtlve(i in thu direction of r. If, therefore, the particle desi'ribea any cloacd path, so that its position, velocity, and direction of motion are the same at the end as at the bffpDnin;;, ^, will be equal to i/f^, and no work will be done on tie vvholc dnring the cycle of operations. H HenO! an indcfiaite amount of work atnnot be generated by a Bporti«le moving in a periodic manDer ondvr the sctioa of tbe force Kmiumed by Weber. 854.] But HcImholtE, in his v«y powerful memoir on the ' Eqni^ tiona of Motion of EU-etricity in Conductors at lU-st*/ while htt shews that Weber's formnla is not inconsiiitent with the principle of the conservation of energy, as re^rds only the work done during a complete cyclical otH^ration, points ont that it leads to the conclu- sion, tkftt two electrified particles, which moTC accordinj? to Wcher's law, may have at first finite velocities, and yet, while still at a finite di«lanc« from each other, they may acquire an inRnite kinetic ent-rgy, and nay perform an infinite amount of work. fc'To till* Weber t replies, that the initial relative velocity of the rticlM in Hdmholtz's example, though finite, is greater than the locity of light ; and that tlio distand.^ at whieh the kitietio energy beoomcs infinite, though finite, is gmaller than any magnitiidt; which weoan pcTX<eive, *o that it mny be physically im|)o^ib1e to bring two moleeulea so near together. The ej[ami>le, therefore, cannot be teated by any experimental metliod. Uelmboitz { has therefore stated a eaite in which the distanoea an not too small, nor the velocities too great, lor experimental verifioa- ioil. A fixed non-conducting spherical surface, of radius a, it ani> armly cliarged with electricity to the siirrocc-density <r. A particle, Eof miuB M and carrying a charge r of ckvtricity, mores within tiw sphere with velocity v. The cicctrudynamic potential calculated trom tlie formula (20) is 4«r««(l-^), (21) and is independent of the position of the ]iartivle within the sphere. Adding to this f, the remainder of the potential energy anting I • CrtUr-M Jouinal. 72 (1870). I f BIrAlr. Maatk. inArwrnittr* Aler Jat Prinrip Jrr Brlnjlanff ihr JEiwrpfe. I t UerhH JU(.aa(<brKcU. April 1872 i I'kU. Mag., D«a. 1672, J^mv. 442 ACTIOK AT A DISTANCE. [85^ from tlic nction of otJier forces, and i imp', th« kioetio etier|*7 o( ti« portidr, wc find as the equation of ener^ 1 («_ J '^) v*+ i wa<Te+ r= cotatL (S») Siac« ihc ecooix] tvnn of the coefficient of t>* taaj be tocreand id- dvfiDiUily \>y inffri-n»in^ a. the radius of the sphcro, while the aarhcf daoflty a remuins constant, the coefficient of i^ may be mnde nfigstiv*. Accelurotion of the motion of the particle would then corrcspuoJ to diminution of its m n'tvi, and « body moving in a closed path and a£t4Kl on by a force like friction, always oppo«it« in direction to il* motion, would continually incrouM in velocity, and that withoot limit. This impossible reault iit n necefmry oontMiucnm oraaMiminii uny formula for the potential which introduces negative terms iato the coefficient of v^. 855.] But wo have now to conaider the application of WeWa thcor)' to phenomena which can be realized. We have seen how it gives Ampere's expression for the force of attractioD between twa elements of electric currents. The potential of one of these ele- ments oQ the other is found by talcing the sum of the valoes of tlie potential ^fr for the four combinations of the positive and oegativt currents in the two elements. The result is, by equation (20). taki&K the sum of the four vulue« of ^-7 it — » Mat - -7- TT' riti d* where and the potential of one closed current on another is -»'//;£^*'^'="'^- H M=jf'^dtd^, ac in Art*. ^23, 524. In the case of closed currents, this expru«non agrees with t which we have already (Art. 524) obtained •. _ ffW/» The«ry oftAe Muetim 0/ Electric Currentt. " 85fi.] After deducing IVom Ampfire's formula for the KctJnn lietwecn the elements of currents, his own fonnuU for the action between moving electric particles, Wcher proceeded to apply hte formula to the explanation of the production of electric corrents by • In the Bbolii of Uib iDTMiiijpiiIoii Wotxr •dopt* tbe dtElndynamln ■jnuui of unlu. In tU> tnuiae <*« dwajr* um (h* •IwitronMciiBtio tjttuta. Tha ^mtUo mH- lullc mih M cumoi b to tha deatrodjwmlr oali In lltr Tstla of VI to 1. Art. RSB WEIlERfi THEORY OF INDrCRD C0BRENTS. 443 P>nfl^cto^f«tric induction. In this he wag cminontly ctcppssTiJ, and we shsll ittdiratc tlie method l>y whinh th« lawi: of indiicccl currcntec may lie deduced from Weber's formuU. But vie must oUterro, tlint the oireumstance that a kw d«d»crd from the pheuo* Mmcnii discovered by Ampere is abt« alito to uccautit for the ph«no< mens afterwards discovered by Faraduy doi-s nnt ^v^ ko much additions) weight to the evidence for the physicul truth of thv Uw as we mig-ht at iiret sappose. For it has beeu shewn by HelmholtE and Thomson (see Art. 543), that if the phenomena of Ampere are true, aud if the principle of th« coQBcrvatioo of energy is admitted, (hen the phenomena of in- duction discovered by Faraday follow of necessity. Now W'eber'a ^law, witJi the various assumjitioos about the nature of electric ^RUTTent« which it involvee, lends by mathematical traafiformatioos to the formula of Ampere. Welder's law is also consistent with the principle of the conservation of energy in so far that a potential rxists, and this is all that is required for the application of the principle by Ilelmholtz and Thomson. Uence we may assert, even Ixiforo making any calculations on the subject, that Weber's law will explain the induction of electric currents. The fact, therefore, that it is found by calculntioo to explain the induction of currenta, leaves the evidence for the physical truth of the Eaw exactly where it was. On the other hand, the formula of Gnuss, though it explains the phenomena of the attraction of currents, is inconsistent with the inciple of the conservation of energy, and therefore we cannot < that it will explain all the phenomena of induction. Id factj ' it Gul« to do so, OS we shall see iti Art. 8S9. 857.] We must now consider the electromotive force tendinji* to Bproduce u curTcnt in the element i/V, due to the current in ih, when i/* is in motion, and when the current in it is variable. According to Weber, Uw action on ^c material of the conductor of which lit" is an element, is the sum of all the actions on the electricity which it carries. The electromotive force, on the other hand, on the electricity in ///, is the ilifferenne of the electric forces aotiDg on the positive and the n<^tive electricity within it. Since all these force* act in the line joining the elements, the electro- motive foi«e on ii»' is also in tliis line, and in order to obtain tho electromotive force in the direction of lU we must iwsolvc the force in that direction. To apply Weber's formula, wo must cu1cu!at« L^e varioQB terms which occur in it, on the supposiliou that the ACnOK AT elemvnl. //« ia in motion relsttvely to i«', aad that the cottnU . both rlcmcots vmiy with the time. The expicaawma tfaos fMiot will contain tenns tnToliiDg r*, r/, ^, r, v', end temM not invn'-- iD^ c or P*, bU of which are multiplied bv ar*. Bxaminin^, %& •^^ did before, the four tkIucs of tarh t'.Tm, ooii cottGideriagr firet ' mechanical forc« which artsr« fnin the mm of th« Foar Talnes, '' find that the onlv t«nn which we murt take into account is iL> iovolring the product rr'«'. If wr then coaader tlie force tending' to prodooe a canvnt in Uu epcond etcraent, artwog from tlie diflerenoe of the action oftlK Snt element on the po«itive and the nt^tire electricity of the stcond dnnent, vrc find that the only term which we have to examine ii that which iavolvea vf^. We may write the four termt indnded a S(r«/). Uitu e'(«+P,e,) and «',(w-fc^«i> Since /+/| = 0, the mechanical force arising front tluwe temu it z«ro, but the L-lcctroRiotive force acting^ on tlie poaitire electricity i is (rr + r, «,), and that acting on thfl negative electricity /, u eqiul and opposite to this. 858.J Let UB DOW snppoM that the first element dt ia moving relatively to </«' with velocity f in a certain direction, and let w dwiote by f'd* and fdt', the angle* between the direction of Taad tlioae of d* and of Ha reepectircly, then the tquare of the rdatiTe velocity, a, of two electric- particles is a*Bt» + t>'' + n-2pc'co»e+2ri;co«/^-2r«i'coe;^'. (2SJ The term in rr' ia the tame as in equatioa (3). That tn v, oo wbidi the electromotive force depends, is 2rpcoe/^. We have also for the valu4- of the time*Tariation of r in this case ' ir <fe .dr dr (2Sl vhere r-^ refers to the motion of the electric particle*, and -jr that of the niatcrial conductor. If we form the sqoare of this qoan- litv, the term invwlring re', on wliicli the mechanical force dcpcndu, ia the same as before, in e^joation (A), and that involving P, on which the electnKBOtive force depends, is drdr i* dt i '.I TBBEfiS THEORY OP ISDl'CED ClfRREKTS. 445 DifilTcntiatinfT (26) vrith respect to t, we 6ad rfw'rfr *'''';/^'*'^j^-*-dtd^ (37) ^o find that the t«rm involvings vi^ i« tho huiwi u before in (6). Utrm whose sti'ii alters with that of v is -7- j- • at as 8B9.] If w^ now calculate by th« formula of Oiiuiiii{c(iuatian(18)), Ite reetiltant electrical forco in the direction of the vecond dement f, aaiiiag from the action of the Srat element <U, we oblAio ^tiid/ir(2cwrdt~3c<MrrcMrdt)mar^tl/, (38) iAb in this expression thero is do t«nn involvini; tho mt« of ro- of the current 1, nnd sinoo wo know that the vnnation of ihe printary nirrcnt prodocett an indactirc action on the Mcondaiy circuit, we cannot accept the formula of Oaa« i» a true expreMioo ,of the action between electric partielea, 860.] If, however, we employ the formula of Weber, (19), we or drdr d ^fC±(l)d4d/ (80) we iot^iBte tbia op rc M J on with raipect to t and /, we obtda tbe eleetioinotive force 00 the weoad eircait d .fCXdrdr . .^ Now, wbeo the Gut ctretiH ii dond, riJrJr. CAdfir ^"^J'rdil/'^'i^iiZr'' JJ^d^d, d*r Bat .)^=-f^d..(Z2) (") dtd/ M, bf Art*. 429, 62i, Baace wa oajr write tiw daetsDMotin lone «■ IW which apcca with wkat we hare altaady flrtaMH^ad ty »3Su (W) 446 ACTIOH A DIWANOBT [861. Om Weifr'i Formula, ex>MMidered at rfnUit^from an AeSitm trtm*mlit4 from ome Etectrie Particle to tAt otier «itA a Cotitlaitt FeUcttj, 861.] In a vory int«n>9iting leUcr of Gatips to W, Wel«er* l» refers io the eleclrodytiainic specuIatioDs with wbich iu- had been occupied Ion; before, and which he would h^re puhlulu'd ifbecoaU tlien have estahlitJied that n-liich he considered the rc«l kefttow of electrodynamics, namely, the deduotion of tb« force- uctioff be- tween electric particles io motion from the considcntion of no actiiw l)«twe«n thorn, not instaotaaeoiis, but propagated in limr, ra a aimilar manner to that of li^ht. tie had not tiacoeedt-d in makiit^ thi« deduction wh«n he gave up his electrndyoamic rcAf«rtihet^ tuA he hud a subjective conviction that it would be necessary in the firiit place to form a ooDnEtent representation of the manner in which the propagation takes place. Three emtnent malhematicimiB have endeavoured to s«p|d; thij ltcy>.loiie of elect rodyoamics. B(I2.] tn a memoir present«d to tlie Royal Society of Gottingen in 1»S3, but afterwards withdrawn, and only published in PoirgeB durfl"<i Anna/en in 16t>7, after the death of the author, Bcrnhaid^ Biemnnn deduces the phenomena of the induction of electric rent* from a modified form of Poiason's equation d*F rf»r rf«r I jtF s? + ;v '^^ +*"'' = ?5ii"' where f is the electrostatic potential, and a a velocity. This equation is of the same form as those which espresa tt propn^tion of waves and other distnrbances in elastic media. Hie author, however, seems to avoid making explicit mention of anj medium throug^h which the propagation takes place. The miitbentatical investiffation given by Riemann luia been ex* amincd by Clausioat, who does not admit the soumlne*^ of the mathematical processes, and eltews that the hypothesis th»t potential is propagated like light does not lead cither to the formula of Weber, or to the known lawn of electrodynamics. 863.J Ctausius ha» also examined a far more elaborate investifpi- tion by C. Neumann on the ' Principles of Electrodynanuoa {.' Neu< mann, however, has pointed out ^ that his theory of the tranemiGsion fl of potential from one electric [larticle to another is quite different V fnim that proposed by Oaun, adopted by Riemann, and criticized • M>rdi IP. IWS. WtHM, U. r. t». KFl-STOSB OP EtECTBOBTSAMICS. 447 thy Clansius, in which th« propagatioa is like that of U^btL There as, on the oontntry, thv grvaleBt possible difference between th« nnsmissioQ of potential, accordiu;^ to Neumann, and the propaga- tioD of li^htk A luminotis body tciwh forth hf>bt in all directions, the intensity of ivhich dvpends on tho luminous body alone, and not on the presence of tliv body which is cnlif^htencd by it. ^An ele<-tric particle, on the other hand, sends forlh a polcntinl. «/ I of h ■ at be value of which, — , depends not only on «, the emitting particle, hill on <^, th« r«cnving particle, and on the distance r between the particle* at lAe irutant of^tMum. In the case of light the intensity diminiahea as the li^t is pro* ''pogftted further from the luminoua body ; the emitted potential flows to the body on which it acts without the slightest alteration of its original value. The lig;ht received by the illuminated iKdy is in ^neral only a 'raction of that which f;ill9 on it; the potential as received by the attracted body is identical witli, or equal to, the potential which arrives at it. Bf-siiles this, the velocity of transmission of the potential is not, like that of li^bt, constant relative to the aether or to space, but nther like that of a projectile, constant relative to the velocity of the emitting particle at the instant of emission. *It appears, therefore, that in order to understand the theory of Neumann, we must form a very different representation of the pro- cess of the transmission of potential from that to which we haro been accustomed in oonsideiing tJie propa^fation of light. Whether it can ever be accepted as the ' eonstruirbar Vorstellung ' of the process of transmission, which appcjired necessary to Gauss, I cannot say. but I have not myself been able to construct a oonsieteDt mental representation of Neumann's theory. 1864.] ProfesMT Betti *, of Pisa, has trented the Buhject in n iffercnt way. He supposes the closed circuits in which the electric orrents Sow to consist of elements each of which is polarized periodically, that is, at equidistant inter%'nl8 of time. These polar- ized dements act on one another as tf they were little magnets wbow sxoi are in the direction of the tang«nt to the circuits. The )>eriodic time of this polarixulion is the same in all electric «ircuit«. Betti aujipowK the action of one polarized oleineut on * .Vnom Ctmmlo, civU (IMS). 448 AT A DIBTAHCB. MiothOT at a distance to take place, not instantaneoosly, but ttfia a time proportional to ihe distance l)etwe«n the elemeats. In (in way he obtains espreaeioos for the action of one electric circuit ua annthor, which coincido with thoee which ore knowD to be true. ClauNintt, however, has, iu tliis caec also, criticized eouie parts of tlie mnthcmaticiil calcuhitionti into which vrc Hhall not here enter. 865.] Tlicre nppmrs to he, in the minds of these eminent mo, some prejudice, or i priori ottjectioa, againitt the hypotiieds of ■ medium in which the phenontena of radiation of light and bnt and the eleotric actions at a distance take place. It is truo that at one time those who speculated ox to tlic miM-ii of physical pheno- mena were in the hahit of accounting for each kind of action at ■ distance by moans of a Hpecia] lethereal fluid, wfaoM function and projierty it was to produce these actions. They filled all spsw three and four times over with tHhers of difTereut kinds, the pro- perties of which were invented merely to ' save apitearances,' so that more rational enquirere were willing' rather to accept not only Nov- ton'si definite law of attractinn at a distance, hut even the dogma of Cotes*, that action at a distance is one of the primary properties of matter, and that no explanation can be more intelligible than this fact. Hence the undulatory theory of li^ht has met with much oppoeition, directed not againtit its failure to explain the pbeno- mena, but against its assumption of the existenoe of a medium in which light is propagated. 866.] We have seen that the mathematical expresaionB for electro- ^ dynamic action led, in the mind of Ganas, to the conviction that theory of the propagation of electric action in time would be foundl to be the very keystone of electrodynamics. Now we arc nnablflj to conceive of propagation in time, except cither as the flight of material substance through space, or as the propagation of a con-"| dition of motion or stress in a medium already existing in apac«. In the theory of Neumann, the mathcmatkal conception called Potential, which we are unable to conceive as a material substance. is supposed to be projected from one particle to another, in a manner which is quite independent of a medium, and which, as Neumaaa has hiniself jwintcd out, is extremely differcot from tliat of tlie pro.i pagation of light. In the theories of Rieniunn and Betti it wonldl appear that the action is Bup[iosed to h« propagated in ■ mannuj aomewhut more similar to thai of light. But in all of these theories the question naturally oocnrs : — I • Vitlaoi to HewMn'* Primofia, &id AUiioa. 866-3 MEDIUM NECESflART. 440 •otnethiriff is transmitted from one particle to another at n distance, what is its condition an«r it has IcFt the one particle and bcrorv it has reached the other ? If this Komitthing i>; the potential energy of the Iwo partietes, as in Neumann's theory, how are wc to eon- ceivc tlii» cncffiy as existing in a point of upace, coinciding neither with the one paHicK> nor with the other ? In fact, whenever energy it tmnHniitl^ from one hody to another in time, there inuxt he a laedium i>r Kub^tance in which the ener^ exists after it tt-avef one hody and before it reaches the other, for energry, as Torrieelli * remarked, * is a quintessence of so suhtile a nature that it cannot he contained in any vessel except the inmost sahstanoe of material things.' Hence all these theories lead to the conception of a medium in whiuh the propufjation takes place, and if we admit this medium as an hypothesis, I think it ought to occupy a pro- minent place in our investifjationii, iind that wc ought to endeavour to coattriict a mental representation of all the details of its action, and this has been my constant aim in this treatise. • L^tmi Aeeadmtcht (Fir«iu«, tTK), p. S9. VOL. 11. Qg I 1 ^fc'iBFtios. cl<ctric, 63. 227, 828. — of liglit. 788. AocumuUlun or onndcnnmi, M, S:2a-SSS, Action at ft diUanM, 103, 641-640, 816- S6A. Aayelio region, ID. US. ^ibet, T82 n. Abr. SbG. B., 4S4, S30. Am|W«^ Aulnf Muis, 483, 90S-5SS, e»S, «8T, 833, 84S. Aoion, 337. Anodes 33". Anson lUill, fiSil. 6M. AaUde bkUnoii, SOI. Atn»it[divric •■•ctririt]'. 221. AUnution, elnotric, 27. 3». 103. — ctpliJnod bj lUtB in a malluiii, I0&. Bu^lsf and UilxHiii, 9!P, 789. Batten, vcluuc, 331 lhcU.W,. 26S. aeS. 4)1 Battt. E., 173, KOI. Bifilar uuMiMioii, tit. BbmuUi, 415. Borf*. J. C, S. Bowl, fliheriial. i:*-I8t. BrLit-. »Vi"*t*unM->», 84T, 7SB, 775, 7T8. >— •Icotmtiilit, 2r>3, BriEbt, (TlrC. and aark. SSI. U7. Bio^e, Sir a (.'., 36U, firoun, Jolm Allan, 182. Bnub. !>a. lufT, Hdnrtuh, 371, 368. (dMtiMtMio), GO, S36. VeMkdMiaM, 60, 87, 1D3. IQS, S37- 771, 774-71W. GiiHtclty, calciilalliiri of, 10^ iSA. — uiivuiirviiiPiit of, 227-22S. — In (itectnunoLiiDlLc Iitaa*B». 774, 775. Cufmdty (eleotronugnotlc) <il k omi. fOt, 760, 778, 779. OUh.>tl». S37. Cation, 237. Cauohy, A. L., S27. Cantuliih. lloory, 88, 7*. (^;1ef , A.. aS3. Ceutrulmric, t>8. UtruuitM. alKtric E7e-fiS4. Ciroular «UTr«nla, 694-70G. — wlid Angle (ubt«id«d by. dSS. dutrgo. (tcctric^ >l. Ciarii. iMiautT. 8SS, 62P. 71i. QwHllioLliiMi uf iilootrio*3 iluantitiw, 1130- esv. Clauiiuf. R.. 70, 2SS, 8$3. C«iH)«iont* of elsctrnaUttic cajiaclty and (niluclton. 87. BO. 102. — tif iKiUititial. H7, UO. — nf r«ti*Unvi> ami oHniluvUrity, 267, — of indund ma^fiiDtuation, 436. — of «lcctnnna^otic inducUcpn, 7S6, — »r Mlf.iniliK^Cion. TStf, 7i7. V-orfomt ri)««, 421, 44(. l^oiU ranittuiiw. 33G-3II. — «l«DtnHiugnetic. 691 706. — mcvMuremont of. 708. — cniuparliun af, 763-767. C<iiii|i*ri«n iif eiinuille*, 229. — ..fc"il>.75S JS7. — ornlHrlniiiKiliia fomM, 3S8. — . of rMUtADCC*. 3l(i-3CS. Connntntion. 26, 77. Condtnter, fiO, 226-SS8, Sir CIiuIm WtiivutMno, in Iiit pH|Vr on 'N*w luatrnmont* and CivceBn,' I'iSl. PM, 1813. btmiijbt tbit att*n^inonl intb I'libtic nolioa^ villi ilua acknonlfit-^ment ot Um orijjiul icvrnl'ir. >lr. S, II iinter Cbri<ti<^ nbu liad dMsribwI it i» I'U l-tprr mi ■ladsMd CuiHDia.' Pkii. Tmn:, 1833. no'ler ILq name of a IKfffirTiiIisl Arnuig*- Bicnl. Sm tlw rnuarlu of Ur. tMirant Clark In the Sttittf <^ Tttegmfk t'a^latm, May », U:i ag2 i IXDrJT. HlN. «rtl««nfcdlU N. fcffciM. U7-1M, US. »t 3SX.MT. US-SOC pri-^d. -f » Ma. 7M, Tsa, I «( CMIT, »^ 3U; 3«X M3. C«iUatt fara^ tic CMi*«tM. ss, ssai u*. Cotvv, ftl, S«h S«S, ;«i. Codnodv C. A, Sa, 71. 315, S3. 373. GMlMib1*lMr, ;».W. Cryul. oMdMte li, WT. — WH***^ iniMHkB at. 435, 43(, 4S3. ■ T W f^fWioa <f %)it ia Ik 7>i-;S7. jJmm,3it. CniSS. CwnM, ikctricv ISO. — bM MMkod «(a{^fln(, T4L — iadacwiSSl — •t«dy, S»2. — tkcnu*l«(rtri«, U»3H. — MMenl, 331. UO. KM, U7, SS3. 748. 748. 780, 771, 77». Camat-atajghar, 7M. C^Ma npoa, U. 113, 481. Cjlind^, deetriftMtlao o( IM. — iMgMtiMUaa oT. tM^ 43^ 43>. — tmnwBMM ik. <H2-«»0. CjtlBdrle oculi, <76-<3l. Itaa>p«l TibntiMv 783-713, 7*1 Owi|Mf', 78*. I>miMI'>mU,1S%S71 Dc^ bcu piT— cM«l«r, 74 1. Dtcnmoit, loguiUiaic, 79C. MUilo^ 4S3, 743. MUanln, J. IL J.. 3. I)«lluuuii>. I'., SSI. UEiwity, cbxtrio. Bl, — 111 A cuncnt, 3sy — nouaraaunt << 313. IHaMsMtwD, tV. 440. S3S. IHMMiric, a& 103, HI, 339, S35 381, HHi^.37<t, 7SI. t»-«a. Willi ■111. •teAw. M. ;s. 71. lU. »-»«, «oe. 733, 7*1. S,ll<. IteA, 1 i^iiiiw ■< iC3-47l. Brwk,M. 31. 1». Xll. X3S, 3«e, 95, »*.l 33*. »-67. •0, 73, 7«, til, SS4, MS; 7<3, 7)1. — M>K7 M. — I»«-,6S. — !LlacttB«.S«. — tMidM«.M7. iinHMfal 70 — .faA,B7. — waiba, 4S. B9, 107. 108, ni. — •w.ss- ghafrwiy—fe ijatf of i BiBB ttl yna ww l T. 73S. nMtDlp&i, 330, SS5-373. BaBtraljtav SS7. 3SS. Klac«nMM anHhiCFiMn, 3S3-27^ 7»». — pobrbuk^ Se7. W4-373. Bkemtaacaaiw fotn^ 475, &S0. S83. — OMMOnauBl. 40S. — ■■■u^lBm. »S5. — Bta mati aai^ 730-780, 4 ' unI Jntti iwu tk anllB MauM«4 7Ba. — raUlLa, 491. Ei«em«aaEthin, djmnieal (1im*t «C S38-577. Elaetranwtcn, 311-320. ElMtfDOMiiv run*. 49, S3. IJl, 34«-3S4. at. SOS, t». EUcuut-hom, 309. eUc<i<>«»p«b 33, 814. EbotnalalH lamnnaiiwto. 21i-B ~ pofarisiaon. B&. 111. — sUncUon. 103-111. rjtuai of iBiUiv ^, A«> Eloctrutouk *UI<^ 540. I£ttas>i>u<>. 734. Bli|i««l. IM, 303. 137, ISO. KUtptio inltrnk 140, 137. 70I_ BMfgj', 0, &, 03O ««, 7'-*^ 7ine. ^^^^^^^^^^^^^^^^^^^^^^^^^46^^^^B HSqiia^uua of ooniluctiTity, 39$, 609. Glut, 51, ^^H ^t-- of coiiibuil;, 85. Glov, oleotric, 55. ^^^^H ^K~ of cbctrio ouRoiU, 007. Grawuuum. H., 526, 687. ^^^1 ^^- ot (oUl eumnlii, 910. Grating, electric clTMit of, 903. ^^^H ^m — 0r«l«lr<Ha«4[n'^>o fweo, 603. Gmri, UeorEiv 70, 84, 318, 439. ^^H ^ — of aUobQiDoliia tator, 693. Gnoti'a function, 9S. ^^^^| ■^arLaplMr.77. ~ tiuanm, 96. ^^^^| ^L^of nagnctiMtloD, 41)0, dOB. Gtoovo, elwtdo «a*ot of. 199. ^H ^^~ of mnatiello Induction, G9t. ■ — of PoJMn, 77. Gran, Sir W. R.. 372. 778. __^M Glinnl-ring. SOI. 217. 2'J6. ^^^H V— of r«ii«taii(w, 397. GulU-pen-'hH, 51, 3U7. ^^^H EijuUilriuni, poinUoT, 113-11*. ^^H lluiiiluai. Sir W. Rowui, 10. 561. ^^^| VunilHy, Al„ liiit diionvnric^ 53, Sn, 23G, Hani imii, (34, ^^^H Harri». .Sir W. Sitow, 36, 218. ^^H 2S6, S30, SSI, S3<, 646. 668, SOS. Uent, ootnluotii-Q of, SOI. ^^^^1 — hii experimmu, 3S. 4i!9, 930, S68. — gcnenlcd by the ourrviit, 342, 2SS, ^^^^| — tai tnnbodi. »7, !3. I±i, i03, 528, ^^^M &2», fill. £»'i, r.0(. 604. — «pMdfltvofolMtridt;, 213. ^^^H — Li* •ju-uUliuM, S4, 60, S8, 107, 109, US, 439, £03, £10, S<7, 569, 615, 762. H«iliO, I3R, 140. ^^H Helix, &1S. ^^^1 F»d, 6'ifl. Helmholti. U., 30^ 431. 543, 713, 833, ^^H Foctuur. G, T, 3S1, S74, 6(8. ^H roltd, R., S36-&>g, 669, Hnlcriotallc electrnmiolen, 818. ^H teirwnn^ 138, HO. KTcnoaugnatfc, i^, 139. S44. ^ IVW, olectriti, ■*). Hix^kin, C'liarlct, 353, 36u. 800, ^^^1 ItctU. W,, olotOiicail mMhiDO, 313. ^^^H Ilotnitein, Kairi, 471n. ^^^^| — deolmiiiagiictic', 5SS-61I>. Huygoni. Chri-tian, 783, ^^^^| — of onirarm fmea, U7'i. KyilntiiHe ram, 5S0. ^^^^| Pirn (wjng. 7(5. HyjKjwliie, 151. ^^^^H iriBoa. H. L., 787. ^^^^H ^L flaiiL aleoUio, 36, 3*. H>^ inociiiit>T«OTi1<lc^ 61, 111, 2E>5, S29, 331. ^ — m«<Mllc, 3S0. FlM/ia. ^^^^H IdlnnUitlc cIrctroiDobin. 318. ^^^^H \-ai»tt^ elmtdc, llll, 155-181, 189. ^^^H — uin^Mic. 318. ^^^H Farc«. cUatmtni^cticv ITS, 580, GS3. — niovinj;, 663. ^^^^H ~ dKlnimolivo, t». Ill, 333, S4J, iMO- Inmgitikrjr magnctlo mBtbr, 380. ^^^^| 364, ZiS. 569, 379. 5uA, 608. loduool cnrrtaita, 528' 553. ^^^H ^•^ nechuiical. 69, US, KlS-tll, 174, SSO, - In a plana ebecl, fiSOdtfO. ^^^H ■ 602. — WoMr'n tboorjr of, ^56. ^^^^| ^H — luiaiiiiruiiiont </, (1, IqiIocoI ■na|{n»tia>tiDii, 4S4-4I8. ^^^^| ^K — acting >1 * iliiUniv, 103. IndaoUon, vleclnvlaUr, 36, 75, 76, tlL ^^^H m _ Bnwtt 83, 117 123, *0(. - — nu^eticv 400. ^^^^| Foomuli, L., 767. luertll^ olcolrio. 550. ^^^^| Foumr, J. B. J,, £., S4L 33-2, 333, SOI- - mmnunU and iirodiicti <£, 56S. ^^^^| H W5. Ini-iUUn, 31>. ^^H InvrreioD, 'loctric, 16^161, IM, 314. ^^^H Ion, 337. 255. ^^^| GitraMOMUv, 210. 707. ^^^1 ^ •- diOnUkl. 3(6. K -- Muiilinv 717. ■ — •Imk1u<1,708l poicliloTld* of, 800. ^^^H IrroauiuiUablo outvc% SO, 421. ^^^^| ^^^^H — ohMTvMfon oC 7(I~7SL ^^^^H Gasov otMrln* iliictiarge in, 36-77, 370. Juobi. Kl. H., 336. ^^H — iwiiMOM of, 36S. Jonklni, Winiam, 546. Sto Phil. Uoj., ^^H Churfe*. J. P, C7. 1834, III. SI, |9l 351. ^M Jcnkhi. rlwiiiini/, 763, 77*. ^H OmHStla, J. M., 3<6, 712. OBIItc* «Ue4naH4«r, 318. Jcichnuinii, K, iAS, ^H OaoHt C. >'., 16, 70, 131, HO, U«. 400, Joiilo. J. p., 344 363. 448, 4fi7, 468, 796, __^M 421. 45(. 45», 470, 706, 733, 744, 851. ^^H Gcomcttic nioan di«*ate^ 691-693. ^^^^H G«o)nob[7 of iHxilluD. 431. ^^^^H Olbwa aod Iluv-Ui, S39, 76». ■K«}rien«ufdMt(oajraaii>lM,'Ml- ^^^M IT" KInetiw, 969-565. ^^^1 454 iwwby: KtTcliliolT, CxtUv, 282. Sie, 439. 7M. KoblnuKh. Ituaolpb. 3a&. 965, TU, 7f 1. Urnf. G.. 17. Ml. LunGllaf rMAot, 412. [jiplaco. P. a. 70. LopUon'i conlKciunti, lSft'146, — «<ioatiOD, Se, J7, SOI. — MpuiBon, I3S. LaBsndn's cMlItoient. 13&. L^nlu. G. W.. Id, 4!t. L«iu. E.. :iijG. a;<<>. fits. Uglit, «lrt;lr.iiu»ftnr|jc tIit>Di7 of, 7Sl -SOB. — - and iiu>L:t»'Li.<ir>. 80^^31. IiiniMlQiuiiv. 04, SI. — •intognl. lA-30. ^ oT electric tbieo. tH, tli. — of iiuicnalc rnn'*, 401. 4S1, 4D8. 1U». 5»0, 800, 60T, 633. Linn orequOiliriutn. 112. ~ or flow. 33.203. — of •loctrio IndndlaD. 82, 117-123. — of niMipiMic lailucUun, 4(14, ifV. It'i, 641L5S7, 702. LituiBUB, C, 23, liouvillc. J.. KS. 17S. LiKlntr. J. B.. 18, 23, 421. IiOmii, L., SOG n. • LoKhliiidl. J-,i. Maitiwci7>it«Ulc (ihenDQieiui, 4'JS, 43G. Ma((iiv(, lU |inip<atliM, 371. — (lir*clH>ii vt BiU, 872- 31*0. — nt^vlio iniiniriil ^f. 384, 390. — MOtrv Bad priDcip*! mlm, 353. — pcrteniiril anergj of, 380. Mutictic kctton uf llsht, fiOO, — itiaturbniiDC^ 473. — (orw, law of, 374- (Itnolian of, S7t, 493. Inlcnrity of. 4U. ~ Iniliicliuii. I<i(>. M^^tio ' nialtur,' 350, — mvMurBiuciili^ 44(i-4<14- — p«lc>. 4SS. — wjircy. 488. — varUll'ino. 472. HafiMiiHii ofiiliipfv 441. — (vrrwLrial, KJA-174. Mwwtintkii, coiuiiDaeolB oC 881. — UidiMed. 424'43U. — Aiop^'« iLeory of, 883, 333. — P<4wnn'ii IhoiBy of. 4SR, — Wawr". Ofirj of. 442, 338. Uairna*' ((>^ 1*«, 251 Uanco'i, IlcBry. nut'i-'l. SG7. MaltblMBii. Aui-.. 3.V2. SBO. MoMiiiviHvnl, Xinxtrj of, 1. — (if riistria l(«otv is. — orolMtraaUdo etfmMn, SSB-SSO. MeanmmMit nf DteotonnotiviB fona potcBlul, 818, S&S. — of naiiilanci!, X3G-3S7. — i)f oiinHtMil vurnaU. 748. — «( tniwlfinl oiiirtBU. 748. — ofooii.. 708, 75»-7a7. » magnecic, 449 VH. Mnliuni, eltctfomafiictic, S98. I uiiiiiiifcr(,iiiii. 6CI9. Mlireiiry, rntiiUnot of, ML .M rula. KwtUooe of, 383. MjohoU. John. 3«. MlUoT, W. n.. 28. Mirror OMthod, 430. MoWulu' cliar)(« «r tloclricily, S59. — (lunnilii, 633, — ■Unilarda. 9. — vortlMf. 832. MolaeuU*. liaa <£, h. ~ alMlrie. 300. — magnvlU-, 430. 893 34K. Moraont. ma^tUo, 334. — of liwnia. 589. Moiaaiitliin. 3. — alvcinikiiiDllc. 578, S8S. MottcUi. <i. V^ 6± Motion, «if iiatiiHU of, M3-945. Movins kin, 300. — ooadu«(«n, COS. — tRu«ai. ees. MnltI|>lo cniiducKoi^ 3T8. S14. — FuiHTtiiaif, P. Mullipliotiaii, MMMI oC 747. 7M. Naatuann. F. %, eodBotcat of aa^aUiia- UoD, 4,10, — niagnatimtion id «Ilfp«rid. 4SB. — tiiODrjr of indund concaiU, 943. NamuMui. C. G . lUO^ S30. 383. Ntehoban'i [t«n>lTln|f Dovbkr. 20ft. Niokal, 429. Una lottho!'^ 114. 348, SOS. f)iwud. a C «*», 47S. OI><i>.l^.?t-. 241.338. Ohm. Uw, ■.iii, 340, 9S». OpMilr. 788, OTatj- sllitwiia. 153. Puling, A.. 304. Parabulaiiit, oonGical, 154. Panmusnxfc (lanie aa FnrannvMilo);, 439. 428, 844. PdUor. A., 248. Poriiidj,! niBcUue). 8. Poiplanrtio Tttfim, ti, 118. Pnmu*l>ilitt, iiBii{wtk>, 428. ni. pbimp*. a L, 342. Phn of Ui* Ttmiiw, 99. Piano curTMt.alw(4, 894-««l. llaKUiy oUlraoU, 131. ^^^^^B^^^^^/jr^j^^^^^^^^4s^^^^ PUtymetfff, clcetrcp-, 22P. Rloiniuiii. BemhitriJ. 431, SII3. ^^^^| Pincl[<T.J"llu». S3l'. itijflil uiJ 1nft-haiiil*d ■ytt««iu of SM* ^^^^H PdIuIh of ti]ulUbriiim. 112. ^^^H P«i-»on. S, J>-, !.'..'.. 431, 487. «74. — ciTa<iiiulr'|><ilaria«d nyi, 613. ^^^^| FauaoD*! equKtioD, 77, 1 49. Rilohie, W.. Hi. ^^H Paiawn's tiEiMrjr of laagiivUim, 137, 430, 431, 441, S3S. BiiCot'i ^J. W.) SctDDFlary Pile, S71. ^^^H DntaUoii of pliuiu of pDlarimCian. 808. ^^^^H ■ — iheary of vavo-prnpayatjon. 7SI. — ma^utlinii, a iiliiiiiuiunuan of, 631. ^^^^H PoUr diUuiliiin uf uutEiielio (urou, 3I>S. Klltilniaiiii. K.. 37i>. ^^^^| Poliuil;. 381. RiiIp ol clooljomiigiiatio dlraoUtm, 477, ^H PoLiriattiDti. dwtTiMtttCio. M. III. li>4, 196. ^^M •~ •kotfolvtio. 3:>T, ^4-2;2. — mMtwdo, 3»I. ^fl — -f IlKhl, SSI. 791. ficaiar. ^^^M — oiiwilw, SI 3, Sc^ln f<tr HLirror iilTHHrrati^rni, 450. ^^^^^| Pol«* «f a magnst, 373. Sn-torial li.iimoDic, 133, ISS. • ^^^H ■ — magiwiie of Uib urth, 469. Hcebeck. T, J . 350. ^^^H PodliTd and motive. liiiDTflntiani about. Selenium, CI. 363. ^^^M K 2S. ST, an, 37. 63, t!H'81. 231, 374. 'i^i. ■ 417. 489. 498. ~ Piiieotial. 16. ti<.'K-[ii<tucti»u. 7. ^^^^H — miuftimiiniii (if, 75S, T7A, TTtt. ^^^^| — coil ofiuuiiuuin, 700. ^^^^H — vkMrio. «5, TO. SiO. SL-niJti« gntoiiuiiieter. 717. ^^^^H Sixic* of otHorratitnidi, 74U, 750. ^^^^H ^ — aasMUc, 3«3. 391. ^r— oCtwu ciivuita. 4'J^). KImU. mntpcilc, 409, 4S4. 485. 806. tH, ^H 070, 094. 0W«. ^^M — of tun <M«*. a9S. Sl-iri«ni(, (;. W.. 33fi. 361. - ^^^H Potential, voctw, 40B, 48!, 590, 617, Sinn, nit^tliinl uf, (55, 710l ^^^^H 667. Sin^tar poinla, 139. ^^^^H Prini^|<«l axca. 29S>. 303. Slope. 17. ^^^B Prublnnu, •IwUflntadc, ir>t-SOS. H— •iMArekiotiiliiLi':, 3<9H.;1SS. ^P^-nu^stie. 131-441. Silvio. A.. 372. ^^^M Stiiilh. AnhitnU. 441. ^^^H Smith, Vi'. K.. 1S3. 316. ^^^H — (iMtromagnttio. <J4T-706. >kvp bubble. 125. ^^^H Proof of tha law of tlw lavatw Squiin, Sa1«ioiii. iragncllo, 407. ^^^^^H H 74. ~ aUcttiolnn<rn''l'c. 070-6^1. 727. ^^^^| ■ PnK>rpEanv,233. ttolMlDldal dirt n Initios. 21. $S. 407, ^^^H Kl Solid Mf-I*. 400. 417-4S3. 4S5. 69K. ^^^H K SpawvariatioQ, 17. 71. 639. ^^^H Spatk. 57, 170 ^^H Qoailric uirfKOT. 147-1G4. SpcfiiAc induetlra oapadtj. 53, 59; 94, ^^^H Quaiitit)'. axpcwdMa Iwrm phjrukal, 1. 111, 32V. 32S, 334. 637. 763- ^^H QuaatitiM. «l*iinliuMi«a of elcatnmu^ — oonductiTiiy, 27^ 637. ^^^H Brticeao «29. — midanot^ 277. 637. ^^^^| Qaatembma, 11. 303, 490. S2S, HIS. ^ baat of eloctririt;, 2Si. ^^^^H QuJuke,G., 316 1.. Spherv, lis. ^^^H Sphcdnal oanij union, 145. 146. ^^^^| SplnricHl Lanuimioi. 128-146. 391. 481. ^^H Badialian. faroM anovmed In, 799- Spin], lomriihiiitc. 781. ^^^^M IUiildoe.W.J.M.. 115. CSI. tJtMliliifl vlectruniftn-, 317. ^^^^H BayordoctruiiwfftinEicdldurbanoa, 791. — gftlvanomotiT. 708. ^^^^^H Bacipneal propuliL o, vlKtmalktic. 86. StokM. G. G.. 24. 115, 761. ^^^H — BlMlTDhilWInntio. set, SIS. filuoar. Q. J.. ^^^H — DtoKDacKv 4'il, 423. Stntiflvd cniMliiDinn. 319. ^^^^| •— dtcirnuui^Detie. 538, iStrMi. rlKtrvoUtio. li>5 HI. ^^^H — kinollc Mi. — dwlrokiuetic, «i 1. «I5. 016. ^^^H ItaoAX iiiolliiid «f. 750. Suult, Hon. J. W.. 103, SOS. ^^^H Rfaianal cUiyK 3S7-43I. SaiUee InloHnl. 15. 21, 75, 402. ^^^M — ilu^atix*li"D, 444. d«n>ity, 64, 223. ^^^H Bcfiltaiitfaer. SIO. Hiirfnur, iKiutiiDUvlial, 48. ^^^^| BoiiUtuw of cuDiluctani, SI, 4TB. — •leotriHnl. 78. ^^^H ~ t^lo* of, 362-300. Suipoidod coil, 721';29. ^^^H — MDnlioiin of Sn7. Supciudan. brtiLar, IS9. ^^^^| — .laalc'i, ^^^H — unit «f, 758-767. — «lM<iiM(atia mcaaura of. 353. 780. — TbonuDD'a, 721. ^^^H r"--" — untfilu, ^^H ^^^45^^^^^^^^^^^?^^^^^^^^^^^ ^^H 1W)lM«feaefB(iMitoarfto«IL 700l TtatriMl MnMli. S33, SaO. 5W. Ot. ^^m _ ti dtniNuicMu, i2ueat. Mt. 74^ 76S, 760, 771, 778. _| ^^^H — of •UctronintJVB (bret^ 858. ■ ^^^B _ of mjgnulic rnUliun, 8S0. ■ —of ^Mi^1»aa^ MS-8dS. — tfam *—■'——'-'. 8. ^^^1 ■ — of vrlodty cf Ugbt utd of •laotrao^ — ikriTod, 8. ^^H ■ natfe dUtarbuM*. 7S7. — obctn-bami. 83S. ^^H H — of l«M(KtV]r •ail nddiud n^stUiB- ~ n«ii»tick 374. SS5. ^^H ■ tkm. Hi. ■ T^t. P. G.. a. »1. SST. 5SS, 067. 7»1. H l^ncDDi ^TMiaiiMMr. TIO. — eloctndjrnaiBk. S9S. ^^^H ■ TuRanu, BedMd cS. 4S4. 710. — lOMtlol, <S». ^^H ■ Td«Knf>l> <*>>H SSS, SS». — of i«iMn««k 7Ba-7«7. ^^^ ■ TiMfoiy m^MtiHttloo, Ut. ^k VandoK. •botratatu^ U. SB. 107, 108. — nik* of Iha tvo i^nK^au, ;88-78«. ^^H — olaHwignBlift MS. ei«. ■ ^^H T«irMtrUlBWfiiotu«, 403-174. Tari»tkaorB»|iB*4i«*hnMBb^«;2. ■ ^^^ Tlnlkn, T«Um RoIniI, 43U. VaHoy, C. V., 310. »7I, 882. X0. ■ ■ Tlwcm. 0*«tn'i, 98. Vtetar, 10. ■ ■ — Ewnduwy 11*. VMtof pototUL 40&. 4S1 S»0,817. <ST.^ ■ ~ CkNilenb'ii. 80. ^^^ — TboauoD'a, 100. mUmmn, 888, 828, 7&S. ^^B — Omm'. 400. br Um ntio «f >leotri« DalUL TU. ^^^1 TlwMj ofamfluld. S7. 760. _| ^^H — of two fluids 3S. — ofd«cWoaMgMacdiatiirituc*,784. ■ ^^^B — of MMgnetio aMtor, SSO. — oriigbt,787. ■ ^^^ — of mpinr molotiBlm, 4S0, a33-815. — oflha cleetrio buiumI. S60. ■ Vttdx, ILS, S0«, 8S0. ■ ^^_ — of iual«ca)v voUcim, Vi-L VTbnlk., timt of, 43«. 7SS. ^^| ^^K _«f KtM««lmab«MH94l03,a41-«U, Voll, 639. ^^H ^^P S4S-SH. Volte, A., 24«. ^^H Toltunour, 337. ^^^| ^^H IlKaMin. EHrWUMii. Vortfa^ MelwuUr, 83S-8S1. ^^H ^K — fInUk imMcn, 41, ISt. 166-tSl. ^^^^1 ^V 178 Water, miMaaoa of, $<5. ^^H V — «XMriw«>l*, 51 . S7, 248. SO. 77^ VTaic prop^Blion. 7S4, 78S. ^^^H Wcbor. W, Ull, S3S, 8«>. ^^H K •- Imwobmbu, 127. -201. SIO, 311. SIS- ■^ to, S73. 7SS. rS4. ^^1 — mmrtliB, SIS, SMt, 4M, 407-41 1^ — Muotd napwUtm. 443-448. 838. ■ ^H — cmU ofmtoamcw, 780-;e3. ■ ^^l — MMtaBM^ 888. SSI. SS8. TS3. — niU»«f clMtm mitt. £37. 771. ■ ^^M — it>Miio.«lMttti9l7, 907, 341. 349. MS. — cleeirDdnMDic fonfcaU, M0-$61^^^^| ^^H W«(llM4in. W . 447- ^^^1 WhaaiaiMMTx RHiW 347 ^^H V — tk«or7 of dectrieily, V, 37, «», 831, — p|~:t™«aiif. »is, 758. 7T5, 77«. ■ ■ S»«. Wl>c«(ll, W.. 337. ■ ^fc — vortax notion. 3)\ 487, 70S. WledeaHm. d, S38, S78l **$, MT. ■ ^1 ThaOMOD Md Tfttt'i Nfttitnl PbOoay Wind, docttfe. U. ■ ■ ptij. 128, 1S9. t4U. 181, SCO, ii», 870. Wi|» 779. ^^ Wccii, 8. ^^H ■ Timm.pnMOBatTibtMiim.iSe.TiS. ■ Ibio-nUvral. B41, 568. ^^^^H ■ TodlxiMr. lU, 140. ^^^^1 K ToRlodl^ BTMitolMiv H8. Zaro Ta»!!)w, 735. ^^^| Zonal hMHinio. 138. ^^H 1 i BSO. ^^^M IPILA^H^ VOL.1 1. ^■: i /■fajotfl'iSS^ctn.-.-ii'/^. Fol.JT- FiO HV. Art 388. Av CyUruiers nu^mttxmi trtuwt^sefy. fiir Uie Del^aUs of the CUrmdon Prtss. [•'ii^ m IS- M I, .'I ,' ' I ■a I hi I : vf vf-.-»i,>-»i-..-.--r, ^r/T-.T-V I,: ..^ Tro, XV Art 434 : = =f — f -i C =? -44 I _-- 4— r— t - , .1^1 1 1 ^, Mr- w ^ - mto J ■^- — 1 Vv^^^^'^ - i ■"7^-Tcr/>!f^''"^3-Y^ ^ f^ tS<^ M i r— f — h ^J^JT'-'£j^xx^*^ry^...- ^^^g^^'^^+'^rn " I „ — T) ^ tii-YinnZ , {{] lit ' m4l '\ \':\ 1 '- 'TTxfciil ' ^MJte"' ~ —■ - — \ -iU (C IXV^c^^^? ,._ _i m \^''^\ ' '^<j x ^^ I i ^ N^^^Vf "^>f"'""t "'~~Z1^ N^ /\ e /T--i/ ^~~h~l 1 I V^^A''^^^ \^^^^ ^ ■ * — t ^ ^S xu ^n-huu: t — 1 \ \ ±T' 1 aTff4- t -i — 4- 1 -A ^ -Tl — ~rHT^ — i~ - .'S/ .'"<- ,Ai;\;jLf'.'' ■■'* ;(,.: V.iTr.d..;!. i-T?,;s l: 1 ■ Tu-^^wtttr t 'OSr\\%rv\^??rZ.' ^_- — ^ — 1 \ \/\ ^tK X-Vti 1 ^\-<^A. i^^Sf3<t-9tX4H- - 1 A / -y^ ■"-^^•%^A''Tl^\ 1 1 mm h ■^1-t . _ ' ~ JuJpVi^V^^V^'.^' i TT _\ \ ^"r^A^ y>_ L i-^^*^ ~ " " \ p^-V^^t^^" \^N \ I i T-A'\j. A'\ \ ^A — \ - — t- - 1 1 I L-A— A V,--T^\ J . :-XLi^^^Y{X^^\'AJ^- 1 ^L Q/iJtdfr m/tynt^re^ trof^tTrttr/f pi^trrd ^n«/ and ft'i^l j if FIG XVK ATt. 496 dfMilvn (V«3S H ^^^1 ■ H AH ^^H ^^ lc/i\\^> I 1 ^ TS^^T^ ^^CSoOvx.'OcycS' ^so^v^ f} 1 . x^v/SQjTK^ j4ThP* ^"j^^ jOttw ^ ' 1 1 \X r 7 1 1 ^^^^^^H "' tht DdiKNiUs :^ Vst Qarrn^en. PitAx ■ .jfefwltih. Pttjs .1 i .1 fiC XV! An 436 ' ^tu-n^rHttm:: — 1 — H \ \ v'''^-'' - ■"^^iT , V^^jk*\ ~\ \ __ ~- — 1 i 1 __ 1 / -J . j 1 L-— *r^^r7\i J^"*^ \ >^, ■\J^^^ \ 1 -~i 1 1 1 1 1 -y^ y X^^'X ^^^ V I 1 / i 1 1 \ L^i>''V^ V-^^Y^ \ ' ^J^— 1- 1— 1 ttuVoVV-iXu--- 1 1 J - 4 f \\ FtG 3twr Art. 496 ^.» -Jiaiindan/ fYue a '■^Jj'^jf^f/.'^Jl'^.r/'':-:-:^. r^-'j/. Flo xvin. Art 467, 702 CircuUw Current^. '/•■■ IVLyittfS i-j' !Jic i'/'in-'l.i<.i r^y.rs. JtQ XUC A« 715 7m> Ct-rcutar GirrenlJ r jl ll I'i 1 ^^^ I^H ES^S^q^^ ^ =^^ m ^^^^^^1 -4 -- 1 1 t fi ^^Bi H- 1 F ^ — ^ T ,i ^-^ m 1 -1 — "^ 1 \ i — t""'' i i 1 _i 1 — 1—4 — \ T \ -i — z^^^ ^^K CtrmUar Cur/vnt 1 Jtilj tBS6. ClarenlTon ^Jress, ©xforli A SELECTION OF BOOKS PUBLISHED FOR THE UNIVERSITY BT HENB.7 FROWDE, AT THE OXFORD UNIVERSITY PRESS WAREHOUSE, AMEN COKNEK, LONDON, ALSO TO BR HAD AT TUB CLARENDON PRESS DEPOSITORY. OXFORD. \Evtry iMi it teimJ in iJmA, untat Mhtraiiie JitiriitJ.J X.EXICOITS, GBAMMABS, &0. Anglo-Saxon. — An An^/i>-Saj:^n Diclienary, based on the MS. CoUMtioQi of Ihe iBie Joic-ph Ilosworth. D.D.. Prnfeuot of Anglo-SiEoii, Oxford. E(Ul«a anil riiliui:»l l>y Prof. T. N. Toller. MA. (To be comulcled dJu. A-I - " b foUT norta.) I j/. <a<^. r«iU I U(l H\VIt>TLlAN(pp.ri,£r6), iSS). 410. By James ClIlNF-SF..— .^ Haiidhock of the Ch'tneu Language. SDRimcri. itiGj. Sua- half bniind. 1/. S/. English. — A New Iinglish Dictionary, on Hisl&rieal Priii- i S< idMctIjiU collccttil by the PhlloloukAl Society, iiiray. LL1)„ PfCTiileiit of ihc Philoliitkal Society ly Sthotut )Lnd inen of Sdcnce. Part I. A — AN' P»(t IL ANT— BATTEN (pp- w". 3SJ-(«*)> Imperi'I liplti: foaniled mainly on VAUeA by Jnnic« A. H. Miiiray. i.L.U„ Prcaitlciit of ihc Fhiiolotte!"! Society ; villi llie aniitancc oFmany Sthotanand inen of Sdcnce. Part I. A — ANT (pp. x*i, 3S)>. 410. III. td. each. An Etymological Dictionary of t/ie English Langnagt, By W. W. Skcal. M.A. Stt^nd EJitian. 1884. 4to. 1/. 4/. Supplement to the First Edition of the above. 1884, 4to. 11. W. ■ A ConcUe Etymological Dictionary of tfu English Lan- gtagt. By W. \V. Skcal. M.A. S/tmJ EJilim. 1885, Croun S™. 51. W. G R E EK . — A Greek- English I. (xitott, by H en ty G corge Liddcll, D.D.. and Robert Scott, D,D. Scvtatb EtUtloo, Rerifcd and Aug- mciiled thtoughoul. iSSj. 410. il.t6s. A Greek-English Lexicon, abridged from Liddcll and ScotI** 4la. edltioD. chiefly for the qm of S^oob. Twenly'fiiil Edition. 1884. Square itmo. 71. Ci/. A copious Greek-English Vocabulary, compiled from klhebcttauthotitiea. iSjo. utoo. it. - A Practical Introduction to Grtek Accentuation, by H. W. CbAodlcr. M.A. Second Edition. 18S1. 8to, ia/.6./. CLARENDON PRESS, OXFORD. Hebrew.— TAir Bwk of Hebrew R&ots, by Abu 'I-Walld Marvin ibn jBnih.otbeiwUccitllcfi Rabbi Vfmkh. Now firetcdilod, withu Apptndli. bv Ad. Nculitucr. 1875. ^lo. il. ji. 6J. A Trtatixe oh tkt usi of the Tenses in Hebrew. By S. R. Drfw, D.D. Scocnil Etlitloo. tSSi. Exlr«fc*p.Sw. ;<. M. Hebrew Aceenluatwn of Psaints, Proverbs, and Job. Ky Willuia Wickri, D.D. iSSi. Uciity 8vo. slifFcovpn, 5/. Icelandic. — An teclandie-EngUsh Dietienary, based on the HS. colleclionK of tb( laic Ricluid Clcuby. Ealu^d u»d completed ^)' G. Visfdaon. H.A. With an Intre^nction. uid Lifc ot Rlchud Cleubj. bj G, Webbe Duent, D.C.L. 1SJ4. ^lo, j/. j/. A List of Englisk Words tfu Efymolcey of tvhieh is HluitrMtd fy tamfttriifiB witi Ittlaitjir. t^jwrcit In the ionn of U ArntHnixiolhtatMrt. Bjr W. W. Skol, M.A. 1S76. »tiichc<l, ». An Icelandic Primer, with Giammar, Notes, and GloXKtry. By Henry Su-td, M.A. Eilra («p. Sto. 31. W. Jntt PiMuitd. Aa Icelandic Prose Reader, with Notes, Grammar and Glotarr. by Dr. Gudbtmnd Vigfuuoii uid f. Vo«k Powell, M.A. iS;g. Extra fcnp. 6to. io». i4. Lattn. — A Latin Dietitmary, founded on Andrews' edition of KicDitd't Liun Dicilonar}'. KviKd. ealiirgtil. and in grot pun trwiiiKn brChiulinnT, Lewia, Fb.D., aiulCliirlesithori, L.L.D. i8;i) 4(0. 1/51 Mei,ANES7AN.— T"**^ Melanesian Lar^nages. By R. H. Codiintjton. D.D., oftbe MdnnFiinii Mbtioii, 8n>. iSj. Sanskrit. — A Practical Grammar of the Sanskrit Lanffuage, arrMif^l with rtfcrcnce lo the Cluneal luignaget of Earope. for tbc we of EuglithSluiknli. byMonivtWilliinu, M.A. Fontth Edition. 1877. Gvo. t>;i. A Sanskrit-English Dictionary, Etymolo^cally and ogicallj anan£cd. wiih ipcdal refcrtttoelnGre^, Latin, Geraui, Anglo. , Eng;Iub, ana other cofjnilv Ii'd<i-HurO{«ui Langpaca. By Montel ^>. M.A. 1871. 4I0. 4/. \^i. dJ. NaUp^ihydnam, Story of Nala, an Episode of the U*hi-BhiinlB: the Suiikric Im. u-itba copioiuVocabnluy. oaduiinproml vereion of Dein Milman'i Tnndation. by MonJo Willuuni, M.A. Second Ediiion. Rciiicd and Ini|iror<d. 1879. 8vo. is*. —^ Sal-untalil. A Sanskrit Drama, in Seven Act*. Edited by Mf>iiicr Willumt, M.A. Second f^liioii, 1371S. 8to. Hi. SVRIAC. — Thesaurus Syriacus : coIlejjeruntQiiatrcmirc, Bcrn-j ■ttin, Lonbach. Anioldi. Aenll, FUM, Koedivcr: cdjdit R. Payne Sfnilh,! S.T.P. F««c I-VL iSW-Sj. UD, fol. cacb, i/. i«. Vol- I. (ODtauiing Foic. I-V, int. foL $/. fs. Fasc. It'll, m. fol. tl. 1 u. 6J. Juii /\Uiiie<J. The Bookof Kalllahand DintHah. Translated from Artbi'c into Sjriac Edited by W. WriEhl, LL.D. ■SS4. 8to. 1x1. 4 ^ 1 OBEEK 0I.AS8I08, &0. Aristophanes: A Complete Concordance to the Comedies >nd FnxgniciitJ. B)r Ilcnfy Donbar, M.D. 410, 1/. 1/. AristotU : T/te PoUlUs, translated into English, with Intro- <)dcIioii. Miit;tna1 AniLl;riU. Kolit, and IiuilMt, hf B. Jowett. M.A. Metlium 8vo. I vcili, tu, HeracUH EphesH Reliquiae. Kcccnsuit I, Bywatcr, M.A. Ai)ptDi3ic» loco nddiUc sunt Diogenii Lacrtii Vits lictncliti, Pulieuloe llip- p(>aBIci De Diacta Lltiti Piimi, EplttoUc Ilcrxliteac. iS;7. Svo. (b. HtrculoHeHsiunt Voluminum Partes 11- 1834. Svo. iw. Fragntenla Heradattettsia. A Descriptive Catalog^uc of the Oxford copEn of the Hervnlaiican HoIU, touttlicr witb lh« («ats of wvcrnl papyri, accompanicti bjr facsimiles. Editn] by Wa1t«r Scotl, M.A., Fellow of Mcrton Colfi^, Oiford. Royal Svo. cltl^, iij. Homer; A Complete Concordance to the Odj-s-iey and HjiTtnt of Homer : to which U added a Concordance to the PantlUl Pauafiet taUwUUd.Oclxuoy.andHyniiit. l!yHcni7Dunbar,M D. iSSo. tfM.U.it. Scholia GraKa in Iliadem. Edited by Professor W. Dindorf, after n new eolUiJon of the Venetian MSS. by D. B. Monro, M^., Provott of Oriel Coltcge. 4 ratt. 8vo. i/. loj. Voli.VandVI. In tii Pnil. Scholia Graeca in Odysstam, Edidit Guil. Dindorfius. Tomill. 1835. Sro. 1^. 6if. Plato : Apdogy, with a revised Text and English Notes, and Digctt of riatonic Idiom*, by Junes RIddcU, M.A. 1878. Si-o. 81. 6if. Philebus, with a'rcvised Text and Englisli Notes, by Edsand Potte. M.A. 18A0. Svo. ;i, fiif. ■ Sophistes and Politicus, with a revised Text and English NotM, by L. Campbell. MA. 1S67. 8ro. 18/. Theattetus, with a revised Text and English Notes, b; L CumphcU. M.A. Second Edition. Svo. to/.&f. —^ The Dialogues, translated into English, with Analyses and InlrtuJuctlcni, by It. 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Piioc rtdnctd 10(4/, 4 Grbbk. — AVwiOT Ttstatntnlum Graece. Acccduiit parallcla S.Scfl)>lurac locn, etc- Wi<Iil Cirolu* LIoji!, S-T-l* K. i9ni<i. v On writina psper, with wi<l« maigin, lo/. Novum Tfilamcntum Grtuce juxti Exemplar Millianum. tSoM, M. 6J. Oa wrilins papor, wllh wide mni^m. yj- Evaugflia Sacra Grattf. Fcap. 8vo. limp, ts. 6d. Tfu Greek Trstmnrnt, with the Readings adopted by the Rcviivri of ihe Aulhotiwil Version : — (l) Pic* type, with MniginnI Kcfcccncci. Demjr Bvo. 1M.6/. (i) Long rrimci lype. F&ip. Svo. 4>. 6if. {]) ThtMine, «n vtilinj* {Uijicr, «il)i ulil« miri-iQ, Ijt. — TAi* /"drtj/flr/A'^-if TVjtawww/, Greek and EnKlish; bcinp ih« AnilioiiKd Veision. iGi i ; lh« Ktviscd Vcmion, iftSi ; and the Greek Tal (ollowetl in ihc Revised Version. Svo. i ji. W. 7"*/ Rniud ftrium is lit joint firrfirly o/lit UnivtriUiti ef Oxford imdCamirMgi. - ' Canon Afuralorianus : the earliest Catalogue of the Book* of llio Xrw Tcvtuncnl. tailed with Nu(» ami a Kauaimile of lh« US. to the AmbiMian Libniy *t MiUn, b>- S. 1'. Tre^ellci, LL.D. 1667, — Outlines of Textual Criticism applied to tlte New Testa- ■WW/, ^-j C E. lUiiiiiioiid, M.A. Fouilb Ediiion. Exua tatp. 8ro. J*. W, Hebrrw, etc. — T/te Psalms in Ilebreio without points. 1879. Crown Svo. 3J. td. A Commentary ok the Book of Proverbs. Attributed lo Abraham tba Em. Edited fiom a MS. m. the Bodlciui Libtarf b; S. R. Driver, M.A. Crown Svo. paper coven, j/. 61/, ' The Hock of Tobit. A Chaldee Text, from a unique MS. ill the Bodlrfin LibniT ; with oilier Kalihiiiicsl Tcxti, I'^ii(:Iiih Tiansla* lioDi. and the IiaU. Etlitnl by AjU. Ncutiiser, M.A. 1878. Crovrn Svo. 6f. Horae Hebraicae et Taimutliene. a J. Lightfoot, A new Edition, by R. Gandell, M,A, 4 toIi. 18J9. 8vo. 1/. 1*. 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DiiiinclionK, acd CUn LitU: of the' Head* of Collccct, Sc, dec, Tiom the Tbicteenth Ccntiiiy to 1 SM^ UATHEUATICS. PBTSICAI. SCXEKCE. ftc. Aeland (//. Jf., M.D.. F.R.S.). Synopsis of the Pathet^cat Seriei in tkt Chifird MKitum. iSfi;. 6to. ». &/. De Bary {Dr. A^ Comparatim Anatomy of the Vegetative OrgaHl t/tki Phantrtctaii and /Viiu. TraKilalta asd Anuolalol by V. O. Bower, Sl.A., FL„S.. and D. II. Scoll. M.A, Ph.D.. K,I„S- Wiib two hundrcil and (otly-Mw voodcutt and la Index, i/. w. W. Royal SroL, half nMrocoo, 4 I I I Mailer {y.). On eertam Variations in the Voeal Organs of tlU PanttiS tint iatv hUinrta ti^aftJ ntiin. TraMl*tod In P. J. BcIL B.A-J and e<llted. «lib oa A)>|iaKliiL, by A. H. Oarrod, M^.. F.RJi. Willi mtca.] 1S78. 4to, popcT curen, -ji. 6rf. Price {Barthoiemew, Af^.,K/?.S.). Treatise on Infiaitesiuial Vol. I. DiffeTtntUl Calculus. Stoond ]ullti«a. %-m. X^M. Vol. U. Inleeral Calcolnt, Caloittu of VMUiiou, and Differential Equalion*. Scmnct Edition, iMj, Svo. iS/. 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F.K..S. Wilh Poitralt. PUte*, and WoodcBlk. » vol*. 81a, )/. 4/. Wcstu'cod {y. O., M.A., FJi.S.). Thesasirus Entomchgicus JlfftioHiu, 01 a D«c;iption of the nut&t Intecl) In the Collection given 10 ^the Univtnjty l>y the Kcv. WiUiam Hui>c. WItb 40 Plaiet. \%^^. Small 'plio, balfmojoccot jt.lvi. ^bt ^citlf iSoolis ci tU Tca%t. o ev VABI01-; Oiieital SciiouAt. and editso n F. Max MCuJtit. [Demy 8vo. eloth.] fVol. I. The Upanishads. Translated by F. Max Mulkr. Part 1. ThrATindogii.upaniiliB'l.The Talavitini'iipiiaiihid. The Aitareya- iunnyaka. The Kauililtaki-biihcnajiia-niianitkid, aadTbc VSfiuancyi4aiiihlu- upaniiJiaii. t<u.6il. Vol. 11. The Sacred Laws of the Aiyas, as tau^t in the SciioaU of A«utaiiilM,Gi»ilam_B. Vluih/4a,aiidBaadhilyiina. Traiulatid by ^m IW. C«or][ UuUa. Pail 1. Ajmnamba and Gaaiuna. i<w. dt. Vol. lU. The Sacred Books of China. The Texts of Can- teci»'^tim- Tnt»Jattil br lamctUcef. Pajl I. The Sb£l King, The K<li- gima porlica) or Lh« Shib Kins, aad The Hiilo Kins. I U, 6d, Vol. IV. The Zend-Avesta. Translated by James I>armc- ttelo. P»rll. TleVendldfid, t0i.6J. Vol. V. Tlie Pahlavi Texts. Translated by E. \V. West Paul. TbcUoiKUliu, Balinuin Yiul, ui<l Shi^ait U-tfal)Ut. txi.6J. Vols. VI and IX. The Qur'An. Parts I and 11. Translated by E. H. ['almpr. *U. Vol. VII. The Institutes of Vish//u. Translated by Julius Vol. VIII. The BhagavadgUA, with The Sanatsi^^aitya, and Tbc AneglUt. Tianilxltd b; Kiihtu&lli Trtmlolc TcUaj;. ten. 6J. Vol. X. The Dharamapada, translated from Pili by F. Max Mullcr: uidTbcSuiU.NiDiu.tnn>hted fram PSli brV.K>i»Ix>]); Uaf Cananio] BooLs of tbc ItuoiUilnx. im. 6J. Vol. XI. Buddhist Suttas. Translated from PAli by T. W. Khyi Daiidh i. Tbc Mahaiarinibtiijin iiiitunU ; t. He DbaDi>a4*kka- Tcn^ - _ ' " '1 tt*. IOI.6J. ^M piwtaltsna SatU: 3. The Ter^m SnllanU; 4, Tli« AkaAkhvvva SntU; S. TbcAilakbiUSuIU) 6. ThcMitbi-tuilauuiaSatlaiiU; ;. The SnIiUnn Vol. XII. The .Satapatha-Br4hmaffa, according to the Text of Ibe Midhnndina ScbooL Tniutalc<l by Juliui t);j^tu>c* I*>'t '■ liookituui II. 121.6J. ^^ Vol. XIII. Vinaya Texts. Translated from the Pili bj T. W. Rbyt Davldi and lltnnana OldcnbetE. T»tt L Tbc PltbnokUo. Tbe M>hAtig),-«. 1-IV. toi.6d. Vol. XIV. The Sacred Laws of the Arj'as, as taught in the Schoob of ApostmiilKi. GanUniA, VjUith/^n uwl BABdhlyant. TrauUlcd bji Ceots Biililcr. Paxil. VidtluAa Bad BautlUyana. iai.6J. Vol XV. The Upanishads. Translated by F. Max Mijllcr. r«( II. The Ka/ia^Dpanbhad. The MivbAka-QiNinuhad. Th* TaiiiiilyKlia apanUhad, Tbc Bnliadlranyaka-Djaiiitbad, Th« .S^vtaJvatua-upuiUfcad, Thi PrAi^a-upMBbad, and llio MaitiiyaMa-llrilunam.«p«niafaad. loi. tJ. Vol. XVI. The Sacred Books of China. The Text* of CoO' fuciaattm. Tninilalcd b^ Jamti Lcggc. Pan II. Tbc Yl Kirj-. Im. &/- Vol. XVII. Vinaya Texts. Translated from the Pali by T. W. Bh)* Davidt ami Htnnann OklenbcTs. Part IL The BUUnan, V-X. Tbc A'uUava^i^, 1-tU. 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S(ce«d CaiUon, 18S1. Eiltafcup. Svo. p. Justinian, Select Titles from Ihe Digest of. By T. E. Holland, D.C.L., and C. I~ .Slixiloctl, BCI.. tt>o. \^t. Alio aold in Part*. In paper oov«r). an foUowa :~ Tut I. InlroducloryTillrt. ii. 6if. P*i( II. Family Law. i>. tti% 111. Pmpeity Law. i: fKf. Part IV. Uw tf OMicilieot (No. l). *i, 6rf. Pan IV. LavorObligsti<m> (No. i). 41. U. Lex Aquilia. f ; Aquilia. The Roman Law of Damage to Property : fl beips a ComiMnUry on the Title of ihc Dtcnl ' Ad Leson Aqnilion (is. i). ^^ Wim M lotioduclion to the Sioiiyof the Coniut Iviii. (.'inlia. Uy Entfai Grueba, Dr. J»T., U.A. Demy Svol i(u. &/. futt J\tUit»eJ. -m Markby ( IK„ D.CL^. Elements of Lauf considered with refer- 1 Twiss {Sir Trovers, D.C.L.). The Law of Nations considered ^ Pari I. Ou the Rlchu anil Putin of Nilioni in lime of Pace. A new Edilian»^| Kcvixd and Knlan-T>l. f88^ Demy 8x0. 15/. ^ Pulll. On tbcHiglitt a»d Dntiei of Naliont k Tise of Wtr. 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Thorold,Af.A.y A Manual of Political Economy t for the «je CpfSehccIs. Third E^ilion. Extra ftnp. Svo. 4/. W. Smith's Wealth of Nations. A new Edition, wiih Notes, by J. E. ThMold Rosen. M.A. ) vols. 8to. 18S0. iii. IVi/soH {y. M., B.D.), and T. fowler, M.A. The Principles ^ iltr^t (Icttoduclory Cliaplen). Sva. jj. &/. Jml ruUiihtJ. X. AJBlT, fto. TMc Cultivation of the Speaking Voice. HuUah {John). SccoDi] EJilion. Eitia fcap. Sio. %s. hd. Oitseley (Sir F, A, Gore, Bart.). A Treatise oti Harmony. Third Edition. 4tO. icyt. A Treatise on Counterpoint, Canon, and Fugue, based DpoQ thai of Chenibini. Second Edition, ^to. i6^. A Treatise oh Musical Form and General Composition. Second Edilirin. 410. 10^. Robinson {y. C, F.S. A.). A Critical Account of thi Drawings ty Mkhfl An^tla and Raffatllo in the Umvcnily Galleries, Oxford. 1S70. CrowTi Svo. 4^. Ruskin {John, M.A.). A Course of Lectures on Art, delivered befori? [he U