Hickstoroidal1881

be “OnToroidal Functions. i XIV.ByW.M,Hicxs,M.A.,FellowofSt.John’sOpllege,Cambridge, : : Communicated by3.W.L.Guarsnen, M.A., PRS! d ‘ReosivedFobranty21,—ReadMarch3,1881 ‘Tue, following investigation’ wasoriginally undertaken asthefoundation forcoptain : researches onthetheory ofvortex rings, with especial reference toatheory ofgravi- tation: propounded bytheauthor intheProceedings.of the.Cambridge Philosophical Society (vol. iii,‘p.276). Asthe’results seemed interesting in’themselves, andas theyalsoserve asabasis forother investigations, more: particularly inelectricity-and ‘ conduction ofheat, Ihave thought itadvisable topublish itvas aseparate paper, especially asIcannot hopetofindleisure forsome timetocomplete myoriginal Hi purpose: ‘Theword “tore” ixusedasaname forananchor-ring, here restricted toacircular section, andby“toroidal functions” areunderstood fiinctions which satisfy Lartace's equation andwhich aresuitable forconditions given over thesurfaces oftorts. ‘The‘first section isdevoted tothegeneral theory oftheemployment oftwodimen-*sionalequipotential Hinesincertaineasesasorthogonal co-ordinates inproblemsof three dimensions. From thiswopass atonce totheparticular casewhere thetwo- dimensional lines arethesystem ofcircles through twofixed points andthesystem of citeles orthogonal tothem. Itisshown that. these satisfy theconditions ofapplica- Dility. Byrevolution about thelinethrough thetwopoints wehavefunctions suit- : ableforproblems connected with twospheres. Byrevolution about thelinebisecting atright angles thedistance between thepoints wehave fimetions assoviated with auchor-rings ortores. Bythefirstsysteni itisalaopossible todeduge functions for” ‘whatmaybecalledaself-intersecting tore,and.bythesecondfortwointersecting : "spheres,” Asecond application ismade fortheparticular casowhere theopening ofa z torevanishes andtheres adouble cuspidal point atthecentre. ‘Thesecond section isAevoted tothedevelopment ofzonal toroidal funetions—thatis,forconditions symmetrical abouttheaxis*ofatore..Itisshownthat,forspacenotcontaining thecritical axis these arothesume aszonal spherical harmonies of ./*Throughout thepaperthoaxisoftoroistakentobethelineporpondizalar toieplanethrvoghitacentre; thocircle truced outhythecentr ofthegenerating circle ofwtorewillboelle bycirealer‘xis,andthecirelebythetnopointsaborementionedtheeitcalcrete,Mpocotsexxt, : 4 B10. MR WyaEHICKSONTOROIDAL FUNCTIONS. f-imaginary argument and(when thewholeofspaceoutside atoreisinquestion) oforders(n+1)/2:Forspaceinsideaforewehave-acorespondiing snalogywithzotialharmonies ofthesecond kind. “The properties ofthese functions arefound. tohave. ‘analogies withthoseofthe ordinary spherical hirmonies, butwithessential differences ‘Thespace outside atoreisdifferent from that dutsidle’a sphere inbeing eyelie;in general, then, thefuctions forspace outside willnotbedeterminate from thesurface - conditions alone, ‘The above, functions aresuitable only when thete isnocyclic function:itisshownhowtoobtainafunctionwhichwillcompletethesolution. ‘The third section deals with tosseral toroidal functions, whieh come into useforthe mostgeneral caseofnon-syinmetrical conditions. It’is-shown''how thedifferent orders andranks depend oneach other, sothat theymay becalculated internis of two. Integral expressions arealsoobtained, asinthesecond section, which areneeded infinding theeoefficients inexpansions inseries.’ ‘Thefourth section briefly notices thefunctions suitable fortoreswitholit acentral _opening. ‘These functions bearthesamerelation totheforegoing functions that cylindric harmonies (Besse1’s functions) dotospherical harmonies, ‘ Inthethsection afewvexamples aregiven oftheapplication ofthemethod, such asthepotential ofaring,theelectric potential ofatoreanditscapacity, theelectricpotentialofatareaudanelectrified ‘circularwirewhoseaxisisthesameasthatofthe——tore, thepotential under theinfluence ofanelectrified point arbitrarily placed, and thevelocity potential foratoremoving parallel toitsaxis,aswellastheenergy of =the motion. \ OFprevious writings onthesubject, ornearly connected therewith, Iamonly 2) acquainted withtwo, InRumtayy’s ‘Gesammelte Werke" (chap. xxiv.) isashortpaperofsixpages,“UeberdasPotentialeines!Itinges.” »Hiearrivesatthesanediffrentialequationas(7)inthispaper,pointsoutthatasolutioneanbeexpressed as+ahypergeometrig series inseveral ways, andthateachfunction canbeexpressed in terms oftwo,which greelliptic integrals ofthefirst.and second kinds. Thepaper is ‘@posthumoué oreandisnotdeveloped. There isanoteonthesame subject, by ‘W.D.Niven inthe‘Messenger-of Mathematies’ forDecember, 1880. ‘Though not bearing onthesamesubject, apaper inaybementioned byC,Neumasy, “Allgemeine Lésung desProblemes’ uber’ don-stationiiren ‘Temperaturzustand eines Tiomogenen Karpers ‘welcher vonirgend zweinichtooncentrischen Kugelflichen begrenzt_ wird” (ELW.Scuaupr, Halle). “Thisisapamphlet ofubout150pages. Heusesco-ordinates analogous tothose inthepresent paper, butthemethod ofdevelopment isvery different. ‘Thefunctions arespherical andcylindric, harmonies ofrealargument, and ‘those ofthesecond kind donotenter. Heconsiders thestationary temperature inw shell bounded bynon-concentric-spheres; inaninfinite medium inwhich aretwo spherical cavities; auidsimilar cases whentheboundaries touch, Itsinterest incon- *Thegreater portion ofthefollowing pager wascompleted before Ibecame acquainted withthis paper ofRasy’ ofwiththaofNevsany's mentioned below, MR.W.M.HICKSONTOROIDALFUNCTIONS, OMeheat netion with thefollowing pages- consists chiellyinthefactthatthepotentialis ‘expresied inaseries oftheform, es.“AerSF.(u), Gale) ae pT andthattheorthogonal, co-ordinates employed areclosely allied. GENERAL THEORY OFCONJUGATE CURVILINEAR CO-ORDINATES INTHREE DIMENSIONS. SPECIAL CASE. 1.TtiswellknownthatifLartace's equationbereferredtoasystemoforthogonalco-ordinates 1, mittakes theform i. . 2(Udg)1d(Vag). 2/Wdg)_ oeown)tie(Gwis)tao(tvae)=? where einen ~ ya (OO (anh 2wae) ++) : yee(Pg(20)?5Yoel? we(e) +5)+0)ri\®=Ptfou? d zwe (yey+) : Letusnowtakeu,rtobeanyconjugate fimetions-of p,2;p,xbeingtheeylindrical co-ordinates ofapoint. Also take wtorepresentaseriesofplanesthroughtheaxis i of=,sothat w=-tan~! y/-r Then u,v,wareorthogonal surfaces and a eee(*)4(2H)(2808(2)?ye : veg) +E)Geyer, tek : Wes : Sothateqnation (1)becomes eee oe 2Beetee LuaPees i SB MPin)Fin(Pde)FTreaNeeaPT eROs a3 ole)+8)) pes Inthiswrite 6=¥//p, then OagO[BoiSal]a(te.2) Maeracaa8: . eoeearrferiieeacaesar “Batybe .: Pag Fa : \ # 4K2 Put 612 ~— MBW. MG.HICKS ONTOROIDAL FUNCTIONS. _Heresince'u,0areconjugate finetions ofp,= ee,aetye=? e ap\t opis dy :: (+)=parepreerop) Ha: sothat iaeebn = (2) ene+p{arti} =0ren) Byputting 2¥0 wegottheequation forfunctions satisfying conditions sym- metrical about anaxis which by-an-obvious analogy may becalled zonal functions. Ingeneral, putY=y' cos(miv+-a), then ’must satisfy . By BW stad~ Bee pp vee ‘When uj»arefunctions suchthat1/(p£)* isoftheform4(/(u)-+F(0)), itispossible to obtain solutions oftheform Y=EX...Yua 008(mie-ba) where " ( ee=(4m?1)fw)XX oe ‘ Phe(dnt1)FWY—0¥ whicharesuchthatX,..areconstantwhen«isconstantandY...constantwhenvisconstant, :Asaninstance offunctions satisfying these conditions wemaytake theelliptic co-ordinates: ‘ p=acosh wcosvy =asinh usind : Here poem ts tePE cos? cosh?x Andthe,equations forthefunctions X,Yare @X [4ndihe+(;saat") x0, Ov (amtoa—(feaeee) Y=0 5 f \ MR,W.M.HICKSONTOROIDAD FUNCTIONS, |O18 ‘Thefirst,produces-finetions- analogous-to-those discussed inthis paper—the second ©) sphorical harmonies ofargument ¥—v,andorder“F ‘Thesurfuces u=const.give confocalspheroids, SinceV/p=Veeashweos8,itwillresultthat¢isexpressed 08thesumoftermsoftheform{AP(x)-+BQ(i)}{CP(i)-+DQ(e)} cos(nw), where’ PY,Qarespherical harmonies ofargument 7--v,.and P,Qarespherical harmonies of oe imaginary argument, : oe Intheapplications that follow ithappens that, vare such thatp%é?isafunction : ofwonly, sayf(u); inthis case weobtain solutions byputting y’=ycos(nv+8), where¢, ey_(opot ga Tew eno EO a Thesolutions ofthis equation form=0 may becalled zonal functions, forn=0 : sectorial functions, and form.ngeneral, tesseral functions:” IfUn U'ns-are two independent solutions ofthisequation thegeneral value of¢isgiven by Vpb=ES{AU 4608(nv$a)005(120+8)-++A’U'a, C08(n-f.a')cos(mie+8)} If@begiven over anytwosurfices «=const, it-isclear thattheconstants canbe = determined intheabove bymeans ofFourter’s theorem, This willbemore fully. discussed inthesequel. 2.Beforepassingontopartioularcases,thereisoneremarkable resulttobenoticed. Ifintheequation transformed asabove, weputp=w cos(fv+-y) then Wsatisfies theequation £ a ae asFemare ee Henceify’beanytwo-dimensional potential function, thenALcos(§w-ty)isn! three-dimensional potential fonction. Since thisexpression changes signwhen w increases by2xitisnotsultable forallspace;butadiaphragmmusthesupposedto : extendfromtheaxisof=to.infinityinonedirection,andtobeimpassable, Though== theresult isinteresting itdoes notseem toearry important. consequences, asthere is notsufficient generalityintheexpression. Wemaychoosetheformofthesurface, < andcertain otherconditions, butallthesurface conditions arenotarbitrary. .Thusletustakeananchorringdividedby-aplasiethroughitsaxis:*Tet.uskeepthecurved 5 surfageandoneendatzerotemperature, thenthedistribution oftemperature at-the- otherendisdeterminate thoughitsabsolutemagnitudeisarbitrary.Toprovethis,©= wenotice thatif(a2) betheradii ofthecircular axis, andgenerating civele respec tively,andrthedistance ofanypointfromtheciteular axis : 614K WeM. HICKS: ON TOROIDAL FUNCTIONS. oe atts. W=log 2}log 0-0 and : AL erarte . = 5 Noy cos (4 ayFelogOYcondiy) Thisistobezerowhen w=0 .~.y=™and A O42. w.t=50y,beCont? in Butnowthedistribution oftemperature attheother endisgiven by Ay. pape? =NlogPatt? taj, toeie leaving only theabsolute magnitude Aatourdisposal, Further, there must. be supposed ageneration ofheat allalong thecircular axis. ‘This example serves toshow __ theartificiality ofsolution given bythisform. 3.Forthecaseofananchor ring, ortore, itisatonce evident that. theproper functions 1,vto'take arethewellknown desgiven by” cmlowbats!vipriclog PEE viz.:e=const. aseples ofcircles through twopoints (4,0)andu=const. aseries of * circles orthogonal tothem, andeach containing oneofthefixed points. Ifthese be made torevolve about thelinethrough thefixed points, wegetfunctions proper for ‘voSpheres (u)s oFthesurface formed bytherevolution ofacirele about alinecutting it(0). Ifthey revolve about theaxisof2,wegetfunctions proper forcircular tores (i); oFfortwointersecting spheres (x). Ttwill beuseful tosetdown here inacom pact form, formuke relating tothese funictions, which willberequired Inter on, Most ofthem areeasily proved andaresetdown without proof. zs ueflogZhe :=Seeepen?| 1st ttm be wd psa 21,W.M.HIGKSONToNOIDAL WuNCTIONS. BIB PAEahbedaa ¥+ m etetewcy Canes eressinhpeeee eeedarepticmer feb) : oeeaeOagg OE v whence thestatement made above that pé=/(w)- LatIt,rbetheradii ofthecircular axis and normal section ofatore (w); 2the radius ofasphere (v);then : Rose : cohw= : sinhxZOE Repeeere ne.) Further, if,theradius ofutoretoapoint P(u,o)make anangle @with theplane. ofthering : ‘ causArtem: :>ahaa rT) : inp VINiubesoresin With theabove values of(u,v) thegeneral equation fortoroidal tesseral functions is OF yg SLrk ar) ‘There isonecaseinwhieh thefunctions usedabove become nugutory—viz. :when« iszero, orthetores aresuch that Rr and‘they touch themselves attheorigin, In thiscasetheproper curves aréthetwoorthogonal families ofcircles, touching, theone settheaxisof2,andtheother theaxisofpattheorigin—viz : 616 ~MROWeM. HICKS ONTOROLDAL FUNCTIONS, (utri(pfa)=a ea ¥MeRee enya . - & depnate n=BES=0 ) 7 and Se : da OHggFO Itwill beshown that between the latter surfaces and tores there isasimilar relo- ____ tion tothat between cylinders andspheres, andbetween thefunctions tothat between Spherical Harmonics and Brsse’s functions, 4.The potential duetoaring ofradius b,centre at(0.2') andplane perpendicular to axis of2,is = . . wes t=Temp epom 2 aa =f +f xalWocea Inthe case where itisthe critical circle and lieve “iov=(yaamat Tngeneral thedistance between twopoints is.(e—z')'+p?+p2—2pp’ cos(w—w'), which expressed inbipolar co-ordinates becomes Gsu-os&)Coews{00881coshwf—cos(v=)—ainhwsinhWeos(w—w')} ear 2 : pee aA MR.W.M,HICKS’ ONTOROIDAL FUNCTIONS. Oe ae ZONAL, TOROIDAL FUNCTIONS. a" 5,Inthecasewheretheconditions aresynimetrical abouttheaxis,gisindependent. =*ofwundisoftheform ‘s ¥ $=AJhi3008(nha) =m oe where yisthegeneral integral involving twoarbitrary constants oftheequationE Ye ayyH Tats =Oae Faint ue From thepotential ofaring, attheendofthelastsection, itisatonce seen that «particular integral forspace, notincluding thecritical circle, when n=0 is - 0 wavsf jhwahwo8 Inthesame way,calling thepotential ofthexing¢,itmaybeshown byfinding that = *: aeeetAnexteenerer oe From analogy with thiswemight assume sift 0 Y=Vial sahwe8? o Sore e Qwth Qn-1 BL andbysubstituting weshould finditpossible byputting p=" or—2*=) toeatinty theequation, Butthefollowing, bymaking usooftheorems already proved forzonal harmonies, seems tobemore direct. Putting, inthedifferential equation, 2 y=v sinh u.P ep aw aartooth wie—(M— Ant PHO. (9) whenco itisatonceevident thatP,is«zonalspherical harmonic ofdegree21, ° withapureimaginary forurgument, Hews, inhis.‘Handbuch derKugelfunctionen,’MpocoLacxxt, 44 a 618 SMR W.-M. HIOKS OY‘ToROIDAL FUNCTIONS. 5 hhastosomeextentconsidered sphericalharmonies withimaginary argument, buthehasnotdeveloped: them, atleast forfractional indices, inaform suitable forappli cation here. Consequently they will behere considered independently and with especial reference tophysical applications. Hereafter, C, willbeusedingeneral to represent cosh 1sinh w,respectively. ERGY: ‘Wehave then ingeneral $=VO=e2801(A,P,+B,Q,) cos(n0-+a) * where P,,Q.aretwoindependent integrals ofequation-(9). Wefirstdiscuss the integral already obtained : Pa( ee ee (tO) (C=S 0086) Itiswellknown thatthisintegral isthesameas no [(C-Seoom Fo. ee thesecondsolutionobtained above;thismaybeeasilyshownbythetransformation (C8cos6)(C—8cos#'}=1orbymeansofthesequenceequation(14)below. 6.Discussion ofPy. =We have ae, _[2nt\e"_S—Conada(S i)(C-Seo6) 3 Whence 28 dP, ‘BRPSP OP ee 8) Similarly from ary a-1 csoa[[(C-S08¥'(S—Ccos800 2 ayFoy ROP ee (18) ‘Combining (12)and(13),wehave (2n41)Pey:—4nOPe(2n—1)P.=0 2... (Id) -This sequence equation mayalsobededuced atoncefrom (10)or(11). MR,W.M,HICKSONTOROIDAL FUNCTIONS. a9 ae Tn(14)put : 3 ae_Cn-9)@n—4) 2 : : ih_Pe=Grctensh). a with. sofPiety B=n, 3 then -| nF) : taoa20hbg 0 ‘ : Atapg=20)Cate: where: =Cnt? Cnt pee 0Gn) Gata Hgania} and ort Iiisclearfromthiathatu,isofthe'form-a,u,—B,¥, wherea,arerationalintegral’ 2algebraical functions of2C';a,ofdegree n—I, andB,ofdegree n—2. Thefirst three values are(writing 20=2) tht =H, ae j=(ete), hay Wecannowshow that«,,8,areoftheform Sta OsCbOO6.pay Forsupposing a,ofthisform, ie.,wanting every other power ofx,itfollows atonce i thatyy,isofthesameform, anditisseenabove thatayisofthisform, whence the statement isgenerally true, andsoalsoforBy : Nowa,satisfies theequation : ‘2 ecmaass omy ees Sy : with e e041 Hionce substituting theabove value fora,wemust have : : ayes Ocpiaguey cece aay Oud oe SL, : a 4u2 620 ‘MR,W..M. HICKS ONTOROIDAL FUNCTIONS. Hence a i Gay(CaaGeneraPlas:Cengrart +6bOheiMly-im) From this Wage (Corteest ota) ng{CnagCnnsh ooHOACans tb) oeHeyer =Sum ofproducts twoandtwotogether, withshe exception ofallproducts where thesubscript numbers aresuccessive, Assume that (—}'a,,=sum ofproducts ofthecsupto¢..» 7together with the exception ofanyinwhich successive subscripts occur. ‘Thea (aor=(—Y"{c,-2(prod. uptocy7together, &e....) Sn ) +... 5 ~™ =(—)*!{Prod. (r+1) together upto¢,_;without suecessive subscripts. } ~ Whenee byinduction theassumption isseen tobeuniversally true. Itmay be thus stated, a,,isthesum rtogether oftheterms - BoP Qnaayt2846°68°°°“Gn—2)Qn—4y allproducts being thrown outinwhich, regarding thenumbers inthedenominators as ‘undecomposable, asquare occurs inthedenominator. ‘We have a= 1 =_n=3)(n—-2)=~ wl) This result isofvery little useforapplication. IFthecoceflicients (a,)areneededforparticular valuesofztheycanbeveryrapidlycalculated bymeansofequation(15), while iftheir general values aretobetabulated, equation (16) will serve tocalculate them insuccession, Further, 28,isthesamekindoffunctionusaineverywayexceptthatitdoesnot contain ¢,;infact28,isthesame fimetion ofcy,€y. »-Cons that ty.i8of¢,,€)-»» nny5callingthisa’,_ywecanthenwrite +MR.W.M.HICKS ONTOROIDAL FUNCTIONS. ORs Aer 4 «rj4;may heexpressed asellipticintegrals, vie,: oa 7ao SLR’ Neca : =[optageevlftgaeVPF 4ae na[Vo=Sem a=J,[ViFantadsFe| where a28 fo) #C48 -(C45 . Bei-e™ vae™ ‘ and sarcev4} ;Hence 7 ARPA) 2.2/oe yBg! a Paseo (Soe4VEaiF)Aeyear1a where wemay suppose thenumerical factor dropped ifwearedealing with the differential equation, butnotifwearedealing with theséutence equation: The value ofP,when u=0 is# Thesestatements areatonceseéntobetrue.Since1becomesinfinitealongthecritical ciele itfollows thattheP,atenotthesuitable functions tousebywhich toexpress fictions which arefnite inspaces containing thecritical circle, i.e,within anytore, Butitistinite andcontinuous forallspace outside anytore, 7.IfweputforP,inequation (9)P,Q’, wefindinthevisual way " P.Q.=BPL+AP Se, Regarded asananalytical solution oftheequation thisiscomplete, butinthisform é itisaltogether useless forapplication, Now Hes" hasshown thatthespherical * harmonic ofthesecond kindisexpréssible intheform ee de * Ueleae : +Kugelfanetionen’ Kap. ji : 62e *MROW. M.HICKSONTOROIDAL PONCTIONS, : Wefirstshow thatthiswithamodifeation satisfies theequation (9). Forputting ".ag onl nae SO SSSSSgy; (C48cosh2 Qe__2n41("_84C oshayg~ . ee)eeseahoe #Q,_/(2u+i\t, _m4if"59d uy“.-(+ya-SEPsinhOF(C++cosh6)-"0 (mt I\25WIC coshB10eae acres 4 CQ. aQ_/2n+}\%) _2n+1("C84(C=1)cosh8 a a \.orsenere ab a= =o. Here alsoasinthecase ofthePfunctions itcanbeeasily shown that 28 aQy_Berl dheWrTOW : 28 Qe,Invi dy=OQe ‘and - (201) Qe AnCQu+(22— 1)Q1=0 Hence asbefore n—2)..2 aaay bet) where é d ={"-em.eeelyJC+Scosh@ harcone saloreeanay, 4 MA.WiM,HICKS ONTOROIDAL FUNCTIONS. (eee Inthese change0into28,then i : Pol ; fee oe2)CH8F28combO if 2 eeeom]caerare ‘2 Again,writecosh@=ee4,thensinhO=tan¢,dO=seolf,andwhen@=0or©, $20orf ; Hence aol c.SeeswaalVC48=C=H aint} ne eR SBVER ce eee OS Also : By(gene omafore arg 2[Fer sintg=palyecitaint gy PaSaleAlPSL.See r =Te— Sih C0 sin?ot 3 Now = pit_(FWaintgg “ql acwansy : dé=[ace ae ‘ and ; yeh =E)aEee Gy s andfinally ? BS eee en ‘The.value ofQ,forusb:acey-andforusvenvts moro.Henoe,Q.iatwuilable: for x space within atore, andnotforspace including theaxis,8.‘TheforegoingvalueofQ,basbeenobtainedfromatlalogywiththatforP,;but) iy ares MR.W.M.HICKS ONTOROIDAL FUNCTIONS. : inthesamewayasP,(forspaceoutside atore)wasobtained fromthepotential offa ring, s0alsomayQ,bedetermined froin thepotential ofapoint attheorigin, for space notcontaining ityi,forspace within atore. Fortheinverse distance ofa point from theorigin is 1 fizaee«VeFoun8 Hence 1 p, v- 4008v=A-Pa AB,Q,)c08(na) Now, firstly, since thisistobefinite throughout allspace notincluding theaxis A,=0. AlsoitisclearsinceC>cosvthat1/,/C+ coscanbeexpanded inaseriesof powers ofcosv,andtherefore inaserie’ ofcosines ofmultiple angles only. Ifthisbe done thecoefficient ofcosnvmust beB,Qe encebyFourten’stheoremes08nbHd aie) ____If wedefine Q,80astomake #B,/2=(—1)*/3, then = cs nBld VIR ieee _— con 8> ={,onan es 28) :Wewillnowshowthatthisexp¥esionforQ,agreeswiththeformerone.Inte- grating byparts anddropping \/2asunnecessary inthesequence equation mm 5vce_["situndsin@2nQ.(=)=-fCrean . also. cyfe nOlC eos8)5 (ref aeronae0 El eed cosn808 + ScNatetescreo) ‘Hence 3H “= ofone ‘cos(n+)Acreeha gfCranes|(CremOy * = ifr conn [Fcosa=101ronere-Cf etiam|eraa! MR.W,M,HICKS ONTOROIDAL FUNCTIONS. 625 ae gia Fi a(yenig(cent8cosE18 4 FON Qarh(OFQre (PrefPERT i sof”sinnin8 =e ea! : =4n0Q. 9 ‘Thesamesequence equation asbefore.-Henceitisonlynecessary furthertoshow. thatQ,Q;arethesame inthetwocases, f e Now ae awv7=[rere Wee . = Terre{AJC+1=26int 2 fw=Fif,ViR—wsine Meer IfC,$beeliminated between N’and#=28/(C-+8) there result theequations w2VX polodey idee1+ i Hence bythesecond quadric transformation —_, F,=(1+h)F and since 2 Weay/E(+k) Q=2PF. Again 2 0 >ge 2840~aviarval ary =P2B,0) Nowns EQ)=r 204.RQ") s dkde" a =nSeethe+(+k)F} Mococtxxxt,ai 626 Mi WM: HICKS ON TOROIDAL FUNCTIONS. ‘whichonreduction andsubstitution for2becomes EQ)=;4; BHF) © QiFoyAREF LFEF) =v? Ee=yi EF) and : epHR) 2 CE) eR”a SpE) deve} Wehave infactproved that . ao. 008nOdO—_* eri 5, lawGan pV, Jasonows By meansoftheidentity . EF+EF-FF=} < or » peleE-F'=;(}-EF) .canbeexpressed inthefollowing manner, viz, pO) Bfwm Fak a,QW=2G,=1) are #(apttaF)} 9,Thefollowing relations between PandQfunctions willbeuseful inapplications, viz: ae ()PaQ—P.Qn sory B)-P.Q—P.Q=5 woeeeeeQM) )PQ PoniQs=(nt) F They areeasily proved, forsubstituting forP,iy Qusxfromtheir sequence equations itfollows that : MR,W,M,HIOKS ONTOROIDAL FUNCTIONS. == 27 (2m+1)(PosQv=P.Qre1)=(2n—1)(P,Q-1 PQ) éq =P.Q-P,Q yy=4{EP—-F@—B)} 3=4(EF-+FE—FF) ;=on 4 ‘Again ‘ 3 28(P'.Q.— P,Q)=(201)£Q.(Pasi CP.)—Pa(Qey—CQ,)} :=(2n+1)(PrQ—P.Qus) pace Ss Ina similar way (y)may alsobeproved, 10,Asbearing onthequestion oftheconvergeney ordivergeney ofseries occurring inany investigation itwill beimportant toconsider thevalues ofP,.Q, when-n is infinite. Taking theexpression forP, e P.={(C—8.005 oyd0 itisclear atonce that. rf Pan<(C+8)P.> Py é : Further sinceP,incregses withu,4°ispositive, henceP,,,>CP, Also from “wo =| ——n 1 Qn<ghge AlsosinceQ,deereasos withv,4isnegative, andtherefore Q.4>CQ. Hence Pari Qeri <P.Qe buttends tothelimit unity, sothattheseries P,Q, isdivergent. But theseries P,Q. cosn(v+a) isconvergent, except whenv-+a=0, i sue : —38 MR, W,M.HICKS ON TOROIDAL FUNCTIONS, Puithor ifw’>'u (PY,QY,herestanding forP(), Q(w)) yC+8. Pan Qen<qrghiQs Hence theseries Byes c+8\sP..<3x(575) andistherefore -convergent. Much more-then istheseries P,Q, cosu(v-+a) convergent. Againifw’<u was PriQeaGpgeOe andasbefore, theseries 2P’,Q, cosn(v-+-a) isalways convergent. 11.Both thefunctions P,.Q., except along thecritical circle andaxis respectively,*make@finite,continuous, andsinglevaluedwhenwisintegral. Thefirststatement hasalready beenproved, thesecondfollowsfrouthe'way inwhich“,4areexpres- sibleintermsoftwosuccessive P,orQ.,andthethirdisseentobeatoncetrueby ___integrating *roundacircuitlyingonanytoreu=constant, when[2820isseento vanish..Nowthespaceis«cyclicone.Hencethe’abovefunctions arenotsuitableforexpressing anygeneral conditions inthespace without atote,though theyare suitable foranygiven surface conditions whatever. Still keeping tophysical analogies inordertoobtain solutions suitable tothisease, wwewillconsider thepotential duetoavortex ringorelectriccurrentalongtheeritical Circle, Thiawould givecyclic functions, butalsocertain surface conditions, Inany particular cagethenitwillbenecessary totakeacount ofthese surface conditionsbymeansoftheP,orQ,.Thispotential. ismeasured bythesolidanglesubtended hythering. ‘The(eolid angle) Xecanbeexpressed intheform;—c.s. denoting cosv,sii6,— _— inv/CaelCteSoe tb__ 348—VIpsin0ORCSat JOS ksine|OreJOme - O4e—/0=e,+E nd) whore = aa sceSYS ne) ce MR.W.36HICKSONTOROIDAL FUNOTTONS, 208 Tocompletethegeneralexpression forywemustthereforeaddaterm == = Ss v3 os i Jnv(“008k1+coso—sinhw a fe Q Asn SrenmneegVarrreag: ‘Weshalldenote thisbytheletter AQ,sothatthesolidangle varies as0\/O—< 1m. SEOTORIAL AND TESSERAL FUNCTIONS. 12,Thedifferntial equation which hastobeconsidered inthegeneral caseis q y nth 4‘ etecee co PEL whichintheeaseofsectorial functions becomes gece Py dottdedAsin? Intherestofthispaperweshalleallntheorderofthefunction andmtherank, Calling thesolution of(7)Yu.»weproceed toshow howy.,can'be expressed in terns ofYum Yas Dropping themforthetime, assume Boks Yen=peA+Ppa) : ‘Thenwriting (4m?—1)/4=2, andsubstituting intheequation which Ysqysatities, makinguseoftheequation forYstoexpresa“¥*,and“¥*intermsofyyandee”shall get : Won Mant avon : aufembry tap22(8)Lew(artomr) Nowchoosef;4,60that ee ‘ : . 2 ae SN2 z: Sone) frrnig—2 7()—0 :: #—Qnb1)6+4/'=0 A ye 530. 2y4R:We HICKS ONTOROIDAL FUNCTIONS, Ifwetry$=AS, weshall findthatboth theequationsi x F’— (2041) F420 C=O 2f’—2nAS=0 . canbe.satisfied simultaneously iff=nAC. Hence, whatever be,theequation YorA(uCytS) * holds. Again, wemayalsodetermine /;4,sothat : vanstee . Inthiscasetheequations forfand¢are “ are ve ay2hUP smn npewg 2AP)20 as 4°-(Qn—1)b+2/" =o . *Which aresatisfied by$=BS, f=—/BC. Hence, .:=B(—nOv.+SY). Thetoroidal functions themselves arey,/S, andthetwopurticular integrals are ~represented byP..#Qe Forthese functions theabove equations become - ‘ .Pasay Anf28Pnat(20+1)CP..} Py) =B.{28P".,.—(2n—-1)CP...} ~—and-similar equuitions fortheQ. “ -# Since thésolutions P,,,ofthedifferential equation aremultiplied byanarbitrary conitant, wema, when ‘weconfine ourselves tooneoftheabove equations, put Aor - B=1,and after solving theequation ofmixed differences multiply theresult byan arbitrary constant. Butifwewish tocombine both formule 80astoeliminate the differential co-eficient intliem, then theP-in-both must bethesame, and arelation willhold-between-A andB,This weproceed tofind. Dropping the(m)asunneces- ‘sary, write 2 Py= Af 28Pe1+(2H—1)CP 4} and substitute therein a P,.,=B,{28P’,—(2a—1)CP,} Whence: MB.W.'M.i1GKSONTOROIDAL: FUNCTIONS. = OBL Pi=A._.BAS(P+5P,— 1, a *a ;4 which since3 : BN veg debe wee : PteSByee FG : becomes es Ky aes, NX Pesttn'=(en— 9B, : Hence, cs es : : AaB Gx Gm=n) ; Ifwechoose el 1 : he temtm—t then. = 2mn) 1 2 these conditions aresatisfied, and theformule agree with those found forthezonal function when m=0., Hence : : . 2SP,,.=(2mp2n41)P...5,—(20+ NOP, 5fe=5 2SP"_,S(2m—2n4 1)P.-H(2n—1)CP,.. [> oP) :Fromthistherefollowsatoncethesequenceequation , (2m+2n-1)P.ng,—4nOP,. +2n—1—2m)P. 0 ss(QR) Inthiswrite et ae and==GitDaaTig2a8).Gnd) Then ee Nesey BOsCees anes Oa+Eaaes=0 whenee, if -ae(2n—1)?—4ni® ergGait nes anda...a’... arethesame fanetions ofcx,&¢,8fq;a.)areofCy .eel (2n—2)(Qn--4). 2iyra:RSshar LayPas=GHGDe—Timede—Als,Gmponte HemePad (28) 7 ‘Theseformule holdforthétwoparticular integrals P,.andQeandthey aa ‘express thetessoral function ofany-order andrank interms ofsectorial functions 639 —-MR-W, M.HICKS ON.TOROIDAL FUNCTIONS. ‘and tesseral functions ofthefirst order and same rank. Inthesame way aswas proved inthecaseofzonal functions, itmay beshown that . Pars:Qua—PasQaier .=@n=1—2m)(Qn—3—2m) .«(=2m), =] 29=GuFTF2m)n—1 Im).BFBm)EE Ieno—PacQas)««28 also that, pkaHnB=2m).(1—2m)PasQan—Pra PiusQua—PusQns=Gigizin) Grim) as 2M) 13.Inthesumewayasrelations havebeenfoundbetwéen successive ordersof|* toroidal functions, relations may befound between successive ranks. Not putting theorder ninevidence, write Ok Proceeding asbefore itwillbefound that,f, mist satisfy theequations sex 21, SmtCspam) we ey ba(nSoo it ; z RMN ge,ole 4ey'=0 J . which aresatistied by , . ; gaa, fat leading totherelations Pay=A.{P—mgP.) inprecisely“ne‘manneritmaybeshownthat * Pi=B,(P-tm&P.) and that é : AB.=5= ABest =GG— mtie . +Tewe put :ypaaeas ayAaamt) then4: * pooaceian) eres AME.W.M.HICKSONTOROIDAL FUNCTIONS. emesis] andwhenm=0theP,,havethesamovaluesasforthetoroidal functions already eeidiscussed. ay Finally then, : a4 2SP..= I2mCPa np(2n-+142m)SP a5, i aS. 2SP'n=—2MCPpat(2N-f1—BM)SPreye fo ey 6 fromwhich thesequence equation follows at:once 2 : (2m+2n41)SPyss+4mOPy-+(2m—2n—1)SPx.=0. s+ +(82) Ifwewriteinthis rmmlm). AD CV" : * ‘then y Piou OmaDt=Aat(8 :HaarttttGye (e)Hie? Bycombining theformule (26)and(31)itisalsopossible toobtainrelations between onderandinktogether, Forinstance, frointhefirstofequation (26)andthesecond ss ofequation (31), weget a m+2n-$1)y=(2HCPF28Pae =(2n+1)0P,,.—2mCP +(2n-$1—2m)SPaya + . . =(2n+1=2n)(CP atSPon.) with three other relations,‘ The four formule are i (260,312). PaceyCPnu—SPay n=O et Cone(20+1-$2m)Pang—(2n-+1—2m)(CP ae+SPpn)=O (624) «P6B.Sta),(2m--1— Ln)(Pay—CPna)=(m+1+2n)SPy= Ese(268.316). (2m-1—2n) Pusat (Qm—1+3n)OP uF(2m—1—2n)SPan y=0 Woarpnowinaposition toreduce stillfurther therelations (29),(30). ~~~ Forputting n=0 inthesecond of(32a) (m+1)P,.=—(2m—1)(CP y+Sania) ¥ i: whenoe“| : : 3 (2m1){PxQuy—PasQa}=2m—1)S(PxiQuaiv—PanisQas) MDCLCLAXXL © ex : : 634. MR. W.M.HICKSONTOROIDAL FUNCTIONS. But(82) = (2m=1)SP x= —(2m—3)SP any—4(M—1)CP ay ‘Therefore the above =S{PiQio~ Pua}. But from thefirst of(32a) SPioQ.o—Poor)=PosQoo—PucQoi=27 ~ Hence: z egQH—1=2ii)(Qn—B—2m) (=20) PassaQuaPasQhanr=2 Or51p2m)(2un—142n)-- (2m)™ and 1m)~ (38) > orac 1=2m) =2m) ‘PaQea—PasQn=neEin) (En) § Inthesame way, orbysubstituting inP'.,Q..—P...Q'au thefirstof(26)orthe first of(31), there follows aa Parsi QaePaaQnari=S(Prng1Qau =PQue) 14,From theformule nowdeveloped itispossible tofindthecomplete integral of —the general differential equation, Butasinspplications theco-efficients are.deter- ‘mined interms ofdefinite integrals itwillbewellalsotoconsider thesolutions from a different point ofview. Ifanypotential function beexpanded inaseries ofmitltiple ~ sine andcosines ofv,w,multiplied by/C—e, weknow thattheco-eflicients must beoftheformAP+BQ. Nowsuchafunction istheinversedistance ofanypoint: fromafixedpoint,Let.uschooseasfixedpoint,tosimplify theexpression asmuch 18possible, apoint ontheaxisof«within thecritical circle, say(w'.0). ‘Then the distance of(w.v.w) from thisis . Wl?*0G4e-88"VCERIOC+e-88coswh* Ea Henceitgeen cara]<raz:beexpandedinaseries,theeoolfciontofcosmweosne willbeoftheformAP,.-+BQus Further;forpointswithinthetorew(ie,>’) A=0, whilst forpoints without, andtherefore including theaxis(u<w’), B=0.Heneé : ETAL"(, cosmesormeded gp vomy,(;VUCOFcoso=8Scosa]=APsOFBQna . according aswu’.Nowif-thefixedpointbeonthecriticalcircleBisalwaysequalto“zeroand(C=8=0) - ‘ * ‘MR,W.M.HICKSONTOROIDAL FUNCTIONS, © Lee, _S0mmi.conni4 io eet| arcs[Jie oe ; hore HereA=0unlessn=0..Hence ‘ Peete | cosme pisse APnatalpeg 0 ae Wehave already found that : : foo Pat|as 2 :|,(CSene2 wwearetherefore ledtoexpect thatingeneral ae ee ; 508Oki een Faea ecaee a |,{C—8cona} “ete which caneasily be-shown tobe.the case. i * Bytakingthefixedpointattheorigin-we have ee é pay=(Penainteomnish :BenffsVicente HereB=0 unless m=0, andthen < {2 # te copnd. fase éoO ee : anexpression which, has’been already found.: ‘These.expressions assingledefinite integrals arealreadyknowntobesolutions of thedifferential equations, andaregiven byHenve inhis’*Kugelfunetionen’ They mayeasily be-pro¥ed-directly, andconnected withthevalues found already bythe sequence equations, andthevalues forPy,Pyy,ee. ‘Thus writing. : ys osmadd eedPA) astJ,(C—Sieona)* 3 thointegral iseasily shown toeatialy equations (32),andtheonlyfurther condition requisite isthatAshall hechosen soastomike itagree with PowPay feNow. ee Bee ":8 Lay Pie) eniis Vee i i ANS : ene [MRWM-HICKS ON-TOROIDAT, FUNCTIONS. Hence : "608Odd . A=1andPy=| en =|,CBee Returning tothegeneral integral, sinceu,w’entersymmetrically, andsinceif 122,u'Su,itfollowsF(7cosmircosnvdiato .% [ey egEPaQaotLPhnQew accordingasusu’,whereLisindependentofwor1.HeneoLmay'bedetermined bygiving.particular valuestoor1’,Supposew’attheorigin,then oo tLaiabcosnro—_ LQ. lina ai.EaSse (8 oon Tofindthevalue ofthisexpand theexpressions under theintegrals inascending powers of$';which isultimately tovanish, ot Then if < - p<m [cosmucos?wndw=0 . : pam f68meexewel= p>m—theintegralisfinite=I(say) - Hence- 05mw 98!cos\*rosetinane 25taa* (coon)+--'}a Fonecmlcosmb(.Ree ) where . . ca, 0.ofain(Ima)? =138Gm)ae _ e221(2n+I)2nt3)... On+2n—1)IC et "cosweely LQ.eS) aa Mie MR.W.M.HICKS ONTOROIDAL FUNCTIONS, Cee | ‘This atonce gives anexpression for:Quyvie! as * cosnedy z, :MSA) whereMisaome‘constant depending. onmin.Thishasnowtobe.found: If,we 4 write = : {conn ea ,U..=8" en * : J,cc—coney itiseasily shown that .. (2412n)Unay4nCUnatQn—1-42m)Uaiy=0 ‘Thiswillagreewith(27)if . ¥ia _f2m+2n—1)... CwtT) :: NYaa=Gy1—din) cs—2m)Oe Hence =Om—1420).. @m41) . N=Gy=1=3n) dm) and .- coe2 a (C=ese) ; where Nisafunction ofmonly. Here again thisisfound tosatiafy - Lone (2-+1)8Qe15—4MOQi g++CM—1)8Q. 150 ne which agrees with(82)if . N=(=)"N_ ‘ and ass ‘ &WN es . Butfromtheknown value ofQuyweseethatN=1/x/3, Hence. ie aH (=) Cn1—2m).amg (coangte —) : an ee Ceareae carste(RA) VEGi=1+2m),FIM)\ooae : ase i SCC aaa cers itis 688 MW. MLTICKS O% TOROIDAt, FUNCTIONS. :Also. ©4uf c asia. := 135...Gm=1) (Ca=142m). Qn) A +S Vth (ay) Since thedistance between twopoints is a Te BICC—08(P=¥)88conew)". I Itfollows thetthepotential for«unitpoint at(w.'.w) is $=!VC=HO=FEL...P_.Q%.,£08 n(v—v) cosm(o—w') ‘ «for pointsoutside thetorew';whilst forpoints inside, itis “. (38) b=}V(CLGO=ePEEL...P.Q...008 n(0—0') 608m(V0—w’) where, when m=0 orn=0, half theabove value forL,.,must betaken. \Whenv=0P,.=0exceptwhenm=0whenP,.=m, whichagreeswiththevalue found insection II. ‘Also P,,,, behaves inasimilar way toP,,,forincreasing n,whilst P.,,,,>P.., when «mis large, asisclear atonce from theintegral expression forP.... 1.Also since|: } 7 QuaceSafe nee|,(C=conoF itislearthatwhenw=0oQ,.=0 forallvaluésofmnAlsoQu.behawes 18Qua forincreasing n. : IV. 4 TORES WITH NO CENTRAL OPENING.. = "15, Inthe ease where the hole ofatore vanishes the functions hitherto considered ecomie nugatory. In-this ‘ease wemust have recourse totheéo-ordinates already roforred to'in (8). Itisnothere intended todevelop thetheory withi thefulness of °thegeneral cuse. The functional differential equation hasbeen shown tobe : Py_pyttl eFrashledae : é MR’W.M.HICKSOWToiOIDAL FUNCTIONS. 6aee InthiswriteY=V/w@when ‘s ‘ae @O dG a ee aaaiee + gate geOaeta Re: theequation ofcylindric harmonics (Brsset’s functions) withimaginary argument. a LetG.Hbethetwoparticular integrals corresponding tothecylindric function J.Y . ofthefrst andsecond kind, ‘Then ee a Gale)=I(iuiV Ha(uu)=Ye(ni) as : Andthepotential fimetion canbeexpressed intheform 3 $=ViEFESYAG Anu)+BeHa(nu)) c08(nv-+a)008(in-+8) Bae Many oftheproperties ofthese fanetiéns tanbeatonce written down fromthe analogous propertiesofJ.Y,Thusi . Safe) Og a =2["cos(musin0—méyio 55 LI"os(uzens8)sin®do ke. ee Soalso ‘Bigs wn. tGesy : ,=uG,.—mG,[ ‘ and“GuuBGG.=0| wid‘equations theHals¢satisfy. s: Moa Thesequence equation hasbeensolved byLosmtxt," soos,tofallyexpress Gy < andH,,intermsofG,,G,,HyH,.Butin-any particular easewhere thevalues site 4: required itisbesttocalculate successively: bymeans ofthesequence equation direct, Ma Inthespace within atorexcanbecome infinite, viz.>attheorigin,andisnever ‘ero;thisisevident.fromthieequation, Z usshs *“StudionterdieHesselchen unctionen' pf : he 56104 Yun,aLaHtoKs ONTOROLDAL FUNCTIONS. 3 (><, Without, itmaybecome zero.along theaxisbutinfinite nowhere, forasit‘approaches, (5theoriginicmustapproachafinitelimitwhichdependsonthecircloalongwhichit©auoves, Now.wlici Wisinfinite,Gisinfinite, HengetheGfanctions belongtospice outside 4tore. We're ledtoconclude that theHfunctions belong tospace within Thismaybeproved asfollows :Amongst many integral expressions known forY,one iggiven byHutve,* viz: pot . saga ts [esmao - CS Phi emaioste a : se KO ieee : "This igeasily verified, forsubstituting inthedifferential equation it.hastobe shown that, <3 ae . Fink? 0—1coshyen! do==0 : . which,oi,integiatiiig thefirstterin-bypartsfollowsatonce.Fromthisformwe -gather that ; . Les fed whenu=0 H=["d0=0 Be ue HO -cwheiee “Hy”isthe proper fanetion forspace within atore. From thesequence ‘equation thisisseen toapply also.to theH,,ingeneral. . : EXAMPLES AND APPLICATIONS. Inthissection Ipropose togiveafewexamples ofthe-upplication oftheforegoing theory, to,thesolution ofphysical problems,16.Potential ofaringthoseaisisthesameas'thecriticalvivele.Let2’beitsdistanige from theplane ofthecritical circle, itsradius, wv’ ite dipolar co-ordinates, eat _-Then thepotential ig)” . oul”reneen kena = HoGAPE PEELipwos0 ‘Thisexpanded takestheform VC=EA.D, 005(nefa)« > +forpoints outside thetore1’, ©«Kugelfaetionen’ p11 “ MI.W:M.HIOKS ONTOROIDAL, FUNCTIONS, << Ht Forapointontheaxis,w=0P=andtheabovebecomerespectively a i HS ‘andmyaeek608(va). ee : Tewilltherefore besnoreconyenient todetermine theAya,fromthissimplified Ttiscleatthat1/4’—cos¥—#"}!cansbeexpandedin-aseriesofpowersofcosines of(r=) andthrefore ofmultiples ofthesame: ; “Hence : 4O= He and of‘ MS oa .: Faeww=EA-0808 Therefore. iio 34S"connO : : wana Veron "7 =48Qe :; £ But < TA=2u8'Q', Hence: itrgenerat the-potentiat-for points outside thetorewis =P(ES)P,Qconn(vv)—4P,Qo} -.ss(86a) Consequently thepotential forpointswithintheforew’is ‘ Mp8 Cae hon, i roroa(&){P,Qcosn(v—v')—4PQ} ssss(36) Boththeseserieshavebeenshowntobeconvergent. }IfMbethewholemassofthering * M=2elp=2mue, as Tefollows asacorollary thatthepotential foramass Mon theaxisis,forallpoints. > ‘notontheaxis, 5 : IM Bain pe : geBEVIS T=Qecosnov) HO}...+(87) 4 Also, pitting Mat0.’and —M-at 0,0, andmaking vzeroandMinfinite, the i Potential for'auniform fieldofforceparallel toaxisis , : BVTeSpnQeainne (88) ‘ mpedeuexn,ho : 642 2 MR: WM HICKS ON TOROTDAL, FUNCTIONS. 17.Bléctrio potential ofa-toreauditecapacity.LetV-betheconstantpotentialofthe tore (u’). Then (A., a.)ust. bedetermined, 80that x 5 : $=JO=IA.P, cosn(oba,) may2=V forallvalues of»when ww’.. Hence .=0 and ‘ 7oy("cosa =2/2VQ and oe wAP= /2VQ% “=VY7o=e{28,P,cosnot2?Ph}we (88) Thisseriesiseasilyseentoheconvergent, since(§10)itislessthanLietey where C+S<C'+8, 7 ~ Tofindtheéapacity oftheringwemusttakethesurface integral of<4,8overit So, qdenoting'the capacity, Aa Pigg dn’4,08du2 avi,MPGeouadn 2 _VBaSy[* Af 8 iP.) Qs .a ehclagecBet VORG|posnade ordropping thedashes, andwriting ; aH . t= ag(Per CP,) a22080,("ft2n41/Pany o)|conae Now. ak roomed[ogtt=vI2. and| : Q,_ 241 : :QuCQ.)aeson whenee;: 2a, IASR1+1) PanP,Q) 403,420%byQL. HO) ‘MR,-W.M.HICKSON!TOROIDAL FUNCTIONS. goaa ‘Thisexpression forthecapacity inaninfinite series ismoreconvergent thane Whenthesection oftheringisnotvetylargecompared withthéradiusofitsby Fscircular axis, 4 es a=ta(9420)verynearly* : : (FyFE’ Ss: or:. seg1 (40M).. =ayn=r{ae— Fe} ‘ where - pagy WoeBaa eae Measured interms ofthecapacity ofasphere whose radiusisequaltoatangent fivm thecentre tothetore, thecapacity is = ere © WhenR=3;theomission ofthetermdepending on©introduces anerrorofabout. 27percent, Bmay beexpressed ititerms oftheangle subtended atthecentre bythetore, viz. ifthisanglebe2a,>pote of Trea088#88(9 ¥atn5 When K=sin3°(aboutr=iygR) ~q="793xcupacity ofabovesphere K=sin 6°(about r=}R) q=-898x... | 18.Wemayfindthepotential also’fortheelectricity induced onatore,put,to earth,byachargedcircularwirewiththe-same-axis asthetore.Forthepotential = ofthewire(w’,v') forpoints’ within (w’)is(36b) : ae . ApS <i epy y.$=/0=e2P%Q, cosnfo—2)=3P.Q} - whilstthatforpointsoutside thetore(u,)dueto.thecharge induced onitis $y=/CZA.P! 008n(v—a,) *‘Theexpressiongivenintho‘Procedingy’ isincotrect.402 { O44 __ MR..W. M.HICKS ONTOROIDAL FUNCTIONS, ‘andthecondition isthat-when u=n) $,+¢,=0 ia 0, AP Pg,andAPy=EPO, -eu‘Whence’ * ©os" P's, Py ap86Yom, WP,008vv)+EPLQ—UPD}«(41) and thegeneral solution when thetore isinsulated andhasacharge ofitsown is found byadding thepotential found inthelastarticle. Alsoifthesectionofthewirebeverysmallwecanfindthecapacityofthesystem *approximately, bysupposing thewire tocoincide with oneoftheequipotential surfaces near it. ‘ 19.Asanexample oftheuseoftesseral functions with constant surface conditions, wewilltake theproblem oftheelectrical induction onatoreunder theinfluence ofa point arbitrarily placed. Welosenogenerality bysupposing itintheplane of(22); ~~ Jetthen itsco-ordinates be(w.v’.0). The potential duetothisforpoints within u' husbeen found attheend ofSection HL, viz, o=t(CIC=2)EDP’Qu.sC08mwcosn(v—v') ‘Asbefore, thepotential oftheinduced charge willbeoftheform >F $=VE=EEA,P.,, cosmwcosin(v—e') + and(the tore being 1) i AaPia pVEOL, PrasQan FS=WOVE =O91,2Bo"(PheQaa—QarPas) 608jweosn(v—e’). (42) When thepoint isontheaxis, allthese terms vanish (§14)except form=0,Ifnecessary, also;theTapacity ofatoreandaverysmallspherecanbefound |approximately from these formule, 20.One more example illustrating theapplication tocases ofdifferential surfuce conditions maybegiven. ‘Take tlieease-of --tore- moving paralldl toitsaxisthrough aninfinite fluid with velocity V.Here theconditions urethatif¢bethevelocity . potential forfluid moving past it, b=-Vet 4, and ee MR.W.aHICKS ONTOROIDAL FUNTION. =—— 6g ‘Thoexpansion for@hasalready beengiven, viz:(forpoints notontheaxis) i ‘ pfCZeREnQ, sinny + re ‘Todetermine jxwenoticethatatthecriticalcircle|(aseverywhereon theplane*: ofay) a: : 1 dopie ag Taking apoint outside thecritical circle avay En'Q. 4 Theeasiest waytocalculate ‘thisistomake thepoint approach. thieeritical circle; iesue, when =V=" lim(C—1)'Q, . = ao: =,lim=f Great a : | ofa :' =fWorkoa4/3 which gives thetheorem warae—1S,eosech*2 |Hence 2 beets 2gvYo—e(AP,sinNeaalainnv) : wher’#0whenutyforallvaluesofv,‘Theterms:ineos”nvwouldmerely inerease 4bytheseries foraconstant, wemaytherefore -without lossofgenerality put«.=0, andthen, ising dashed letters todenote differential coefficients, aes 8 aewe#-8{5JotaPen)+VO-AANP”=n} sinnv . 0when um ee osEES(A,P.-11Q) +2(C—0) (AP—nQ')|sinne=0 Ramsar APs tAPASP. +20) —s 5 lub) QO), —6.4.200 4) 646 MR. W.M.HICKS ONTOROIDAL FUNCTIONS, ‘Now itiseasily- shown that PrytP.—(SP.+2CP",)=0 with asimilar formula forQ.,wemaY-hence write the’above equation (AnyAe)Png(AaAaB =Qing=Vea with initial equation (A.A)P2-AP=Os—Q, ‘Togetafirstintegral ofthiswrite thesuccessive equations inorder, multiply those containing P’,,,, P’, byBP’.andadd, weget (Ae ADR PAP PUP Qe PVP Qi Pi) : HPQ PQFSert) . aM ANHOageeSee A AA — ° Put 7 yt, We(APLQPeay Sen then since : ™ lpy —p ; FeaU/W oP MeMeth peas) 1 A ‘SpylOF+a) ba)e} ey Henee (utDita oyete. ite . AenAg tentke and Ota gosttteAan te2hTatty ‘A,isundetermined totheextent of«;butsince thevelocity potential must-be finite everywhere, «must bechosen sothat theseries ZA,P,(u) shall beconvergent, Ttwill first benecessary toprove that A,isfinite when nislarge;«mustthenbe ‘chosen sothat A,vanishes forninfinite, andlastly, itwill remain toshow that with MR. W.M.HICKS ONTOROIDAL FUNGTIONS, or thisvalueofatheseries 24.2.0) convergent, fromwhich theconvergency of@ willflow’atonce, Now é ret wpMeGyOD ; ‘ Senseseal5HoN a . ey=CQ | ne lgtQey Cgwasnds}$e-s ! beeced Boththeseriesontherightarefinite,hencesoalsoare~8),and377-**2, and A.tendstoafinitelimitwithincreasing n.,Itistherefore possibletogiveaa'value which shall make thislimit zero, Itisgiven by otha :osEM—an=0 whence ata, igorrta : Aegan, Sta esvegearn(AB) Lastly itremains toconsidet theconvergency oftheseries ZA,P,(u). When nis verylange—A.tendstothelimitO's, whichis (tI ta/Qey Qe<Gatnee ep) be Alsosince <u, P,(u) <P...Hence theseries under consideration is. Ayte [QugPe (Qualys<@en Ryo & é Thosumofthefirstsetofgermsis<°45"'s 29..andofthesecondctia :C49) ont :<“V'sE 40. bothofthesearefinite:Hencothesum£A.P,isfniteanda Sortiori thesumZA,P,(u) sinnv, 3 _.Finally thenthevelocity potential forfluidmotion duetoatoremoving. parallol to itselfthrough afluidatrestat-infinity is btv8y /oxestAP,sinne where Ayisgiven by(48)and ott ‘WR. WM. HICKS ON TOROIDAL FUNCTIONS. :-ba i gee : SEA :Scat ipe andwhereP,Q standfor“Pt,den) ‘Wemay nowfindtheenergy ofthefluid motion. ‘This is,thedensity ofthefluid Beingity,gy :de T=—[anpndo. gi? \‘ and ) wa t=O 7 ab_yiiy &ao Von Pome _ “T=—2natVsi"erate : =—820VSEA.P," cr” nL) (L208 (4I)e—oon41 But . ffSehae=2v20, =o aly _ooTS—F @VSEA.P. 1QQ) But Wa Wr=—208Q,, Pe OVSSIAPQ. oe (AM) whichismoreconvergent thantheseriesfor¢. Ja similar manner may befound thevelocity potential foranymotion oftrans:lation,orthemagnetism indticedinauniformfieldofforce. MR,W.M.HICKS ONTOROIDAL, FUNCTIONS. Ce | [September, 1881.—At thesuggestion ofoneoftheReforeosIgiveafowadditional =nomerical iustrations,. Thefirstistheratioofthedensityofelectricity atapoint a‘on-a-tore furthest fromtheaxistothatatapoint nearest theaxis. ‘Thepotential a ‘hietothedistribution ofelectricity onthetore isgiven by(39). ‘The normal foree e atanypoint ofthetoreis : apbu__ Ce : whilst forpoints furthest fromtheaxisu=w, v=0, andforpoints nearest u=v',, ram Putting these in,remembering that 2SdP,/du=(2n-1)(P,.,—CP,) and (2n-+1)(P.41Q.—P.Qu41)=2m, itiseasily shown thattheabove ratio is” 1ri hd i (a rerarNEMon(shtptat}$HQ—Q)—20~1)200, Ifthe first. sequence equations inQheadded together there results 4(C=1)550Q, =(21+1) QuQ)+WA whence 4(C—1)22Q,=Q)—Q, : Further, putting (—)"Q.=yu thesequence equation foryis > (20 Utes ANC, A(201) g-4=0 whence asbefore : (CHS). (CHS. 4.H=—(QQ) - Finally then theratio ofthedensities is any : 1 yy ee=(e) :opty, 3 Ifterms higher than P,beneglected thisis a(ueeyEamer lave notheenabletofindafinite expression for31/P, and3(=)'/2.cbut when : theratioofrtoRisverysmall, thefirsttwoternns aresuflcient, InanyothereweMDCOCLARt. de 650. MR_W. M,HICKS ONTOROIDAL FUNCTIONS. wecaneasily findthelimits oferror produced bynioglecting terms after agiven ono, ‘Thos suppose allafter P,beneglected, then §10 piasties eeev7G80” |whenee itfollows that = : valwa oe : SaVP.> ipp<a—ePP, Similarly itmaybeshown that, being odd~ :+ (oyypey BIa2BalVPayUP, me pees ey<:se(ie) “ie, Forthetwocases of=sin3°and¥’=sin6?(corresponding very nearly toR=10r und5rrespectively) theratios are“5171.and °2656. . ‘Theratio ofthevelocity ofthefluid atthecentre ofatore tothat oftheture itself’ when itmoves withont eycli¢ motion, parallel toitsaxis, iseasily found. ‘Thepoint is =givenbyw=0v=whichmukesPam, ‘Thevelocity ofthe,tuid =9 . aN? axtud.a(—y therefore ratio =—1685(—)'wA, ~ Inthe table below aregiven thevalues intwocases ofa,Ay,Ay,TY(the effective mass ofthefluid measured interms ofthefluid displaced), andV’,theratio ofthe velocity atthecentre, tothat ataninfinite distance when thetore isheld atrestin the stream. SSeof esfw A rjow| : wine|voces|ote —ew000. wy|tosis|rao|ones|Sort or|neous|risz|Lex oS Suppose thetore held inauniform field ofelectric force parallel toiteaxis, ‘The potential ofthefieldis . eee ou ai2AC—e)EnQ, sinww Mi,W.M.HICKSONTOROIDAL FUNCTIONS, ree a Hence,supposing the-tore-to-be at.zeropotentialandtohavenocharge,the= potential ofthedisturbed fieldis,dashed letters denoting functions ofwv’, i Res Be Coin (a-2P)sinno i | 4 ‘Thedensity atanypoint ofthetoreis : Voyage "fpr Woo Pe]. ato =ere{Peng,“refainnw SA (C—e)!3 sinne ii Now atthepoints where theosculating plane touches thetore=randp=R, ~ whence CHc=S8rore=1/C. s=8,C : ‘Thegreatestdensityon.asphere similnty tnflyeticed is*‘Theratioisthen vist —_ayisyp sas ree=elemter,t} ‘Thevalue ofthisratio fortheeases already: considered are forF=sin 9, “075 wBasin 698 When thedirection oftheelectric fieldisperpendicular totheaxis,itspotential is : . Scns. 4=p con.api r=Hpcosw=pin Hence clearly thefunctions fortheexpansion ofthis.are thetesseral functions, Pay Qj...andtheconditions, sincethepotential holds forspace outside thetore,arethat Heeee EEREeoWEALD, o98n=9 when wu! forallvatnes of i ee ae : 4 oz Mn. W.M, HICKS ON TOROIDAL, FUNCTIONS. Henoe e : FAPa+DaalOe"deo or APM QaI=0 and ee egAEMFant1)(Qo-%—CY.) But s aa.BeBC -CXad) From thefirst of(32) 5 waBP..=Phaci—OPoes eeA(Pa,) “Which enables ustowrite theabove inseveral ways, Asbefore, thedensities of electricity induced atpoints (w’0.0) and(w'.r.0) ateeasily found.) {.653. z Rinse esr!| XV. Polacanthus Foxii,«largeundescribed Dinosanr. fromtheWealden Formation’ = intheIsleofWight, =ia ByJ.W.Hotxr, PRS, : Received Jannary 3,—Real Jannary 27,18 . [Phares 70-76.) Fontheopportunity ofstudying theremainsdescribed inthisnoteTamindebtedto athecourtesy oftheRev. W.Fox, ofBrixton, IsleofWight, who lastautumn gaye inefreeaccess tohisrichcollection offossils obtained inthat locality, £ Much shattered bybeing very hastily dugout,andsince much damaged bythe accidental breakages andthedissociations scarcely avoidable in.theabsence of suitable plice fortheir safe-keeping, there isrisk ofthese remains becoming: before long losttothepaleontologist... Inview ofthisnotimprobable eventuality Iventure tooffer totheRoyal Society these notes, inwriting which Ihave been reminded that itwastothis Speiety thelate Dr.G.A.Maxrett, now more than fifty years sinco, communicated hisfirstdiscoveries ofIguanodont andHylwosaurian remains,‘Theremains ofPolacanthns werefoundbyMr.Fox’in1865inabedoffblueshaly. clay, which occurs near themiddle ofthecliff, ashort distance east ofBarne's Chine, Thebediseasily recognisable bythelargequantities oflignite which itcontains. ee Professor R.Owen, towhom Mr.Fox showed some ofthese fossils soon after their discovery, suggested fortheanimal indicated bythem thenaine Polacanthus—many- spined—P. Fonii, andthisname Mr.Foxadopted inanaccount-of hisdiscovery read. byhimatthenextmeeting oftheBritishAssociation. Abriefnoticeofthediscovery witharudewoodent alsoappeared about thesame time inthe“Tlnstrated London : News.” Boththese cominnications have onlythevalue ofpreliminary notices. by vee: persons without anatomical training; and nodescription ofthefossils sufficient forthe : useofpaleontologists hasyetappeared. x Mr.Fox's MS., readatthemeeting oftheBritish Association, cannot nowbefound, andhispaperdoesnotappearinthe“Reports.” Anabstract whichLmadeofitin1869givesthefollowing listoftheparts hebelieved hiehadsecured. 3 “Sacrum andpelvis; 7lumbar, 7anterior dorsal vertebrae with their’ ribs; 20 ©caudal vertebre; 2femora; 1tibia with fibula; 3-metatarsals, phalanges, and 3 ‘unguals;20to80largederinlspines,andasmanyscutes.” j : ‘Theéeattered remains which last-autuinn Isucsveded inbringing together again do