a : Pan ate . ae ar: a I eek owe SER “ «8 Lf . vere te ~ . ne ‘ ed Os Rom et: ete tye ais (ss SERS : ‘ ‘ wats wie Of . , : . pis kA \ . sone . . se , .» “est wats sates » ; arate te i Pyeve et nat, AmDaTh) . ah - ‘" ee ve BALD, ‘ ae he {hha eee’ aca vist ‘ ne ae Ta a Geicinstens : . au ves PAV eee, ‘ : weenie SS vs ves Peg eR Ree eye BS A Toe SUM ge eae Salty : : eet iM ate aes wy Yraliterels TANTS et eae oe. i ig + . os i kee Lao a . . ne . 2. . Pater tiie) F . : . : vee Viet s : i ANA R OMNES Rie i at ace es Ee er . ovat tes PB ‘ \ < Otek eke 1k <4 . Ss . eae . ra . . Pa 4 z < i ery a . iy ES i Nc ue cvs yt Nets ks J Md . 2 tons > wee pat 8h om met im ™' ; ; : ESI K Rete nae e Nin evaaalea es Ra Seiten hoa 7 . . . o2 We ® e's ' ‘2 . pa eed . . ete . RAC ea ad te anaes : . oe ’ eae trig +. Ao 7 aes sre 65 ' Ste ys te oe ae ‘ . ir 2 . : fe : van : 7 : Z 7 Vee sae * utes m an. rea protects : ‘ - Woe ee ose i F SOUS gets co te 7 atee its Sees rhe : ie COE OR . eae ret . See ATES Se ee poate & Ai : UE SES oe : Sey iS on : : YESTOR wos : sp : J he ents aim Sopeees rho cs : é 5 : ‘ OPE Ze Plenty ‘ ‘ F : fsa i / , PES : , ¢ ‘errs. Cad ‘ - . oe a Pee Tete ieee rat A : eee) Pac fA emer Se ue : lees SREP LS OLED AA dg Lee hd nie a eh eines LN er etna) oe ee I ATE amet Seta be ob re eT nae ea rere S, EFT ER sg pe deed are Std Lettie + ere rer * ¥OFIL ES Ly Pitt ici aeeraes 55 beware" 2 Pe ware eee ons teense Eee Pee PEE trial " Ree i pipaeraie S + ae ppb we oF wee LLG tee OF peer e yg heir err oat ¥ ieavetoee (tee eee f- iets FE, eae Prete. evens Ore ee Fee eg wren Peeps Digitized by the Internet Archive in 2009 with funding from University of Toronto http://www.archive.org/details/transactions14camb VALET. XVII. CONTENTS. On some General Equations which include the Equations of Hydrodynamics. By M. J. M. Hirt, M.A., Professor of Mathematics at the Mason Science College, IEF TSTTHBYANENIN aoecosoouaededaonan apse on nb8ab00007605E0e8 000 SOB ERE OA dado dune EE UEo a Be acescEUncCnaTe Teer On the measurement of Temperature by Water-vapour pressure. By W. N. Suaw, M.A., Emmanuel College (Plate I.) On the pulsations of spheres in an elastic medium. By A. H. Leany, M.A. ...... On the curves of constant intensity of homogeneous polarized light seen in a uniaxal erystal cut at right angles to the optic avis. By C. Spurcr, B.A. (Plate IL)... On a certain Atomic Hypothesis. By Proressor Kart Pearson, M.A. Communicated by H. T. Svearn, M.A Some Applications of Generalized Space-Coordinates to Differential Analysis :— Potentials and Isotropic Elasticity. By J. Larmor, M.A., St John’s College Observations and Statistics. An Essay on the Theory of Errors of Observation and the First Principles of Statistics. By F. Y. Epaeworru, M.A. Communicated by Aa Nise dbs, (EnaAUGi stop Dad Lb NB aye coccbemanobecocte (dee Does GaE hace Gunceo CSoCEAOSECE aetrereecreere tt On a new method of obtaining interference-fringes, and on their application to determine whether a displacement-current of electricity involves a motion of translation of the electromagnetic medium. By L. R. Witeerrorce, B.A., Scholar of Trinity College, Cambrid gems (latest lTTeiQVemV i) rs ccerene se oo -clsckioncte cei beeistetaeties cisneisiseisweriesecatceseinsre On the mutual action of oscillatory twists in an elastic medium, as applied to a vibratory theory of electricity. By A. H. Lrany, M.A., Pembroke College ........................ On a class of Spherical Harmonics of complex degree with application to physical problems. By E. W. Hopson, M.A., Fellow of Christ’s College, Cambridge Table of the Exponential Function e* to twelve places of Decimals, By F. W. Newman, Emeritus Professor of University College, London ...............secceceeesceeseceasecsseves The Equations of an Isotropic Elastic Solid im Polar and Cylindrical Co-ordinates, their Solution and Application. By C. Curer, M.A., Fellow of King’s College, (QWINSTHC IED onnis boodieboonocd Aon sod ans OS DOGO GHMDADObeDoAoGDEe— aes nao acnocn ace adSboRNadENedadoacodpeosaaRbaa On Solution and Crystallization. By G. D. Liverne, M.A., Professor of Chemistry IU CMmU MV ersitymotm@ambrid eu. atecsepcac-accerereesceelt cena esto seeeaeac- eee ccm e aes On Solution and Crystallization. No. Il. By G. D. Liverne, M.A., Professor of Chemisinysans thes Umiversity, of (Cambridge: s2a..sceeecaresce cee ciesecctieiss ec ce= ceria ls riee: Systems of Quaternariants that are algebraically complete. By A. R. Forsytu, M.A., Hass) Hellows of Wninity College, (Cambridge rite «cei s:00/ 015+ + ocujoois oe ainaie sis iesicleaeseeiee sea On the stresses in rotating spherical shells. By C. Curren, M.A., Fellow of King’s Claire, Crvinlortles) <5, coaanaed 200000009 06003005006000000700nGaDHoGDod FepORaH so b-boagnGese-.0nq0qnq00000 On the Binodal Quartic and the Graphical Representation of the Elliptic Functions. D3 Jeo CASTE? aiocacsconoaumon deapagmannce ce 4cbeaEebopsruSsnocaads0cro bone raonccnche odoaanccmaeches PAGE 30 45 ADVERTISEMENT. Tue Society as a body is not to be considered responsible for any Jucts and opinions advanced in the several Papers, which must rest entirely on the credit of thei respective Authors. Tue Soctsry takes this opportunity of expressing its grateful acknowledgments to the Syypics of the University Press for their liberality in taking upon themselves the expense of printing this Volume of the Transactions, I. On some General Equations which include the Equations of Hydrodynamics. By M. J. M. Hu1, M.A., Professor of Mathematics at the Mason Science College, Birmingham. [Read October 29, 1883.] In a paper written by Clebsch, “Ueber die Integration der hydrodynamischen Gleich- ungen” (Crelle, Bd. Lvt.), he has shown that if the symbol p of equation (II) of this paper be constant, and if m be an odd number, then w,, u,-..u, can ‘be expressed in the 1+1 : of the symbols f,, f,...f, vanish. The object of this paper is to obtain results which apply to the case of m even as well as n odd, to remove the restriction regarding p; and also to enquire what equations correspond in space of m dimensions to Helmholtz’s Equations of Fluid Motion, and what form of Art. 2, where however corresponds to a vortex line. The following is an abstract: 1. Let z,, z,....«,¢ be (n+1) independent variables, U,» Uy++U,p be (n+1) dependent variables, and let gq stand for V+ | = where p is a known function of p, and V is a known function of the independent variables. Let the following equations be satisfied du, du, du, du, dq dé + dn, + Ms da, * nt dz, dt ee pe ew A ee retain cc tae AM Sc, ceeds (I), du,, du, du, ng rai Us Te + U, da soa ar Ue, as da, | dp, d d d = + —=— — — = inecoobaddereosnacsaeiiananscocdoo ( dt as (pu,) + Ts (pu,) +... + cE (iif) = Dee ceneresocco: (II), U,, Uy ++ U, being symbols for any definite functions of the independent variables satisfying (I) and (II), let their values be substituted in the equation df, of df of _ ay tM ae: i saabor On gee Soognnnoondedosapeasaoer (III). Won, SLY, Parr I. 1 2 Mr HILL, ON SOME GENERAL EQUATIONS WHICH Let P,, P,...P, be n independent integrals of this equation, definitely selected. Then if A stand for the determinant 1 2 1 dx, da, (oem dx, dP, dP, dP, | GLOSS Th dz, | dP. dP. MO. ify eS Te it can be shown by means of (II) and (III) that d d d d\/A (Git ae tae, t siatantate +0. ge) (5) =o 2. It is shown that if A be an arbitrary function of the independent variables, and Fis Fo-- Ff, arbitrary functions of P,, P,...P,, (not #), then u,, w,... u, can always be expressed thus :— 1” da a ape Pie 1 ae eae 1 _dK dP, aP, dP, tpt da ae Aa de tee te gan dK dP dP dP "da Figg Toast eons He le 3. It is shown that if n be an odd integer, if f= oe, and if 2, & -..6, be the square roots of the coefficients of &,, &,...&,, in the determinant = which is equal to ap Ze NS ola Es ae Es hala aes igi Enis ae =) ae Bs. the signs of the roots being properly chosen, then 6 f) = du, E, du, on du, = du, E, du, Ee du, AG = p da, * p da, * eee iagpedaais p de,* p dx, Sh reise ai da,’ é (*) _£%, £ E, du, ee E,du, _ &, du, a, E, du, He —, du Si\p)~ pda, * p de,*""* p dz, pda, * pd," p dx,’ dé, dé, dé, _ at aa tise aa 9 But when n is even, 51 =I 0; and this is the sole representative of Helmholtz’s Equations in this case. INCLUDE THE EQUATIONS OF HYDRODYNAMICS. 3 4, In this Article it is shown that if a, be the coefficient of oF in the determi- dz, nant A, and if 2,,A,...A, are functions of P,, P,...P, (not t), subject to the condition oA, On, an, : : age ; : SP. fap tt spo the following set of equations similar to Helmholtz’s Equations hold good, whether m be even or odd. é ( ie NBor stake ont *2) = Sn a Noto SEDO DS = =e son HE Vin + Aon ar ooo Nin du, p p z RR eee eee eee EE EH EEE EH EEE ERE EE HER EEE HEE TEETH EE HEE TEES EEE EEE ES EH EEE SESH EE HEHEHE HEHE EE HEE EEE EEE EEE EEE Ee + 38 + cost I CRE 6 (he oe AG ae eee eee) Agta Egg ee $n, Me, NZan + Mabon F222 + Ayan Ue, RRR REE Ree EEE He eee ERE EOE EE HEHEHE HEHE EE EEE OEE EEE HEHEHE HEHEHE HEE EEE EEE ESHER ESSERE HEHEHE EEE HEHE EEE EEE Ee r) (s +245, fee + ee) Mt thoy toe FAO Me, Eee ASin + Aoton tes $A,2,,, Lu, ot p p dx, p Gln. d d dg, itis + Motor + wee FAAS) Fee # dag. Otis +s op sh oee + AG.) + 1 & d ar dz. (1245 Fg % q+ oe + Man) = 0. 5. It is shown that if n be odd and X,,2,..., be the square roots (with properly chosen signs) of the coefficients of X,,, X,,...2,, in the determinant A which is equal to an NP Ai Ah.- = yg Me Soe 1k nets of, _ 8, where ,. = SP, = oP. > then gE, = Ajay, a NyMoy ae O00 ar NZar> E. = AO a A,2o9 qisietelen shy Ay2n2>° y on = AG in oF NV Bon a O00 Van” In the case when v is even, it is shown that JB=A.JA. 6. Adopting the language of ordinary Fluid Motion, it is shown that the vortex lines Eg, fa always contain the same particles, being the intersections of (n—1) loci whose equations each satisfy the equation (II). The same argument shows that the loci dz, dx, dx a. +..+XrX24 = ee Pode Ge padiy. ping Ain Pte es 2°21 noni 1712 Aa, +A are always the intersections of (n—1) loci whose equations satisfy the equation (II). —_—- “= 4 Mr HILL, ON SOME GENERAL EQUATIONS WHICH 7. It is shown that (n being an odd number) the forms for u,,u,...u, can be reduced to Clebsch’s forms, so that u,dx, +... +u,dx, =dK+P., ,aP, + Pas GOP ot a. ote eee dz, da, _ dz, and that the lines € Ties —— e ) lie on the loci P,=constant, P,= constant, . fo constant, Pus constant, ... P,_, = constant ; the equations of which satisfy the equation (II). 1. Taking the notation in the abstract of this Article, let the reciprocal determinant of A be the determinant IER: iy a (ae ROE (i fee nae : sila td d Oka. d Putting for brevity Di for = ag t oe u, aa +...44, ey it follows that DA = D (G)+ = Fibs. 2 = Dt ™ Dt\de a 5 (ae ea alas +a, (DB) a 2(B)s ce 2B) % De\da,) +“ Dt\da,) + * ™ De\de, =| Seip mivisieisie nie sioiom(olelsialeinelsietniateia/eiv eisie sel» ola\n.e/anisiels|~inieleimniainsina 7) ‘n) +4 i ae!) mae io) = + 54 (at) +4 Di aan) + + 8s De de 2 D (dP,\"_ @P. &P. dP. dP. HEN wilas)= dtde, * dade," da,de, +" = dodo But since P. satisfies (II), r Differentiating this with regard to z,, TP, , du, dP, @P, , du, dP, TP. du, dP, &P, 0: didz, * dz, dz, + “‘da,de, * dx, da, | ”*deda,*"""* de, do, | *dz,da,_ ‘pg be _du, dP, du, dP, du, AP, Di\da,/ da, da, de, dz, '""""" dx,’ dx,’ INCLUDE THE EQUATIONS OF HYDRODYNAMICS. DA du, dP, , du, dP, du, dP. Pi a = fa (Fe dx, ine da, aitigaeteat ae du, dP, . du, dP, du, dP. M12 (zm dx, ian dx, aidoci ces + ae, ae AD ¢ canodoboccoss ous acgodacHansdbdoriaocdcauasoDaeaD du, dP, , du, dP, du, dP, Tin le dx, dz, da, ees dx, da. )f lee du, dP, | du, dP, du,, ) a (im dx, de. dz, Sein’ dx, dz, cine (ze dP 2g My ae: ey 4 au, — S\dada, dx,.da, 2" dx, dz, ar qoooocgderacacogusbannoocarmagpe sdo0e bee eee eee e eee du, dP, . du, ait du, =) Aen Ge dx, dns ai Ohio. (hips = Ieecine sicisinieecieis sieieeleiasleseleeelniaselelevcinie)sisicleciesiee aleisiei=(s du, dP. du, Py ey dP, ) a (ae dx, = Lesa e dx, dx, du, dP. , du, qk 4 On = = (um dx, am in aan dx, dz, a SpovedoadgousadocdancoscooonrS0ncencosodEecodr COOeG " du, dP du, qP, du, =) ; oe (aa day dite Cin Ati dz, dx,/\’ DA du du, du, wr = ae A dz, 0 + ween == dx, 0 du, du, du,, Cae COE A ade oa ica 0 sta cteteleiainioleieieintesieieiete ete siatsleatelcleretsielstetelctelsielelelals du, du, du,, Soa des (eS Ap onot + ae, A; DA_, du, , du ¥ = ae (aaa oie: * da, ; But equation (II) in the abstract may be written Dp, du, 4 ly 42%) <0 Dt (aa TEA A a Hence eb (=) = 0, ol 6 Mr HILL, ON SOME GENERAL EQUATIONS WHICH It will be shown in the next article that this equation may be written 2. Now suppose that instead of «a, @,...@,¢ being the independent variables P, P,... Pt are taken as such. Let differential coefficients according to the new system be distinguished by the symbol 6. dP, & dP, 8 dP, 8 8 d Then +E aaa aA tee OPTS: dt. dt ‘8P, But since P. satisfies (IIT) r dP. qi. ; Tee) Pace, a U, Fo Une $d dP. dP, dP\ 6 Bi dé t (« dea a dx, ete a i bP, dP, dP. aPNYS + (“ae + Us da, sterayateisisle + Uy, =) SP, Fade lolalrelelelalafecielele btetelsye Stele eisteteM-taiale Meiers eteie aetatelaia (ota ftp rn eB) 2 («, du, 2 dex, seneee U, dx, FS P, da DE ROR Rnd 6 uP. 8 mie (a SPM doer aE Oude sp) GP. ie ae. 6 dP, 9 + U, ee SP, 45 a, SP, ci eeteiale sie + da, = SP sia o's.0(aje cisn nleleio's's pitlieicie\clo\ale'els ulelole' « eleis/e/6M vie o10[n1s\01a)=/=fs GP. iP Wak. ie aP. 8 +4, ta nae diz. spe ae + du, =) d x d Nee} d aie <2 det da “dee “da,” “. What has been denoted by zp may now be written é pe? 6t" Also equations (I) of the abstract may be written Le hea (= rd ae ir 2h) 8 da, \da, 5P,* de, SP," da, SP,/’ buy da __(aP, 8 AP, 8 AP, BY) ét n= (ae: OP, ai aP: da, .0R ey. REET E RHE EE HEHEHE EHEHEE HEHEHE EE HEHEHE EERE EES pene INCLUDE THE EQUATIONS OF HYDRODYNAMICS, 7 du, dg _ (2 8g , dP, 8¢ hao dP, 1). oe Pa des GFA da,oP, da, OP)? bu ou ou og + Oya Bet a By Te Hn Ge 4 SP bu, bu. ou, Sq “a Sy On BE uy aa OI inaneanumnenasaesananesspe: (V) bu bu, du,, bq Gas ay tT Sue ap eo »T San ét a one It is now necessary to find the value of 5 Di dead, we. eee ame de ae 9 ae Cy Sa ent s aae ap. (= cia = da, — dx,_, dit, ie dec,, dP... dP... aP as ge. ges ieee as dP. dP, dP, dP aay ieee ie Let the reciprocal determinant of (—1)™z,, be (Steve Sood (Sy eos. Soiaeresce Be | Renee ie ee Beet [Spanyere ees B...| > pe ee ocs: (e ORE oBiey alee nocd jo 3h Sec ccer rece sscn er cc senses ceccessceseesee see sesees Ben.) du, dP, , du, dP, du, dP, du, dP, , du, du, dP, = {8.(ae ede de, Stee Be Bap ae ze hie toe) q da, , F . Adamvaa du, dP, _ du, dP, du, dP, du,dP, du, dP, du, ) +B, oa Gas dx, a dx,,, dx, re + de... de) st ee lea dx, ade di, oa dex, da, | TTT TAN ora hetn\a\oicta: a o)sin ofeien|nic\slelelejels|e's\nia(njo\e\s\ni¢.e/eluie(n\e\e|e,0(6/0'0)¢\5\0)u\s]0]6.8i8\¢/eiale/n.ejees.cisnicie-e'sle.cie)ejcjeie/eie\eieisivie(eteccieic@ ose a cm ce sae du,dP,, du,dP_, du,dP.__, dutde. dw, aPo Oe, CUP, BB ae at Paes dx, te Fe. dx, le. +B, elae dz, dx, da acca ya dz ) st 2 6-1 8 (G Pray du, dP,, a, rai 2 oe) B du,dP_, , du, dP,, se) eae ae Og) co dx, Lurie 7 al oe as ae dx, = dx, a rue dz, Mr HILL, ON SOME GENERAL EQUATIONS WHICH iS ( dud Pon, du,dP,,, du, 2) du, dP, , du, dP,,, oe i +48 raa (get dz, ‘de da de, dey) fer bP ee. i dx, 'da,, dz, '''*dz,, de, — du, dP... , du, dP. du, aP,,, du, dP... au aP..,, du, aP,., + Brn a Cee dx, La a dx, i ar age a le dx, + ae, , ae, a hes =) SPF ota n notea wan ceaoeceiccemecte es seine ste seca nlaseateas cuemeceee Pe boing ayia eolaran cial colle wate ele aoe CER du,dP, , du, dP, du, dP, du, dP, , du, dP, du, dP, Bua Get Ge de ie ae Fa) tot B ma Gao ine Gide ae =) du,dP, , du,dP,, du, dP. du, dP, _ du, dP, du, dP. * +8, w (Get de, * de, dx, * A dae Gan) tot Baa gat du, * dz, du, * dx, de, - 5 {1} = Fe (Aya) tt (8 FEE ta et Ban et ABs Gat) Hone Git 0 Pca nyiabonvcienesiciisis sec cineud.cesiepsainisinjeieieie oie\olainisia(slnis ein ieje\neisie.ss\cle vice einln\e\elale pleesieeisls\c,0,c\s0e\0\e0s(ois\«\ejein/=is sin(e/siS(a\e\e/» olelofeleibiarn\vlelel + get 0+ a 0+ +e 1)"a,,)+ +) (8. a ate +B +B get Byes oe +. +e 0 +) 0+ jet Ot.. tT (B, gt ne, wg Beas gett A Baan) tga TY aah gee Et aieheteimieimin alain’ ale inte aiolotaiereielelalalacajeiele)pi= hie =tatelsicloleie\le/aie/=(alelane s1s(ala}ate|etxratata ofa =! chosen with proper signs, then 2,,=£&£&=%,,. & is the Pfaffian {2, 3, &—, having the same relation to the suffixes r+ 1, r+2, ..(n—1),n may be denoted by 2, 3, has to the suffixes Observing further that EE = ze = ‘& 1D is we eewnee it follows that eS &e a eS & ie 4,5... (n—1), n}, {r+1,7r+2,..., 1, 2, eee aoe a Bees aa ee 2 ae aS ace wy ToL saree cas the Pfaffian (r—1) that &, .. (r—1)}. BO faa en Ser Shee 5° Daas ca ee a See erie er ir FE 1 Op du, oe (dU, ) 7 &t (ED 2 (| ét + de, = +2 2 Ge £8) . O& (1dp , du eee Or ac ae a mn OVEN Didi &. cb du) £, du, E, du, Use) ae ae a p da, Mr HILL, ON SOME GENERAL EQUATIONS WHICH 16 But FB t ba tos + EB = 05 “ Es 5 at Case Stree St aes = 0. But =()) an seo PS e =0; ei ce p en 9 . 5 8) _&,du, , & /du, E, (du )= E, du, E, du, E, du, | : g(a) eget Gah Ea) + i oe + 8m ae: a ena Sa dz? . 8 (&)\_ &, du, , & du, E,du,_ &, du, 4 i, E, du, 3 ()= a pene ae Ary te ee Ey, da, pd (& )= Edu, , &, ity bn du, _ &, du, 4 6 du, rey E, du, Similarly, 2 es ét pdx, p du, Pip adap dz, p dx, p ax, where 7 is any integer from 2 to n. Thus there are n equations analogous to Helmholtz’s equations of Fluid Motion (a being an odd number). But if x be an even number then &,, &,, ... determinants of odd order, all vanish. In this case consider the value of oz ay Ey es ay ale ‘ae oe ~ Ot & &o Es es Eon eS aes es ee Es, Pe Be Ens ae Qn0) ae It is=— d du du du Mu ; | En T+ bag ata 24] bi, * Ee, * Ei a fnge, + fe de, tS ge * vy d 1 d das du, du du du, du \+ £a gat + Ene t bis dar + | + Ender $6,541 Tat + bngat t | tba gat t Ea ge' + Eu ge En, En Ei Fe ns mn ROO EERE HEHEHE EEE EH HEHEHE ESET EEE HEEEH HEHE HEEH EH EHH EEE HEHE HEHEHE HEEH SH EHEEEH SETHE HHOEHH HEHE EHH HEE S ES ide ee — similar determinants. £, being square roots of skew symmetrical In the determinant which is expressed, multiply the constituents of the second row Ht and so on, and add to the corresponding constituents du, “4 a by ———, of the third row by ar, dz, Treat the unexpressed determinants in a corresponding manner. of the first row. INCLUDE THE EQUATIONS OF HYDRODYNAMICS. li The result is = du, rs en oF rao ae dav, Ex, En. Es Bt pease Ea Ess Eas soeere En ES Es = sieS SS |, du, du, du, du, du, du, du, du, du, Neel “Eugen + #"do, siete ee ant alas tbs Ta Use lens aS + bie go, + ue dr + Be slays Eu E., Es Ea fs os | ae a a — similar pairs of determinants. . . du du,\ (= The first determinants of each pair are together equal to — = bo oe) (=). wy vn The coefficient of du, . : Spe = in the second determinants of each pair 1s eee ce cee vere nccssscerscscesessccscs and this vanishes unless 7=s, when it is =—5; a) (= a) (=) _28p (=) ot dex. dz, p ot 2 This is the sole representative of Helmholtz’s Equations when n is even. Tt still remains to show that Ha + = + + a = 0. n It is known that if m be an even number, and |G, --++++++ A» | a skew symmetrical | werner reece eevesse | Wiese el (ise Aare 00 a determinant, then its square root which is the Pfaffian {1, 2, 3,..... (m — 1), m} = 0, {3, 4, + (m—1), mtorr t Oy, {+ Eo 1, My Dy B55. ee +a,, {2, 3...m—l}. Vou. XIV. Parr I. 3 vo 18 Mr HILL, ON SOME GENERAL EQUATIONS WHICH Further £ = (2, Sanus n—1, n), E = (8, 4 «0. n—1l, n, 1), &, = (4, 5, 2.0... n—1, n, 1, 2), &= (rb sss 3 bacco r,—1), &.=(ra+ 1... Py Uessgoce r,—1), Bary +L os. tye cece ¢ r, —1), | ee (ee ener Reece n—1) It being supposed that 7r,, r,, 7, are any three integers between 1 and n inclusive, in ascending order of magnitude. Further if two of the suffixes in any Pfaffian be interchanged the sign is changed, 2.e: (1, 2, vee 1, +8... m) =—(I, Digs Sia ta Titel 270) Ep = (KI (ry TFL eee ng Lyorssh loki... figal Der Be r,—1), £,=(—1)r (r,, 7, +1... Ng dy 7D saves Mh, Hs eed r,—1), £,,=(— Ie) &,, La. Mie llcg Desens r,—1, 7, +1..6.. r,—1); therefore the term which contains £7, in &, is (—1)"-"-1&,(7,+1...2, my Sop phan Paar oe eel Eee jean | ina aa eal seinen ues Ll) ~ E. m £, is (—1) eS ee Oe ae Le BOG el Rs el Recor el ymepepce p=) Ene, in &, is (-1)e—nn-Vg, +1. lm 1, 2... -1, 41... dD; therefore the term which contains E,”, in &, is (—1)e-M-1O-rs-9 E, (1, 2...7,—1, 7, 41...7,—-1, 7, 41...7,—1, 7, 41...0), E,,, in &,, is (—1)s-e 1+ -n.-2)—-D ES (, 2.7, — 1, 7, + 1...7,-F, 7, 41...7,-1, 7,41...n), Ene, Wr Ey, 185 (1) te a OE (Ui Zee Tamaly Tauck Loo Tg aly Fact «waked, Geatali nD) Now since n is odd (— 1) -1F (nr) (5-8) — (— 1-1 IP = (— 1) nF = (— 1)etnitr? = (— 1)ntntrs ? (= 1)"- ry—-14+(n—r,—2) (7,—1) — (- 1)te-%e-1=(r- 1" = (— 1)rerrarnt =: (- Prater tnt = (- Lehi, (- 1)®-15+7:-1)4 (n—=,—1) (r,—2) — (- L)ietent? = (- 1)nitrstrs? — (- 1)ntratrs, INCLUDE THE EQUATIONS OF HYDRODYNAMICS, Eg Now every term in day pacisce pee: Socboe 4 contains one first differential co- dx, dx, dx, efficient of one of the quantities &,. dé, d. his ; =i ehae The only place where Fer, can occur is 1n , and there its coefficient is da, dz, (11s | Be. H— 1, ALT... nal, oe l,... r= 4) 7, +1 ..m). The only place where aera can occur is in = and there its coefficient is the same. Te rT The only place where _ can occur is in = 3, and there its coefficient is the same. rT T3 1E dé dé But OSrars T3171 ns — (). . day, He dir, + dzx,, 95 dé, dé, dé,, therefore ie Sh ooaos ar ae aE nocade + a 0. 4. It will now be shown that equations analogous to Helmbholtz’s Equations of Fluid Motion are satisfied if instead of &, &,...... £, there be substituted the quantities 12 oe A NnBns Nyaa + NgMag F vsveee TUNG else AiGin tb Nglen TK seeees +n, Where De Nereis o- A, are functions of P,, P,...... P. not t, subject only to the condition Dyan ye dP, dP, eeeeee $P = These equations will not reduce to the identity 0=0 when m is an even number in general, but they may do so for special values of 2,, A,...A, satisfying the condition to which they are subject. we aP, aP, dP, He oe. ee as es Since Gp (— 1) TH] vere eee neceecneeeeeseeceeeeenerees eee ees P ae dP,_, dP, Ee ee A RE Te TNE CN Wire cee Tah RS Seat soe aa dis, dP, dPiedP, dP,, da, dee ae. dx, 20 Mr HILL, ON SOME GENERAL EQUATIONS WHICH Therefore i a,,(—1)™ iP ae. dP: TP. dP, dP, dP, dP, aode. ERT Na Ea ES dx, dx, Cin, cating dG Gis ae dz, Sal Bee te aceon. seadauch en ussiotart oa uemetarber ert eee Tek fo S| |qoscosesacagaset sncnasnasoosoangccostropasdhpicor dP... aqP GF. dP. GI CP OP ee 7 eS ed Cael dak. i ip dz, dEdG, dz, da, da.dr, — dxida,; dP... dP... dP... dF, dP... dP... dP dP. = 7 Sibel aie = ae: ie So i, ae Z aR CPSA AGP. dP. \aP., dP dP, dP. dx, ween d 1 o da,,, wee eee dx, | da, eeeeee dz, dx, ween da, dP, dP, dP, GP. dP, dP, dP dP, de, walaalatl da, d ae ovsteee de, de, da,, de,,, SE OE dz, Ea iacoteiniors veel plenbebe eee os ennine mee nnoncsamass seceaene te taee oo tee ee eeeee ners eeee ee eeeeeeeeenees Conese eneeetenes aP. i ae iN ae ldP,_, dBi * aR aR de, fo: de, de,,. sono0e da, de, = 8 da, da, Peri “dz, d*P as an ae ark. (ios aP..s dP. akan | aes Pea eye Tae. sae os ter au GE de weer eeooeseeeeeseensesecvessosccesesessouesssesesecee | qjg- §q§é-§ é|weesessceseeservessesess sees Ceeseeeesesesssgenesesse | dP, ri gaa Oe dP, | Fey aM CPS CP. d’P, | ie.) ae ie Va dz, | dor,dix, °"""* dz,_dx, dz,,da,-""""” dx,dx, Consider now Mat ae a set ROE d a E74 MEER ae Me aglanes C “F ‘ (ihe 2b alculate in it the coefficient of F dzdzx, Since dn (— 1) contains only second differential coefficients, in which one of the 2 differentiations has been effected with regard to z,, the terms containing a ae occur nly in ite and Z a ony ™ aa, ™ da," The determinant in the first of these which contains it is Ei ae ge ge ap, a es Fe cages no aE dP ah... GP, rd a dae eer pee as UR at ccue genie eee cs aa | dP 1 dP r+ dP r+ dP dz, ee ee ee dz; dx... Oe dz, ? &P, PP OP, a’ P, &P, dada. dz, dx, dz, da, ~~ dx,dz,, dx, dz, dP. dP. dP. dP. INCLUDE THE EQUATIONS OF HYDRODYNAMICS. 21 and the determinant which contains it.in the second of these is (Mien iicie. dP. dP, dP, dx, eee ee ee eee eee) dx, di, ween dx, ap dP re Tae UB pee acer Tas ae ces | dP. AP 44 dP as aP 4 | se vee coe eee ae ae d’P, Ge Che fle Gels Chee. li da,dax, ~~ Ghee Ghe, Chie, (hp, dx, dx, aP,, ds dP. Gye Cie aoe ST a eat ee eee es | Therefore the coefficient of = is | aP, dP, dP. dP, dP, dP, |+(-—1)°**7*"" (same determinant). ee TemneRe es eae 2% diss ae. de Ges. ge ai. GPs: es en ee an Fa ap. (WE on CH Pn We CHE. dP... ee Tie dase a ia es ae. dep Ghent) — (diaedhs, ae. ta ee ea area ceekace. ) = PP Pes n—1,n of suffixes. : Nr aSn-1 2 2 Hence the coefficient of 45 Xr,5, -- 7,8, _T283 Pa-18n—1 P,, P, ... P.3,,7 tall terms ‘which can ‘be suffixes oY" atte dh a unsichoaentinons «teardiongtcsx eve smeatererscete nets pay tO _ F288 T—-18n-1 + (Pes, Peg P,2,* tall terms which can be suffixes) Now the sum of all these products = dP,, aP,, = ali, © dx, dP s, aP,, dx,’ da, dP,, dP, dx, ” da, aP,, @P, agit dz, Pp get Olen ae dz, da, et So dP, , | Tada da For consider the products; each product which is itself the difference of two products. PPP e eee eee eases eeee eer easersesereeseseeereereessereerseeeee eee ee! = +all such products as can be obtained from this by interchanging pairs in & is obtained from this by interchanging pairs of ete e eee ee eee ee ee eee CTeCe ree ee Teer eee eee cee eee ee eee ee ee ee | stews eee ennne ne ee nee n—-I is the product of 5 terms of form Fo INCLUDE THE EQUATIONS OF HYDRODYNAMICS. 25 Each of the last products is the product of two constituents of the determinant. 7,8, T8q 1, -18y— . n-l ' 3 Thus any term of the form eae oes P gt Pat contains 22 products, each of which is an element of the determinant in question with the right sign. Moreover there are n— ; : | —— such terms. Therefore the coefficient of ,,5,Ay,s,---Ar,.s,., contains |n—1 elements QF ze of the determinant; and since all the terms in that coefficient are different, the coefficient must be equal to the determinant. Hence in the expression for & there is no term Ay, Ap,s, ++ Ar, 1,1 Unless all the af : “numbers 7,5, 7,5, ... 7,8, be different, for if any two were the same the determinant ao a would vanish. Therefore r,s, 7,8, -.. 7,.8,. must be n—1 of the numbers 1 to x. 2 2 There will be n cases to consider, viz. when they do not include 1, 2, ...n respectively. When they do not include 1, the coefficient of d,,r,,...r 2a—-1n is the determinant 4,, ; the coefficient of Ay. Ax,s,+--Ar,.s,., IS £¢@ where + is the sign appropriate to 11? — 2 2 the permutation 7,s, 7,8,...7,_,8,, considered as derived from 2345 ...n—1.n. < pia Te Therefore &, contains the term Gye ag Magee Meg teed Nese, Mesa oen rg gaa gee) = an ae 2 2 n—1-n where 2, is the square root of the coefficient of ,, in the determinant A, ee ee cee pete ate SO When 7,s,7,8, ... 7,_,8,, do not include 2, the coefficient of 2,,2,,... A, 18 the determinant per ee eH a GPs Pe. “ apie Saat te Laan ETE Meal, da? “dz,” da. dF, aP: aP. dz,’ diz, | dz, dP, dP, ae Gr Ot) da Worm SV, Parr i 4 26 Mr HILL, ON SOME GENERAL EQUATIONS WHICH And in like manner it may be shown that &, contains the term 2, the square root of the coefficient of X,, in A, and so on. 4 Where A, is Hence E=VYM, $A, HAG%y, + oe + AYA 111 8°31 nny and similarly, EF, = Vy Bq Ht NgLog + gM go +--+» FAA 112 2 3°32 nn? eee eee eee e ee eee eee eee eee eee ee ee eee eee eT! ‘Ss = Ay%n +r Aon ar Non Fee ar AnBans A,, A, «+ A, being the square roots of the coefficients of 2,,,A,. ... »,, respectively in A. Now observing that &,&, ...&, are the square roots of the Ay, Ay» ++» A, are the square roots of the coefficients of &,, &,,... &,, in the determinant | coefficients of d,,, X,,--- »,,, n the determinant ey on ht) Ei Man No me Mn a a ea ish Ny Ay sd Xeon Sai EN Pia he Lo Ane Se: Ran du, du of. of é | =—-—> eS Ie and that &, cea and that X,, SP ~ SP, it follows that the relations between &, &,...&, and 2,,a,... x, through the quantities U,, Uy... U, are the same as those existing between 2,,d,... r, and P,, P,....P, through the quantities f,, f, ..» fi» dé, dé, dé And since in proving the equation i nee T dee t, see + Fs 0 in Art. 3, the quan- tities u,,u,...u, might have been supposed perfectly arbitrary functions of «,, 7,...2 - it follows that n? Oe eH eg SP. she yas ry 5 Hence &,,&,,... &, have been expressed in the form of the last article. Now consider the case in which n is even. JE= nee aoe 2s Sen 3 oe ap Ene, Ent, eee Enna fete IOI) } The coefficient of Ayo,Ar6, +++ Avngn 1S 2 tn 78, 18s tebe : ; : ; ; : Py, Ly ..P** +all terms which can be obtained from this by interchanging pairs 4 - n-1,n of suffixes 19, _ 1282 ae A . . . . . t(P PP... P** +all terms which can be obtained from this by interchanging pairs 1% 664 tn—-1tn of suffixes) F vcascctevotsesvansossseesveresen'y sas seonleenneemabele Manat reodshin sn hten singe renee AageGUTOntOODUANSGNS: INCLUDE THE EQUATIONS OF HYDRODYNAMICS. 27 _| aP,, aP, = ie eres oe Te dP, dP,, ieviag Sake UIST yiae ers ra aP, aP, da, A aipn bun ae TAcdOe JOC ROSORCRO= 005000000 i dP. §3 ak, ie CR ae eg ig dP, dP, ra aI Ot ais aP,, dP, rep pies ee Te Therefore the coefficient of Aye,Av,5) ++ Ars, Will be + A, if 7,8,7,8, ... 7,8, be all different; the sign will be the sign appropriate to the permutation 71,8,7,8, ... 7,5, considered as 23 derived from 1,2,3,4... 2. In all other cases the determinant will vanish; LEE = A (Naghgg te Actin ee E Ney Arye, 022 Aras +--+); . /B=A./A. 6. To prove that, using the language of ordinary fluid motion, the vortex lines B-G ae = always contain the same particles of fluid. [Analytically, the sets of 1 2 n yalues of #,,z,... , which satisfy these equations always satisfy (n—1) equations of the form ©=constant, where © is an integral of the Equation (III).] If such a vortex line lie on the locus O(z,,#,... #,t)=0, then it is known that dO dO dO bdo, + * de, O50) Ties tee” Substituting for ££... &, their values from the last article, and expressing the differential coefficients of @ in terms of the independent variables P,P, ... P,t 8@ dP. 80 dP. 80 dP, (A214 + Aga + {ere +t) (SB dae +3P dz, +... 5P “~*) 50 dP, , 80 dP, 50 dP. $ (Ryyg + gag F vee Ate) (SP ae: aE SP, da, +. LER ie cncncnccccccccccvercccscccenescacccevocesossssvsces atataletalevalatateleielerec NOY? 1888. 0° ~02 ~ 015 (— 0-18) ~ 0-22 | —-20 5° —02 — 0:19 10° —03 — 0°26 15° — 02 — 0°30 20° — 03 — 0°34 25° ~02 ~ 026 A glance at these tables of corrections shews that the various thermometers differ very considerably in regard to the variation of the secular error, the first example for instance shewing an increase of secular error of only 0°2 in ten years, while the second shews an increase of 0°45 in four years. The tables may be also taken to indicate that the method of taking a freezing-point and deducting any additional secular error thus discovered from all the readings does not give a strictly accurate table of corrections, the Geissler thermometer being the only instance in which the differences between the Kew corrections in January, 1882, and December, 1880, are approximately the same at different temperatures throughout the range. Correction of thermometers by means of water-vapour pressure. It is the purpose of the present paper to suggest a method of referring the readings of any thermometer to a standard scale without the necessity either of direct comparison with a standard thermometer or of subjecting the instrument to any considerable altera- tion of temperature. It is generally accepted by physicists that the saturation pressure of the vapour of water in vacuo depends upon the temperature and upon nothing else. The question has been the subject of many experiments, but has been regarded as settled since Regnault Mémoires de l Académie, T. xx1.) published his table of the This table expresses the relation (Relation des Expériences, pressures of aqueous vapour at different temperatures, between the pressures of aqueous vapour and the corresponding temperatures as referred to the standard scale of temperature adopted by Regnault. This relation being perfectly definite we might, by repeating Regnault’s experiments with a given thermometer, use Regnault’s table of tensions in order to refer the readings of the thermometer to Regnault’s scale of temperatures, This would be practically using an instrument adapted to shew the Vou. XIV. Part I, 5 54 Mr SHAW, ON THE MEASUREMENT OF pressure of water-vapour at different temperatures as a continuous intrinsic thermoscope. Such an instrument, which may be called a water-steam thermometer, is figured by Sir W. Thomson in his article on Heat in the Ency. Brit. Vol. Xt. p. 568, and its advantages for the purpose are there pointed out. They are mainly that the indications of such an instrument would depend on the temperature only, and not on the state of the glass envelope or other varying quantity at the time of the observation, and the different specimens of the substances used in the manufacture of such a thermometer are for the purpose perfectly identical in their properties, and in consequence two such thermometers, made quite independently, would give at once the same indication for the same tem- perature without any previous comparison. The practical comparison of an ordinary thermometer with such an instrument would be in many respects identical with the comparison of a mercury and an air thermometer. Such a water-steam thermometer is, however, a difficult instrument to construct and to manage. We proceed therefore to describe how the water-vapour pressure may be measured without employing such an instrument. It was first enunciated by Dalton that the pressure of water-vapour in a closed vessel, in presence of water at the same temperature, is the same, no matter whether there be air present in the vessel or not. Experiments with a view to the verification of this law of Dalton’s were undertaken by Regnault, Ann. de Chimie, 3rd Series, Vol. xtv. 1845. By a slightly modified arrangement of his apparatus for determining the tension in vacuo he determined the tension in air and in nitrogen gas. The pressure in air was observed for thirty-four temperatures lying between the limits 0°C. and 38°C. The results obtained shew a tension in air less than that derived from the table for vacuum by amounts varying between 10 mm. and "74 mm., the mean of the differences between calculation and observation being ‘44mm. The results obtained with nitrogen gas are very similar, the mean error for the first set of observations with nitrogen being ‘56mm. ‘These differences, though very irregular in amount are considerable, and always in the same direction, and might be held to shew that Dalton’s law is only approximately true, but Regnault, at the time he published the observations, himself suggested that they might be due to some constant error which he could not then discover, and in a subsequent paper on the tensions of ether and other vapours (Mémoires de Académie, T. XXxvti.) he adduces reasons in favour of that suggestion, attributing in fact the diminished pressure of vapour to the molecular action of the glass side of the vessel upon the saturated vapour contained in the air producing a condensation upon the glass, the slowness of diffusion preventing the pressure reaching its maximum value by consequent evaporation. This method of accounting for the discrepancies between calculation and experiment in reference to Dalton’s Law has been subsequently confirmed by experiments of Herwig (Pogg. Ann. CXXXVIL) upon compression of vapours, where it was found that the pressure of the vapour could be increased beyond the point at which a deposit was first formed on the sides of the vessel, and the vacuum saturation tension was the increased pressure and not the pressure at which the deposit is first formed. We may therefore conclude that Dalton’s law is strictly true provided the air is saturated in such a manner as to avoid the molecular action of the sides of the containing vessel. TEMPERATURE BY WATER-VAPOUR PRESSURE. 35 By Dalton’s law the pressure of saturated steam is the same whether air be present in the vessel containing the water and steam or not. If then e be the pressure of steam in air in millimetres, A the density of dry air at 0°C. and 760 mm. pressure, t the temperature of the moist air, a the coefticient of expansion of air per degree centigrade, d the specific gravity of steam referred to dry air at the same pressure and temperature, f the density of the steam at the given temperature and pressure, i.e. the mass of one cubic centimetre, e _(l+at) f aa Heise a Of these quantities, as ¢ only enters as a factor of a small correction it may be read on any thermometer; f may be observed by causing a known volume of the air to pass over a substance which will absorb the whole of the moisture (and nothing else) and determining the increase of weight so caused; a and A are known constants, and we can therefore use the equation given to determine e if d be known, The specific gravity of steam, however, referred to air at the same temperature and pressure, is a quantity which has not yet been completely determined for all temperatures and pressures. ‘Theoretically the specific gravity of steam referred to hydrogen may be determined from its molecular weight, and the specific gravity referred to air may then be calculated from the known specific gravity of hydrogen; this theoretical value would of course be constant for all temperatures and pressures, and equal to 0°622, and Regnault (Ann. de Chimie, 3rd Series, xtv.) undertook several series of experiments to ascertain if such were the case. The first two series of experiments were made on water-vapour in vacuo, and shewed that the number quoted 622 was, within the limits of error of experiment, the true value of the specific gravity provided the fraction of saturation of the vapour experimented on did not exceed 0°8. The third series of experiments was made with air artificially saturated with moisture at a known temperature. A volume of this air was made to pass through two drying tubes of sulphuric acid and pumice by means of an aspirator whose capacity was accurately determined, the gain in weight of the drying tubes gave the quantity of water-vapour contained in unit volume, and this could also be calculated from the known pressure of saturation at the temperature of the saturating vessel assuming the theoretical value ‘622 for the specific gravity of the vapour. A comparison of the results obtained serves to shew whether this assumption of ‘622 as the value of the specific gravity is justifiable or not. The series comprises sixty-eight experiments on air saturated at temperatures varying between 0°C. and 27°C. The following table (V) shews the mean percentage difference between the observed 5—2 36 Mr SHAW, ON THE MEASUREMENT OF and calculated values of the mass of moisture per unit of volume at the different temperatures. TABLE: V. | Temperature of Number of | Percentage Corresponding error of saturated air, experiments, difference. Thermometer reading, 0° 9 “44 0°-065 14° 9 [iz 0°12 fi i 86 0°12 20°5 21 45 0°07 24°°5 22 ‘90 0°15 The differences thus tabulated lie all in the same direction, the observed weight of moisture being in each case too small, They may be accounted for in four different ways, (1) the pressure of aqueous vapour in the saturated air may be slightly less than that given by the table of tensions in vacuo, (2) the density of vapour in saturated air. may be less by about 1 per cent. than the theoretical density, (3) the temperature readings of the saturated space may be slightly inaccurate, (4) the differences may be due to the incidental errors of the experiments. The irregularity in the numbers for different temperatures tends rather to shew that either of the first two suggestions is insufficient completely to account for the observations although part of each difference may be due to one or other of these causes. The amount of thermometric error required in each case to account for the differences is tabulated in the fourth column of the table. Regnault gives no details as to the method of correcting the thermometer - he employed in the experiments, but his known familiarity with thermometers of every kind and their errors makes it highly improbable that he could have overlooked these in the case before us. In reference to the possible experimental errors it should be mentioned that Regnault made a number of experiments upon the dessicating power of the sulphuric acid and pumice used, with the result of being completely assured as to its efficiency. The pre- cautions adopted in order to secure the observations against the possible sources of error that are suggested by the arrangement of the apparatus are not definitely stated, The construction of the apparatus for a repetition of Regnault’s experiments is so simple that it could be arranged in any laboratory, and the observations themselves require only the ordinary amount of care in manipulation. We may use them to deter- mine the temperature of the saturated space and compare the results so obtained with the temperature given by a thermometer enclosed in the space, and thus correct the thermometer for that temperature. We have only to adopt such a corrected value of the specific gravity d for that temperature as is given by experiments with previously cor- rected thermometers, provided that the value of the specific gravity as given by such experiments is always the same for the same temperature. I have had occasion in the course of some experiments undertaken for the Meteoro- logical Office to repeat the experiments of Regnault in order to assure myself of the efficiency of the absorbing substances which I proposed to use on employing the chemical TEMPERATURE BY WATER-VAPOUR PRESSURE. 37 method for determining the pressure of aqueous vapour in the air, The investigation involved the question as to how far the method is liable to. unavoidable error, and the results will therefore serve to shew to what extent the method suggested above may be relied upon for giving the value of a thermometer correction at the temperature of the saturated space. I purpose giving the details of these experiments for the information of those who may have to use a similar method, as I have been unable to find any satis- factory account of the sources of error and the methods of avoiding them. The apparatus, which shews the general arrangement, consists of three distinct parts: (1) The aspirator (C) for causing the passage of a known volume of air over the dessicating substances in the drying tubes, (2) The weighed drying tubes (B, B) for determining by their increase of weight the quantity of moisture in a known volume, (3) 1946 2702 0016 0018 Corrections. -— 12 —°3 | Tension of Vapour calculated from Tube I. U, 16°66mm, 4] TEMPERATURE BY WATER-VAPOUR PRESSURE. 0 @ 00-— | $9.91 €¢-9T 0 G 90. — 08-1 L-6T ect $0. — OL-9T 99.9T 0 ¢ 70-— 68-21 CELT $ €& | W— | GOST | SB2T it UL 80: — G8.8T LL8T GG | Fl.— | 68ST | S38T L @ 90. — SL-61 SL-6L 6 @ CG: — 9G.8T FELT &S G Sone C¢.-81 LLST Cie | wete—= | TPS 6Z.81 0G I Cg: — GG.8T EZ-8T uy “‘queumtiedx Gy “I9JVIN7VS JO *T aquy, jo ‘aouaroytq|* “dure uroay uLOdy moneing ; p[Bo “UU *p[Bo “UU Ur UOISUAT, Ur WOISUaT, Z 80-61 aoe €100-+ | &Z@00.+ ZSIG- lice ra 2 Si GG.8T = O100-+ | §100-.+ Cog. Wat 5 II L61 F861 = 8100-+ | 9100-+ ZOLE- eal iT) Os OL F-1Z 00-0 = 8100-+ | ¢000-— G8Lz- aE |) TUL . 6 G1 #9-06 — 6100: + | 2200-.+ OLSZ: S VOr " 8 pry omoydsoyg prey onmyding 0-12 18-12 6000:— | c000-— | T000-— 008: . zs c ft CF-1G 16-02 9000.+ | Gz00-+ | 2000. + T&6z. ON, 9 G.1Z 6-16 21100-+ | §&00-+ | €700.+ FLOE: 2 ws G 1-16 90-16 8000-— | T100-+ | ¢000.-+ | FF6z- al 7 F 8-16 G0.1Z 0OF00-+ | 0800-+ | $z00.+ 916Z. “4 oS. g GGG €6-06 8100-+ | 2TLO-+ | 8000. + 866. V j 1:6 G0-1Z 8Z00-+ | St00.-+ | 0¢00.— CL6G- q | pAme| Tf pry ormyding pry omoydsoyg (z) (2) : 3 : *SOTUUBAS UT “qu Ioyeitdse Jo | 103BInyRBS JO ASN ee ora ‘T equy, “IORI Faye -Le dx TIA Gav, Vou. XIV. Parr I. 42 Mr SHAW, ON THE MEASUREMENT OF In the first seven experiments the two phosphoric acid tubes were placed first, and in the last three the sulphuric acid tubes were in that position. The columns giving the increase of weight in the different tubes shew that these nearly always gained a small amount, but that amount is very irregular and is about the same whether the phosphoric acid tubes or the sulphuric acid are placed first, and the calculated tension is in nearly every case within 1 p.c. of the tabulated tension. This being about the same error that occurs in Regnault’s observations we may take it that the first tube was sufficient to completely dry the air passed through it and that the increase of weight in the other tubes was due to some other cause, The different connections were made partly by glass and partly by india-rubber tubing, and this suggested itself as a possible source of the observed differences. A number of observations were therefore taken with a view of determining how far this might be the case. I. A glass tube about 5 feet long was mounted as a connection between two mereury cups; and air, first dried by passing through a phosphoric acid tube, was passed through the long glass tube and then through a sulphuric.acid tube and the weight of the latter determined before and after the passage of the air. There was accordingly nothing but the glass tube between the two drying tubes. The results were as follows for four experiments. Taste VIII. | | Increase of weight of | Date. the sulphuric acid tube | in grammes. July 18 +0029 yD) +°0002 so 20) — "0004 | S a, a —-0002 It appears therefore that after the tube had been first dried no further increase of weight of the sulphuric acid tube occurred, and that glass tubes may be used as con- nections without any fear of error. II. The glass tube was replaced by an india-rubber tube 6 feet long, and similar observations taken, fifteen experiments being made. The sulphuric acid tube always gained in weight although every precaution was taken to keep the india-rubber tube dry between the experiments. The least amount of moisture was obtained when a second observation was taken, immediately after the completion of a first. The increase of weight of the sulphuric acid tube generally amounted to about 15 milligrammes. An india- rubber tube cannot therefore be used with any security for connecting two drying tubes. We may therefore conclude that the increase in weight of the drying tubes after the first in the table of results given was due for the most part to moisture derived from the india-rubber connections. For the last three observations in the table, p. 12, these connections were made as short as possible so that the amount of india-rubber surface TEMPERATURE BY WATER-VAPOUR PRESSURE. 43 exposed to the dry air might be small. With the apparatus in that form the drying tubes gained very much less in weight than before, and we may give the results obtained from these three observations as an instance of the accuracy which may be expected in correcting a thermometer by observations of this kind. The first column of the subjoined table gives the temperature of the saturater as determined by a thermometer corrected by a table of Kew corrections. The second column gives the temperature as taken from Regnault’s table of pressures, the pressure being calculated from the weight observations. Thermometer 87400 . From weight . pee observations, Dak 19°34 19°30 — ‘04 18°55 18°49 — 06 19°08 19°08 — 00 I have since arranged another apparatus which is the same in principle but which is slightly modified in some details, the air after leaving the saturater is passed through two glass globes and its temperature is again read immediately before entering the drying tubes as well as in the saturater. The following table gives the results obtained. Thermometer Temperature (C. L. C. 21) in saturater | calculated Difference. corrected by Kew Table. | from weighing. 14°34 14°21 — 15 15°74 15°45 122-29 | 12°29 ‘00 With regard to the second observation it should be mentioned that the air in that ease had fallen in temperature to 15°27 or 0°47 below the temperature of the saturater before entering the drying tubes, so that it would appear that the air was then more than saturated or the calculated result is really 0°12 too high instead of being, as apparent at first sight, too low. It is possible therefore that in this observation the air contained a small amount of moisture in excess of that required to saturate it at the temperature at which it passed into the tubes. This excess of moisture may have been held in the form of a very light cloud or the air may have been supersaturated. It appears then that from this method of correcting thermometers we might expect results accurate within one-tenth of a degree, and that is probably not wider than the limit of accuracy of the thermometer as corrected by the Kew tables. The errors are at present all in the same direction and part is probably constant, and therefore a more extended series of observations would enable us to determine the value of the constant 6—2 44 Mr SHAW, ON THE MEASUREMENT OF TEMPERATURE. error at each temperature which is inseparable from the method and give the thermo- meter corrections within less than the tenth of a degree. If that be so, the fact that the apparatus is of simple construction would make it a thoroughly practical method of determining the error at any temperature. It also has the additional advantage as com- pared with the Kew comparison method that the experiments can be undertaken in the place where the thermometer is used. CavENDISH LABORATORY, Nov. 26, 1883. III. On the pulsations of spheres in an elastic medium. By A. H. Leany, M.A. 1. THE motion due to the pulsations of spheres of the same period of pulsation in an incompressible fluid has been investigated by Professor Bjerknes of Christiania*, by whom the following results have been obtained. If the pulsations of two spheres are in the same phase of vibration, there will be an apparent force on each of the bodies, which varies according to the law of the inverse square of the distance, and tends to make the spheres approach one another; but, if the pulsations are in phases differing by half a com- plete period, there will be a force tending to repel the spheres from one another, and varying according to the same law These results have been experimentally verified, and similar effects have been shewn by some experiments, described in the Journal of Telegraph Engineers for 1882, to hold in air. An apparatus shewing these attractive and repulsive effects, together with several other “inverse analogies,’ to use Dr Bjerknes’ phrase, between electro magnetic effects and pulsations under water of spheres and cylinders was exhibited at the Paris exhibition*+. 2. These phenomena, together with several others of a kindred character, may be explained by the following general considerations. Suppose a periodic force of the nature of surface tensions or pressures to be acting on a sphere, whose centre is fixed in space, and which is itself pulsating with a simple harmonic motion. Then, since the magnitude of the force which acts upon the body varies as the superficial area, it is clear that the effect of the force will be greatest, when the surface of the body is greatest. If therefore the force is a simple harmonic function of the time, and has the same period as that of the pulsations of the body, it is clear that its effect during one complete oscillation will be to urge the body in the direction in which the force acted when the area of the sphere was a maximum. For, considering any two instants the time between which is half a complete period, it is clear that the force at each of these instants will be the same in magnitude and opposite in direction; so that the resultant effect will be to urge the body in the direction which the force had when the superficial area of the sphere was the greater. Thus we have only to consider the effect of the force during that half period when the sphere is greater than its mean value; ie. than its value at a time midway between the instants of greatest contraction and expansion. Let 7 be the time when the sphere is greatest. Then, if 2p is the complete period, we shall only have to * See the Reports of the Proceedings of the Scientific + See La Lumiére Electrique, 5th Oct, and 9th Nov. Society of Christiania, 1875, and the Repertorium der 1881, and Engineering, 1882. Mathematik von Kénigsberger und Zeuner, 1876, p. 268. 46 Mr LEAHY, ON THE PULSATIONS OF consider the force between the instants rth and r—2. But if 7’ is the time when the 2° force on the sphere is zero, where 7’ lies between 74h and rf, it is clear that the force at the instants r’+a and 7’—a are the same in magnitude and opposite in direction. Also, if r’—7 is positive, the effect of the force at the time 7’+a will be less than at the time r’—a; since the area of the body which is acted on is less at the former instant than at the latter. Thus the action on the sphere during the period between the instants 7-2 and 7 will exceed the action in the opposite direction during the period between 7 and r+f, and the resultant action during a complete oscillation will be the same in direction as at the time 7, when the area of the sphere was a maximum. A similar result will follow if 7’ —7 is negative. We have therefore, in order to determine the direction in which a periodic force of the character described urges a pulsating sphere, merely to determine the direction which the force has when the area of the sphere is a maximum. Now, in the case of two spheres A and B pulsating with their centres fixed in an incompressible fluid; it can easily be seen that the change of pressure due to the pulsations of A increases with the time differential of the velocity along the radius vector from the centre of A. The action on B due to this change of pressure is of course greater on that side which faces A than on the opposite side, and the force will therefore be a repulsion when the velocity due to the pulsations of A is increasing, and an attraction when the velocity is diminishing. Now when the volume of A is greater than its mean value the velocity is diminishing; hence, if the pulsations of A and B are in the same phase of vibration, there will be an attractive force on B when its volume is greatest, and the general effect of the changes of pressure due to A’s pulsation will be an attraction towards A. Similarly, if the pulsations are in opposite phases, the effect will be a repulsion. But if these changes of volume are executed in a medium having properties similar to those of the ether, in which the vibrations producing the sensation of light are supposed to be propagated, the results which have been given above will not continue to hold. For, in this case there will be no flux at the surface of B, if the displacements are not large; and the force will not depend upon the velocity, or upon the changes of velocity, but upon the absolute deformations. If the waves of displacement are long, compared with the distance between A and L, the medium will be compressed as A expands; and the effect at the surface of B will be a repulsion if the volume of A is greater than its mean value. Thus the effects produced in an incompressible fluid will be reversed if the oscillations are performed in an elastic medium, and like phases of pulsation will give rise to a repulsion and unlike phases to an attraction on the pulsating bodies. This way of looking at the problem appears to indicate, that, if spheres are pulsating in an elastic medium, the period of pulsation being such as to give rise to waves which are long, compared with the distances between the spheres, the results obtained by Professor Bjerknes will be reversed. If the distance of B from A exceed a quarter wave length it is evident that this result will not SPHERES IN AN ELASTIC MEDIUM. 47 be true. Supposing for example, that the medium under consideration is the same as that in which light-vibrations are performed, we shall have, taking the approximate velocity of propagation to be 200,000 miles a second, a wave length of 200 miles corresponding to 1000 vibrations in a second. Thus for all distances of A and B at which any sensible effect can be observed we can take the phase of the vibration to be the same; but if the distance exceeds 50 miles our argument doves not apply. 3. These considerations are founded on a principle which seems to underlie many eases of differential action; namely, that if a body be acted on by a force F, where F is a simple harmonic function of the time; and, if the action of the force on the body due to variations in the position, magnitude, or shape of the body be expressed by FF’, where F’ is also a simple harmonic function of the time of the same period as F’; then the effect of the force on the body will be to urge it in that direction which # had when F” was a maximum. This principle can also be extended to the case where F’ is any periodic function of the same period as F, provided that F#” has only one maximum value during the complete period 2p, and also satisfies the condition #”(t+a)=F" (r—«), where 7 is the time when Ff” is a maximum. The truth of this principle can be established by the same considerations as those employed in § 2; since we shall have F’(¢) diminishing, as the numerical value of ¢—7 increases from zero to p; the complete period being 2p; and the whole of the argument at the beginning of § 2 will apply. As an example of differential action which can be treated by the principle just enunciated may be mentioned that of a body placed in a field of force, where the force at any point has for components ; - tb - Wt : : Lsin™, M int Nein where LZ, M, N are functions of the co-ordinates of the point. Let the body be constrained by some independent cause to move, so that at the time ¢ its position is such that the force acting on it has for components L sin M in N in™, where L, M, N are periodic functions of the time, of period equal to 2p, which have only one maximum value during that period, and satisfy the functional equation f(t+2)=f(r—42), where + is the time when f is a maximum. It will then be found that the action of any component L ine will be to urge the body in that direction in which the component acted when Z was a maximum, 4. These considerations do not however give the law of the action of the force, either in this case, or in the case of a pulsating body which was mentioned in § 2. In order to completely investigate the mutual action of two pulsating bodies in an elastic medium it will be necessary to find the displacement at any point due to their joint effect, and it will be found that the law of attraction in the case of unlike phases, and of repulsion in the ease of like phases will be that of the inverse square of the distance to the first order of approximation. The term of next order of importance will always be a repulsion and will vary according to the law of the inverse cube. In the following work I propose to establish these results. 48 Mr LEAHY, ON THE PULSATIONS OF General expressions for a periodic and steady displacement symmetrical about an axis. The equations of motion of an elastic medium* are a pd _A+2Q 4. ade p(dy 4dB\) 7 sin 0 7a = 7 sin 07, +S (ap =n a a ag Va ees de. p 2) 7 sin 0 7 = - sin 0 ale 7 (1), do A+ 1 dep ey d@- =p sind dd = (ee dé} j where uw is the displacement along radius vector, v is the displacement along the tangent to the meridian tending from the pole, and w is the displacement along a parallel of longitude tending from the fixed meridian; r, 0, ¢ being the co-ordinates of a particle in its undisturbed position; % and » being the coefficients of elasticity of the medium, p its density, and ¢, a, 8, y being defined by the equations a 1 d ; saa ee Pr he Geeeerar ao sin 8) + a rsind dd "ae Ss r) {ie (rv) — = (rw sin ay} | Corer Beas sin) — Te asis'eetisidenatazoer eee eal y=sind {75-5} | The particular values of u, v, w depending upon ne , which are propagated with the velocity if must make the terms vanish which depend upon Thus we must have i (rv) — 5 (rw sin 6) = 7? sin ott dr’ o (ow sin @) — Mt = sin du d 1 dy do~ ar") = sind dp’ en - d* (dy Substituting in equations (1) we get >,(— -)=0, etc. Thus since is essential] of q hE y; dr dy _ & oe diy _ a LET - ER periodic, we must have —+ = 0, aq" ag or a=B=y=0. dr * Lamé’s Elasticity, Art. 84. SPHERES IN AN ELASTIC MEDIUM. 49 These conditions give dF dF ; dF : Ree rU= 9? se ee 6 setae siuseasins wet exticeiteseeaites (3), where F is a function of the same period as u, v, w. Hence e=y'P, where godnd ) 4 1 d sin 0 d )+ 1 a’ V> a” dr) * 7 sin 0 Al a do) ' sin’ dd’ and equations (1) reduce to a Witlloa. ceomtcecenseeenstouteiissiseeeensdaneay. (4) Next taking the values of wu, v, w which depend upon a and which travel with velocity we : , we have e=0, or a wu sin 0) + 3% 7 (rv sin 8) + — dé ae ny 0. ‘ dM dN) 2 = Hence we must have r sné@.u= acl) | ; dN dL SN ae al sipoasinohocaacddasesonoanpsadadadandaand (5). _a_ aM ~ dd adr Substituting in equations (1), we get d[@L p(_,, 28nd d/M\ 2cotOdM_ 2 a alee Shore 2? a) d? op rs d@\sin@ r dr r sin’? dd _d(@M M/_,,, 2cot@ dN ar { a? ae (v2M— Sino ie)t d{[@L wi_., 28nd d/ M ) = 2cot0 dM 2 aN) aad = _Hoay | {vil ie ead, (an r dr sin dps | dr ca ae d (’M_ py oy — 2 00t8 = d (aN yp v2Nl; dd = dé =a a ain'@ db)/\ dd\de pS? aide fe tas 4 cot d a Ie ea dete det db 7 sin dp pec na cond dy a Ve dr? de® dd? sin’) de where v1 These results give aL _w (is, 2sind d ir 2cot@ dM 2 a df p\™ ® dO (a PF dr rsin?d dd aM wlio, 2cotd dN “dé Vee sin’ db ae Ns Vou. XIV. Parr I. 7 50 Mr LEAHY, ON THE PULSATIONS OF 3. In order to get a displacement symmetrical about an axis, we put w=0 and This will giv d = c rie 2 Made ay “= dr sind do db rsind dr where F satisfies the equation a’F 2X+2u (a°F 2 dF 1 d dF 5 FT ie ar (oats a ane ae (sino do )f Susidecte nese teeenee (2), and J satisfies the equation aN pe d'N , sin 8 chy al a AY De et ne aoe Now, whatever be the forms of / and N, they can be expanded in series of Legendre’s coefficients A,P,(u), where w~=cos@ and A is a function of 7 and ¢ only dP,, : sin @ de s . Also, since can be expressed in terms of Legendre’s coefficients, F and N can also be ex- pressed in series of form B, sin 0 de’ where B, is a function of r and ¢ only Pat Fo5 4 Pin) ad N= 5B,sing Ele (H) and the equations (2) and (3) reduce to aA, N+2Qu (A, 2 at 2 OE) 4} ne) Chel ip Cle r Lies ad’B =" d°*B, _n(n+1) B dt ar 2 ni Tr Let us now suppose the displacement to be atmmene and steady, of period equal Qer wv —. P o a pp »_ PP We shall get, on this supposition, if k?= y+ Qu , v= 4. = Te (ikr) ert, B =F. (thr) e'*, where f,, F, are the solutions of the equations d’f. 2 df, n (ee a’F,, n ane = d (ler) * ker (kr) * +h- ~ (kr) ee Wee d(h FACT al aS sie (hr)? The solutions of these equations are f,(tkr) = A,r" (; 5)" vain Aloe (2 d ) e—tkr T d r nth n+1 FF (thr) = Boa” (7 5) etl 4 B lynt (- =) en-ihr be > SPHERES IN AN ELASTIC MEDIUM. 51 Hence, if f, and F, have the meanings given above ; < = =P, (2) [eee Ft (ikr) eee Dar (ihr ) eit | y= 32 Ee) af a (re ae ete AEN a with two similar terms obtained by changing the sign of 7, will give the complete value of a periodic and steady vibration, which is symmetrical about an axis. 6. The series f, (chr) and F, (thr) can be expanded in powers of kr and hr respec- tively. ? 1d d A a — For putting r°=z, we have are 2 therefore Sf (ikr) = A,2?. am (£ / * eit z sqddoonBebe aA eRe HEReemESBerecee cys F. (ihr) = Bez aD. ez d J with other terms obtained by changing the sign of 7. : : - 2n—2s! na ST! Hence, if when n—s is negative, =.) means (-1) ‘sx Onl? ey Age ye eg ee ene | J. (thr) = a aa ae os 7c n— ric Desi * s+2!s— aie r) Se ks > , ~B on | 20 2n — 2. ay Uae pe . F. (thr) = (—1)"B,. thr fon mb +(—1). po (thr)e+ roa (ihr)**1,..... the highest odd powers of ikr and of chr in the series being (tkr)"*? and (thr). Hence we get u,=(-1)"4,P(u).°™, ee EDS aS HSSree (ikr)* — eee —— area FINE PA) th ee ay ee any ee any..| | Re ne af nea 1 prea! Gane 4 + (3), Pei. 2. a So aM Gs an Gat oA a mer 7. Let us now suppose that the medium extends without limit. When 7 is very Ad Wor ; ; ; ae great r7+1 G. 5) etkr = (ik) * ter Hence at a very great distance from the origin the dF, (u) coefficients of P (u) and se 16 in equation (5), § 5 become On bo Mr LEAHY, ON THE PULSATIONS OF i (4 / , - De? (pt+Ar) +D, er(pt ia ik : an, n(n+1 w= — (C,etoerbn) — Ce ott) 5 —— v, Be = {Cet letthr) 4G ’ei(wt—kr)} 4. th “ {Dy ei(ptthr) _ J) ‘ei(pt—hr)}, r n The terms e*(?**) and et(?t+4”) represent disturbances travelling inwards. Hence, if the disturbance is to be zero at infinity, we have C,=0, D,=0; and the amount of the displacement at any point in a medium extending to infinity and bounded internally by a surface vibrating in any assigned manner is given by two functions u and v; of the form given in equation (5), § 5, where Ffalior) = yy" (=) emi, F (ihr) = B ake ( 5 eWihr, To investigate the disturbance produced by the presence of a small fixed sphere on the axis of symmetry, if there is no slipping at the surface of the sphere. 8. We shall now investigate the disturbance produced by the presence of a small fixed sphere on the axis of symmetry, where the disturbance if the sphere were removed would be given by the expressions u=displacement along AP=>P, (u) {ie + ae rt eivt, r v= displacement perpendicular to AP= > -— i = | Sat Sh ee where f,=Ayr" (- s) en ikr F,=A,/rt1 (- sy en ihr the waves being supposed to be long compared with the distance between the origin A and the centre of sphere B. ‘Let radius of B be b, and let distance between A and centre of B be c. In order to express the conditions that the displacement should be zero at the surface of B, we must find the resolved parts along and perpendicular to LP of the displacement given above by w and ». SPHERES IN AN ELASTIC MEDIUM. 53 Taking the coefficient of P (y), , s Y 7} U, = displacement along BP= u, . eee +0, see Me eee a be (1). UV, = displacement perpendicular to BP=—u, aU, i J Taking the leading terms only of expressions for u,, v, given in equations (3) of para- graph 6; and omitting the time factor e’?!, we get Qn! Ak—Ah fain no HE (u) 2B pee — (+1) (c08 '+-") Py) } Qn! Ak—A/h aaa {(c08 os O + — See +(n+1)P (hz). sin ot | u,/=(—1)". tc HS penn ())} v/=(—1).%c. | Qn r We shall have to express wu,’ and y,’ in series involving Legendre’s functions of pu’, where pw’ =cos@ and their differential coefficients with respect to 6. This operation can be facilitated by the following transformations. By a known formula*, sa | P. (He) = - ot — dc” r? therefore, differentiating + n times with respect to ec, Ps 1 1 2 ein a) _ an fl- (n+) RW) e are = oF?) Pu). 5} rehitiore, (3). Differentiating this result (first) with respect to 0’ keeping r’ constant, and then with respect to 7’, keeping 6’ constant, we get = fain ner oP al (#) — (cos 6” +=) (n+ 1) P,u)} om 3 faa) 24) LANCE py! 94 NOSES py | »..(4). ss a Go: ‘\e dP, (4) +(n+1) sin &P. w)} gt ret ante (n+1)(n+2) dP,(y’) 7 4 Mt Vm + 2)(n+3) dP,(u \ ir est do" 2! de ot 31 de’ ae Reducing the values of w,’, v,' given in result (2) above A means of relations (4) we get uy! =(-1)"te. a PEG) ee yt nwa he Be dP 9 ee * See Maxwell’s Electricity, Vol. 1. art. 132, equation (28). “==, n n Mr LEAHY, ON THE PULSATIONS OF ny 2n! s( oa a n! nt Ak—nA/h or U, ‘=(- 1) Le on (n 1? =e Le: co a tien (w'). cn . os oe = ae * 2n! $C1 1". m+n! dP,, (u’) Pia A,k—nA,h a nvenceces (5). al . “o" (n * (nt) mS Se m! dé’ : cn » cn? In order that the surface conditions may be satisfied, we shall have to take w,’+ (u,’) for the whole displacement along BP; and v,'+(v,’) for the whole displacement perpen- dicular to BP; displacement due to u,, v, to be ze where (w,’) and (v,’) are to be of such magnitude as to cause the whole ro at the surface of the sphere B. These terms (u,’) (v,’) must be of the form given in equations (3), § 6, namely, where the constants B,, B,’ have to Uy, an ( m? v, + (v,') =0 These equations of condition will give Bea ly 2n!(A,k —nA,/h) . P, (u) B,,.(y eren Pn) _ 2 i 4 os ‘ — ee ee, tye Tue ore | be chosen so as to satisfy the conditions ta) x , when 7’ =8, 271 m+n! pr Q7 (n 1? ents [2m +1- 2n! =A" Bo (= 1p 2n! (Ak —nA,'h) “Qn — m1 2! {mk?+(m+1)h*}" ¢ m(2m+ 1) k*+ 4 (m+1) ht + (m +1) (2m — 8) Wh? 2 (2m —3) {mk* + (m + 1) h*} v|, Q"7 m+n! [pie Zee (lye al an [2m to the second approximation. It is n approximation, for, if we neglect the get (u,’)=0, (v,’) =0. Substituting in equations (6) and we get . 2m — 2! {mk?+ (m+ 1h” ec” (m +1) (2m +1) ht + 4mk* + m (2m — 8) Wk? 7? 2 (2m —3) {mk* + (m+ 1) h'} ¢ ecessary to determine B, and B, to this order of squares, it will be found on substitution that we “ths writing C, for (—1)"k.A,, CO,’ for (—1)".th.nA,’, Qn! (C,—C,’) # s Ki, ie ge a i ee ee ors gist 2"(n!)* co" 2. m! {mk? + (m +1) h?} 2"? (pumas _ 2n! (C,-0") rae m TEPn, (pe) _ m+n! [(2m-+1) {mk*— (m—2) h*} r? —m (2m—1) (k?—h’*) b"] Upahes 2"(n!)*c"" de 2.m! {mk*+(m +1) h*}r"™ c mm=1 © (6), SPHERES IN AN ELASTIC MEDIUM. 55 The displacement, whose components are given in equations (7), will be that produced by the presence of B in the field of vibration, if high powers of kr and hr are neglected. 9. Let us now consider the displacement produced by the simultaneous pulsations of two small spheres in an elastic medium, the waves in which are long compared with the distance between the spheres; the centres of the spheres being supposed fixed in space, and the displacements such that no slipping takes place at the surfaces of the spheres. Let the radii of the spheres at the time ¢ be given by the equations r,=a(1+u, sin pt), r,=b (1+, sin pt), where u,, wu, are small quantities. en : The displacement at-any point due to the pulsations of A alone, if we neglect the disturbance due to the presence of B in the field of vibration, will be compounded of u, = displacement along AP = = {f, (kr, — pt}, v, = displacement perpendicular to AP =0, where fe (kr, — pt) = A cos ( pt — kr, +) 3 5 A and g being determined by the boundary conditions. Similarly the displacement due to the pulsations of B alone will be compounded of u, = displacement along BP = = {f, (kr,— pd}, 2 v, = displacement perpendicular to BP= 0, B cos( pt—kr, +8) . T, where Ff, (kr, — pt) = B and £ being determined by the boundary conditions. Putting f, and f, into the exponential form, we have fs iA cosa—A sina (2 =) ee) _tAcosatAsina/l d ) giltri—p) a 2k 7, dr 2k r, a 1 1 a a 1 1 =). es iB cosB—Bsin B (7 d ) e-tn-29 ~B cosB8+B sinB (- d ) ae) 2 2k r, dr 2k 7, Of, 2 2 2 2 ul ior) Mr LEAHY, ON THE PULSATIONS OF The equations to find the constants are obtained by expressing that uw, must be au.sin pt at the surface of A and wu, bu,sinpt at the surface of B for all values of t. These conditions give A {ka cos (ka — «) — sin (ka — a)} = au, ka sin (ka — a) + cos (ka —a) =0 B {kb cos (kb — 8) — sin (kb — B)} = Bu, kb sin (kb — 8) + cos (kb-8) =0 5 whence the constants can be determined. The functions u, and u, will give the whole displacement to a high degree of approximation at considerable distances from the spheres, the radii of which are supposed to be small. But it is evident that, in the immediate neighbourhood of the pulsating bodies, these values for the displacement cannot safely be taken; for, the surface conditions u=au,sinpt and v=0 when r,=a, and the corresponding conditions when r,=0 will not be satisfied. We shall therefore have to investigate the disturbance at the surfaces of the spheres A and B, in order to find the terms that have to be added to complete the solution. This investigation would be very difficult in the general case; but, since in the case under consideration we take the waves to be long, so that the lowest powers only of kr need to be retained in the neighbourhood of the vibrating bodies, we can by suc- cessive applications of equations (7), § 8, obtain a solution to any order of approximation that may be required. Taking account only of the lower powers of ke, ka, kb; we shall add terms u,', v, of the forms given in the equations just alluded to, so that at those points, whose original distance from the centre of B was 6, the displacement may be compounded of au, sin pt along the radius vector and zero perpendicular to it. The conditions at the surface of A will not be satisfied by these values, but, if we transform to the centre of A as pole, we can add terms which will satisfy the conditions at the surface of that body. This series of operations will in general have to be continued indefinitely; but, if an approximate solution only is required, and if it should appear that the surface conditions are satisfied for both bodies if a certain order of small quantities is rejected, we shall have obtained a complete solution to that order of approximation. Neglecting all the powers of ka and kb except the lowest, we have, by (2), A=au,, B=by,, snae=1, sns—1, therefore equations (1) become Mh (Lata NG: au /\ dNpe 7} (ker — pt x5 SEETe & — ) et (kri-pd) — se a) erin Iie P9 = ) 2k \r, dr, r, dr, (kr, — _ — uy 1d i (kr, -t) Du, 1d —i(kr,—pt) Sf, (kr, pt)= A 3 rr) ri) — yA & an)? ?P SPHERES IN AN ELASTIC MEDIUM. 57 whence we obtain to a first approximation Geer bu, . i — LC A me CaM, CO. | oie ctencbea elasjelaiatelals ololaoiete'os 1 9p? ’ 2 Op? -! “~ with two other terms obtained by changing the sign of 7, Introducing the condition that, at all points which were when undisturbed on the surface of the sphere B, the displacement is to be bu,ie” along the radius vector and zero perpendicular to it; we shall find that, in order to find the whole disturbance, we shall have to introduce terms w,’, v,/ of the same form as those given in equation (7), § 8, so that Dg Ete g—int Ane, mil (2n + 1) ees (n+1)h?}r.2—(n+1) (2n—1) (?—h’) b7] BD?" uy = 10 5 ie RTE, es) 2 (nk? + (n +1) 2} 1," e = | pg Oa ness ¢_\n bP, (My) (20 +1) {nk*— (n — 2) hé} 7? — n (Qn — 1) (K? — h*) B® oO [ v, =1C 28 e'P = ( 1) dé, . 5) (nk? + (n +1) h?} ae oe J 0 DUC BAT BRO US Ed 45 p CORORB ASA aaecodae seca (5), if ~,=cos@,; 0, being the angle which r, makes with BX. Again, using the condition that at the surface of A, uw should be equal to au,ie~” and that v should be zero; we shall have to add terms wu’, v,/ in order to counteract the disturbance caused by the displacement uw, along BP. These terms will, as before, be u/=— too Me “SP (u,). n[(2n +1) eee aie Doe hs) alana 2 {nk? + (n+ 1) Myr, (aa ae u eintS dP, » (oy) (2n + 1) ie h?} 7? —n (2n —1) (hk? — 1?) a? a | i 8, 2 {nk? +(n+1) h*}r"™ Cr J The disturbance produced by w,’, v, at the surface of A and by w/, v/ at the sur- 1 face of B will be found to be of the third order of the small quantities - and 2 If there- fore we neglect the terms of the third order, we shall have a complete solution for the displacement at any poimt. This displacement is given by those parts of the terms U,, U,V, 3 Uy, U,, v,, Which do not involve small quantities of the third order. At the surface of A the displacement will consist of ab® Uy Bh’r, + (kh? — ) a’ 2c? (8 + 2h*) r? aus. Ou, soto 4. P , ~int | u =displacement along AP= Op 2 te oR P, (u,) te? (m,) te 1 v = displacement perpendicular to AP Bly AP, (H) sa ipt ite, 3 (kt +B) rn — (BB) a? AP, eo 2 dd, 2 Sea te~ tpt | with similar terms obtained by changing the sign of 7. Similarly the displacement at the surface of B can be obtained. Wor, XLV, Parr td, 8 58 Mr LEAHY, ON THE PULSATIONS OF 10. Investigation of the mutual action of two small spheres pulsating in an elastic medium, the waves being supposed to be long compared with the distance between the centres of the spheres. Let us suppose the centres to be fixed in space, and assume that no slipping takes place at the surfaces. Under these circumstances the displacement can be found as in § 9, and if small quantities of the third and higher orders be neglected we shall have for the components of the displacement at the surface of A, abu, 3hir? + (kh —h*) a® 2% * (k* + 2h*)r? _ Bu, dP, (u,) . a abu, 3 (ke +h’) r? + (ke — h*) a IP, (Hy) int 2c" ~ dé, 2c* 2 (k? + 2h’) r° dé, : with two similar terms got by changing sign of 2. eu se) (u,) ie tt + P, (u,) ie~#*t, When r,=a we shall have du dv du 6, = = 0; dé, =(, and hence the dilatation «= ae (Teas u=— te, y=0, Also the force on the sphere A resolved along AB in the direction AB =[{(re4 20 5 ) cos 8, — H(: i + 4p —7) sing de, S ee = & A B where do is an element of the surface and the integration is performed over the whole of the sphere. dP, (u,) 1 Since [p. (#,) cos 0,do =0 and sin 6,do¢=0, unless n=1, we need only attend to the terms in w and v involving P,(#,) and oe 1 Hence the force in the direction AB _ _ 3b*u, = i" (X + 2p) k? cos*O, + wh? sin’ 2ac* Jo KP + 2h* Introducing the condition (\+2y) k*=h® and taking account of the double sign of 7, we get 127r,? sin 6dé. cagiley Force in direction AB at the time t = — Spee a) as La ait iy eee |) 20+ 5 ac The resultant foree on A can be found if we know the forms of u,, %. If u,=p,, u,=p,, Where p, and p, are constants, we shall have during a complete period = SPHERES IN AN ELASTIC MEDIUM. 59 1237p (r+2n) 0 P _ ie a’p, (1+ p, sin pt)’ sin pt dt Foree on A in direction AB =— = 2X + 5p ac __ 2b" (A+ 2p) a 5 ACN =a) PiPa Ge vetrceteereees (2). This is a repulsion when p, and p, have the same sign and an attraction if they have opposite signs. In either case the force varies according to the inverse square of the distance between A and B. 11. Second approximation. We shall now include the terms of the third order in the expression for the dis- placement. As in § 10, it is evident that the only terms which will euter into the dP. (u,) re The coefficients of these terms in w,, u,, u,, v, have already been considered, and the action arising from them has been given in equation (2), § 10. We have next to resolve the displacement given by w,', v,) in equations (5), § 9 along and perpendicular to AP. If the resolved components are (u,') and (v,') we get, if the terms of the fourth order are neglected, » _ tba*u, aa Bh?r + (ke — he) b : dP. (H,) 3(V +h) ro —(k?— Ne (u.)=—9,¢ P |(cos@.— 7!) P *» (Pa) « (k? + 2h?) sin8, 16 }] expression for the force on A will be the coefficients of P,(u,), and of Rude 2 (k? + 2h’) r,* hs + (1). ae Shtr2+ (kW) b* r\ dP, (1) 3(h-+K re (PRB { | “=>, e-P {sim OP, (u,) (e+ rs + (cose,— A) 8, al: 2 (B+ IR) re \ J These terms can be expanded in series involving Legendre’s coefficients of , and their differentials with respect to 6,. For by equation (3), § 8, we get P,,(,) anal! / r, , (w+1) (n+2) Be athe = (1). enh + (0 +1) Pil) 4 ED pig) +f nO therefore BIO. ba (f,) 1" 1 [aP,(u,) n+l dP,(u) r, rm .. ios Le, 3 dé, c 4 th) (n+2) (dP,(u,) GP, (u,)) 7 _ | aegis \t + ete. hae (3), P.(u,) aN aoe Patil 2 (nm -+1) P,(u,) —(n —2) Plu) and re (cos 6-7) = (= 1) ‘on [?. (H,) “i 3. 1! = C n+1 ue +5 91 13 (n + 2) P,(y,) +2 (n - 3) P,(u)} b+ -.| Neogene sce (4). Also differentiating equation (2) with respect to @, keeping 7, constant, we get 1 GP, Cie) _ (yy, n(n+1)r, (1 dP, (u,) 4, mt? dP, (u) r, per dé, = crt? 2! dé, 31 d0, : (n + 2)(n +3) dP,(u,) 7? t + 4! de, C aha a Podepee casas were els 8—2 60 Mr LEAHY, ON THE PULSATIONS OF in 6, dP 1 1 therefore as =f =(-1)". 2 = a Ee {P, (u,) — P, (#,)} 2 3) . +P Pn) — Pra) 2+ OA D9) (Pu) — Pela too [nner nlO) 1 aP, (uy) _ , n(nt+1)r,f 1 aP,(u,) and at (0086, — 7) Gg = Oa Es 48, : aE (n+ 2) Ee +3(n = 3) oh are maa 1+ 3) Sg 44a 4) Ses | If all the terms of a higher order than the third be neglected, we get by the above equations : iba’u, _. 3h? (u, ) — ys é ae 72 +22 PR (,) Ofo sislho | dr,dy,.” (- i yt P+1 Y nm.n 1 oe P : (m) | | ses eecccecceceee (WV Je qe ()- (—1)" ae ps a) | J Pp pte dr, dy. r r, 1 To find Wy ve must substitute a, for r, in (v) and then take (v) in conjunction dv, with (i); we obtain yp _ pao TA, (- 1)" |p+1 al) Be NV oe ot a, = =s | ara — P(e) + ee (a) nant... PP, ) 5, It remains to equate to zero the coefficients of the various zonal harmonics. We may note however that equation (vi) gives us a means of solving a still more general problem; namely, if instead of a simple pulsation as given by d,, we had a vibration of the atom symmetrical about a pole coinciding with the central distance, i.e. if d, was a function of the form $.P,(u) + bP, (u) + $,P, (u) + ete. 76 Pror. PEARSON, ON A CERTAIN ATOMIC HYPOTHESIS: A vibration however about a pole not peculiar to the atom such as this would not be one of its free vibrations; it would however be the pole about which the free pulsations of the second atom would create forced vibrations in the first. In order to see more clearly the nature of these forced vibrations, we shall for the present follow the more general problem, and assume d,=%, bP, (x); d,=%, $,-P, (¥). It will always be possible to return to our original problem by putting all the ¢’s=0, except g, and ¢,’. Equate the coefficients of P,(m) in (vi), A : pn 0 =) 2 $, i a, ’ A, ve po mC Ke 1 si a= it Hence, $,=—- a B,=—¢ .a,, by symmetry. Equate the coefticients of P, (x), (-1F" pt) pyaar BNE he sh AB $b, =< ao 4 BEY {R04 4 +2) (PHD —(p+3)(p+2)(p +1) 2 ete! cjolteisoeene (vii). By symmetry, ; (—1)"" p+1 Pp a)" A A oe = = 1 —! 1 —? b= apt Bt BAY" [t+ 42940) —(p+3)(p+2) (p +3) 4+ ote} Beton 1g" (viii) We have here types of the two sets of equations which enable us by continual approximation to determine the value of the A’s and B's in terms of the ¢’s and ¢’'s. We see generally that e (— UP ie Ae : os 2 [p+ — ,+ terms of the order € ‘ 1a" : a eg ees ; ‘ Jer (2). B, Ip+1 ¢, + terms of the order = Now ¥ is greater than a,+a,, and hence, the atomic radii being supposed nearly equal, “: and “2 will be less than }. Therefore for any considerable value of p we should be justified in putting athe rt] , pt? tl» pt For example, in the case of A,, and B,, the terms omitted would be, roughly speaking, stsoth of those retained. Pror. PEARSON, ON A CERTAIN ATOMIC HYPOTHESIS. 77 6. Before proceeding to calculate the values of the coefficients, let us determine the form of the total kinetic energy of oe fluid. Supposing the density of the fluid unity, we require the value of taken over the surface of the two atoms. Now at the surface of the first atom 1 | P n= ease [4° YIP pw) + a es (2) n(n=1)...(n—p +1) P.,( »)). p=0 n=p \Y d Ce : 4. a, and rahe bP, (x); | [2 (n) do, = SpE ? 1 dw Se 27 A,b,(—1)p \p therefore - ake ie do, =— , “Cp 4Ayar? Feet yi B, -; ports YB -(+) +(pt 2 (p47 = (p+ 8)(p+2) (p41) acto}. Now for the series in B’s under the second summation we can substitute from equa- tion (vii). We find for all values of p>0, ai (¢,- ee 4,)=(2)" {, = Op N+ (p4 2)(p+1) ete} pP a, Hence 1f, dp, a= 2rA,¢, (—1)? |p aa > do =— &, (2p +1) a?" ae __ 2m, & fp, — Fes jo |3 7 + etc} 7 Aa er wb teat , go rdaBat 1" ptt ‘ p @p+l) oh pap + 1).a7* . 25 =~ 21a, Ag — “tbe {B, a a4 |2 “ete os: 2ma? | = (—1P"|p— 127 A,d, =, pp Pl) to aR aaa grr Bros We may therefore write the total kinetic energy of the fluid as follows, remembering the values of A, and B,, ~“I wm Pror. PEARSON, ON A CERTAIN ATOMIC HYPOTHESIS. K, = 2rab2 + 27a, ¢,” 2Qq7ra lo B B, _ 2a 1B, - 4 I2. 2 ete 7 __ 2ra,'by {4, 5. ae. Be ate Df Yea Ge WaT. _ ys? bp: 270," | 5? (-1)"" |p— 1294, ¢, *: p@p+i) ts an o¢,*. 27a? 42 (—1)* p= 1 27 Bd, ; >, p@pel) to Se wihochiewws haa aaa (ix). Now we have supposed the velocity of the extremity of any radius of the first atom to be given by d, See: (“), hence we may represent the increase in the length of the radius as given by =¢,.P,(u), and ¢, may be termed a generalized co-ordinate of the atom. 7. If we wish to calculate exactly the kinetic and potential energies of an atom we must adopt some hypothesis as to its nature. Suppose, for example, it were an uniformly elastic spherical solid, then we might deduce the internal displacements at any point corre- sponding to a normal surface displacement = >¢,P,(u), from these displacements obtain the forces of elasticity, and so, by Clapeyron’s Theorem, the internal work. [The displace- ments are found from equations (xii) and (ix) and the corresponding forces from equation (x) of a paper on the “Distortion of a Solid Elastic Sphere,” Quarterly Journal of Mathe- matics, Vol. XVI. p. 877.] On calculating such internal work, it will be seen that it contains not only squares but products of the generalized co-ordinates; every co-ordinate ¢, occurring in two products, namely, ¢,¢,, and ¢,¢,,,. Remembering this special case, we shall not assume that the ¢’s are the normal co-ordinates of our atom, but suppose the potential energy V to be of the form: V= 3% (ted: + 2Ppahy Pa): Similarly the kinetic energy of the atom will be supposed in like manner to be given by K, = 3 Doty’ + tor bob) The like letters dotted will be used to denote similar quantities for the second atom. Hence, if Z be the difference of the kinetic and potential energies for the whole system, L=K,+K,+Kj-V-V’, and the equations of motion will be of the type HE = GE SME et Moe (x). 8. We may here stop to note a very curious possibility; since K, is a quadratic function of the ¢’s 3 : 27ra,° gus " © Qara,$ 4,” . (p+1) @p+l) *~* (p+l) @p th) Pror. PEARSON, ON A CERTAIN ATOMIC HYPOTHESIS. 79 so far as the terms independent of y are concerned; it follows that the vibrations in the 3 fluid appear to increase 2, by ae, Now in the expression for the kinetic energy of the atom, the 2’s are what express the mass of the atom. But if all the 2’s 27a? 6. (p+1)(2p+1) fact that the vibrations were taking place in the fluid. Hence we are not compelled to suppose our atoms to have any mass at all, they will still have an apparent mass owing to the vibrations taking place in the fluid. It is a strange possibility that atoms, the supposed foundation of all matter, may be the one thing in the universe immaterial; and were zero, they would still have an apparent value, 1. , owing to the ‘that their supposed mass may arise purely from the fact of their existing in the ether! A like train of argument applies to linear motion of atoms in the ether, they too have an apparent mass, which may be their only mass. It must be noted however that in order to obtain the vibrations it is necessary to suppose them or thew surfaces possessed of a potential energy similar to that given by the V’s above. The notion of an atom as a perfect vacuum (even as to ether) in the boundary of which in some manner potential energy is resident seems at first sight startling, but it does not appear altogether incon- ceivable, 9, Suppose now y very great, so that the atoms have no mutual action, then: dL = 4cra,° hy : 3 : dé, (p+ 1)(2p+1)~ Ay Pp + Hpobo + Morb, + My Py + ete., dL ~ dd, = TP + PoP + PrP; + Pro P. + ete. Pp Hence the equation of motion gives :— q=2 q=n - 4a, = malt ——, > z =0. $, ps aF (p ae 1) (2p Je mi aF a Lng Pa ae ne Pra Pa us T,Pp We may write down as many other equations as there are ¢'s. Assume a vibration of period = and we obtain the following determinant from which to calculate x’, v T,— 1 (rd, + 47a,’), Por — 2 Hors Pog — 2 Mog » ac | =0. 4c," Pry — 2 Hos te (a, ae 6 : ) D Pie 2 Hyp» 4cra,* Pao — 2 Pgs Por — Pay» Ta tr (4+ a), tee This determines the free vibrations produced by a “single pole” disturbance, if the atom is disturbed and then left to vibrate freely in the ether. In the case where the Wong SOI, IAN 1, 11 80 Pror. PEARSON, ON A CERTAIN ATOMIC HYPOTHESIS. free vibration is to be a pulsation, all the ¢’s except ¢, must vanish, and we find n, =a/ sa ent 2b 5 Lo r+ 47ra,*’ that is what we have considered as the peculiar vibration of an atom, that which finds its analogy in the single bright line of the spectrum in the case of a gas or vapour at low pressure and not too high temperature. ; 10. Let us suppose for a moment that the natural pulsation of our atom does coincide with the single bright line vibration. Then the period of the light wave will be given by 2a xf oe where p, the ether density, has been introduced to preserve the order Qo of quantities. Now 2, (if it be not zero) will vary as the volume of the atom com- bined with its density, and +, will of course be of the third dimensions in space but may only vary as a,°; for example, if the potential energy is resident in the surface, it will also contain some factor representing the mechanical structure of the atom—for example its elasticity. Hence we see that the period will very probably be a function of the density or radius or both of the atom. It is a noteworthy fact that “there is a connection between the atomic weight of the metalloids and the region of the spectrum in which their lines appear under similar conditions” (Lockyer, p. 143). A careful ex- amination of the position of the ‘single bright line’ and the atomic weights of various substances might possibly throw some light on the value of A, and 7,, and so on the mechanical nature of atoms. Does the spectrum order of the single bright lines in the cases of the elementary gases in any way follow the order of their atomic weights? or of atomic diameters as calculated in the Kinetic Theory of Gases? (See Oskar Meyer, Die kinetische Theorie der Gase, p. 206). 11. We may here note why it possibly is, that the natural pulsation is of more im- portance than any of the other free vibrations of the atom. These latter may occur about any pole or any number of poles in the same atom, while in another atom they may be taking place about an entirely different pole or system of poles. Suppose two atoms to collide and then separate and pass beyond their sphere of mutual action; each will vibrate freely about the point of colliding contact, but the position of the corresponding poles in each will have no relation whatever to the poles of the vibrations produced by the contact of another pair of atoms. The free pulsations in all cases will be the same and from any small volume will have a common direction; the free vibrations will however have no common direction, and so produce nothing like the effect of the free pulsation upon the ether. 12. Let us now suppose the atoms to be so close as to enforce vibrations on each other. We shall look at this problem in the following manner. When no such influence existed we calculated the free vibrations of a single atom, we now ask what are the free vibrations of a system of two atoms acting upon each other? Such free vibrations are when con- sidered with reference to one atom alone forced vibrations, they occur only owing to the mutual position of the two, What are really the free vibrations of a system of atoms Pror. PEARSON, ON A CERTAIN ATOMIC HYPOTHESIS. 81 mutually reacting? Here we shall find a new set of periods differing from the old, in that they are functions of the distance between any two atoms. Now the expression for A, is a quadratic function of @ and ¢', which may be ex- pressed as follows :— K,=4¢,. [2na, aE Boy +8 “ + Qi = _ 4 & # a + J _, bub, {8 pa + higher terms in i oy ¥ ea Doi pier e 2 F ache + ef p=l ry es 4b. Ta +. a p=l pH=0 s=0 BSS ge ee \ tos p, dp, lyn Ginn ®) PO =O Ge = _ bh. . oP ‘ot pal = 9 + _ b4, |e tee + a rig 7 (& ) a =; $, p, ly” om we alt 2Qara +276 loaepeyt +} } Font 3 P= . 27ra Ss ah 2 Xe +2? p+) pth | 18 We note at once from this that the terms due to the mutual influence— (a) Bring in products of the ¢’s, but that these products become of small importance when the order of the ¢’s is considerable or the distance between the atoms sensible. (b) Affect also the coefficients of the squares of the generalised velocity-components, but when the order of the component is considerable to a very small extent. In other words the mutual action affects most the co-ordinates of the lowest orders, in particular the natural pulsation. 13. Proceeding as before to form the equations of motions, we obtain a series of linear : : 3 : 1 relations between the ¢’s, the coefficients of every term being functions of ~. Assuming ¥ g values of the type Ccosnt for the ¢’s, we obtain linear equations between the C’s and n’, Eliminating the C’s by a determinant, we arrive at an equation to determine n* The 11—2 82 Pror. PEARSON, ON A CERTAIN ATOMIC HYPOTHESIS. : : : : 1 ; coefficients of the powers of n in this equation are functions of me and therefore its roots may be expressed in the form S, =, -* = = Cte. y 277, Ve : : , will be one of the vibrations of one or other atom S, 0 If y be great, n, =, and considered as vibrating freely. If y decrease, we shall have to consider the next term = if this however be still small, the period will differ but little from that given by §, Still from one pair of atoms to another y will vary, and thus we shall have a number of periods grouped, as it were, round the natural period. If these vibrations correspond to light vibrations, we should expect to find the spectrum lines spreading out into bands as the atoms of a gas are brought nearer together. Finally, let the atoms be brought very close together, then we shall have to consider a long series of terms SS Sei Loe Melts altering n, very considerably from S, In other words the vibrations given by any two atoms will differ very considerably from the free vibrations of either. Furthermore the vibrations given by different pairs will depend on their distances, and so vary from pair to pair. Hence there will be a continuous series of periods between certain limits, built up out of the vibrations of systems of atoms at different distances. If these vibrations correspond to light vibrations, we should, when the atoms are close together, expect a continuous spectrum. Supposing for an instant that a chemical combination consisted of two atoms in a very close union, we see that the free vibrations of the two would differ very materially from the free vibrations of either alone, hence we should not expect the spectrum of such a combination to be identical with the spectra of its two components superposed. 14. Hitherto we have been dealing only with two atoms, we may now ask what difference is introduced into the above results if three (or more) atoms are all within their mutual range of action? The calculation of the value of K, for three atoms in the most general case of polar vibrations presents considerable difficulties; we shall content ourselves with examining the nature of the pulsations of any number of atoms mutually reacting. It is not difficult to shew that in the case of three atoms :— » eee Sar ee iy 4rra,* ‘a ae 4rra,” a,” Kk, = 27a, "7 + 2rra, fbi?+ 27a sb, 4 aay = $b, a Pigs = db, bo + —* babe +terms with higher powers of =, Pror. PEARSON, ON A CERTAIN ATOMIC HYPOTHESIS. 83 Forming as before the equations of motion, we find Ya 4 ebrblhes Uae ty CRN) oe oe 0 meats = 0, with two similar equations. Now assume $,=C, cos nt, ¢,/=C,cosnt, $,’=C,cos nt, where nt is written shortly for nt+a. Then 2,2 4, 2 , 2 CL (7,—n’ {4ra,? +2,}) — C, Sri nr’ —C, ames n? = 0. Vie Ys1 eet ae Wes 4cra,? +,’ a 4ra,*a,” 4rra,7a.? 4a," Q, — = > Q, = Sear ts Q; = eas as: Yes Ys1 Viz Dain al Then Osta (ja 2a) + .Qn + C.Q.= 0, gl ae eof Wel similarly 0,Q, + C7, iz 7a) ts CQ, =0; 2 5 (alae C,0, + €,Q, + C7, & = =) =i 3 Eliminate C,, C,, C,, and we have the following well-known characteristic equation to deter- "(5 -3\(5 =) hoa)o 2 *-3) ToT To DT ND Re a n 170 (52 n ill ae Py a — QT (=-3)- 5 a (Fs- ga) +20,0,0,=0. 3 V, n eel mine — :— n Therefore the three values of n are the lengths of the semi-axes of the quadric a2 2 ca Hee ee: Q, — YZ + 2 0, be oe = UU 3 Jr, Jar” To Jrote 1 re and —, -3, 2 are separated by the quantities 3 15. Let us consider some particular cases. a. Suppose the free pulsations of the three atoms not to be nearly equal, and let » Y,, ¥, be in order of magnitude, so that following the analogy of the spectrum the bright lines of the three atoms would be grouped from the violet to the red in order. V~,, V, io 9) rss Pror. PEARSON, ON A CERTAIN ATOMIC HYPOTHESIS. We find approximately, 2 2,,2 2 te | 1 = 1 ee V,Vs Qs Vy, 2 2 a3 2 2> n Vs ToT Vy — Vs ToT) Vv; =, Vy 2 2.3 2 2,2 1 = 1 3 vs VY, 1 Vy Ys ae ero ’ 2 2 7 2 Q> Me Ve ToT) Me —%, TT) Ye Ys 2 23 2 2,2 1 _4t = 1 Us Vy ee 2 2 (Ef, 2 2 2 29 Ns V; T) To Vs Vy To To V; — Vv, and therefore greater than Hence the following quantities must be in order of magnitude, 1 1 1 2,2, {7 and = : 1 = Q,25 1 1 1 Q.Q, = te a eee a aa! 2 2 D ’ 2) 2) aia 5) Ns Vs, Ys 7 Q, 2 nN, Y% TY Y n, V; TQ, where a and - lie between the limits given, but which is greater is not determined. If Vs 2 we draw a figure corresponding to the spectrum, we have the following result. violet an, »; Ng c Vs Ng red Ea ean WERE Here »,, v,, v, represent the bright lines corresponding to the free pulsations of the three atoms, a, b, c denote the position of the lines corresponding to the quantities which SE li separate the roots of the cubic for 7 The thick lines n,, n,, n, denote the spectrum given by the system of three atoms, according however to the mutual distances between the atoms, n, may be right or left of »,, i.e. may take up a position marked by the line d. We note that the position of the lines n,, ,, n, within the limits thus determined will vary according to the distances of the atoms from each other. Suppose now the molecule of a certain substance to be composed of three different atoms, then when that substance is in such a condition (as a gas or vapour at low pressure and not too high temperature) that its molecules are vibrating freely, its spec- trum will be of the kind here described, and will in part consist of two bright lines, the one nearer to the violet and the other nearer to the red than any bright line due to its component atom when they are pulsating freely. Pror. PEARSON, ON A CERTAIN ATOMIO HYPOTHESIS. 85 We may express the surface displacements of the three atoms as follows : = i eis. — 1 = pe $, = C, cos n,t — C,. Q, (= - =) cos n,t — C,. a: (= = =) cos nf, T Y; 2 0 VY, 3 Ly MY =F Oil tL \ ¢, =- C,.=3 (- >] cos n,t + C,. cos n,t— CO, .—% (- =) cos n,t, Tae NUS YE Tope, Hs ” RN. DP cae: = $, =—-C,. 7, (= - -3) cos n,t— C,. @, es - =) cos n,t + C, cos nt. 3 2 To Vv V, It follows that ee OW ates = Gat vi\t Ae ¢, =—7,C, sin n,t— mC. (5 - a) sin n,t—n,(, = - =) sin n,f, with like quantities for ¢,’ and ¢,". : : ee alee : Since, when vy, is more nearly=y,, Si a te large quantity, we deduce the follow- 1 2 ing theorem: the more nearly the pulsation-periods of two atoms agree, the intenser will be the vibrations of the system compounded of the two. For example, in the sup- posed spectrum given above, if the lines »,y, are close, then the lines n,n, of the combined spectrum will be very bright, and the line », will be very much to the left of v,, while n, will be to the right of »,. On the other hand, when », and », are very different, the intensity of the vibrations will be diminished. [Cf. Lockyer, Studies in Spectrum Analysis, p. 140, for a somewhat similar analogy in the case of vibrations damped by the encounters (? approach) of dis- similar molecules. ] B. Suppose two of the atoms to be alike, or mes and vy, and », to be not nearly ee) equal, then the discriminating cubic becomes: ” 1 1 2 1 1 1 1 " 1 1 mand’ Ga~5) (Ga ga) OP + (Fa— ja) 2" (Fa Fp) + 20,0,0,=0 1 3 2 Vs nv 1 v3 = ==>-—> === 5F == == Ts Sy z° us The following are the spectra of the molecules composed of two like atoms and one unlike—when, First, the pulsation of the two like atoms is more rapid than that of the unlike, Secondly, the same pulsation is less rapid. 86 Pror. PEARSON, ON A CERTAIN ATOMIC HYPOTHESIS, violet n, Y, Ng Vs Ng red | V, > Vy: violet Nz Vs ny vy Ny red Vy < Vy We see from the values of n,n, that the corresponding lines will be farther from v, than n, ts from Y,. y. Let us consider the case of three like atoms within the sphere of their mutual influence. The discriminating cubic now becomes 2 2 2 m: — Q, + Q,' +, m+ 20,005 _ 9 T) To iy i ) where m= (5. : mae V, nN ? Hence we see that m has three values of the forms a, 6, and —(2'+ ’), ehh ay BE ie aes n, V; Lee a n, ma ve a B ’ 1 gt Kage pS Burm We have then the following spectrum: violet tm Ny Vy nN; red In the case where the three atoms are at equal distances from each other the lines n,n, coincide: violet ny yy Nz red Pror. PEARSON, ON A CERTAIN ATOMIC HYPOTHESIS. 87 Here we have se ie aes 1, Ce TS eg ae 1 Ls age naa. Tee The line , however will probably be now much brighter. 16. Second set of cases. Suppose we have only two atoms mutually reacting, then (dara? +) by + Tb) + ae ay $, =0, Bue (erat Bae) ih oe =O Assume in =C,cosnt, o, = C, cos nt, therefore {r, — (A, + 47ra,°) n°} C, — Qn’ C, = 0, {tT — (Ay + 47ra,°) n*} C, — Qn? C, = 0. Hence, with the old notation, By er we! ae ee poe ons (a Gaal = or we have now a discriminating quadratic instead of the cubic. provided that v, and v, are not nearly equal, Dy aie al Ola» wavs We find approximately, Let v,, v, be in order of magnitude, then (y, > v,) (Oy as) ae and n,,. 18'_-<\ vi, or n, and n, lie outside v, to »,. Hence we have the following spectrum for the combination :— violet ny Vy Vo Ny red Suppose we substitute for n? to find the relation between C, and C,, it easily follows that = aa Quy g¢, = C,cosnt—C,. ae i 008 1d . o.= — Carte cs nf + O, cos nt T) Vou. XIV. Part II. 12 88 Pror. PEARSON, ON A CERTAIN ATOMIC HYPOTHESIS. where as before We note from these values that while the period of the free vibration of the first : 2ar Qa : : : ; ; atom is changed from — to —, yet the forced vibration having an amplitude varying v n 1 1 with t will not rise into very much importance unless y, be nearly equal to »,. ay, uly Aha e - a , the discriminating quadratic becomes 2 1 We then have a theoretical spectrum of the following kind :— violet ny Vy Ns red In both cases, that of two atoms and that of three atoms all equal, we get a double- lined theoretical spectrum, but with these distinctions: the line n, is situated in exactly the same position in both, in the three-atomed spectrum it will probably be much brighter; the line , in the two-atomed spectrum is replaced in the three-atomed spectrum by a line , still further removed towards the red. What, if any, is the difference between the spectra of an element in two different molecular conditions (e.g. oxygen and ozone) ?* In the case of two equal atoms we find ?,= C,cosn,t +C, cos S ¢, =— C, cos n,t+ C,cos n,t where nt =nt+t a, nt = nt + 4,. 18. Third Set of Cases. Suppose there to be p mutually reacting atoms of different kinds, In order to simplify the notation, as we are treating only of pulsations and not polar vibrations, we shall signify by ¢,, ¢,..., the several pulsations, by A, ... A, the * It may be shewn that the energy lost in the dis- hold them together, Thus disassociation of a diatomic molecule ought to be accompanied by greater generation of heat the nearer the single bright line is to the violet end of proximately. This is positive, if C,>C,, which is also the _ the spectrum, e.g. the fundamental line of hydrogen should condition (Art. 33) that the force between the atoms should __ be nearer the violet than that of carbon. 2 : 2 association of a diatomic molecute="1 (Cs'= G1") @ )Q ap- Pror. PEARSON, ON A CERTAIN ATOMIC HYPOTHESIS. 89 coefficients of 44,7, $¢,°, ete. in the kinetic energies of the respective atoms, and by : : : : ; Qa T, ... T, the coefficients of }¢,”, 4¢,",...4¢,” in the respective potential energies ; ne — 1 will be the periods of the free pulsations as formerly. Then 2 2 L = 320162 +450,62 + PErb2 +3 Me 6.4, rs Here as previously we only retain quantities of the order Ef Applying the typical equation of motion and putting ee we find : (r, + 47ra,°) 6, + 7,6, + =Q,.6, = 0. Assume ¢,=C. cos nt, ete., hence, C, {r,— n’ (r, + 47ra,*)} —XQ,_,C, = 0, : : aol Eliminating the C’s by a determinant, we have the following equation to determine i: i 2 Gre Qi» Qi “Eo a oa Q1»> ae 1 shee all Qa» 2 (. 2 *.) ? Qos; Q, p> OF 0.33 aoe ane eee eee (53-3) Assume ele p, and suppose pw is small, then pa ae 1 U\ ye il 1 1 Bb. T,T. +++ Ty eels ceceee ae Pp 2 1 Tl elt) al 1 1 a Qi T3T% Tp & = =) (73 = =) sislejeiexe = aan =) ll Soe Vo alae 1 1 = Qs 5 TPA vate ip (7: — 53) (3 = = siv'avlve (53 — 2) + ete. Hence, if the v’s are not any of them nearly equal, 1 1 Qe VY Vy as Vy Vs. a Sa es Pay Tar | 3 ete, nN, VY, Tig 24) an = T,T, Diath and generally 17 90 Pror. PEARSON, ON A CERTAIN ATOMIO HYPOTHESIS. We have then a means, given the free pulsations of any number of atoms, of cal- culating the pulsations peculiar to their combination, or of deducing the theoretical spectrum from the known single bright line spectra of the component atoms. ; : i : Adding together these expressions for na We obtain the result r _ se — a . n, 1 From which we deduce the following conclusion: 2 2 3a Aa, n 2 29 s s or in words, If any number p of different atoms (we shall see later that they need not all be different) be mutually reacting upon each other's pulsations in any manner what- ever, or if some of them be reacting and some not reacting, the sum of the squares of the periods of the pulsations given by the p atoms is always a constant. The constant of course being equal to the sum of the squares of the periods of the p atoms pulsating freely. If the p atoms enter into a chemical combination (form a molecule) altogether or form several different chemical combinations (different molecules) the result is still true. This seems somewhat suggestive when applied to the bright lines in the spectrum of a compound. Further we have by considering the minors of the above determinant when n=n,, (6: errs OAS pe ( T (= -<:) T, (Cea) Pe VY, Hence we may write $, =C,cosnt— >. en cos n,t, {r= 2 to p} d,=C,cosnt— d. ae cos n,t, fr=1;3 2p} d, =C,cosn,t— >. Ces .cos nt, {fr=1, 2, 4... p} ¢, =C,cosn,t— =. GF ply Ye cos n,t, {r=1 to p—l}. Ty(¥, — Vp) The above equations then determine the periods and intensities of the pulsations of the combination of p atoms. 19. The whole question of deducing a compound spectrum from the single bright lines of given elements depends on the consideration of the above symmetrical determinant and its minors. The cases in which 2, 3, 4,... all p atoms are of like kind, may be deduced a : ‘6 re ay ete. in the above determinant. V, ea Ue i teh by respectively putting | Pror, PEARSON, ON A CERTAIN ATOMIO HYPOTHESIS. 91 20. We may note the following general property of the combination spectrum, if v, ... v, ; 4 lies 1 ; ths be supposed in order of magnitude, then ees and’ = is > ty or the combination 1 i Np Vp spectrum will (theoretically) have two bright lines, the one nearer the violet, the other nearer the red than any of the single bright lines of the elements. 21. One very important result of the above investigations is the following. If all the atoms are alike, =T,=..=7T, and y,=v,=...=y,, and we may write the discriminating determinant eee teen eee eneeeeoe ORONO ae ae iu 1 il i : : ; where m= (——=) . We then note that in this equation for m, the power m?* does Vv not occur. Hence the sum of the p values of m is zero, or Ve Veen eps 00 4 Arr? Anr® Aer” or a ai aa iataiiee, nl Pia nN, n, Ny v that is: If owing to the effect of pressure or temperature an element gives p spectral lines instead of a single bright line, then the sum of the squares of the periods corre- sponding to the p bright lines is equal to p times the square of the period corresponding to the single bright line. 22, Again, if A, , denote the determinant obtained from the discriminating determinant by omitting the first 7 rows and columns, then (Salmon’s Higher Algebra, Lesson vr.) the roots of the equation A, ,=0 separate the roots of the equation A, ,,,=0. In pr other words, if there be a spectrum of q bright lines due to q mutually reacting atoms, and if another atom be introduced into the sphere of mutual reaction, then the first set of qg lines will separate the second set of g+1 lines. 23. It must be noted that in all calculations as to the theoretical spectrum of a com- bination of atoms, we must not expect always to have as many bright lines as atoms in the combination ; for first, some of the atoms even may have natural pulsations which corre- spond to ultra-red or ultra-violet rays, in which case, since the combination always has two periods, one greater than the greatest and one less than the least period of the free atoms, it follows that the combination will have at least two invisible (so-called heat or chemical) rays, it may have more, Next, if all the pulsations of the free atoms corre- spond to light rays, yet since the combination has at least two periods greater and less than these, it may have one or more periods which correspond to either chemical or heat rays, to Pror. PEARSON, ON A CERTAIN ATOMIC HYPOTHESIS. > 24. Lastly, we may note that all the periods calculated for any combination of atoms depend upon the central distances between the atoms. If the atoms are changing their relative distances the periods also will change. Now suppose the distances of the atoms of a combination to vary about a mean position, then about each mean there would in any interval of time be a series of others differing slightly from it, there would be the appearance of bands in the spectrum; in other words, a combination of atoms which are oscillating about their mean distances would give a fluted spectrum. The oscilla- tions of the combination of atoms could arise in various ways, as by the action of one combination upon another (one molecule on another), or even by impact if such became frequent owing to close proximity of the combinations, as in the case of a gas at consider- able pressure, or in a solid or liquid body, 25. There is however another statical or average method of looking at the pulsations which may possibly give rise to a fluted spectrum. Ifa great number of atoms be at each instant within the sphere of mutual action, and we suppose their distances not to vary, then there would be a very great number of pulsations corresponding to all the different central distances; the very complexity of these lines may make it impossible to distinguish them from bands, and so a fluted spectrum arise. This would be more or less the case when we consider an element not in a molecular state and whose atoms were very close together. In the accompanying figures let the atoms be supposed at rest and uniformly distri- buted, or else suppose the average number of atoms in a given volume to be uniformly dis- tributed over the volume at their mean distance. (i) (ii) (ii In the first figure the mean distance is one-fourth of what it is in the second, and one- fifth of what it is in the third figure. If the radius of the sphere of mutual action be a little more than four times the mean distance in the first case (=/18 times the mean distance), we find 3+7+7+9+9+49+4+74+7+3 atoms in the plane of the paper, and about 340 atoms in the sphere of mutual action in figure (i), 7 atoms in figure (ii), and 1 atom in figure (ili). In the last case we should expect a single lined spectrum, in the second a 7 line spectrum, while the first case would present us with 340 lines. The densities in the three cases are as 1 : J; : z4;, or supposing the temperature to be the same the pressures are in the density ratio, For example, suppose the spectrum of some element to have at Pror. PEARSON, ON A CERTAIN ATOMIC HYPOTHESIS. 93 ordinary pressure some 300 lines; hence, the temperature remaining the same, at gz the ordinary pressure it should have a spectrum of not more than 7 lines, and before the pressure reaches ;!,th it should give only one line. This is not unsuggestive when we consider the spectrum of iron. Here of course we suppose that the ultimate atom of the element (which is supposed to consist of uniform atoms) has only one free pulsation. 26. This method of considering the arrangement of the atoms of a body may perhaps be useful in explaining some peculiarities of the spectra of solid bodies or of elementary bodies whose atoms are all uniform, but when we come to the consideration of bodies composed of unlike atoms we are forced to consider the atoms grouped together in com- binations or as they are termed molecules. The fact that a mixture of two elements gives a totally different spectrum to their chemical combination forces upon us the notion of atoms united in groups. We are then led to consider the forces which must exist between our pulsating atoms when they are united into groups or molecules, these we shall term chemical forces; the forces between groups of atoms or molecules will be termed molecular forces; while the forces which act between groups of molecules at considerable distances will be termed gravitation forces. Can our theory of atoms as spherical bodies pulsating in a fluid medium throw any light on these three kinds of forces? In order to investigate this point, let us return to the expression for the total kinetic energy of atoms and fluid. We shall consider it only so far as it refers to pulsations, leaving out of account all polar vibrations. From equation (ix) we have ; F Qra2d K, = 200562 + mas? — 2s Po {z, ato ete. etc} Y ae ee a Qrra,2,, A A 2 ETO 4, 14/2 eto} Y 9 where 4,=— ¢,0,, B,=—¢,a,, and generally (-1)"" |p+1 p ay" | B B. } = a—— A,++ (4 B,- 1) 2 1) = — ete. = AAR (AY (Rot) H(t (ptD ete}, a pice) pee P aye | xs A, 9 Be 0= Pee B, +E (%) |4,— (p+) +(p+2) (p+) 2 — ote. pt2 a, 27. We proceed to approximate to the values of the d’s and B's. First Approximation. Odea aja? A,= Qa? Pp» B= ‘Qe Po? Gea. ep Ienton = fe se Poe B, Bont d, OR aja? Z| im “By! 0? B, = ca po 94 Pror. PEARSON, ON A CERTAIN ATOMIC HYPOTHESIS. Second peeks: 3 5 2 gar, a, shies alsa . a, a . a,@, ; — a, ae =, eel 8s pf eee at at IP A; = y d+ “Dry 5 Po» A,= 3 pp Qry! Po» 5 8 87 _ &% drs a, nae a, a ; ald 3 See Anes yt Got Qn? dy B,= j Third Approximation. a. 2th 22 9 pj’ "ey 2 Po Dry $+ a ‘a, B,= Des a ce by: Lastly we may note that A, and B, are of the order 7 and hence appear only in K, in terms of order pag which we may very reasonably neglect. a 28, For possible future use it may be as well to put on record the value of these : : : 1 constants in the case of polar vibrations so far as terms of the order a3 ; 3 a,°a 5 4,= 74+ = “$s + “oe , + a ge4 w dy + 3 Pan CP fa Het 5 Oa a,” B,=> p, 1r Da $y + ay % ze ee. the ? ds, a‘ Gta PON A,=— 5 $.— Se bs Se es a‘*,, ata? ata? BiG eee 3y" Po Dy Py; aa tas Pata se aja? ; A,=54 st Sy! Po» By= 549s + 87 py: a’ ; a. i A= "== Pe B=— 755 bu &e., &e, 29. It remains to substitute the first set of constants in the expression for K,; arranging the terms according to powers of y we find: - A 4: K, = 2na,2 + 20a," + arte Bedi 4 7M, “a, 4ra,*d a (a, $e + 4495") + gas 4 (abe + 4,44") iim! ae 37ra,‘a," is bobo + aes a, + a,',”) a “a, (a, +.4,") a body + ete. = 2a,’ b? + 2a; p+ U, say. Pror. PEARSON, ON A CERTAIN ATOMIC HYPOTHESIS. 95 If we suppose the two atoms to héfve linear velocities qg,, qg, in directions h,, h,, then as mentioned in Art. 2 we must add to the above value of A, the expression @ ; aa. fl 8.3 — , bool aa! Hee — 79,924, 4, ais () 27 by 9,4, by dh, () 2 + 27db,q,4,74,° (;) + higher powers of - Now if q,, gq, are small compared with ¢,, ¢,’, those terms involving only products of the forms q,g, and gf, may be neglected in the value of K, as compared with those which involve squares and products of the ¢’s. In other words we shall suppose for the present that the motion of the atoms about their mean positions or in space has nothing like the magnitude at which a point on their surfaces is moving. (We might express this more concisely by saying, that the linear velocity of the atoms has no magnitude like that which by our theory is supposed to produce the phenomena of magnetism.) We have to introduce a term 4m,q,°+4m,q, into the expression for the total energy of the system. We shall also suppose some system of forces to act on the two atoms capable of expression by the force function W, so that the force in direction k on the ae ht: dW : : ; : first atom is given by aE’ °° that the equations of motion for ary variable y say, take the form, d (=) _dL _dWw dt\dy/ dy dx~ ; 30. We have now to consider two totally different cases which correspond to the following circumstances :— (1) Two atoms which affect each other’s pulsations or atoms in the same molecule, (2) Two atoms which do not affect each other’s pulsations or atoms of different molecules, ra! 3 = Ena ae) pape, + Ky’+ 27a,$6.°+ 3° Gs sl +U =K,+ KJ/-V-V'+U, if we suppose the coefficients in K, and K, altered so as to include the terms which denote apparent increase of mass. Now let y be a coordinate of the first atom then the equation of motion may be written djdK, ,dV_dU dW * (=) Ber am at ay dt dae) Case (i). Let two atoms be mutually influencing each other (i.e. atoms of the same molecule), but uninfluenced so far as pulsation is concerned by any other external forces, then W=0 or the atoms will appear to move as if they were not in a fluid medium but were with altered mass and subject to a force function U. Vou. XIV. Part II. 13 Now L= K, oF 2na,*$," ate vr ae 96 Pror. PEARSON, ON A CERTAIN ATOMIC HYPOTHESIS. If y=h we have the equation q a (a) aoe dt\dg,/ dh’ to calculate translation so far as influenced by pulsation, for the velocity of the atom in direction h. Case (ii). Let the two atoms not be mutually influencing each other’s pulsations (ie. atoms of different molecules at some distance). The pulsations of the atoms will now be looked upon as completely independent; these pulsations will in fact be considered either as absolutely free or as the forced pulsations due to other members only of the molecules to which the atoms under consideration respectively belong. In order to find the force between two atoms of different molecules, we shall then assume that in the expression for the force-function, the pulsation velocities are fully determined by molecular construction as in Case (i). We in fact neglect oa and a so far as they depend on intermolecular distances, but retain ae as giving intermolecular dh force. 31. I. Atomic Forces. The forces between atoms of the same molecule. The force function between two such atoms is given by Aqra.7a,? TA, = , [So bobs + (a9, ar anh, 2 4qra,‘a,¢ , ; na zd Pp ts By = (a, db, ar a,%, ‘) eo bbs 37a,‘a," ' 47a, "a, + Set (a) +4,9,") + 5 * (a, +.0,) $d, + ete. Y ° Now ¢, is of the form Rcosnt+S8 cosn,t, and 0 1 2 DS FR eink chew AS R' cosn,t +8’ cos nf. e the product ¢,¢,' will vary with the time, but it must be remembered that 2 = inde = are infinitely small—each pulsation of any atom being gone through billions 2 a times a second, if we are to judge from the analogy of light vibrations. Hence we shall be justified in taking the mean value of the product ¢,¢,, or the sum of the mean values of its terms. If m, be not equal to m, the mean value of cosm,t.cosm,t=0; if however m,=m, the mean value of cos*m,t=}. It follows that the mean value of Pi vitae = pelel Poo > Oe eS ) Pror. PEARSON, ON A CERTAIN ATOMIC HYPOTHESIS. 97 2 2 and of foe es , 2 oa R248" Oil taor Now R, R’, S, S’ are functions of ; which have already been calculated to a first degree of approximation, Le. when the first term of U was included in our equations of motion. In order however to see more generally the character of the force function we shall calculate its value in this simple case on the supposition that its second term is also considered, The equations of motion may then be written: 27a, ‘te A4mraa bo + arr §.=0, $, (A, + 4rra,*) + 7.6, + e $, = oa Pa,) ap Ty + Qd,’ = 0, $,' 2 a Pa,) = 7) Po ar Qh, = 0, 2 where P= ar and the other symbols have the meanings already assigned them. Let $,=C,cosnt, =C,cosnt, Wal Then, {r, (= --,) — Pa, 0,-QC, =9, 12 V; (laa re fr (7s -5) 4 Pa,} O06 0; or, Ta p - =) — Pa, +7, é — =) - Pa} = QF. nv D, ‘i Nev, Hence we find approximately, v, and v, being supposed not very nearly equal :— lt Seal OP yap 4. Fa ()* vv. 2 (De ie ugh Di is Wht eee) 2 2 aL wet Pa, wy © / n," Vy TT ee v,” T) wow (¥y — ¥) 2 3 6] {2 Ua Q es } (4 1 7 2 2 2? TA Dap Tey) ey he) 2 3 a’ Q Let ae Q VU, Ys == (Ol! 27 pI—ve t,t) (Yeh) ae O} 8) 1 oO 2 1 We have at once dy =— (n,C, sin n,t +n, C, sin n,t) — e. = ? dg, =—(n,C, sin n,t + n,C,' sin n,f) 13—2 98 Pror. PEARSON, ON A CERTAIN ATOMIC HYPOTHESIS. and U= + Q nC, C, ats nC,'C,’ os P ae (n,°C’ ats n,’C,”) AP a, (nJC,? =: n," C,”) a = 2 4 = Q° vv, Cry," a ay 4P (a,v7C,* ar av, C,”) 2 vZ—v2| T 4 — GB ty _ (Cyy,* (vt = 20,1) _ C."v,! (,' = 22,") 27,7) = V, Dh T; T) be = 7 5 = ’ tic al (Oss ee Ory where oes 87a 1a," ; Pos = re = : = 3,3 2f72 2 "2 Pee hia, a, (a, C, + GV, C; ) T 0 UF ° ao oe 4,4 24 2 2 12,4 2 2 _ 128:r“a;*a,” __v,'v, {om (uv, — 2v,") _ C,*v,* (v, — 2v, 4 ’ 2 2\3 ' ? ToT es) 0 and as before v,> v,. 32. Now unless we make some assumptions as to the internal construction of atoms it is not easy to predict anything with regard to the nature of w, and w,. We see however (ey y 2 G 2y 2 that yw, will be positive if = is > 0 0 » ‘If C,=Cy and +,’=7, this will certainly / eosin asl pate s LM A,+47a°’ 7? A, + 47a, improbable, that if »v,>v,, 7, will be >7,, and hence that »,°r, will be >~»,*7,’. In like manner p, consists of two parts, the first of which is always positive and the second Cr»; (vy, =a 2v,") C,*v,* (v," a 2v,") To be the case; or again if C,/=C/ it seems not , since pv, = negative or positive according as is greater or less than T) Y2,, 2 If C,=C{ and +,=7, the latter part of yw, will certainly be negative; as again if = 0 Ce ; ae : : , be >—2~2 ; but whether #, as a whole is positive or negative we cannot determine till T) we know the relation of the potential and kinetic energies of an atom to its radius. We have however obtained the following result :—The force between two atoms which are mutually influencing each other's pulsations, i.e. the ‘atomic force’ between two atoms, varies partly as the inverse cube, partly as the inverse fifth power of the central distance. In order that two atoms may combine to form a molecule, the force between them must be attractive; hence as a first condition for a molecular union of two atoms it is necessary that uw, be negative, or OH 4 H, T ie But if y becomes very great we have d, =— Cy, sin vt, ¢, =— Cv, sin vf, Pror. PEARSON, ON A CERTAIN ATOMIC HYPOTHESIS. 99 or C;'v,’ represents the ‘intensity’ of the free pulsation of the first atom, while (,’v,* is the like quantity for the second. In order then that the atoms may form a molecular union, it is necessary that the product of the potential energy coefficient and the intensity of the free pulsation should be greater in that atom whose pulsation has the greater free period. We have of course only calculated the lowest powers of z in the force function, but Y these will be the most important. When y is of any magnitude compared with the atomic radii the first of these is really the all important term, and we might say that: Forces of chemical combination vary as the inverse cube. 33. In order to throw more light on these forces, let us suppose the atoms equal ; then we find: Bigdl apQea, , er Oe mer I peh reel | 12 Se ee nN, Vv, T) T) and ¢, =—7,C, sin n,t —n,C, sin not, ¢,= 7,0, sinn,t—n,C, sin n,t. Hence ( 2 2.4 3,6 5 P. 2 2 U 3 ae \ CF a C,’) at ae (C? Ur C;’) ar = (CP = C,’) ar = (Co a cp is = (C? ot C,’) 9 0 0 2 24 3-6 4.6 y 2 =— 2 (¢2_ 9) 2% (924.02) = (02-02) — (2 a Pa) 20, A) Hence if C, be not equal to C,, two equal atoms within their sphere of mutual reaction act upon each other with the following forces: (a) A force varying as the inverse square of the distance, which is attractive if C, C,. (b) A force always repulsive varying as the inverse cube of the distance. (c) A force varying as the inverse fourth power of the distance and attractive or repulsive according as CO,’ < or > CO, (d) Vv V, 1 2 words a molecule which gives lines towards the violet end of the spectrum or in the ultra violet will be a stronger chemical combination than one which gives lines further removed towards the red end of the spectrum. 35. It may then be questioned whether the nature of the equilibrium in a diatomic molecule is statical or dynamical? There are two kinds of statical equilibrium con- eceivable, either the atoms are in contact or it is possible that the repulsive force between them may at some distance greater than the sum of their radii. be equal to the attrac- tive force. If the equilibrium is dynamical we must suppose one atom describing an orbit relative to the other; in this case we may compare a molecule to a planetary system and the action of one molecule on another to that of one planetary system on another. In one case the disturbance due to the approach of two molecules will be from the statical position of equilibrium, in the other case it will be from the mean orbit. When the atoms are equal, and supposing C,=C,, we find the force between them ~ 20% 0e2_428 sr) , T) T). a and this vanishes, or there will be a non-contact position of equilibrium, if 24, 2 6 Oy," _ 4 Gy — 2Pa,=0, ai 3 tt) 0 ne 1 a 2 that is, if a , —1=0 ee y @ 14 Ny ) 4a, ( ‘dara, : 2 T. ) (remembering VY, EEE | Now y must be greater than 2a, for such a position; hence, : 2 4 (1 a y er: 4a,’ 4cra,? Ay ii a a E= AG) Ay 9 or (1 ae) (a + Gia) ~2>0, gs> B12... Whether or not this is possible depends on the internal construction of an atom concerning which we have made no assumptions. It must be noted however that if the atoms be supposed to have no mass (A’s=zero), this non-contact position of equilibrium becomes impossible, 36. Let us now turn to the case of a relative orbit. In order to find its form we may suppose that forces equal and opposite to those acting on one atom are imposed on the Pror. PEARSON, ON A CERTAIN ATOMIC HYPOTHESIS. 101 other. This really amounts to no more than altering in a certain ratio the constants of the force function, and hence to find the relative orbit we may still assume that the foree function is of the form ee ;:—supposing that the atoms are not similar or, if they are, that C,=C,. 1 F Let one and @ be the angle y makes with some fixed straight line, then we have : du mn au et (ers a3 to integrate det U= ia (Qu, — 4,0") where fh? is a constant. 2 Or ultimately, (3) =a? + bv? — cu’, eels oecaresaot where a, f and & are constants. de Hence a8 + ==. where w=cos ¢ and a, §, « are other constants. It follows that ¢ =am (46+ 8), or u=cosam(aé+ B). Since w=0 when a8+=(2n+1)K, it follows that the form of the orbit thus given is not as a general rule a closed curve; in other words there is no permanent union of the atoms to form a molecule. Hence in this particular case of two unlike atoms a dynamical stability of the molecule does not seem probable. Whether the atoms of a molecule are in statical or dynamical equilibrium seems a question of considerable interest; it is possible that they change the nature of their equilibrium as the molecular state of the body is changed. For instance, the atoms of the molecule in a solid body may be in statical, while the atoms of a molecule in a rare gas may be in dynamical equilibrium. 37. We pass now to the forces between p unlike atoms forming a molecule. Here we are certain that the first term of the force function will be of the form Aqra,a U=% aa “ $.4,=20.4,6, where ¢, and @¢, are the pulsations of the rth and sth atom respectively, and are given by 2 2 Vy V, ._ — Cn. Qrm¥m Pr 5 1, SIN Hi A G- 7; be Cn ana Pa Un Vs nO , =—n,C.sinnt+ > é,=— n,C, sin nt +> n,, Sinn, E. 102 Pror. PEARSON, ON A CERTAIN ATOMIC HYPOTHESIS. Now we need only the terms of most importance in the product; these are oe Ole Dave —n2C2sin® nt jt, — nC? sin’ n,t Query! 18 v, Ee v T, (v,” TF T, (v, — v,’) Ke Q,02 ve e Oe s ve = 2 (v2 — », Hence ‘ : 7 Te eae (toe wi; +) ; 2 (v2 —v,) T, T, We sce at once then that the force between any two unlike atoms in a molecule varies inversely as the cube of their distance, and that if the quantity 7v*C* is greater for the atom whose free pulsation has the greater period, then the force is attractive ; if however it is greater for the atom which has the less period in its free pulsation the force is repulsive. 38. The expression zv°C®, or the product of the potential energy coefficient and the ‘intensity’ of the free pulsation, may be termed the chemical intensity of an atom. The Telok —viC2r ; T,—v, CU, expression See, or the difference of the chemical intensities of two atoms Tv vp yp? 1 2 divided by the difference of the squares of their periods, may be termed the chemical coeffi- cient of the two atoms. If the chemical coefficient is positive the two atoms will attract each other, if negative they will repel. Its sign thus determines the possibility of two atoms en- 4,44 tering into combination. If /, represent the chemical coefficient, the quantity a Ree 1°72 measures the strength of the force between the two atoms. The force between them _ 647r*a,‘a,* F 1 ; c ray ee ae \2-— 3, tending to decrease y, and is therefore attractive if F, be positive. TT. 12 It might perhaps then be better to adopt as our definition of the chemical coefficient the ot Mt arate HE abetg product of peter oR cores intensities and the quantity Sires *, A knowledge difference of squares of periods cate then of the chemical coefficients of every two of a set of p atoms would enable us to determine the possibility of p atoms uniting to form a molecule, and also the stability of the molecule so determined. Again, if r other molecules were added to the p already combined it would enable us to determine whether this action would destroy the com- bination of p atoms. 39. Let us consider the simple but very suggestive case of a third atom being added to two already in combination. 9 Let a ; = be the periods of those in combination. Let also v, be > v,, and therefore 1 2 for combination J,1,. (a) If the ‘chemical intensity’ of the third atom is less than those of the other two (1,J, and >J,), it is incapable of uniting with them to form a molecule because it is repelled from both. (c) If the chemical intensity of the third atom is greater than the one and less than the other chemical intensity of the already combined atoms (J, lies between J, and J,), it attracts the second atom and repels the first. Then in order that it may remain in the combination we must have AGS AL ; soft ——8 <—* (where the A’s represent chemical affinities). 13 23 The first element will then be turned out of the combination, if We shall then have the second and third atoms left together attracting each other irrespective of whatever changes may, owing to external disturbance, take place in their distance. In other words we have a new stable compound of two atoms, Tf, on the other hand, —B > Ag Viv Ys1 the first atom will not be forced out of the combination, but we shall have two mutually repelling atoms united by means of a third which attracts both. In this case the forces uniting the various atoms of the molecule are not all attractive; it is obvious that the molecule requires less to disassociate it, and we may say that we have an unstable compound of three atoms. Such theoretically unstable compounds seem to throw light on the nature of easily decomposed or even explosive substances, The other cases may be treated more briefly as they present no further novelty. Case (ii). v,v,. Remembering that since the first two atoms form a mole- cule and »,>v,, therefore I,>J, we find: ay E>T and T, and therefore > J, The third atom will attract the first atom and repel the second, A Al a Tf, —-—2>—4 ; ; ; 5G , Tyee | the third atom will enter into a combination with dln AR f the first atom and turn out the second. and aS | Vis Ys2 J Vou, XLV. Parr I. 14 If, == 19 Yer Naa an unstable compound of the three atoms will be formed. An. Au! . Pe Tee If — As, then, since y,, can never get very small, — An will be less than Ay 8 3? ? 13 DS Ga) 3 8 $2 $1 32 21 and the third atom will be foreed away from the compound of the first two atoms. If (as by the application of some mechanical force to bring the third atom and the molecule of the compound together, perhaps, for example, extreme pressure) y,, could, notwith- standing the repulsive force on the third atom from the diatomic compound, get so small that —4e 5 4a 82 21 would be that all three atoms would be disassociated. Yet it seems probable that even this would only be possible in the case of a limited number of molecules, and we might almost venture to state it as a general law: that, ‘no element in its disassociated state then the result of adding the third atom to the diatomic molecule (i.e. when its own atoms are not in molecular union) could possibly so break up a di- atomic compound of two other elements that all three elements would appear disassociated.’ This law might be very easily generalized. (c) If Z, J, and therefore I,> J. Stable union of all three atoms. (b) I,TJ, and J, 27a,‘n?C_C’, cos (2, —2’,), 1 3.8 pe, =X > —— (an? + an?0"?). at We see that mw, is always positive—i.e, two molecules always attract each other with a force varying inversely as the fifth power of their distance. 43. But with regard to ~, we see that its sign depends on the difference of phases of the atoms of like kind in the two molecules. If these differences can be anything what- ever there may or may not be a force between any two molecules varying as the in- verse square of their distance, and that from molecule to molecule this force may, if it exists, be attractive or repulsive. We should then have the following result—the molecules of a gas would be in part attracting and in part repelling each other. In this case it seems possible that a quantity of gas might be obtained whose particles had a disasso- ciative tendency :—in other words, that a gas on expanding would lose internal (molecular configurational) energy, but the experiments of Joule and Thomson seem to show that a gas always does some however small amount of work in overcoming the cohesion of its own molecules when it is expanding. If this be granted we are compelled to consider that the differences of phase of two atoms of like kind in two different molecules are not purely arbitrary, and to assert that the differences of phase of any two like atoms does not generally exceed oe If we get over this difficulty by supposing that in any finite portion of a gas such as it is possible to experiment on, we shall have as many repelling as attracting molecules, and these in all varieties of difference of phase, we find that in the expression for the internal energy of a gas (due to molecular configu- ration) there will be no terms whatever varying inversely as the first power of the distances, and this internal energy will, varying entirely as the fourth power of the distances, be a very small quantity as shown by Joule and Thomson’s experiments. 44, But even this assumption does not free us from new difficulties. These molecular forces must in the cases of a solid or liquid become the forces of cohesion or of viscosity, if we do not suppose in either case the atoms of different molecules to be as near each other as the atoms of the same molecule, and so the forces between different molecules to become in reality what we have termed chemical forces. Yet if these chemical forces become the chief forces of cohesion in the case of a solid it is difficult to understand how the molecules are preserved in such a case. All the molecules would be forced into chemical combination with one another, and there would seem no reason why a solid should always present us with a gas composed of the same kind of molecules. We are compelled then to consider a solid as composed of molecules whose distances, if small compared with the gases, are yet great compared with the atomic distances. That the atoms of one molecule influence those of another either by slightly affecting their respective pulsations, or still more probably their respective distances, the continuous spectrum of a solid forces us to believe, but if the molecules are to remain distinct we must suppose the principal forces between them are what we have termed molecular forces and not chemical Pror. PEARSON, ON A CERTAIN ATOMIC HYPOTHESIS. 107 forces. If this however be so how can the forces between a number of molecules be partly attractive and partly repulsive? Under such a state of things cohesion seems im- possible. This leads us to the conclusion that in any substance the difference of phase between two like atoms in different molecules cannot exceed 5 ; or rather it will be sufficient if for every pair of molecules 2a,'n,*C,C’, cos (4, — @’,) is positive. In the case of monatomic molecules we must have the difference of phase of the atoms always less than = In the case of diatomic molecules we must have 4,2 / , 4. 2 It ! a,‘n,’ CC’, cos (4, — a’) + a,'n,’ CC’, cos (4, — a’) always positive, and so forth. 45, Now it must be acknowledged that these seem very arbitrary distinctions, and one is inclined to demand a general law which would include them all. The difficulty lies in this: What is it that determines the ditference of phase between two atoms ? It is obvious that we may consider the Cs as all positive, since any negative sign would be brought into the argument by alteration of the phase, the condition then that the differences of phase shall not exceed ~ is sufficient although not necessar . We may then p 5 g y y ask: Is it necessary that two like atoms in different molecules should have any difference of phase? May not the state of pulsation in every molecule both as to amplitude and phase or at least as to the latter be the same for their like atoms? Is there anything which at all confirms this view? Suppose we attempted to find the result of a number of molecules whose pulsations of like period had every conceivable variety of phase disturbing the ether at any point. Then if one molecule produced at that pomt what might be termed a positive disturbance, another could be found owing to the difference of phase giving an equal and opposite negative dis- turbance, and so the total effect of all the disturbing molecules might be almost zero. In the Undulatory Theory of Light we are compelled to suppose at the same instant every point in the plane face of a small wave to have the same phase. It is hard to conceive how such is possible, if at the same instant every molecule helping to form that wave has its corresponding pulsation in a different phase. For example, if one molecule produced a disturbance given by D sin (nt — Ba +x) at any point of the ether, another owing to its difference of phase would produce a dis- turbance of the form D sin (nt — Bx + x’), and the total value of such disturbances would be zero if we gave « all possible values. 108 Pror. PEARSON, ON A CERTAIN ATOMIC HYPOTHESIS. Hence we have suggested to us the following conclusion :—Like atoms in like molecules are pulsating in the same phase. We can then deduce the following law of molecular force. The force between two molecules consists in its more important terms of a part varying as the inverse square and a part varying as the inverse fifth power of the molecular distance; both these components are attractive. 46. We may draw the following conclusions :— Every term of the intensity of this attractive force varies inversely as some (period)® of the molecular pulsations (i.e. has a factor n*). Hence those substances which give bright lines nearest to the violet (supposing a,‘n,’ to increase with n,) will have their molecular forces greatest. In other words, they will do more internal work in expanding. Or again, when in a solid state their forces of cohesion will be strongest or they will be what might be termed ‘tough’ substances. According to this theory then the toughness of pure metals would be in the same order as the frequencies of their single bright lines. Again the relative orbit of two molecules so far as they are uninfluenced by other molecules will be very approximately (retaining only the first term of the force-function) a conic section about the other molecule in the focus. It may be noted however that Clerk Maxwell’s experiments on the viscosity of gases led him to suppose a repulsive and not an attractive force varying as the inverse fifth power of the distance between two very near molecules, a result with which unfortunately our theory does not appear to coincide (cf. Clifford’s Lectures and Essays, I. p. 241, however). Let us return to the statement that like atoms in like molecules are pulsating in the same phase. Supposing the atoms not in molecular union with others or monatomic molecules, we are led to the result that all like atoms when they are not portions of molecular compounds are vibrating in the same phase. 47. It is however not necessary to suppose that the phase of the free pulsation when altered to a forced pulsation is itself not altered. It is sufficient if the alteration be a function of 7 and vanish when ; becomes insensible. The following example will perhaps cast light on the meaning of this statement. Let us suppose four atoms two and two alike, and that they are about to unite one of each kind to form two diatomic molecules. Let the free pulsations of the first like pair be given by , = D, cos (v,t + 4,) | $, = EL, cos (v,t + 4,) $, =D, cos (v,t + 4,) } es $, = E, cos (v,t + ,) be the free pulsations of the second like pair, each pair being in the same phase. Then the effect of atom ¢, on atom ¢, is as follows :— (1) to alter the free period in a manner given by the following equation :— 2 2,2 Lb Le oe, eee / 2 2) Ns V, 0 Pror, PEARSON, ON A CERTAIN ATOMIC HYPOTHESIS. 109 (2) to alter #, say to C,; (3) to alter a, to a,+ Qp, say. Then we find $, = C,, cos (n,t + a, + Qu,). But to this we must add a forced pulsation with the altered period of the first C, Qv,7v, atom, namely: = 2, cos (n,t+a,+ Qu), T,(v, —v,) the intensity of which is also a function of Q. We thus have finally = C,Qv,'v,." Tv’, (v," — v,) If y were very great however the second term would vanish, n,=v,, and C, must $, = C, cos (n,t +a, + QH.) ma cos (n,t +4, + Qu). become = £,, or $, = ZL, cos (n,f+ 4). Thus ©, can only differ from #, by quantities of order Q. The same reasoning gives us ir 2,2 ¢, = CZ cos (nt + a, + Q'v’.) = _CO'"Qv iv, ; = cos (nt Wie) T (v>—v, ) a were Cn) Hence so far as these terms are concerned the attractive force between the molecules Qnra,n,'0,0", 00s (Q'u',— Qu,) 2 2 HY / is given by and since Q'y’,— Qu, is always very small (of the order =) it is less than and the T 2 ) force as given by each pair of terms will be attractive. In order that the pulsation periods of each molecule may be the same, it is neces- sary that Q=@Q’. In order to obtain a wave of light from a number of different molecules it appears not improbable, though perhaps not absolutely necessary, that pu,’ = p,. There is still another poimt to be noted: the chemical affinity of the molecules formed from the atoms ¢, and @, is a function of D, and £,; the chemical affinity of the molecule formed from the atoms ¢,' and ¢, is a function of D and £,, or as it would be perhaps better to say, of C,, C, and of C/, C. Now when a substance is under con- stant conditions we cannot suppose the chemical affinity to vary much from molecule to molecule, and hence we must have C, and C/ nearly equal to C, and C,. Hence we may say that when the molecules of a substance are subject to much the same external conditions, not only the phases, but the amplitudes of the atomic pulsations of the molecules are for the corresponding atoms of different molecules very nearly alike. 48. We have seen that the action of one atom on another is a vanishing quantity when the atoms are at any distance great compared with their linear dimensions; if however a great number of atoms are all tending to influence another atom even at a great distance in a like fashion, especially if they are all tending to force upon it a pulsation of equal or nearly equal period to its free pulsation, then the sum of these vanishing 110 Pror, PEARSON, ON A CERTAIN ATOMIC HYPOTHESIS. individual actions becomes so magnified as to be of importance. One atom in the sur does not in the very slightest affect the pulsations of an earth-atom, but an infinitg number of such atoms in the sun may affect the pulsations of an earth-atom. We cannot in that case treat the action of individual atoms upon each other, we must rather look at the outcome of the infinite number of atoms in producing a wave, and at the effect of this disturbance on an individual atom in its neighbourhood. Even then it seems im- probable that it will be much affected unless there is a period of the wave of equal or nearly equal magnitude with the free period of the atom, In this case we must suppose the amplitude of the free pulsation of the atom—or in the case of the atom being in molecular union its principal pulsation or the pulsation corresponding to the free pulsation but with somewhat altered period—to be most affected. Such a change of amplitude however will affect the chemical intensity of the atom, and thus the chemical affinity of two atoms. Hence we note how it may happen that waves of different period may tend to alter the constitution of a body either as to outward appearance or molecular construction; waves—termed for convenience chemical, light and heat waves—may affect the body in different ways because they alter the chemical intensity of its atoms, the chemical affinity between the atoms of the same molecule, and lastly the molecular forces between the parts of a body. 49. If magnetism and electricity consist in the oscillations of the ultimate atoms of a body, they will not directly affect the pulsations, but by altering the atomic distances they will indirectly affect them. When however the atoms are pulsating freely, it is impossible any longer for alterations in the mutual distances to affect their pulsations. Hence if once a body is completely disassociated, magnetic or electric action according to our theory ought not to affect its spectrum. 50. We may note then that in order to fully grasp the ‘state’ of a body we must know not only its atoms and the particular manner in which they are united into molecules, but also the exact condition of the ether in the spot where the body is situated. For instance if P be a body, and S a closed surface surrounding it, then the condition of the body P depends at every instant on what is happening in the ether over the surface S. Its so-called mechanical chemical and physical properties vary with what is happening in the ether at the surface S. For example, the molecular theory of gases supposes the molecules of a gas only to be affected by the close neighbourhood of other molecules, when they are supposed to exert force upon each other, In the above theory the condition of a molecule at any time must be looked upon as a function of the disturbance in the ether at the spot where it may chance to be. If we knew the exact characteristics of the disturbance of the ether at that gs neighbourhood at the given time, we should be able to determine every property of the molecule without reference to other existing molecules, 51. III. Gravitation Forces. If this theory of a continuous medium producing by its disturbance all the phenomena of the physical world be at all a true one we must expect it to give some explanation of the Pror. PEARSON, ON A CERTAIN ATOMIC HYPOTHESIS. 111 so-called. principle of gravitation. If two bodies P and Q are apparently drawn together by a certain force, it can only be because they are both disturbing the ether, and the effect of their disturbance can be represented by that force. Whatever “force Q exerts on P” must be the result of the state of the ether as produced by Q over a surface enclosing P; there is not such a thing as a force “inherent” in either P or Q. Now the only method whereby P can affect the ether is by-its motion, whether of translation (in- cluding oscillation), rotation, or vibration, Now the motion of translation of two spherical atoms produces between them a force of a magnetic character, a sort of inverse magnetism if the translations or oscillations of the spheres are forced upon them, but a direct magnetism if they are free translations or oscillations of mutually reacting spheres. Hence the force of gravitation cannot have its origin in the translatory or oscillatory movement of the atoms. With regard to a motion of rotation it is difficult to see what effect the rotation of a sphere could have in a perfect fluid, if on the other hand we suppose the atoms ellipsoidal it will be: found on calculating the kinetic energy of the fluid due to two ellipsoids rotating about their axes that no term occurs in it of the order —, in other words two rotating ellipsoids do not mutually act on each other as if subject to a gravita- tion force. We are then thrown back on vibration or pulsation as the cause of so-called gravitation, and, as we have seen, the pulsation of spherical atoms does introduce terms corresponding to the ordinary gravitation potential. If two atoms be pulsating with periods not mutually dependent, we have seen that the force-function is given by 2 3 3 Td, LG be A F it U SS Doo a cee (a,6.° a5 ah, ’) + ete, uf Now if the atoms bé at sensible distances, the second term may be neglected as compared with the first. The first will be all important, at the same time the gravitation term will only be the first of a long series, the rest of which may be safely neglected as insensible. 52. We see then that two spheres with pulsation velocities ¢, and ¢,' have a potential given by es! tna,a2bby. > Suppose ¢,- and ¢,’ to have terms m,8, cos (mt+a,) and m,@, cos (m,t + 4,), 2 : ; then the potential will vary with the time, unless —, 7 the pulsation periods, be so 1 2 small that we may take the mean of ¢, and ¢,/. We then find as a first condition for a gravitation potential that two atoms must be vibrating in the same period, for if not the mean of cos(m,t+4,) cos(m,t+ where p would be defined as ‘density.’ There seem however many objections to such a supposition as that:—the portions of the attractive force arising from the same period occurring in both elements are the same for all periods;—foremost among them being the fact that as one element changes its state, the number of atoms giving a definite period may grow less and vanish ultimately altogether. We are then led to reconsider the expression =D/D,Svbv' 2 oi If we were to retain the one element the same but vary the other, the first set of constants D, would remain fixed, while the second D/’ would change. There would then be no inconsistency in defining =D,.D, as proportional to the density of the one body which gave the set D/. The only result would be that the density of a body if measured by means of the attraction of another body would vary according to the attracting body we might select. For example, let the earth be taken as standard, then if we determine the relative densities of two bodies by means of their weights, these relative densities will be in the ratio of 2D/D, to =D,’D,._ But if we proceed to determine them by means of the attraction of any other body with attraction-constants D,”, their relative densities will be in the ratio of > DID ALO SDD xe. and this by no means need be the same ratio as that given by their weights. It is probable that the attraction-constants for most solid bodies which have been taken as standards may have been nearly proportional, so that D D D r 5 p ih; Tipe Ww DL DYED: s = etc., and that the density of any given substance as found by different methods may be nearly the same. Such a result however, seemingly affecting the very basis of physics, is perhaps sufficient to shew that these pulsations are not sufficient to explain the phenomenon of gravitation. On the other hand it may be noted what very different results have been obtained by different methods for the mean density of the earth, ranging from less than 5 to 66. Even in Baily’s experiments the average mean density as obtained from a set of experiments with one kind of balls on the torsion-rod differed from that obtained from experiments with balls of a different substance. 57. We are not however compelled to consider the ‘mass, whatever that may mean, of a body as varying. All we must suppose is that the force of gravitation is not that of the product of the masses divided by the square of the distance. It is a certain function of the constitution of both bodies divided by the square of the distance. This function may be in the case of most solid. bodies almost exactly equal to the product of what 116 Pror. PEARSON, ON A CERTAIN ATOMIC HYPOTHESIS. we term their masses, but it is in itself a function of their relative atomic conditions, and can in certain cases be conceived as vanishing. For example, in the case of two gases whose molecules are vibrating freely and which have no common pulsation period. 58. We may remark at once that such a theory does not suppose it possible for a body to be without weight, because a solid body like the earth (or a liquid) with a continuous range of beats will attract all other bodies whatever may be their atomic con- dition. Nor is it difficult to understand how it is possible that change in the atomic condition may not affect the weight. Let C, be the attraction-coefficient of an element of the earth for the period =. Then the attraction of that element of the earth on an atom (a,) pulsating with that period : 2rra,n7 B. is (as above, Art. 55) :— aM, rae Now suppose in some (say solid) state we had m equal atoms all influencing each other, then we should have to sum the quantity »/°B,C, for a practically continuous set of values of s. On the other hand, if the solid becomes completely disassociated we may suppose its m atoms to be pulsating with some period 27/n,. In order that the weight may be the same we must suppose :— >n7B,C,=m .nZBC,, 2 or that n, BC, = sos A or that the attraction on one atom pulsating freely is the mean of the attractions on all m atoms pulsating with mutually enforced periods. Now if we remember how x, is formed from n,, differing first above and then below, and the like holding probably for B,, B, and : , Bn 1 C,, C., the above relation seems perfectly intelligible. If B,, C,, ete, vary as a? the con- 3 dition would become = =— a relation proved before to be true. (Art. 21.) Again, supposing the solid or liquid to pass into that state in which its molecules are freely vibrating, and that these molecules consisting of p different atoms are then disasso- ciated. The weight of the p different atoms freely pulsating will be the same as that of p Pp the molecule if PaqvehiCs— 2a,'n, B,C. 1 went yey 8 where ee ae nN, r Vv, —Y, 3 1 i Hence, if Bio—,, and C, as —, a, Tr Vv, we must have 5{= 25 which follows at once from the values of as etc. (Art. 18.) Pror, PEARSON, ON A CERTAIN ATOMIC HYPOTHESIS. 117 59. The above are only taken as hypothetical cases to shew how it might be possible that the weight of a body or a mixture of bodies might remain unaltered notwithstanding molecular or chemical changes. Such changes alter the period and perhaps the amplitude of each atom’s pulsation (B,), but if this amplitude varies as = 7 they will not alter Tia? the ‘weight.’ We have supposed however that waves of like period to the pulsation affect its amplitude. In other words, if an atom pulsating freely were placed in a disturbance of the ether of its own period that the amplitude of its pulsation would be increased. The absorp- tion of light and heat forces this conclusion upon us. In such a case our theory would compel us to believe that the gravitating power of the atom or its weight had been increased. It is possible that such increase of the amplitude may be very minute, even when a great number of atoms are put together, still, however minute, it is capable of presenting to our senses all the different stages which iron passes through, for example from an ordinary temperature to a white heat, before it begins to change its atomic condition. Some trace, however slight, we should expect to find of this alteration in the weight. In the above theory a red-hot iron ball ought to be heavier than one at ordinary temperature. If no such trace, however minute, can be found, it would seem an additional argument against a theory of gravitation as due to pulsation of spherical atoms in the ether. Pulsating spherical atoms as the basis of physical phenomena seem capable of throwing considerable light on chemical and molecular physics. We may have to look further for an explanation of the principle of gravitation, either in the shape of the atoms themselves, in their method of vibrating, or still more probably in the nature of the medium in which these atoms are vibrating. If the ether be not a ‘perfect fluid,’ if there be something of the nature of ‘skin friction’ between atom and ether, it is possible that a rotatory motion of the atoms may be the origin of gravitation, Be this as it may the explanation of gravitation even as of magnetism and electricity must be sought in the translatory, rotatory or vibratory motion of the ultimate atoms, Appendix. We have supposed each free atom to have in its pulsation only one period. The free period is given by the equation T,—(47a,° +2,)n? = 0. Now it is possible that 7, and A, are functions of n, as for example if the atom be looked upon as a solid elastic sphere. In this case let T=X,(n), A=W, (n). x, (n) — 4774,’, then we have the following equation to determine the free periods of the atom: 1—, (n). nr? =0. Let ,¥,, .%,, ++ ete. be the roots, The number of these roots will probably be infinite (as in the case of an elastic sphere). If then we suppose every gas or vapour of an element when completely disassociated to give only one bright line, we must suppose v. 118 Pror. PEARSON, ON A. CERTAIN ATOMIC HYPOTHESIS. only one of the roots to give a period which falls within the light spectrum, and the existence of the other roots will not be disproved by the phenomenon of single bright line spectra. In the case of two atoms the equations which determine the alteration of their periods due to mutual influence are 2,2 {r, — (A, + 42ra,°) n*} C, — pure n°C, =0, 2 2 frac (Qt dict Nits Gee eae, 1 or xX; (r) {i -y, wo} C,-—QC,=0, xa(0) {e—Fal)} C= QC, =05 or x. (0) x00) {a-ha He OF = - , 1 : 1 1 Now Q* is small, and hence 2 cannot differ much from —,, —,, etc, or from v Vv 11 21 = = » ete. Let 5 ils 2 therefore Xr (M1) Xa) m (aay Wy CM) + {- é Vs wd} Te ar m= = ( , 1 Xs (»,) id “yy, (v,) + 1} » Xe (,Y;) | pe st Vv, wn} Caf Now in the case when there was only one free period we found n= @ : (1 Pee ee TT. = zs =) 2 1 Hence in our results we have merely to replace Ga-s2) TT. \ =a =e V. v 2 1 by the expression x (”,) Xe (.,) {3 oe vy (,) 15 1} - :, a vy (of “1 where it occurs in the vibration of the first atom; and ih al mails & ys 1 by xo) xa) ade (n+) [at od] where it occurs in the vibration of the second atom. Pror, PEARSON, ON A CERTAIN ATOMIC HYPOTHESIS. 119 We find C,= ee Xa) {F2— Ya (ot We have then to sum for all values of »,, and we can at once write down the general expression for the pulsation in either atom. The result may be at once extended to any number of atoms. It will be noted that it does not differ in character from the result in the case of only one free pulsation. About every free pulsation we may draw conclusions very similar to those we have formed concerning the single free pulsation. In the case of an elastic sphere both 7, and A, are transcendental functions of Na, €, , where ¢, is a certain function of the elasticity of the sphere. Hence there is an infinite number of roots corresponding to an infinite number of free pulsations. If we suppose the free atoms to pulsate with only a single period or a definite number of free periods, we cannot suppose their nature to be akin to that of a solid elastic sphere. Indeed such a supposition does not carry us further towards a final solution of the problem; we are compelled to suppose an elastic sphere composed of atoms, and we only get one step further back in what would then bear the aspect of an infinite chain of atoms of different orders. We naturally ask the question, whether it is possible to conceive a spherical portion of the ether in any way differentiated from the rest? The only method hitherto suggested by which atoms might be considered as differentiated portions of the ether is, so far as I am aware, the vortex ring of Sir William Thomson. It is not easy again to see how our spherical atom could be considered as built up of vortex rings or to be differentiated off from the rest of the ether by any kind of vortex motion. On the other hand, if it be not a difference of motion which distinguishes an atom from the rest of the ether, we are compelled to suppose two primary substances, ether-substance and atom-substance, or to start with a dualistic physical conception. It does not simplify matters to suppose that the one ether substance can have two different states, or that atoms are akin to steam bubbles in water, for the very notion of change of state seems to point to an atomic construction, and we should be explaining our atoms by means of an ether which would in itself be atomie. March 11, 1883, NOTE. This paper was written before Mr Leahy’s interesting memoir on the pulsation of spheres in an elastic medium had been published, and was printed before a copy of that paper reached my hands. It appears to me that to treat the ether as a ‘perfect fluid, however far such may be from the actual state of affairs, leads to not uninteresting or unsuggestive results in relation to chemical and molecular forces, There are of course Vor Ve PAR Uy 16 120 Pror. PEARSON, ON A CERTAIN ATOMIC HYPOTHESIS. very considerable difficulties in imagining the propagation of light under such circumstances. If the ‘jelly’ theory of the ether be accepted with the additional assumption that it acts as a perfect fluid for displacement velocities incomparably less than those of light vibrations, it would not seem satisfactory to obtain atomic and molecular forces from pulsations whose periods we have identified with those of light waves. On the other hand it may be noted that Sir William Thomson (Lectures on Molecular Dynamics, pp. 277, 278) finds it probable that the vibrations of the molecules which produce light have a velocity commensurable with that of those translations which we consider in the kinetic theory of gases. Hence it would seem necessary for the same magnitude of velocity to treat the ether both as a perfect fluid and as an elastic solid. That the ether itself must by its constitution explain gravitation is a proposition not to be lightly rejected, nor is it satisfactory to suppose two media, the one carrying light vibrations being an elastic solid, and the other explaining gravitation and chemical forees being a perfect fluid. It is difficult to understand how these media could be superposed. If we, however, assume space capable of a varying curvature and explain rays of light by waves of space-distortion, we are able to fill space with a perfect fluid such as we require. An atom would then have to be treated as an element of given curvature capable of transferring its position in space, but of such a curvature that it could not be penetrated by the fluid medium. The vibrational motion of such an element would appear as normal pulsations at the atomic surface in the fluid medium and as waves of space-distortion (light-waves) in space—both being vibratory motions of the same period. April, 1885. VI. Some Applications of Generalized Space-Coordinates to Differential Analysis :—Potentials and Isotropic Elasticity. By J. Larmor. Read April 27, 1885. ie 1. THERE are two well-known methods of dealing with physical questions of con- tinuous differential analysis. In the first and more ordinary one the differential equations satisfied by the quantities involved are investigated directly from the relations and properties of the system; and their transformation from simple rectangular to curvilinear coordinates may be effected either by direct transformation of the quantities and their differential coefficients, or by an independent investigation with the new variables. This latter process is easy where only differential coefficients of the first order are concerned, for these obey the laws of all vector quantities such as forces; but when, as is usual, differential coefficients of the second order occur it involves the use of vector differentiation or some similar process, which is of a more complicated character. In the second method, which was invented by Lagrange, and developed and applied by Gauss, Green, and others, the equations are expressed as the conditions that a certain quadratic function of the differential coefficients of the first order, integrated over the system, shall retain a stationary value when small variations are imposed on the variables. In statical questions this function is the potential energy per unit volume at the place, and its integral is the total potential energy of the system. When its value is known, the equations of the system are obtained at once by application of the Method of Variations. Now this quadratic function, being a purely scalar quantity, will usually be expressible in a form which does not closely connect it with any special system of coordinate directions; and in any case, it may be transferred without difficulty from one system of coordinates to another by means of the vector laws obeyed by the fluxions. The second method of procedure is therefore an easy and straightforward one when it is wished to express the equations in terms of special systems of coordinates. The first method involves a detailed examination of the internal properties of the system, and is therefore well suited to the clear exposition of the relations of a system whose internal structure is known, and to their expression in terms of the more simple co- ordinates. And the two are of course complementary to each other. The object here proposed is to illustrate the use of the second method by its general application to some problems of common occurrence. 16—2 122 Mr LARMOR, SOME APPLICATIONS OF GENERALIZED 2. The character of a system of space-coordinates is completely determined by the nature of the expression for the square of the distance between two neighbouring points in terms of the differentials of the coordinates of those points: this will be a quadratic function of the differentials, with coefficients which may or may not be functions of the coordinates themselves. There is another function related to this, and in a manner reciprocal to it, viz. the expression for the square of the resultant force or flux at a point corresponding to a given potential function, in terms of the rates of variation of that function with respect to the coordinates. These two expressions can be made the basis of the whole analysis, when the relations considered are of an isotropic character. 3. Suppose the position of a point to be expressed in terms of &, 7, €, which are given functions of the rectangular Cartesian coordinates «, y, z by which it may be originally specified. These coordinates will retain constant values over the surfaces &=constant, n = constant, G— constant yecerkneAeseeee nee (1), respectively. The square of the distance ds between two consecutive points & , ¢ and €+d&, »+dn, €+d6, in the space will be given by ds* = Ad& + Bdn? + Cd&? + 2Ddnd&+ 2EdEdE + 2FUEAN «0... .e cece rece ee (2), where A, B, C, D, E, F are constants or functions of & », € which are determined by the nature of the coordinate system. If the system is determined by a triple series of parallel planes, the coefficients will be constants; and if a, 8, y be the angles between the directions in which these planes intersect respectively, i.e. between the axes of coordinates, we have ds? =d& + dn? + d&° + 2dndé cosa + 2d&dE cos B + 2dEdy cos +. If the system is determined by a triple series of orthogonal surfaces, D, E, F are zero, and, in Lamé’s notation, s B= a hace ( a) + ye (5 Ay Seats dos ee (3), C= oe where h,’ = (ey + (a) 46 e) | dé? dy’ | do? ae Ee hone Sule viele Sunetele isrenies aeisan a ite ante racreg (4). so that As examples, there are the common cases of rectangular, polar, and ellipsoidal co- ordinates. SPACE-COORDINATES TO DIFFERENTIAL ANALYSIS. 123 If the system is determined by the triple series represented by (1), whose curves of intersection, drawn through the point & 7, € cut at angles a, B, y, respectively, and if h,, h,, h, are defined as above, then d& dn? qe 9 dndg 2 ds* = a ae as 2h cos 2°38 aap Tee SB +255, 0s Byes eit (5). 4. The properties of the coordinate system & 7, € will in every case be completely specified if the coefficients of the expression in (2) are known. Comparing it with (5), we find that the element of volume is A F EL Vile ABI “eB. decyl het. rs hye 1 =i = --i-||~¢@ {D+ gq (FL. Na, (0 7 Gene * hhh, 8Vd vol....(27), on integration by parts as before. SPACE-COORDINATES TO DIFFERENTIAL ANALYSIS. Comparing this result with (23), with which it must be identical, we see that 2V or Calas av av ¥ da dy? ~ dz is equivalent to @-! hhh E (o74{...3) + FOF) +e @FL...)] A Pate 12.8 where {...} denotes the respective expressions within the {} in (26). The equation y*V =—47p is thus transformed to the new coordinates; and the result agrees and may be compared with that obtained by the method of flux in Thomson and Tait’s Natural Philosophy, 2 Edn., Appendix A,. If the coordinates form a rectangular system, we obtain Lamé’s equation diay. do —ln GV dehy dN. hhh, EG 7) tm ate 4) alah 7) |= 47 denceeectes (29). If in the general case (28) & », € are potential functions, i.e. are such that 9. they satisfy Wises) WSUS. WAG = O.pcogacodoassacosespodonanrenaedecsce (30) throughout the space considered, then £, 7, or € substituted for V in (28) makes the expression vanish; and by virtue of this simplification the expression for y*V reduces to Mee (h, .. fh, .,,€V he, av = A a OeIEe Gree en ae Pv 2 nV? gia ae Pony os a) rae a ne mE) apap 2 av + iE (cos a cos B — cos ¥) aa : : 2 6 VV 4 CV : EV ie. to @* [ie sin’a. dé +h,’ sin’B. ay +h, sin*y. de av av + 2h,h, (cos B cos y — cos *) ae 2h,h, (cos y cos « — cos f) dedé + 2h,h, (cos a cos 8 — cos y) ai wesacies tas (iL)) When the coordinates form a rectangular system, this reduces again to Lamé's result dV AV AV i 2 2 2 vVeh, age ths dt ts Ta The more general result (31) may be expressed by saying that if, with the conditions of this section, rag (4V, av avy q d hk sale (ae esse eo a D>). then c V=f (ze: = $)¥ | 17 Vou. XIV. Parr II. [28 Mr LARMOR, SOME APPLICATIONS OF GENERALIZED It may be observed that it is always possible to choose a coordinate, say &, so that it shall preserve a constant value over any bounding surface, and shall satisfy the equation vE=0; for it has only to be taken proportional to the potential of a free electric dis- tribution on the surface, supposed conducting, in free space,—or it may be in the presence of other charged bodies, but in that case there will be discontinuity at the places occupied by those charges, if they are in the part of the field considered, 10. If we employ the other form (2) for R’, and denote the determinant AFE we have at once i 1 2 7 v=z_|R Sdacl =o f[[e4 | oop (Ge) + a +2(EF— AD) 5+ wet =| GEAHAE. so. ccsnncceuss (35), and it follows as before that y’V is equal to dV dV dV AFE AE x4 alee oes y2g3/dVdVdV) dai) F BD dé 1 dey 1D) dn dé dn d&| dé dV dV dV ED¢@ EP oD =O dé dn dt and corresponding simplifications may occur. | Ben nOCae (36) ; Il. 11. We now proceed to apply a similar analysis to the dynamical theory of an isotropic elastic solid. We have first to determine the quadratic expression for the energy of deformation per unit volume. Following Kirchhoff*, let us take an element of the solid, and let ft, g, h denote its three principal elongations, and F, G, H the three tensions which act on it per unit area in the directions of those elongations; we may conveniently take the element to be a right solid with its edges along these directions. In any case F, G, H represent the complete system of forces to which the element is subjected from the action of the contiguous parts, for with this specification there are no shears. Assuming, as usual, the truth of Hooke’s law for the displacements considered, we have, from the isotropic character of the solid, equations of the form F =af+ bg+ bh G = bf ag + Dh} sstecderccssscenesvsnsercaaesensroe (37), HH =bf+bg+ah * Crelle’s Journal, Ba, 40; Gesammelte Abhandlungen, p. 247. SPACE-COORDINATES TO DIFFERENTIAL ANALYSIS. 129 where a, b are two constants which specify the elastic qualities of the body, viz. a+2b is the modulus of compression, 4(a—b) is the modulus of shear, 20? | | Bo Monae he Gee eSB): a+b | a— is Young’s modulus. The potential energy of deformation per unit volume at the point considered is Bh (Eife=te Gig CP EW) eae cals acl aranooeeeate ace naisiaeee ewes (39), and is therefore ta(frtg +h’) +b(fgotgh+hf), or, say V=ta(ftg th) + O—a@) (fo FGHANP) .....cececcccceceecscees (40). The equations of equilibrium will be obtained as the conditions that the variation of the total potential energy | Vdvol.+the part due to external forces shall be zero. 12. Suppose now the position in space of a point to be given by the generalized coordinates $, y, W. Consider in the undisturbed solid a point ¢, x, W, and a consecutive point +8 yxtn wt+s; when the solid is deformed these points will assume new positions du du du , | Pt UE ae ba ae | ane ) do dy” dyy | l t | K+v+ and xtv+7+ ata? 4gye aaa ee (41). w uid: dw +wtb+ 7 E+ zs pews Cee EtG at ay & | The square of the distance between these 49 will therefore, by (2), be given by =4 (f+ +5 du eats a +B (n+5 sent dv gitaytt dv r) ag* dp dy" dy eee ee, + ape ger aS) (b+ Ge e+ Goat See) 9 (CBs eaten) (Ente Ee dina 8) ae Titan ay S) (7 +gitg,t + iy) oe (42), where A’, B’, C’, D’, EH’, F’ are the values of A, B, C, D, EH, F at the point o+u, x+y, Ww, 17—2 130 Mr LARMOR, SOME APPLICATIONS OF GENERALIZED consecutive to ¢, x, W, and therefore dr By | df GMP Be acts al Saltire Coc caaeeere (43), =A+0A, say, with similar expressions for B’,... F”. Therefore, neglecting squares of small quantities a a" ‘ du dw dv w= 4s+2(49 Ba +304) "+2(...) *+2(...) 2 at aes? ae E v) g dv dw dw dv du du 2 +2 [Boyt get +D(G tae )+z8 Tet Pay teh ng 2}..f 642}. b en ee (44), where YH, is the value of Y before deformation, Since LY and ¥, differ by a quantity very small compared with either of them (i.e. of the second order), we have Fe ae) ee oevaestianmmaeinccee humane oon See ticueiaasdiane (45), where e¢ is the elongation, per unit length, of the solid, in the direction of Y@, at the point ¢, x, wv. We may therefore write (44) in the form Dore = AP + Wn? + OS? + Wl + QHCE + DHE... ceeeceeee ees (46). 13. The values of f, g, h are the maximum and minimum and the stationary maximum-minimum value of this quantity e; the value of &, being given by (2), or, what is the same, (5), viz. I! = AE + By + Cet + 2D + QECE + QF En. These values may be determined in the ordinary manner. Substituting from (2) in (46), we have (A@ — Ae) & + (45 — Be) n° + (O— Ce) 2° + 2 (D — De) nF +2 (L—- Ee) +2 (ff — Fe) En =0......0.044. (47). Differentiating with respect to & 7, ¢, and remembering that for the values in question the differentials of € are zero, we obtain (A—Ae) E+ (ff-—Fe) n+ (EH —Le) F=0 (ff —Fre) & + (AB —Be) n + (D—De) C= fF vvervvevereneereeeeeennes (48), (£ —Le) £+ (M- De) n+ (C —Ce) F=0 which lead to the eliminant A-Ae ff-Fe E- Le HF —Fe W—Be D—De |= wircsscsscssesssscnscceeees (49). G-Le D-De C- Ce SPACE-COORDINATES TO DIFFERENTIAL ANALYSIS. 131 The values of f, g, h are therefore the roots of this cubic equation in ¢ Therefore (ftgth)|AF E\=|AF G\+|4 FLE|I+|ALFE F BD FBD FSD SS BD EDC| |£EDE EDC ZDC = 4 (BC— D*) + 36 (CA — £”) + © (AB-— F’) + 239 (HF — AD) + 2% (FD — BE) + 24f (DE — CF)...(50) —(fgtght+hf)|AFEF\=|A $e L4+4|A Ff Bl+|4A Ff B Ji 18 1D) F BD fF BD FLBD EDC| |£DC £DE ED E = (BC — D’) A+ (CA - LZ’) B+ (AB — FH) C +2(E ff -—AD)D + 2( FD -BWE) E+ 2(DME-CF) F...(51), and the value of V as given by equation (40) may be written down. 14, Having thus determined the value of V, the equations of internal and surface equilibrium may be deduced by the Method of Variations in the ordinary manner. If &, X, VW denote the components (oblique) of the surface force, per unit area, which acts on the boundary of the solid, taken in the directions of the lines of inter- section of the pairs of surfaces yW, wd, and ¢y respectively, and so as when positive to tend to increase the values of ¢, y, Y; and if W denote the potential energy per unit volume at any point in the solid, due to the action from a distance of external systems, —then the energy condition of equilibrium is that sUs 3 [Va Toner 5[ wa roe fo’ Su + XB! dy + WC? Sw) d surface = 0 ......(52) for all possible consistent variations of w, v, w. Therefore, with the notation of § 10, we have [[]>" 8(V + W) dédydy - {fo Su + XB oy + WC! 8w) dS =O oeceeeeee-n-. (53). Now V is a function (quadratic) of the differential coefficients of u, v, w with respect to ¢, x andy. If A, B,C, D, EL, F are constants, which corresponds to the case of oblique Cartesian coordinates, V is a function of those differential coefficients alone, with constant coefficients. If A, B, ... F are not constants, the terms 0A, DB, ... YF will introduce w, v, w themselves; so that V is now a quadratic function of u, v, w and their differential coefficients with respect to $, x, w. Thus we have dV ddu , dV dbu dV dsV OV = ti dp to dudy t+ do apt dp dx db dV dV dV u +7, + va éu+ AR ow miatuielaield/aieie(éveteteraialataiataisinvathial aia (54), 132 Mr LARMOR, SOME APPLICATIONS OF GENERALIZED And on substituting this value in (53), and integrating by parts in the usual manner the terms which contain differential coefficients of du, dv, dw, we have, writing down ex- plicitly only the terms depending on du, [+ oS — 5u. dydyy + [fra Ee asa + [Pa oe Bes dae [fost Sud8 a dp aay iz Cad of gid. d [x3 dV d fos GV\_ g3dV_ yu dW) ={{I de ar 7 qu Say > au Pay 2 d = | me du oy db J bu. dp dy dip “as “dy dy SE Rreyaenee Goveeubatbayes CP ehatel COUN = (0) aSagonssodanocconugodaa one a0da900 (55). Now as the forms of du, 6v, dw are quite at our disposal, subject to conditions of continuity, we can choose them so as to make either the volume integral or the surface integral in this equation zero at pleasure; and we may also have any two of these variations of w, v, w zero at pleasure. This equation therefore cannot be satisfied unless the quantity under each separate type of volume or surface integral is zero; and we thus obtain the internal and surface equations of equilibrium, 15. To express the latter in their simplest form, it will be convenient to take the solid considered to be the element of volume dddydyw of a larger body. The surface element dS is now an element of area of a face of this solid, and is, by § 4, (Ga) Pepe (j ; oa) ids ter Car ode according to its position. The force ® of (52) is the force exerted across this element of area by the matter on the other side of it, per unit area, in a direction such as to increase @ without altering y or , ie. in a direction parallel to the intersection of y, w. If we adopt the notation that 7, represent the component force per unit area acting from outside on that face of the element which lies on the surface @ in a direction such as to increase yy without altering the values of ¢, x, ic. in a direction parallel to the inter- section of ¢, x, we have therefore from the surface integrals the relations -1 dS 1 dV 7 Ley = (ey Setar d a6 iy. iv | rn eerie. or (56), q = ct 9) (peas . ees Be ES) . du as dx J with similar expressions for the other surface-stresses, SPACE-COORDINATES TO DIFFERENTIAL ANALYSIS. 135 From the volume integral in (55) we may now obtain the equations of internal equilibrium. We may shorten their expression by making use of the values of the stresses just obtained in (56). In the first place, we remark that, by (43) and (44), dV_1dAdV 1dBdV dF dV du 2d dvd 2 dd dB a de ne a eee (57). We have then d ( ,3/d3\3 d dS\3 3 (dS 3dV ydW _,) a =) ys} + 1a! = Th} + tat (3) Tas} aria eile 4 with the similar equations | + --(58), Degas | {ps (aS Vay 2dWw ot ae (a =) Toe} + +718 (az) oh a ae (Fa) Dy { — Saas dx faa d 1 (dS d dS Oh ap so) ar 1adWw | a aa) Toy} + ale aa 20s ae (ca) Po} - 9 Ge ASSL 7 the values of a 5 — ay being most easily determined by means of (57). 16. In the case in which the coordinate surfaces $, y, % form an orthogonal system, the formulae become more simple. We have now D=0, E=0, F=0, 1 and, if we wish to revert to Lamé’s notation, we have | Resear (59). gl al Lh, | ses SE BS ye OF es J By (44), ao. Uie=(A Gat ida) + BE +40B) a + (CF. + 400) & du gw oes 4 ue du dy ) ; = = SF || Esl condencuce 0), +(B yt Og,) + (Capt 4 gy) E44 Gt Bag) en (60) where DL? = AL + By’ + CE. The cubic equation whose roots are f, g, h is now Q— Ae SS é i | Ff an Be Df HO eee tteeieee (61), | & D @— Cc | and therefore frgeha eee ats SE ae ce Pee ae, Pier em ee ron Ce (62), : — mz ro _ IF? 2 ; fot gh+hf= a =F aS = pou ad sjssstellaeinids Yate ccdacee (63). 134 Mr LARMOR, SOME APPLICATIONS OF GENERALIZED We have now, by (40), a 2B " he Wee ;2a-2 +o) Pecans (64). Therefore, by (60), du dv . dw 42 yA 10B 13 Vat0(tat apts ata Btso ay awd? data -1 dv dw , dw du Pa o) laa dy dpdy dy dd du /§¥B UC\ 1dv/’C YA\ 1dw/tdA IB +h o3(3 aaa ae gag ba ) wre wa vA dB +4 (— f= Waa 5) ee ae ee ei eee wherein for brevity, as before, a a E dw dw du ht dv meats We have thus obtained the expression in general orthogonal coordinates, for the potential energy of deformation of a strained isotropic solid. 17. By (56), we find ss = ll ia as =a (S45 a) +8 (q+ +55 +0 (+ ey : dV GSTS Son RE NOLODSOTOnNC (66) ; du dv = = 4 Bh 4 4-3 3 (a v) {abe + + BiA- a = Ty, , by ee with similar expressions for the other stresses. SPACE-COORDINATES TO DIFFERENTIAL ANALYSIS. 135 To find the equations of internal equilibrium, we first observe that the values of dV dV dV du’ dv’ dw 0A, DB, dC enter into (60) that dV i da dv tdB dV 1d aV du Ado 7du + Bdé dv * Odd ~dw are obtained in a simple form by (57). It is clear from the way in which dp dx ay 1 dA 1dB 1 dC A dg + Bag bx * Cag T yy Majalavelela slalelolorstaterctatelalclaiststcietctareniere sine (67), with similar expressions for dV pa, a dv dw * Therefore we have the equations for this case d d 3 d dg {(BCA)?* 744} + dyl(CAAY Teel + ap {(ABA)? Tyg} | BC\idA CA\3 dB dC py oH =1|(2E) ag Te (Gr) ap Be (SG) Gp To] ABO =o | d d 3 ig (BOB) Tx} + dy (CAB) T xl + gy (ABB) Ty} ; ag + ...(68), 3dA dB AB\i d i | 3 ) dy toa) aye T+ (G) ey To] BON | d d 3 dg (BCC) Toy} +7 (CAC) Poa} + +a {(ABC)3 Tyy} BC\itdA CA\3 dB dC eae hae -3|() ee on lap le (4 =) iy 1 | ~(ABc)4 ie =U in which the values of the stresses may be substituted from (66). 18. The equations now determined are those that apply to the most general cases of curvilinear coordinates. In any special system, the analysis might be very much cut down by dealing at once with the special values of A, B, C... which will often be either constants of functions of only one variable. Without here going back through the analysis, but merely by taking advantage of the general results already obtained, we may now apply the theory to some special cases. Ist. The case of oblique Cartesian coordinates possesses some interest. The coefficients A, B, C... are now constants, so that NA, YB,... are all zero, and the analysis of §§ 12—15 applies with this important simplification. The potential energy V of the strain is a quadric function of the differential coefficients of u, v, w, with coefficients which are constants: it is given by (50) and (51). The stresses are given in terms of its dif- ferential coefficients by (56). The equations of internal equilibrium are given in terms Vout. XIV. Parr II. 18 136 Mr LARMOR, SOME APPLICATIONS OF GENERALIZED of the stresses and the applied external forces only by equations (58); for a — dV aire Jp ate now each zero, and the coefficients are all constants and can therefore be taken aw outside the sign of differentiation. 19. 2nd. As examples of orthogonal systems we may take the well-known cases ef columnar and polar coordinates. (z) Columnar coordinates 7, 6, z. Here ds =dr+rd& + dz, and therefore A=1, Ber, C= and pA =0, DB = 2ru, nC=0 and the value of V is at once written down from (65). It is however to be noticed that u, v, w are the increments of r, 0, z respectively, and that v is not here the increment of rd@ as is sometimes the case. The values of the internal stresses may be written down by (66), viz.: T.. ma 4b (S 4 “0S, ie »(* SE i Tao = 69 4.a( 4%) 40S, Te: = 3 (a—2)(r ae (70). een ee The equations of internal equilibrium are by (68) d d d aw dy Wl) oa (Ts) Ur fe (rT,2) — To ar Sis 0 d is dw = (r°Dy¢) + (res) +7 f (Ta) aie: ah DN Die (71). d dW: oo ee =f 5 iad | (8) Polar coordinates r, 0, o. Here ds* = dr’ + r°d@ + 7° sin*6. do”, ws. (72), and therefore y « Deas be B=r", C=7'sin’6, dA =0, DB=2ru, DC = 2r sin’6u + 2r’sin 6 cosy wherein u, v, w are the displacements of r, 0, w respectively. By (65) the value of V may be at once written down. SPACE-COORDINATES TO DIFFERENTIAL ANALYSIS. 137 The values of the internal stresses are, by (66), ee a 4b (+ “\40(fe+% +tan 80), Tro= 3 (a —b) (2954 2) ] Tp .. (‘or:) ae (“e+" + tan 00), Ta, =3 (a (a : ap int sine) | (73). Ta, =b 46H 42) a(S +— es =4 (a-B)(rsino 4S) | The equations of internal oe are, by (68), L@ sin 6 T,,,.) a a es sin OT;) + @ Ta) —rsin 07% —rsinOT,,, —7° sin 0 a 0 | © (sin 82o,) + | c Los i i (7°To,) — 7? cos OT4~ r*sin dN = 0 (74). = (r* sin’ OT.) + do ae sin?9 7.5) + = (r* sin OT...) — 7? sin 0 = = | 20. The equations thus obtained for these cases, (a) and (8), may be simplified considerably by performing the differentiations. They are then the same as were obtained orginally by Lamé by transformation from Cartesian coordinates, and applied by him to the consideration of the elastic yielding of a sphere under given forces and other similar problems. They may also be obtained from first principles by a process involving vector differentiation (see Webb, Messenger of Mathematics, 1882, x1. pp. 146—155) which might be extended to the general theory of orthogonal coordinates, at the cost however of increasing the complication and the number of terms. The general equations (66), (68) also lead to the equations which apply in the cases of ellipsoidal coordinates and elliptic columnar coordinates and such other orthogonal systems as have been used. These are the systems of equations which would naturally be applied to the case of elastic solids bounded by confocal quadric surfaces or by elliptic cylindrical surfaces respectively: but their degree of complication is such as to render them of not much practical value except in the more simple forms of strain, in which, for example, one or two of the three displacements are zero or very small. In these latter cases the general analysis is clearly much shortened by introducing the simplification at the beginning of the work. oes bo VII. Observations and Statistics. An Essay on the Theory of Errors of Obser- vation and the First Principles of Statistics. Communicated by J. W. L. Guaisner, M.A. [Read May 25, 1885.] CONTENTS. PAGES Divisions of the subject (exhibited by the accom- panying tree) . = : . A 139 Distinction between real and fictitious Megan (Cp. pp. 161, 167) . a 139 Exponential law of Error not Peis to tity curves extending to infinity . . 140 Proof of law of Error for Finite facility-curves . 141 Simple Induction and Inverse Probability . - 142 Reproductive curves . 142 Knowledge of Modulus anil camara by some writers (e.g. in case of ratio of male: female births)... 5 ee 143 Real Means sought by Simple Tadaesion a 144 Remarks on Mr Galton’s method of Leite tine the number of elements ina binomial . = 145 Real Means sought by Inverse Probability . - 145 Probability-curves (treated inversely) . “ “ 146 A priori Probabilities (cp. pp. 149, 157). 5 ; 147 Most probable value of Modulus (inferred from observations) not Se*+(n-1). - 2 148 Corrected value of probable error in same case. 149 New formula for reduction of observations not known to have same Modulus = ; : 149 General case of finite facility-curves (inversely treated) . : = - : - Inyersion Proper . ° : : Criticism of the Method of ie Squares as distinguished from Inversion Proper. Cp. ps l6te. - . Facility-curve of average more 7 divergent ites that of primary . : . The mean of two jesse not Sess: the most probable value : 5 ‘a A Statistics proper. Subjective Means . o Rationale of Laplace’s Method of Situation, a of his assumption that the mean error is the measure of disadvantage 2 4 c The “ most advantageous ”” ietineeaatiel from the “most probable” value . 5 A Laplace’s “ @ posteriori” method of reaver “ ob- servations déja faites” . . . “Most advantageous” value of mn Ti ccitereed from observations . A . 5 Difficulty of distinguishing aaieeare from reat Means The three principal Menu Sara By F. Y. Epneeworrs, M.A. PAGES Mr EDGEWORTH, ON THE THEORY OF ERRORS OF OBSERVATION 139 Observations and Statistics. a | a Observations Statistics (Real Mean) —_ (Fictitious Mean) b Single Mean Plural Mean c e Finite Facility-curves Infinite Facility-curves + Probability-curves _ — Probability-curves d | Data very numerous Data not very numerous (Simple Induction) (Inverse Probability) e é Reproductive curves Irreproductive ie wid lind |, | | Weight given Weight not given g | g | | Identity of weight Identity not given _ given h h Symmetrical Asymmetrical curves E.g. abedefgh denotes the following problem: Given a not indefinitely numerous set of observations grouped about a single real Mean according to probability-curves, of which the weight is unknown, and which are not known to have the same weight, to find the best value of the real thing by way of inverse probability. OBSERVATIONS AND STATISTICS. THE object of this paper is to distinguish and examine the different cases which are presented by the problem*: What is the best Mean? (a) The first distinction which offers itself is between observations and statistics. According to the definition here proposed, observations and statistics agree in being quantities grouped about a Mean; they differ, in that the Mean of observations is real, of statistics is fictitious. The mean of observations is a cause, as it were the source from which diverging errors emanate. The mean of statistics is a description, a representative quantity put for a whole group, the best representative of the group, that quantity which, if we must in practice put one quantity for many, minimizes the error unavoidably attending such practice. Thus measurements by the reduction of which we ascertain a real time, number, distance, are observations. Returns of prices, exports and imports, legitimate and illegitimate marriages or births and so forth, the averages of which constitute the premises of practical *This problem inyolyes the perhaps more important one: What is the worth (in particular, the precision) of a proposed Mean? 140 Mr EDGEWORTH, ON THE THEORY OF ERRORS OF OBSERVATION reasoning, are statistics. In short observations are different copies of one original; statistics are different originals affording one “generic portrait.” Different measurements of the same man are observations; but measurements of different men, grouped round homme moyen, are prima facie at least statistics. This delicate distinction does not generally correspond to any strongly marked difference, either in the* external agencies which cause the phenomenon of quantities grouped about a Mean, or in our formula for determining the Mean. Accordingly the definition here suggested is not expressed in ordinary discourse. The popular or the technical meaning of the terms will be employed here according to the context. (b) A second division is between cases where the given observations or statistics are grouped about a single mean, and where there are several such quaesita. (c) A third division is between facility-curves which are either finite or probability- curves, and those which extend to infinity. The importance of this distinction lies in the proposition here submitted that, where facility-curves extending to infinity occur, there the exponential law of error does not hold. That is to say, if there be n facility-curves, Y=f@, y=h@, &e., and # be putt for 1,2, + 2, + &e., where z,, z,... are values taken at random from the respective series, then the values of , = 1 2 E do not in general range under the probability-curve (= e a) when the curves f Nore extend to infinity. Let us take the case most favourable to uniformity, where the facility-curves are symmetrical and identical [say y=f(z)]. By the usual reasoning it is shown that the sought expression, the ordinate proportionate to the number of times that # commits an error of the extent z, is equal to 1 a) al n : ={, I. fi f (2) cosaede} cosaa da. 2 This is inferred to be equal to ——e =) where < is the mean square of error for the 7 2 facility-curve f(z). The steps by which this imferencet} is usually reached are two. (1) The expression within the brackets being expanded in terms of a it is assumed that the terms above the second may for purposes of integration be neglected; when small, because they are small, and, when large, because the whole expression is then small. (2) The thus truncated series is for purposes of integration approximately equated to e 4", Both these steps seem precarious. They have been suspected by§ eminent autho- rities. They are proved to be erroneous by eensprcioNs examples. The family of facility- curves which is of the type {@)= | e—* cosaada, wTJo p. 556, § Glaisher, Monthly Notices of Royal Astronomical So- ciety, 1873, Ellis, Camb, Math, Journ., tv. p. 182. * See p. 166. + Cp. Todhi nter’s History of Probabilities, Art. 1002, + See Todhunter’s History of Probabilities, Art. 1002, AND THE FIRST PRINCIPLES OF STATISTICS. 141 where ¢ is any positive quantity, does not produce the probability-curve as that under which the values of # range, except in the particular case when ¢=2. Mr Ellis indeed in his proof of the law of error does not take those dangerous steps*. Yet upon his own showing his course of reasoning is theoretically insecure and practically misleading. The example which he gives, the case where f(z) is 4 e-* between infinite limits, seems even more damaging to his theory than he admits. For the final form in that case is = il () cos az which is known to be of the form e” (a+ bx+ ca? + &e). 0 The following reasoning shows that the received conclusion is admissible when the facility- curves f(z) do not extend beyond finite limits. Supposing first that the elemental facility- functions are all identical and symmetrical and that # is the sum of s errors taken at random; the sought expression, the ordinate of the curve under: which the values of # range, Say UWzs, Where z is the extent of error, is the =th term of the multiple Lf (2) fad Gok Sf (2) 7+ f (w) t7 +f (0) P+ f(a) +f (20) 1? + &e.+f (z) tre]. Observing the formation of the coefficient of ¢# in the (s+1)™ power of the expression +b within the brackets, we have (1) wsii4,2= | FT (2) Ue+z,3d2 [where +b are the limits of the -5 elemental facility-curves y=f(z)]; that is, supposing only even powers of z are involved; for otherwise the above integral will have to be separated into three parts. Assuming such a tendency towards a limiting form that the effect of proceeding from the s™ to the (s+1)™ power is indefinitely small (for that part of the result, those values of « with which we are concerned) we may write the left-hand member of (1) wt The right-hand member may be expanded into ae ot: &e., F (2) dz Urs + OZ is ke and the terms above the second neglected. For since, by hypothesis b is finite, if the linear unit be taken properly, the mean powers of error for the facility-eurve y=f (2) above the mean square may be neglected. An appropriate unit would be sb. Accordingly an approximate solution of (1) is afforded by a solution of the partial differential equation du_ edu du (2) a= 9 SE (where $ == zie 2'f (z) dz), if 3 dat &c. are small. (x-aP The complete solution of this equation is Ae s . The symmetry about the origin shows that a=0. And the condition of a facility-curve, that its Sapte should be equal to unity, gives A= aes The sought final form is therefore a: #2 .t This reasoning is TSC vase easily adapted to the cases where H is the weighted sum (as it may be called) or * Camb. Phil. Trans., Vol, yu. p. 210. + Ibid. p. 215. + «* not exceeding sc?. 142 Mr EDGEWORTH, ON THE THEORY OF ERRORS OF OBSERVATION weighted mean of s observations; and where the elemental curves are not* identical and not symmetrical. (2) The next distinction is between cases where the number of observations or statistical returns is practically infinite, large enough to completely exhibit the law of facility according to which they are reproduced, and cases where the number is not large enough to form the basis of a valid induction, where the data can only be regarded as samples taken at random from the facility-curves under which they range. The first method is followed by Quetelet when he determines the character of the curve under which the data range by taking as the probable error of that curve the point which just divides into two halves the number of observations or statistics which le on one side of the mean, by Mr Galton in his method of quartiles and deciles, and very generally in all graphical methods of statistics. The second method is pursued by mathematicians in the discovery of the mean, when the facility-curves under which the observations range are known to be probability-curves, and would doubtless be more extensively employed but for the mathematical difficulties incidental to “Vignorance entitre ow l'on est de la loi des erreurs de chaque observation+.” The first procedure is pursued by Airy in his Zheory of Errors when [§ 33] he says, “If we investigate the value of the modulus, first by means of the Mean Error, secondly by the Error of the Mean Square, we shall probably obtain discordant results. We cannot assert @ priori which of these is the better.” The second procedure is pursued by Mr Merriman when regarding} the given observations as having resulted from a source of error consisting of a probability-curve with a certain unknown modulus [Method of Least Squares, § 24] he determines by inverse probability what is the point and what the modulus from which the errors are most likely to have resulted. In short the first procedure is of the nature - of ordinary induction. The second procedure belongs to that more methodical investiga- tion of causes by way of Inverse Probability which Laplace in his introductory essay on Probabilities describes as “le principe fondamental de cette branche de lAnalyse des hasards qui consiste & remonter des événemens aux causes.” (e) The next division is between reproductive and other facility-curves; reproductive facility-curves being defined as such that if a couple or any finite number of values z,, x, &c. be taken at random from the series represented by such a curve, then the sum or any arithmetical mean y,«, + 7,7, + &c.+ sy ranges under a facility-curve similar in type to the primary curve. The general type of such a facility-curve appears to be Wepre 5 = e-* cos az da. T 0 (f) The next distinction is between cases where the weight or measure of divergence for the facility-curves involved by the problem is given, and where it has be to inferred from the data. Laplace and those who have followed him, §Herschell and ||De Morgan, seem provisionally at least to have assumed that, when we are given the number of times + Laplace, Théorie, 3rd ed., p. 337. || Lncyc. Metrop. + 7 In this case c? must be treated as a function of s. § Essay on Quetelet’s Probabilities. On Least Squares, p. 143. AND THE FIRST PRINCIPLES OF STATISTICS. 143 that an event has occurred in each of two alternative ways (e.g. births male and female), then we are given the modulus according to which statistics may be expected to diverge from the given ratio, the weight as it were of the given observation. It is assumed that the case may be compared to that of black and white balls, drawn at random from an urn containing as many balls as there are births. But this is to assume that all the births are independent. The more appropriate conception is surely that we are given indeed the ratio between the mass of ivory and of ebony in the urn, but that the number of balls, of independent causes, is not given, but to be inferred from the data. In short the case is similar to the case where we are given a set of observations z,, z,, &c, and have to infer from these data the modulus according to which they diverge*. The statistics of death rates considered in the light of this analogy show a divergence many times greater than that assigned by the current theory. (g) The next division is between cases where the weight of the facility-curves is, or is not, given as identical. Mr Glaisher+ has pointed out the confusion which results from ignoring this distinction. (h) The next and last distinction is between symmetrical and asymmetrical facility- curves. These divisions are exhibited in the accompanying tree, the branches of which have been completely developed only on one side. The work of examining the 256 species infime which are constituted by these eight principles of division will be greatly abbre- viated by the facts that some branches stop short, some, though existent, are barren, and some, though real and important, are so exactly parallel to others that they do not require a separate description. In examining the different species the following order will be ob- served, The simpler cases will have precedence of those which are more complex and difficult. Accordingly the first member of each division will be considered before the second. The first species that will be considered is the simplest; namely that which is constituted by taking the first member of each division. It may be denoted by the symbol abcdefgh. Next will be (or would be, if it existed) considered the case abcdefgh, where f denotes the presence of the second member of the eighth division. The ramifi- cations of the class abcdefy being thus exhausted we shall (or would if possible) consider the class abcdefy subdivided into two species corresponding to the positive and negative values of h. When abcdef is exhausted we proceed to abcdef, and so on. abedefgh. This is the case of a complete set of observations or statistics grouped about a single mean under a probability-curve whose modulus is known beforehand. One instance would be presented by throwing up m coins n times (as Prof. Jevons did) and recording each time the preponderance of heads and tails, with a view of discovering whether the coims are perfectly symmetrical or ‘loaded’ in favour of either side. Here we may assume the modulus to be given beforehand, namely rls = though strictly the modulus * When writing this passage I had not seen Prof. Lexis’ Massenerscheinungen and other original writings; to which I have endeavoured to do justice in a paper ‘‘ On the Methods | of Statistics,” published by the Statistical Society, 1885 Vou. XIV. Parr II. 19 (see pp. 11, 18, 32). + Memoirs of the Astronomical Society, Vol. xu. pp. 102, 103. 144 Mr EDGEWORTH, ON THE THEORY OF ERRORS OF OBSERVATION is here dependent upon the sought mean. In fact the case is not usefully to be sepa- rated from abcdefgh. Of both it may be remarked that in determining the centre of the curve no one odd symmetrical function of the data has (apart from considerations of convenience) an advantage over another. To enquire which function would be oftenest right in the long run would be to introduce an element of Inverse Probability which is not appropriate to the present class (having the attribute d) abcdefyh and the other species of abedef are either non-existent or unimportant. abedef. This case may be thus distinguished from abcdef. In the latter case we are given the law of divergence by experience prior to that which constitutes our present data; necessarily then,—since our present data are supposed extensive,—experience of a still higher degree of certainty, such as that which is embodied in the rules concerning games of chance. In the latter case we must determine the modulus from the obser- vations. The former case is that which, as before remarked, mathematicians have too readily supposed. The latter case is by far the most usual in all important observations and statistics. As before indicated, in the present roughly inductive procedure, there is no ground for preferring before another any one function of the observations, which if the series were really complete might be equated to the mean or to the modulus, For = e, instance, if the curve were y= e then theoretically (1) c= Se, where Se is the one Jae sum of the errors taken positively on both sides of the mean, (2) c’=2Se’, (3) c'=4Se', &. In short we assume here that the apparent errors may be considered as real. The ramifi- cations of this class present no new difficulty. abedé is subject to similar remarks in so far as we know beforehand something of the character of the irreproductive curve under which the observations range. This case brings into view a difficulty which was hardly worth noticing in the preceding cases, namely that we must suppose given not only the complete curve of observations, but also the knowledge at what part of such curves the sought real point usually occurs; in what manner observations group themselves about the real point. This might be taken for granted in the case of symmetrical curves. But in asymmetrical curves the question arises, Is the real point for example at the centre of gravity, or the centre of area, or at the longest ordinate of the given curve? Probably the most important case under this heading is that of what Mr Galton* calls binomial curves, only the term need not be confined as it is by him to symmetrical curves. abedzfgh is exemplified by Mr Galton’s binomial of seventeen (or fewer) elements with alternatives even and known @ priori as in games of chance. abcd2fgh is a similar case with unequal alternatives; for instance the (discontinuous) facility-curve or series which is presented by developing (19+,%,)" and taking 18 ordinates proportioned to the terms of developement. In this case it will be remarked that the real thing sought, the ratio from which the series results, is represented by the point at which occurs both the centre of gravity and greatest ordinate. * Phil. Mag., 1875. AND THE FIRST PRINCIPLES OF STATISTICS. 145 abcdéfgh is exemplified by a binomial curve such as Mr Galton supposes to be pre- sented by the height of a man or wall. The only difficulty here is concerning the determination of the modulus as it may be called, or, as Mr Galton puts it, the number of independent equal causes from which the curve may be considered to have resulted. If we knew all about the series we should know (1) the number of elements, (2) the extent of each and therefore (3) the total range or interval between maximum and minimum and (4) the probable error. Now we are not given (1) and (2). But we are given approximately (4) and more conjecturally (3), from which we may deduce @ (1) and, if necessary, (2). Mr Galton takes as the value of (3) the mean of the series of observations; for instance, in the case of measurements of different men, he would take the height of ’homme moyen for the total range of error ; OP in Fig. 1. And this in fact constitutes a superior limit to that range. An inferior limit, and generally a much more accurate value, would be the interval Q’ between the measurements which are widest apart, between the tallest giant and the shortest dwarf; QQ’ in Fig. 1. Thus in the case of a wall consisting of 17 courses of stones each having a probable error of 1 of an inch, Mr Galton takes for our (3) the mean height of the wall 51 inches. But a much nearer value would be the difference between the greatest and least height of the wall 0 as given by actual measurement, a difference in the case supposed not likely Fig-1. to exceed an inch. abcdéfgh. This case is exemplified by observations presumed to range under an un- symmetrical binomial curve the proportions of which are not given beforehand. The real point sought would. presumably be either at the centre of gravity or largest ordinate positions theoretically identical *. abed. At this stage, ast before explained, Inverse Probability makes its appearance. The following is the simplest case. abcdefgh. Given a set of observations 2, a, &c, and given that they have been generated by divergence from an unknown point « according to one given law of error, a probability-curve of given modulus c, to find the most probable value of «. It is found ._ GHP to be the average of «,, x, &c. by putting ape where 1 (x —ar,)?+ (w—ary)?+ &e. Fee as P= 720" This solution is sanctioned by the highest mathematical authority. There is however a philosophical difficulty which does not seem to have been noticed—with respect at least to observations in time and space; for the analagous difficulty with respect to numerical statistics has been insisted upon by +Cournot, §Ellis, and |/Mr Venn. In the preceding proof it is tacitly assumed that prior to the data one value of the quaesitum is * Dr Macalister’s determination of the parameter ap- + Above, p. 142. pertaining to the law of error which he has discussed in the + Exposition de la théorie de Chance, ch. vin. Proceedings of the Royal Society (1879) belongs to this § Camb. Phil. Trans., ch. Vu1., Remarks on jaw of category. His method is analogous to that of Airy noticed succession. under (d), page 142, || Logic of Chance, ch. Vv. 19 —2 146 Mr EDGEWORTH, ON THE THEORY OF ERRORS OF OBSERVATION as likely as another. Correctly the expression P should be affected with a factor p, a function of 2, representing the @ priori probability that any value of 2 is the true one. Now if p is not constant, as it is usually assumed to be, the maximum point is given not by the above written equation but by the following po +P P=o. To this difficulty* two kinds of answer may be made. (1) It may be pointed out that, as m increases, the influence of p (being supposed finite) becomes ever relatively less. In considerable numbers therefore the received solution is accurate. To this it may be rejoined that, as the number of observations is increased, the case becomes merged in the (preceding) case where the observations are so copious as to reveal their law of gene- ration by mere inspection or at least unmethodical induction. The more refined method only becomes accurate where it is not wanted. This rejoinder may perhaps be rebutted by the allegation that the inverse method, though not available when the number of observations are very small, nor wanted when they are very large, may yet be useful for intermediate numbers. The objection may be compared to the fallacious reasoning+ in the ordinary proof of the law of error that certain terms of a development may be neglected because, both when they are very small and when very large, they are unim- portant. (2) This view is confirmed by the second answer which may be made to the objec- tions against the use of d@ priori Probabilities. It is not true that we have no experience about these probabilities. We have a rough general experience that one value of the measureable occurs as often as another at any rate between the limits with which we are practically concerned. For instance we have some experience that one statistical ratio occurs about as often as another. For if the fact were widely different we should not have failed to notice it. This is certainly rather a rough foundation on which to build an exact science. Yet it is of a piece with the materials of not the least important science to which the calculus of Probabilities is applied, sociology. For in estimates of social utility there is continually assumed some such axiom as that things between which no difference has been noticed may be treated as equalf, To this defence it may be objected that the proposition that one event is as likely as another is self-contradictory; for that a double event is an event, and it is repugnant that both the single and plural event shall have the same probability. This objection has been made with special reference to statistical frequency—e.g. as against Donkin’s axiom that an unknown probability may be treated as }. The objection however is equally available against the assumption that values of space or time are @& priori equally distributed between certain limits, as against the assumption that statistical * See Article by the present writer on @ priori Proba- | national wealth or in mean life is advantageous; or that an bilities, Phil. Mag., Sept. 1884. equal distribution of wealth or power is prima facie a good. + See above, p. 140. See upon the principle implied some remarks by the present + For instance when it is assumed that an increase in | writer in Mind, April, 1884, and Hermathena, 1884, AND THE FIRST PRINCIPLES OF STATISTICS. 147 ratios are @ priori equally distributed between 0 and 1. The objection in all three aspects may thus be illustrated *. | Fic, 2, Fie, 3. Let OP be an abscissa=1, and let the horizontal line QR be the facility-curve repre- senting the @ priori even distribution of x between 0 and 1. To find the distribution of the values obtained by raising each « to the power n, put €=2", (=e do=* ge"), Whence the facility-curve expressing the distribution of — is 7a (where k is a con- stant) represented by the left curve. Conversely, suppose that the values of & were equally distributed, and let the line QR’ represent that equal distribution. Put &=2" and transform to the curve y=ka""; represented by the curve on the right. It seems sufficient to reply to this objection that what is here asserted is not that all values are equally distributed, but only those values in whose distribution after copious experience no unevenness has obtruded itself, Take for instance statistical probabilities about deaths from different causes, or births, male and female, of twins, triplets and four at a time. In the immense variety of the Registrar General’s returns we perceive no prevalence of one fraction, no clustering about one point. And therefore we assert that the @ priori probability of single—not plural—social events may be treated as a constant. Upon the whole the two arguments—that @ priori probabilities may sometimes be dispensed with and are generally given—taken together, and corroborated by the practical success which has attended the calculus of probabilities, appear to establish the foundations of that calculus as rough but solid. abedefgh is non-existent. abedefgh. By reasoning on a par with that which has just been employed the solution of this case+ is found to be the weighted mean of the observations, abedefgh. Given a set of observations x,, , &e., and given that they have been generated by divergence according to one and the same probability-curve from a single * See note §, p. 148. + Cp. Merriman, Part 1. 27. 148 Mr EDGEWORTH, ON THE THEORY OF ERRORS OF OBSERVATION point, but given neither that point nor the modulus, to find* primarily the former and as a secondary end the latter. By parity of reasoning the central point and modulus are given by the maximum yalue of Ppo, where P and p have the signification above assigned, and o signifies the @ priori probability that ¢ would have a particular value. - , d , Whence two equations Tgp Po = 9 and © Ppa =0. Whence, if p and o are constant, AL Y and c= J2 E —x,) ate e —2x,)* a | + n This solution is exposed to the same objections as that of the last but one case, and admits of the same defence. The function o though unknown may be neglected when the observations are numerous. And it is not entirely unknown; for we have a rough general experience that one modulus occurs as often as another. It must be admitted that the rejoinder to this second plea has in this case a peculiar force. For (considering the importance of the quantity c’) it may be plausibly maintained that we should treat as equally likely @ priori all values not of c, but of c’. And again it seems natural to treat as equally likely all values of h the reciprocal of c; as Mr Merriman} done. Considering the pro- has And there is another at least equally plausible view. bability-curve as the limiting case of what has been called a binomial curve, we have oy cx af 4, where V is the number of elements. And, if one value of JW is as likely to occur as another, then h®? not h should be what may be called the independent variable. The evidence which tells equally for all The correct position seems to be that our rough general experience assures us not that one of these views is true but that none of them differ very materially from the truth. Whichever of them we adopt, we shall obtain the same value of wz, and if m is considerable very nearly the same value of c. These conflicting§ views seem equally plausible. of them may seem not to tell much for any of them. It may be remarked that the value here found for the modulus differs from that assigned by Gauss|| and many eminent mathematicians in that they would substitute (n—1) for n in the above expression for c. A careful consideration of Gauss’ argument will show that what he seeks and finds is not that value of ¢ which, taken in connexion is put to 7 (the ordinate of the new curve) =f[p ()1¢' (8). * The case is still a case of single mean according to the definition of class b. + The value of the parameter h in Dr MacAlister’s curve of error as determined by inverse probability (cf. above, ae , 1 ; p. 145) is found by putting has and, in the value of ¢ above given, for x the geometric mean, and for log =] [senile + On Least Squares, p. 143. (x-2,)* § Attention to @ priori probabilities is required in the transformation of facility-curves. The general rule for transforming y=f (zr) to a facility-curve in ¢, where r=¢ (E), But it must be remembered that, if the values of x are a priori equally probable, the @ priori probability of & is not constant, but « é or rH equation in £ the most probable (@ posteriori) value of &, we must multiply 7 by 76 ; whence we obtain f' [¢ (é)]=0 for the maximum, as was to be expected. These considerations should be borne in mind in studying Dr MacAlister’s paper in Proc. Royal Society, 1873, || Theoria Combinationis, § 38, followed by Airy and most writers. Thus to find from the AND THE FIRST PRINCIPLES OF STATISTICS. 149 with the proper value of 2, constitutes that system of 2, ¢ which makes P a maximum, but that value of ¢ which makes P a maximum, account being taken of all the possible values which x may have. It is as if any one, given z=F' (xy), should determine not the # and y which make z a maximum, but, integrating / (zy) with regard to between given limits, should determine the y which makes the resulting function a maximum. But surely the primary view of the quaesitum in this case is that value of « which makes P a maximum, account being taken of all possible values of c; and the next most natural view of the quaesitum is the system of values for « and ¢ which makes P a maximum— alternatives which yield the same value of # A third quaesitum is the probable error z,+2,+ &e. incurred by putting z= 7 But neither to this question is the answer given by Gauss. The probable error is found by taking for the ordinate of the facility-curve under which the values of « range the expression | Pah 2 | | Pahde; 0 —woJ0 supposing the f’s to be @ priori equally likely. Whence it appears that the probable error of # is given by quartering a facility-curve of the form A y a n+1? (Sx? + na”) 2 not, as the followers of Gauss suppose, by quartering a probability-curve whose modulus ie is ee: In short the value assigned by these mathematicians does not appear to correspond to the maximum of anything in particular. Nevertheless here, as so often in mathematics, genius may have flown on to a point whither method afterwards constructs a road. It will presently be submitted that, though the most probable value of the modulus is sp*+n, yet the most advantageous value of the same is greater. abcdefgh. This case differs from the preceding, in that the weight is not known beforehand to be the same for all the observations. In this case put 2b pS eS]: = Ce — Tapas Cy C2 X p X 0,0,0,. ag we Whence c, = 2 (« — a,)’, C, = 2 (4 —2,)", &e. = &e eo (a@—2,)+ ns (a — 2)’ + &e. = 0. C, C, 1 1 ——. + ——_. + &. = 0, (x — x,) & — i) Whence an equation of (n—1) degrees, whose coefficients are symmetrical functions of the given observations, which might possibly be evaluated by a calculating machine. Approximative 150 Mr EDGEWORTH, ON THE THEORY OF ERRORS OF OBSERVATION values of x are afforded by the limiting functions of this equation; the first approximation being as might be expected the average. This solution is exposed to the objections against the inverse method which have been considered with reference to simpler cases. Nor does this case admit of the defence that the unknown functions p...¢,¢, become extinct by the repetition of observations. Each new observation introduces a new @ priori probability. Nevertheless there are two im- portant sub-cases where this defence is applicable. (1) When the observations are divisible into a few groups and all the members of the same group are known beforehand to have the same modulus. In this case P is of the form 1 1 1 _@=aPt+@-mF_ (= 4,?+(e—% F + i po,o,€ ec? ec? C, Differentiating with regard to c,, c, and a and eliminating c, and ¢,, we have a cubic equation for « which affords as rigid a solution as any of the preceding cases. (2) The second favourable sub-case is where the moduli may be presumed not to be very widely apart. For in this case we may be confident that the influence of o,¢,... will become extinct. We have at the same time confidence in employing a derived function of low degree instead of the general equation. abedé. Analogous reasoning is applicable to curves other than probability-curves, pro- vided that we know beforehand about as much of their character—the family at least if not the parameter. The only important instance appears to be the binomial of Mr Galton. This when consisting of many elements merges into the probability-curve. When consisting of few elements it becomes a discontinuous small curve not amenable to a special method analogous to the preceding, but falling into the general class. In general in class 2 the distinction between f and f is, not between knowledge and ignorance of a parameter, but, as the family of the facility-curves is in general unknown, between knowledge and ignorance of some mean function of error (as the inverse-mean-square) which is taken as the measure of a facility-curve’s precision. Let us begin with the simpler division f and consider an important sub-class under this heading, namely abcd@fgh. This case does not, like the case of probability-curves (e) admit of a perfect solution by way of inverse probability. But it may be shown that an approximate solution is afforded by a certain weighted arithmetic mean. Any of the facility-curves (not supposed identical) may be put approximately in the form * H e- Met - Patt. Re, Accordingly an approximate solution of the inverse problem is afforded by the maximum value of Y, where log ; (=Z) = M,(«£-2,)?+M,(a-2,)+ &e. + P,(x@—2,)'+ P, («—2,)* + &e, + R,(«@-2,) + R,(a —2@,)" + &e. * Especially, when the curves may be regarded as probability-curves deformed at the extremities; as in the case of binomial curves. AND THE FIRST PRINCIPLES OF STATISTICS. 151 (This reasoning is justified by mathematical precedent. It is allowable to treat a binomial of many elements as a probability-curve, and by reasoning similar to that which has just been employed to deduce that the weighted arithmetic mean is the most pro- bable value.) An approximate solution of = is afforded by its last limiting function equated to zero: namely R,(#—w#,)+R,(#—«#,) + &e.=0. Whence «=the weighted arithmetic mean R,x,+ R,v,+&e.+Rh,+ R,+&e. It is to be observed that the weights here are not the same as in Laplace’s solution; though they are apt to vary in the same direction as his weights. This may be seen in one important case: where the facility-curves differ only in respect of a parameter, being each of the type y= : ii ive In that case Laplace’s ; ; : 1 : P 1 weights, the inverse-mean-squares, are proportional to @ is proportional to rr When the number of observations is considerable, the approximate value above found tends to coalesce with the correct value. For, since the observations are supposed to emanate from one source of error and to range under a finite facility-curve, as their number increases, it becomes increasingly probable that they will lie, not level, but in a heap. Considering then Y=/(a#—~«,) x f(w—-,) x &., we see that there must be certain central values of x for which the majority of the factors f(w—~z,), &. are relatively large and the minority relatively small; but that for values of w nearer the limits of the facility- curves the majority of the factors are relatively small. Accordingly the effect of the continued multiplication must be to make the curve Y=0O spring up as represented in Fig. 4 (where each Y is supposed to be multiplied by such a factor that fYdx=1). At Fie, 4, the same time Z = log : ee ) tends to become infinite outside narrow limits. Therefore all the differentials of Z outside narrow limits tend to be infinite. Therefore the values of « r—-1 for which a= 0 and that for which =, da da =0 lie close together. The difficulties which beset the direct solution of the general case under consideration have induced Laplace and his followers to resort to an indirect procedure, which may be Wit. SGN, W2ssem JOE 20 152. Mr EDGEWORTH, ON THE THEORY OF ERRORS OF OBSERVATION distinguished from the *proper-inverse method as the Method of Least Squares. This (1) The first is to put for the quaesitum a weighted arithmetical mean, not as the most probablet, but as the most workable value. This sacrifice of precision to practicability seems quite justified by mathematical precedent. celebrated method consists of two steps. (2) The second step is to determine the constants of the assumed form, the weights of the arithmetical mean, by a principle which may be thus generalised. To test the accu- racy of a method of reduction, i.e. function of the observations which it is proposed to put for the quaesitum: take at random any number of sets of values Gh Ga Moran aston statement é., ’ ’ ’ Peale Date at SS Ca a” a” ” CHR Rati ccr ferns oc pe ae which as it were temanate from the same source of error, each e according to the law of error indicated by its subscript, the law of error for one of the observations. For instance the first set is thus formed; e, is taken at random from under the facility-curve y= f,(); e, from under the facility-curve y= f,(«), and so on. Calculate the value of @ for each set and arrange the values so found under a facility-curve; then the divergence of the facility-curve so resulting may be taken as a measure of the inaccuracy of the method of reduction @(z,, x,,...). Accordingly, if the divergence of that resulting facility-curve should be greater for 0, than 6, then @, is a less accurate reduction. Laplace has employed this principle in at least four other cases besides that which is now under consideration. The first instance is his reasoning about Bayes’ theorem, where the process called by Mr Todhunter§ “assumed inversion” will be found to involve the principle just enounced. A second instance is in the problem of mean marriages also noticed by Mr Todhunter |}. to test the accuracy of his Method of Situation by an application of the principle in question. A third instance is in the second supplement where Laplace professes A fourth instance is in the first supplement where Laplace argues upon his principle that the inaccuracy of his weighted mean is of a negligible order. It is here submitted with great deference that this indirect method has no theoretical cogency and does not add anything to the presumptions in favour of the weighted arith- metical mean which are obtained from the inverse-proper, or d@ posteriori, point of view. To show this it may be sufficient to consider the instance in which the method of reduc- tion, the @(a,, «,...) proposed to be tested, is that which the inverse-proper method assigns * See note, p. 157. + Cp. Glaisher, op. cit. p. 10. + The inverse-proper method which has been hitherto pursued in this paper may be thus contrasted with the (second step of the) method of least squares now under consideration. The former, regarding observations as hay- ing emanated from a source of error, remounts to the | source by the principles of inverse probability established | by Laplace at the beginning of his introductory essay. The latter begins as it were at the source, and, taking at random any number of sets of errors emanating from the source, operates on them in the manner described in the text. Accordingly the former (or at least a particular species of it to be distinguished presently, see below, p- 164) is described by Laplace as @ posteriori, relating to ‘observations deja faites’’; the latter as @ priori relating to ‘‘ observations non faites encores.” —Théorie Analytique, third ed., p. 333, national ed., p. 365. § History of Probability, pp. 554—556. | P. 603. ‘| Pp. 6—11 (of the Supplement), noticed by Mr Tod- hunter at his p. 610. AND THE FIRST PRINCIPLES OF STATISTICS. 153 as the most probable value of 2, that is if Fig. 5 denote the curve Y=f(@—2) x f(e—@,) X...... 5 Zi tq Zz Fie. 5. then @(z,, z,...) is the point at which the longest ordinate occurs; and the curve Y repre- sents truly the inaccuracy of @, the number and extent of errors we shall commit if every time we obtained the given set of observations we put @(#,, w,...) for the true value. Now let us consider, if we put ourselves at the source so to speak, if in the lan- guage of Laplace and De Morgan we want to construct a method of dealing with the obser- vations “before they were made*,” how we should obtain a facility-curve corresponding to Y. A little attention will show that we must take, not at random every set of s values which flow from the source of error, but only those particular sets the constituents of which are situate relatively to each other at the same distances as the given observations «,, 2... But in general this last procedure is not the same as Laplace’s. In fact it is only the same in the case of the probability-curve; in which case the inverse-proper requires no aid. The inconclusiveness of the fundamental principle upon which Laplace’s}+ “assumed inversion” rests may be further exhibited by example. Consider the method of reduction 1 1 x," + a," + &e.\” PSS 43 oe ee ; n odd, weights of observations identical. According to the principle assumed, that value of » is best which is such that, if we take at random sets of s 1 1 ; : ota” + &e. observations, and form with each set the mean ( : a range under a facility-curve of least divergence. By a well-known theorem} when s is con- ) , the values so obtained 1 1 1 "+a," + &e. s siderable the values of 4 may be regarded as ranging under the curve Ke so all wae where 9 = mean — twice 7 th power of error for the original curve 2 ae : Te xc” 2 T * De Morgan, Encycl. Metrop., § 89. + Mr Todhunter’s phrase. + Todhunter, p. 570. 20—2 154 Mr EDGEWORTH, ON THE THEORY OF ERRORS OF OBSERVATION 1 1 2,"+2," + &e. s The error then of , may be represented by st where 7 corresponds to any Js” . Fl : : as assigned value of 2 i —— eda, e.g. $, fy, &e. Now, in order to transform the facility- 0 T ie oe x," + 2," + &e. curve in & representing the frequency of the values to one in &’ represent- 1 1 1 1 ; 1 : oe ing the values (z,"+2,"+&c.)", the rule is* to put €=£&™” and (since dé =p ‘) to ie : multiply after substitution by : eis ". What was ae in the primary curve becomes in 2 = (+1) : n Vs ar Js > approaches () xeZ™. But the error of the method of reduction the derived curve (=), that is s x¢x 7". When n is large, this expression x, +2,+ ke. = cT Thus s ? vs a a n .— (at pared .) ; s and n being considerable, the method of reduction (28 +0, + &e is better than &, +2, + &e. a since the probable error+ and all errors up to Z=1 are less for the facility-curve repre- senting the former values than in the case of the latter. The two curves may be denoted by Fig. 6. But we know by inversion-proper, if the facility-curve under which the Fic. 6. observations range is a probability-curve, that the most probable value is See The method of assumed inversion is therefore at variance with the method known to be right. The incorrectness of the principle under consideration may be more generally demon- strated by pointing out the correction of which it stands in need. Let O(z,, #,,...C,, ¢,) be the method of reduction which is to be tested. Let ®() be the facility-curve which * See note § p. 148, the most probable value. If we consider the species which + This example relates to that species of the method of | aims at the most advantageous value, to be explained below, Least Squares, preferred by Mr Glaisher (Memoirs of | the example here adduced will not be so evidently at Astronom. Society, Vol. xu. p.101), in which the quesitum is | variance with the result of inversion proper. rer o AND THE FIRST PRINCIPLES OF STATISTICS. 1 is formed by taking at random sets of 2,, z,... &c. (from under the facility-curves of the observations, namely y=/,(x), y=/f,()...) and substituting in @ the values of «,, a@,.... Let (z,, x,...) be the most probable value as given (by way of inversion proper) by the solution of the equation f fle-a) Xfi (oa)... = 0) or a =0. If now our attention was confined to a single given set of observations z,, 7,, &c., the problem of finding the best member of the @ family would not have much meaning. w(z,, Z,...) is the best value, and whatever value is arithmetically nearest this is best. It is only when we take account of the variations of value which z,, «,, &c., may assume consistent with their emanating from the same source, that the problem becomes signi- ficant. Now for any particular set of 2,, 2, &c., the frequency with which an error of the exact extent +e is committed by putting 0(«,, z,...¢,,¢,) for the quaesitum is F (@—e)+fF (x)dz, between limits including all the positive values of F(x), where F(x)=Y. In order there- fore to find the frequency of the error e in the long run, we must integrate F(6—e) f (a,) f (a,)...f(#,) dx,da,... dx, | F(a) da : between extreme limits of z,, 7,...a,. The result, say y(é, ¢,, ¢,...), gives the law of facility for the error committed in the long run by using the method of reduction @. Now there is no ground for believing that the result of this corrected process is, in general, or in the particular case when 0=c,x,+¢,2,+ &c., identical with the result of the method of least squares, at least when the number of observations, though it may be considerable, is finite; for, if the number be really infinite, of course* any method of reduction will be admissible. Indeed it does not seem to be contended that plurality of observations makes the second step of the method of least squares more secure; but only more practicable, by allowing the law of error to be brought to bear upon it. The discrepancy between the method of least squares and the proper method may therefore be illustrated by a simple example. Let there be two observations 2, and #,; resulting from two facility-curves, y= }(1—2*) and y=#?h (1 —/A’2*), extending respectively between limits + 1 and i (1) According to the principle of the method of least squares, we have to find the facility-curve under which ie,+ pe, ranges; where e,, e, are values taken at random from under the given facility-curves respectively, and 2, w are undetermined multipliers whose sum=1. To effect this operation we may proceed thus. First find the facility- curves under which Xe, and ye, respectively range. They are respectively : E - a , between limits z=+AX, and Hm) Oo | 0° L Xr h WV’ ie be —|1—-—,7?|, bet limits y=+7. =i 7A | etween limits y i * Cp. Glaisher, op. cit. 156 Mr EDGEWORTH, ON THE THEORY OF ERRORS OF OBSERVATION Then, to find the range of values y+2z, proceed upon the principles laid down on fo] a former page*. Integrate the expression Syl No BS) lo h’ between proper limits, and having regard to the sign of the odd terms in z. (The integral breaks up into three parts: from 5 to e, from e to 0, and from 2X to e) The resulting expression will be the ordinate of a facility-curve in e, }H (1— H’e’), representing the frequency of the error Xe,+me,. This curve will evidently be least dispersed when H is a’ maximum, or dH =0. (2) The inverse-proper method puts Y= 3 (1 -(e—2a,)"} x Sh {1-h? (w@-2,)}. The frequency of error e incurred by putting #=)a,+ px, (X+m=1) is for a particular system of a, and 2,, # [1 — {A —1) 2, + wa, — e}] x h [1 —h? fra, + (u— 1) a, —e}*] [Ydx ; where Y is integrated between its extreme intersections with the ordinate. To find the frequency of the error e in the long run the above expression must be multiplied by 7; (1 —2,)h(1—fh’xZ) and integrated with regard to 2, and «x, between limits +1, +} respectively. There is evidently no correspondence between the expression in : w obtained by this procedure and that of the method of least squares. The value of the method of least squares will be more fully appreciated by marking the different degrees of its applicability. (2) It is theoretically right in the case of only a single observation being given. Suppose we had to make with a certain instrument an observation not capable of being repeated. The frequency and extent of error incurred by taking on each different occasion a single observation as the true value of the thing measured would exactly be represented by the facility-curve obtained by continued measure- ments of one and the same thing; provided at least we assume that the @ priori+ probability of the observed value being the true value is constant—a proviso which removes the only advantage which might seem to remain to the “assumed” as compared with the proper inversion. It should be observed that this case does not cover Laplace’s application of his method to Bayes’ theorem. If m white balls and n black have been drawn, then taking m:n as the real proportion of the balls in the urn is doubtless analogous to taking a single observation in space or time as the true value of the thing measured. : : ee, m But the analogy fails in this respect that, when the real ratio of balls is different from ag pe Se : . WD, ae the real modulus also is different from what it was in the case of Fi being the real ratio. Whereas in our illustration we had supposed an instrument of constant precision, * Above, p. 141, + In one sense of the term, see aboye, p. 148 et sqq. AND THE FIRST PRINCIPLES OF STATISTICS. 157 (8) A second case is that in which the facility-curves are of the form oe e@. In T this case the method of least squares* may be said to be accidentally right; agreeing as it does with inversion proper+. It is submitted that the apparent verification of the method of least squares by the practice of observers is due to the prevalence of probability- curves in rerum naturd at least in an approximate form. (y) In a third case, the second, the insecure step of the method of least squares may be dispensed with. This is the case} where all the facility-curves are identical. In this case, if we may assume that the solution is an arithmetical mean, we may assume that it is the simple arithmetical mean—by a principle of symmetry which seems justified by precedent. As there is no reason why one weight should be different from another we may treat all the weights as the same. Nevertheless this reasoning must be admitted with caution. How do we know that it is desirable to take account of all the observations that, e.g. the best reduction of #,, #,, 2, is not (the symmetrical one) either ate or “2 ale or ay There will be given below § instances in which the mean is the most improbable value. The instances no doubt are of infinite facility-curves; but this property may be supposed retained while the curves are slightly deformed so as to become finite. In such cases the average might not be the most accurate arithmetical mean. (6) The second step of the method of least squares seems not to be indispensable in one other case: when the facility-curves consist of clusters||, the members of each cluster being identical. By parity of reasoning with the last case the solution may be written A(x, + 7, + Ke. + £5) a nN (¢, +4, + &e. + 2'y) 8 s' + &e. Now since by this formula we restrict ourselves to taking account of the average ot each cluster of observations the problem seems reduced to the following, Regarding each x, +a,+ &e. : : : : oe average =. —— as a single observation ranging under a corresponding facility-curve to find the most probable value (or at least the most accurate arithmetic mean). ane a,+a,+ &e. P Now the quantities ——*——— are known by the law of error to range under probability-curves whose moduli are the respective inverse mean-squares. Whence the solution of the modified problem is the same as the solution of the original problem afforded by the method of least squares. Nevertheless the modified problem is not the original problem. We have no security that considering the whole series of sets @,, a,...: x, 7,...&c. the arithmetical mean above found is the one which is the least inaccurate. * Meaning, as throughout, the indirect method above | sum of squares (-log Y). + abedefgh. characterised; for of course the direct (inyerse-proper) + abedefgh. § See p, 159 and note. method may in this particular case be called a method of || The case considered by Laplace at the end of Art. 23, least squares, since its solution is the minimum of a certain | chap. tv. 158 Mr EDGEWORTH, ON THE THEORY OF ERRORS OF OBSERVATION For instance, let the facility-curves be of the form Je~#. Is there any reason for thinking that the weights prescribed by the method of least squares, namely sVR, s'VR, are better than the weights prescribed by the approximate* inverse-proper solution; namely sR, s'R’? (ec) Last is the general case in which it is here contended that the method of least squares is neither theoretically correct nor practically useful. abedéfgh. In case of asymmetry the method of least squares, though equally practicable in virtue of Poisson’s extension of the law of error, becomes more insecwre—as Poissont himself seems to indicate. abed2fg. When it is not given that the weight of all the observations is the same, then the analogy of probability-curves negatives Airy’s{ principle that we should treat the observations as all of the same weight and confirms the general principle, though not the particular practice, of De Morgan§, Peirce}, and Glaisher |]. Here for the present may be terminated our discussion of class d. It will have been observed that in the discussion of these problems we have tacitly assumed that the quesitum is the most probable value of the unknown quantity. Given a set of observations x,, 2, &e., we have shown that in the long run the real point, from which a set of observations at those relative distances results, occurs at different points with a frequency indicated by the facility-curve Y=f(«—«,) x f(#—~«,).... And we have taken for granted that the function of z,, x,, &c. which corresponds to the maximum ordinate of that curve is the right method of reduction. This point of view is doubtless natural and obvious and recommended by high authorities. But the more philosophical view, theoretically more correct if not practically different, is that we should select not that value in terms of @,, ©, &c. which is most frequently the true one! but that one which is most advantageous, account being taken not only of the frequency but also the seriousness of error, that value which minimises the detriment of error. Now this quesitum is of the nature of a subjective mean. Its discussion had better be deferred till we come to a. abe. The case of facility-curves, other than probability-curves, extending to infinity appears to be rather of theoretical interest than practical importance. The general type of reproductive facility-curves abce, an — I e cosakadz, T —-2 presents the remarkable property that, when ¢ is less than unity, the average of s obser- vations of the same weight ranges under a facility-curve not less, but more, divergent than the primary curve. If the weight (the constant k) is different for different curves, the ; : t c 2 weighted mean g,x,+9,7,+&c.+sg, where g x the inverse mean ;— th power, is according t—1 * Above, p. 151. + Astronomical Journal, Cambridge (America), Vol. 11. + Connoissance des Temps, 1832, §§ 8,9. In order to | p. 161. render the method applicable, it would have to be granted § Encycl. Metrop., § 136. that each curve of error has the point of no error as its || Memoirs Astron. Soc., xu. p. 103. centre of gravity. For the case where this is not granted “| Cp. Glaisher, Mem. Astronom. Journal, xu. p. 101. see below, p. 168. AND THE FIRST PRINCIPLES OF STATISTICS. L}) to the method of “assumed inversion” the arithmetical mean of, not maximum, but minimum, accuracy, when ¢ is <1. It would be interesting to compare this conclusion of assumed inversion with that of inversion proper. The comparison is easy in the case where ¢=1 (the bounding case between convergent and divergent averages). In this case according to the assumed inversion there is neither gain nor loss in proceeding to an average”. According to inversion proper for certain relative situations of the observations the average is a most, for other data it is a least, probable reduction. If there be two observations L,— @, 2 whose law of facility is = — the most probable value is 1%, provided that But for higher values of the distance between the observations the average is a position . dl db —— = F’ (a1) yet [i F’ (la) yd + terms} outside the at of integration which vanish. y being symmetrical about the point 2=0, = vanishes when / = 0. To find the second term of variation, we may either differentiate at once with regard dE eae ; to J, or first transform a *® thus (by partial integration as to @): i 7 7 =- [Fe-oy] +] F@-)F de -_|ra-oy]+ +f. FQ-0) © de, be pointed out presently, to abed. + Cp. Laplace, loc, cit. * This theory is illustrated by Laplace’s reasoning at p. 365 (national ed.), p. 333 (third ed.), Théorie Analytique; though his reasoning there relates, not to abcd, but, as will AND THE FIRST PRINCIPLES OF STATISTICS. 161 (which vanishes as it ought when /=0, since xy =0 for e=0 or +*, and dy has da opposite signs for « above and below zero). Hence (A Ose) (ae Ue de> [P@-99|-_[Pe-ay] bali eae dy, .{) m dy 1 | F(a ~Nipde+| F (12) de. The portion without the integral sign vanishes, and the remainder is positive (since every element under the sign of integration is positive). Therefore # is a minimum when /=0. Therefore the central point of the curve is the sought Mean. abedefgh (the case of asymmetry) considered as a variation from the preceding case is seen to require a deflection from the centre in the direction of the preponderating side of the curve. dbedeg hardly occurs. There may be two groups of statistics, each of which has a different modulus. But, if they have the same mean, no new problem arises; and, if they have different means the case rather belongs to class @; to which we proceed. abedé: under this heading four cases deserve special notice: (a) The case of any symmetrical disposition of the statistics. By reasoning similar to that which has just been employed, the centre is a critical point. It is one of minimum detriment if the arrangement is a simple continuous facility-curve (zero at the centre and extremities) Fig. 8 (1). It is very likely to be one of maximum detriment, when it lies in a trough between two elevations as in Fig. 8 (2). (1) Fic. 8. (2) O O (8) Where the statistics may be arranged under two probability curves, or more generally any simple continuous facility-curves of type \ feel i (). not having the same centre nor modulus; and practical exigencies require that one and the same Mean should be put for the whole system. Let us first consider the case in which the centres of the two curves are very close together, as in Fig. 9. OV Fie. 9. * For, if F(») xy, were finite, the detriment incident to any subjective mean would be infinite: cadit questio. 21—2 162 Mr EDGEWORTH, ON THE THEORY OF ERRORS OF OBSERVATION If we suppose the qusitum to be for a moment at the point equidistant from the two centres it will tend to move in that direction in which the increment of detriment is least. We can determine then in the direction of which centre it will move by comparing the values of the quantity 2s (J=0) (see p. 162) for each curve. The relative magnitude will depend upon the nature of the function of detriment (as well as upon the moduli of the curves). If the curves be of the form = hen a T it may be shown that, if the function of error is a power (or sum of powers) less than 7 d 2 = . d,E ah a continually +. Therefore dE and the mean gravitates towards the centre of the weightier curve. the second, then is largest for the largest h If the distance between the centres is finite, we have to consider for which curve = is greatest, 1 being the half distance between the centres. (y) A third subcase is that of statistics diverging by a very small range, a very contracted facility-curve. In that case we may be allowed to write for the detriment F(l), F’(0)xe the error taken positively and multiplied by a constant. The typical mean is then such that =(«—2,), the quantity within the bracket always positive, should be a minimum—Laplace’s method of situation; of which this reasoning is sub- | mitted as the rationale. Jt is submitted also that this reasoning throws light on the fundamental assumptions made by Gauss and Laplace on the function of detriment. If we suppose the function £ to involve e in the first power (against which there seems no valid objection) then Laplace’s assumption* of the mean error tends to be ap- proximatively correct; but Gauss’ mean square of error, if #' is a function of 7, is preferable. (6) Another general remark is that the Mean describable as the Greatest Ordinate appears entitled to great consideration; especially in the case of amputated curves such as the anthropometrical statistics of an army (men below a certain height or strength having been rejected); or+ where the Arithmetical mean and even the Median are liable to be displaced by the admixture of returns foreign to the type with which we are con- cerned, yet not capable of being specified and set apart. Next in order comes the subjective mean pursued by the way of inverse probability. But at this point it seems advisable to return to a problem in Observations which we had to postponef, because it involved as a proximate end, a secondary quesitum, a fictitious or subjective Mean. Let us look back then from our new point of view at the problem abcdf, Let us begin with the case of a single observation§, partly because it is the simplest, and partly because in this case only (as above contended) is Laplace's “4 priori” method of least squares perfectly satisfactory||, In this case only then will * See Gauss, Theoria Combinat., Part I., § 6. § See the present writer’s remarks in Phil. Mag., Nov. + The use of the Greatest Ordinate Mean is well ex- | 1883. emplified in the statistical writings of Dr Charles Roberts. || See above, p. 156. + See p. 158, above. AND THE FIRST PRINCIPLES OF STATISTICS. 163 our exposition of the most advantageous value coincide in principle with the ‘most ad- vantageous’ first considered by him; namely in that second definition* of the quesitum (in the case of “observations non faites encore’) which has been a stumbling-block+ to many. Suppose then (as above) that we had to content ourselves with a single observation made with an instrument. Suppose we had our choice between two instruments; and that the accuracy of each could be tested by long previous experience. Let that ex- perience be expressed by the two facility-curves in Figure 9, which instrument is to be Fic. 9. preferred. If we take as our end the most probable value in the sense so clearly defined by Mr Glaisher, undoubtedly the instrument whose curve is more contracted at the centre (though more expanded at the extremities). But it may well be that the other instrument would be in the long run more advantageous, account being taken of the extent, as well as frequency, of error. In the notation above adopted, that imstrument is to be preferred for which? 2 | F(x)y is the least. This agrees with Laplace’s Jo second view of the quesitum in his method of least squares; except that for the general function F he puts the particular function 2 (the mean error)—an often legitimate approximation, as we have just seen. When the number of observations is plural, we must part company with the method of least squares; which can more clearly be contemplated from the present point of view. The principle of its second step as explained§ above is to test a proposed method of reduction @(x,, x,...), as we would test an instrument which presented a series of errors respectively equal to (is Besa) Oe finned, es METERS Zo... 5 Wh YDS Bpoooe , &c. are random sets of values taken from under the facility-curves of the observations). Accordingly this generic principle has two species according as we prefer that measure which affords the most probable value, to which our attention under a former heading was confined, or, what now comes into view, the most advantageous value. It is here contended that, if the generic principle could be accepted, then the most advantageous species would be the philosophically preferable. Mr Glaisher's strictures on that || species do not seem deserved. * Théorie Analyt., ch. 1v. § 20, p. 318, third edition. trical. + Cp. Glaisher, op. cit., p. 101. § P. 152. } Assuming that the function of detriment is symme- || Memoirs of the Astronomical Society, xu. loc. cit. 164 Mr EDGEWORTH, ON THE THEORY OF ERRORS OF OBSERVATION But the generic principle cannot be accepted. And therefore we pass on to inversion proper; which also is divided into two species, according as the quesitum is the most probable or the most advantageous value. It is remarkable that Laplace in his general account of the @ posteriori, or proper inverse method seems to have confined himself to the* most advantageous species. The procedure there adopted by Laplace is submitted here as the most philosophical treatment of the inverse problem—with this modification that, whereas Laplace employs a particular function namely «# (mean error) to express the detriment of error, the general undetermined symbol # is here recommended. The inverse investigation of the most advantageous mean is exactly parallel to that which has been given under+ dbede except that Y (Laplace’s y’) now denotes, not the facility- curve (obtained by simple induction from the very large set of statistics a, a, ...... ) expressing the frequency with which members of a group of real things have each value xv; but the facility-curve (obtained by inverse probability from the not very large set of observations @,@7, ...... ) expressing the frequency with which the real thing behind the observations has each value #. We find by inverse probability statistics for the occurrence of the cause of the given observations; and we take the subjective mean of those statistics. The theoretical superiority of the most advantageous method is accompanied with a practical difference in the determination of the modulus. We have seen that the modulus for a set of observations (supposed to have diverged according to a single unknown law of probability) has different values with a frequency expressed by the curve where Se? is the sum of squares of apparent errors. Whence it follows that the most probable value of ¢ is = But it is probably not the most advantageous value. For jrom the asymmetryt of the curve y it is probable that the most advantageous is greater than the most probable value. Thus the received solution though it has no pre- e m—1’ : ee * . . ° . 2Se : tensions to precision, yet differs in the right direction from +. The ordinary formula nn is to that extent correct, but not for the ordinary reason. Having now discussed the application of the simply inductive subjective Mean (ad), to the inverse real Mean (ad), we may return to the point§ at which our exposition of pure statistics was interrupted and consider the inverse statistical mean. abedefyh. The simplest case is that of a not very large set of statistics considered as random specimens of a whole series which ranges under a probability curve of known modulus * Théorie Analytique, p. 333, third edition, p. 365, t Above, p. 161. national edition. § See above, p. 162. + Above, p. 160. AND THE FIRST PRINCIPLES OF STATISTICS. 165 but unknown centre. One way of treating the problem is—knowing that the centre of the real series of which the given statistics are specimens is the most advantageous Mean—to Fia. 10. Ly LyX determine by inverse probability the frequency with which the centre of the real series has each value z, and to take that value of the centre which most frequently, most probably, occurs. But, it is submitted, the more philosophical procedure—theoretically preferable, though not practically different—is to regard the given statistics #,, #,, «, ... as indicative of a whole series of groups of statistics whose size and centre is indicated by the height and position of the ordinates of the probability-curve * 1 ( case SE pease ¢-——— —) 802 é 8 5 NV 1se We have then to take the subjective mean of this series of series upon principles similar to those already explained. This distinction becomes clearer when the two points of view are no longer coincident, as in the next case. abedéfg. To exhibit the difference just indicated suppose the facility-curve under which the (complete) series from which our samples are selected ranges is changed from the form of Fig. 11 (1) to that of Fig. 11 (2), while the quantity [f(z) 2 (between extreme limits) is kept (1) Fie. 11, (2) constant. Then, if we are employing the arithmetic mean, the worth of the mean in the case of observations is unaltered; but in the case of statistics is impaired. For the central * The area of the curve whose central point has the | The length of each curye’s central ordinate is the area abscissa x is erf. é r) = divided by s/c. c 166 Mr EDGEWORTH, ON THE THEORY OF ERRORS OF OBSERVATION point of a series such as that indicated by Fig. (2) has a representative character in a less degree than the central point of Fig. (1). Hence in the course of a series of series*, if we continue taking the arithmetical mean, though we shall as often be as near the real central point as before the supposed change, yet our Mean will no longer have the same worth, It appears then that the excellence of the Mean under consideration depends upon two qualities: the precision with which our Mean of the given samples approximates to that point which, if the series from which the samples are taken were indefinitely prolonged, would constitute a certain assigned Mean of that series, a certain function of its terms; and the representative character of that point. As to the second quality there is nothing to be added under this heading to what has been said+ under the heading of Induction. The first quality is an excellence common to Statistics and Observations. And indeed at this point it is very difficult to determine how far an element of Observations enters into inquiries commonly called statistical; how far the sought Mean is purely “subjective,” or also an “index”+ according to the just distinction drawn by§ Dr Bertillon; from whom the former term has been adopted into this paper. Accordingly the following remarks on the worth of Means in respect of precision cover certain cases of Observations transferred here from their proper place. It follows from the discussion at pp. 153—8 that it is im general hopeless to enquire what is the best Mean in respect of precision. All that we can say is that certain Means are better than certain others. For the purpose of comparison Means might be classified as (1) The Arithmetical Mean, (2) Two Means which compete with the Arithmetical in respect of convenience, (3) All other Means. The third class may in general in the absence of special reasons to the contrary be rejected on the ground of convenience. The battle of the Means becomes triangular. It is to be decided by comparing the first with each of the others, and these with one another, on the ground both of precision and convenience||. (a2) The prerogative of the Arithmetical Mean, that it is the more accurate value in the case of probability-curves, extends of course to curves in the neighbourhood of that family, and probably far beyond. Take for example the facility-curve y= = (h—2z) (2 positive both ways), Fig. 12. The arithmetic mean of s observations ranges under a probability-curve whose modulus squared is }h*>+s. The Median of s observations ranges under a probability- * Above, p. 165. § Dict. Encycl. des Sciences Médicales. Cp. Mr Venn + Above, pp. 161, 162. on the Law of Error in Logic of Chance. + The subject of the writer’s paper on Methods of || Saving of labour in the working; a consideration Statistics is: “* What is the worth in respect of precision of | which of course enters into every practical approximative an assigned index-mean” ? operation. AND THE FIRST PRINCIPLES OF STATISTICS. 167 curve whose modulus squared* is 3h?+s. The former is therefore more accurate than the latter. There is however one case in which the Median has the advantage in respect of Fic. 12. precision: namely, when the apex of the curve is very high and its extremities very much extended, as in Fig. 13. oie Noe Fie. 13. For in this case the mean square of error, which is proportional to the modulus- squared which measures the precision of the Arithmetic Mean, is likely to be very great. At the same time the inverse-square of the greatest ordimate, which is proportional to the modulus-squared+ appertaining to the Median will be very small. On another ground the Median has the advantage in respect either of precision or of convenience, or perhaps both. Suppose that the data are measurements made with a measure not minutely graduated, e.g. the numbers of men whose height les between each two consecutive inches. To obtain the real arithmetic mean (of statistics taken from under a continuous facility-curve) it would be necessary to adjust each of the given figures by smoothing out our data into a continuous curve. But to obtain the real Median it would be necessary to operate upon only one figure by somewhat hypothetically dividing the central compartment. Where no reduction is thought necessary, the Median has an obvious and not} unnoticed advantage in respect of convenience. It has also an unsuspected advantage in that the weight of the Arithmetic Mean, the inverse (double) mean square of error, is ascertained by an often very tedious process; whereas the weight of the Median the (double) central ordinate squared is very easily ascertained. Considering that it is frequently, if not generally, more important * Laplace, Théorie Analytique, Supplement 2, sect. 2. characteristic property of this Mean—that it minimises the + Ibid. Mean Error—but also employed it: not only by way of ~ Galton in Phil. Mag., 1875. Fechner in Abhand- | simple induction (our d) in the Method of Situation (Mé- lungen Saxische Wissenschaftliche Gesellschaft, Bd. 18. | canique Céleste, 111, 39), but also by way of Inverse Proba- Fechner, complaining that Laplace had neglected the | bility in the important passage already referred to: on the Median, seems to ignore that Laplace not only deduced the | reduction of observations déja faites (above, p. 160). Vou. XIV. Parr II. 22 168 Mr EDGEWORTH, ON THE THEORY OF ERRORS OF OBSERVATION to know what the precision of a mean is, than that it is more precise than another, this last advantage in favour of the Median seems very important. (8) Comparing the Arithmetical Mean with the Greatest Ordinate, we are met with the difficulty that the greatest ordinate has no weight; or at least that it has never been weighed. For the operation which Laplace has performed in the second supplement for the Median has not, as far as I know, been performed for the Greatest Ordinate. In view of the remark at the end of the preceding paragraph, the Greatest Ordinate appears in this respect to labour under a considerable disadvantage. In respect of convenience it has a considerable advantage over the Arithmetical Mean; and (y) a less marked advantage over the Median. In respect of the necessity of adjust- ment the Median and greatest ordinate may seem on an equal footing. As possessing a definite weight the Median has the same advantage as the Arithmetic Mean over the Greatest Ordinate. abedzfyh. The remarks made under the preceding head may be transferred here mutatis mutandis. The following are special to this head. Here come in (by courtesy) observations about which, though we know the (unsym- metrical) facility-curve under which they range, we do not know at what point thereof* occurs the real thing indicated. The difficulties connected with the process of adjustment are greater here for the Median and Greatest Ordinate; but so are they also for the Arithmetic Mean. The weight above assigned for the Median requires a slight modification. By an extension of Laplace’s analysis+ it may be found that the Medians of several sets of s observations taken at random from an unsymmetrical facility-curve range under two proba- bility-curves whose centres are distant from the Median of the parent facility-curve by a quantity of the order = and therefore neglectible; and that the weight of these co- incident curves (inverse modulus squared) is still in Laplace’s notation 2s @ (0); only that @(0) means now the Median ordinate not the Greatest ordinate?. To exemplify this heading, let us take the following problem. The ages at death of a number, s, of men of and above the age of 66, distinguished from the general male population of the country by no other attribute§ than one which is the subject of in- vestigation, have been registered. And it is required to determine|| whether the difference between the mean age of the observed class differs from the mean age of the general population to an extent which is more than accidental, which is indicative that the observed class belong to a different category in respect of healthiness from the general population. To facilitate calculation let us suppose a stationary population. * Above, p. 158. | § E.g. abstinence from liquor. + Théorie Anal., Supplement 2. The p’s of Laplace are || See Introduction to a paper communicated by the to be put each equal to 1; his k equal to 3. writer to the British Association, 1885, + As Laplace says, speaking of symmetrical curves. AND THE FIRST PRINCIPLES OF STATISTICS. 169 (2) If we employ the Arithmetic Mean, we must find the mean age at death of the general male population at or above 66 from the Life Tables, say for England 76 (more accurately 75°8). Then we must multiply each figure in the column d,, by the square of the difference between the age corresponding to that figure and the mean age 76; and divide the sum of squares of errors thus found by the number of persons alive at the age of 66. The quotient 90 (approximately and without adjustment) multiplied by two and divided by s is the modulus squared for the probability-curve under which the arithmetical means of random selection of s lives would range. Hence if the arithmetic mean of the observed class differ from 75°8 by more than two (or three) times the quantity if =, it is practi- cally certain that the observed class has a more than accidental difference in longevity from the general population from which it is selected. (8) If we employ the Median, we must find the median age of the general male population, i.e. about 75°5 years. The proportion of men dying at this age out of the fod number living at 66 is about a Hence* for the modulus squared of the prebability- curve under which the median of s random selections from the general population would 184 ‘ ‘ i range : s or ——. The weight of the Median is thus about half that of the 75 aa ae Arithmetic Mean. (y) If we employ the greatest ordinate we must compare the age at which the greatest ordinate of the set of s observations occurs with 72 the corresponding age for the general male population. But we seem unable in the present state of the calculus of probabilities to determine the weight which should be attached to any assigned difference between the two greatest ordinates: how far such a difference is significant and not accidental. abedzfg. The subject special to this head is the relative weight of data emanating from different facility-curves. Weight is of two kinds; that which depends on volume only, and specific weight. To the former category belong statistical returns which are the means respectively of ,, n,, &c. constituents; all the constitutents emanating from one and the same facility-curve. Where n,, n,, &c. are each large, or where the whole number of given returns is large, this case is easily treated according to the methods applicable to pro- bability-curves and the analogy of the two preceding heads. As to specific weight, it follows from the discussion at pp. 152—7, that the system of weights which consists of the inverse-mean-squares, though a good, is not necessarily the best system. In general there seems possible no more precise determination of weight than the following. In the combination of data more weight is to be assigned to that which has emanated from a higher source: a higher source being one which in past ex- perience has yielded results less wide of the mark. This principle is equally applicable to Physics and the Moral Sciences, * Above, p. 167. 22—2 VIII. Ona new method of obtaining interference-fringes, and on their application to determine whether a displacement-current of electricity involves a motion of translation of the electromagnetic medium. By L. R. Witzerrorcs, B.A., Scholar of Trinity College, Cambridge. (Plates III. IV. V.) CONTENTS. PART I. PART III. General sketch of the experiments which have been Theoretical and practical details of a new method of made to decide whether an electric current involves a | producing such interference-fringes. motion of the electromagnetic medium in the same direc- tion, and of the consequences of such a connection. PART Iv. Description of some experiments similar to the above PART II. which were made upon displacement-currents of electricity, Summary of the methods which have been adopted for | and in which this new method of producing fringes was producing interference-fringes suitable for experiments of | employed. this nature. PART I: AccoRDING to the view of the electromagnetic field taken by Maxwell, there is a certain “ether” or “medium” pervading the whole of space, with which the molecules of ordinary matter are in some way associated or connected, and which is the seat of all electric and electromagnetic forces. It is necessary carefully to distinguish between a portion of matter and that part of the medium which is at any given time associated with it. The equations of electromotive force given by Maxwell: : 5 NA GY EY OS phe weep oe ek cama, : ~ CHL Gv Fe eae have for their experimental basis observations upon moving conducting circuits, and in the case first considered #, ¥, 2 are velocities of portions of conducting matter. It is next assumed that the equations are true generally, but it is not definitely stated whether %, y, 2 still refer to the velocities of the portion of matter at the point (xyz) or to those of the associated medium. If we adopt the second of these alternatives we must assume at the same time that if any portion of conducting matter is set in motion the medium associated with it will move with the same velocity. Mr WILBERFORCE, ON INTERFERENCE-FRINGES. 7a That the second alternative is the one which agrees with experiment is evident from the following considerations. It has been proved by J. J. Thomson (Phil. Mag. April, 1880) that, on the assumption that light is an electromagnetic disturbance, its velocity in a substance which is moving in the direction of propagation of the light is increased by half the velocity of the sub- stance if the first alternative be taken, by half the velocity of the medium associated with the substance if the second be taken. The distinction is not formally made in his paper, but it is obvious on considering the different meanings that the fundamental equations have in the two cases. Now, if air be the substance, it has been established by Fizeau (Ann. de Chim. et de Phys. 3™° serie, t. LVIL, 1859, also Comptes Rendus, xxxut.) that the change of velocity, if it exists at all, is very much smaller than this. Thus the first hypothesis is rendered untenable. Fizeau further found that if water be the substance the change of velocity is about half the velocity of the water. The conclusion is that the velocity of the medium associated with moving air is small compared with the velocity of the air, while in the case of water the substance and the medium move with sensibly the same velocity. A repetition of Fizeau’s experiments with the employment of as many different fluids as possible would probably throw some light on the way in which the refractive imdex of a substance depends upon the intimacy of connection between the substance and the associated medium. The important questions of what changes of properties in the medium give rise to electric polarization and to electric currents have not yet been answered. The first experiments made for the purpose of deciding whether an electric current consisted of a motion of translation of the medium were undertaken by Roiti (Pogg. Ann. Vol. 150, p. 164, 1873). He was compelled to employ an ordinary Arago’s interferential refractometer, which, as will be seen, is not a very suitable instrument for the purpose, and he could find no change in the velocity of light passing through a solution of sulphate of zinc due to a current of electricity flowing in the same direction. Further experiments on the same lines were made by Lecher (Rep. de Phys. 20, p. 151, 1884). He also seems to have employed a refractometer identical with Arago’s in principle and general arrange- ment. The substance that he used was a solution of silver nitrate through which he passed a current of about six ampéres, but his results were also negative. In these experiments the current was electrolytically conducted, and it is clear that the case of ordinary conduction cannot be treated by this method, since substances of good conductivity are opaque to light. There is however according to Maxwell another kind of current which consists of variation of electric “displacement” in a dielectric, and it seemed antecedently probable that such a current should be accompanied by, if not equivalent to, a motion of the 172 Mr WILBERFORCE, ON A NEW METHOD OF medium. It must be remembered that by the term “displacement” Maxwell meant nothing more definite than a certain change at each point of the medium about which no more is asserted than that it is a vector one. The simplest vector change is a change of position, and the experiments that I have undertaken have been made with the object of deciding whether this “displacement” was an actual shifting of the medium, and if so, of what magnitude. A certain probability is given to this view of the question by the fact that electric “displacement” obeys the same law of distribution as the motion of an incompressible fluid (the motion of the medium being generally assumed to possess the same characteristic), but it cannot at the same time be denied that Roiti and Lecher have by their experi- ments considerably discredited it, for, as we consider all electric currents as moving in closed circuits, m the case of a circuit of which part is a conductor and part an electro- lyte, or in the case in which we have the plates of a condenser connected by means of an electrolyte, we can hardly imagine that the medium should have a permanent or temporary motion along a part of the circuit, and none along the remainder. There is however one consequence of this hypothesis of a connection between currents and motion of the medium which I am surprised to find has not before been noticed. If an electric current in a conductor is accompanied by a movement of the medium in the same direction, remembering that the velocities in Maxwell’s equations of electro- motive force are the velocities of the medium, we see that an electromotive force will be produced in the conductor if it be placed in a magnetic field, which force will be proportional to the strength of the field within the conductor and to the velocity of the medium, and will be alike perpendicular to the lines of magnetic force and to those of flow of the current. This electromotive force has been observed by Hall (Phil. Mag. 5, Vols. 1x. and xX. 1880) in various metallic conductors, but his results cannot be uniformly explained by the hypothesis of a moving medium, since in that case the substance of which the conductor was made would have no influence upon the phenomenon except through its magnetic permeability, which was far from being the case. Again, if a “displacement current” in a dielectric be accompanied by a movement of the medium, we deduce at once the equations which Rowland (Phil. Mag. 5, Vol. x1. 1881) by assuming Hall’s effect obtained for the propagation of an electromagnetic dis- turbance in a dielectric along lines of magnetic force, and arrive in the same way at the fact of the magnetic rotation of the plane of polarization of a ray. Consider a current (wu) flowing along the axis of w, and let the magnetic induction (c) be along the axis of z, then, if the current (w) be accompanied by a motion of the medium with velocity (#.w) in the same direction, we have, for the electromotive force, 12 = (h) Q=-c.c.u, 0; According to Hall 2 for gold is in C.G.S. units 6.6 x 10™. OBTAINING INTERFERENCE-FRINGES. 173 Hence =—6§.6 x 10™. Thus if the current used were one ampére for each square centimetre of cross. section of the conductor the velocity of the medium deduced from this would be 6.6 x 10™ centimetres per second in the direction opposite to the current. The change in the velocity of light due to adding or subtracting half this velocity is too small to be experimentally recognizable. From the numerical results given by Rowland we see that if we deduce the value of (a) from the magnetic rotation in heavy glass we obtain a result not widely differing from the above. There are doubtless various causes that combine to produce Hall’s effect, as has been pointed out by Shelford Bidwell, and no importance can be attached to the value of (w) deduced from it, but the fact that the existence of such an effect follows from supposing a motion of the medium to accompany an electric current appears to me to deserve notice. A calculation of the “velocity of electricity” has been given by Boltzmann (Kaiserliche Akad. der Wissenschaften in Wien, math-naturw. Classe, Jan. 15, 1880, pp. 11—13, the note is translated in Phil. Mag. 5, Vol. 1x. p. 307), in which he assumes that a conductor carrying a current is charged with electricity of a certain density, that the electromagnetic force it experiences is the result of Hall’s electromotive force acting on this electricity, and that the current is its motion along the conductor. It will be seen that Boltzmann’s method has nothing to do with that given above, and his assumptions have been well shown by Hall to involve the supposition of an enormous mechanical force on any con- ductor carrying a current which acts in the direction of the current even when there is no external magnetic force, and therefore to be utterly untenable. PART II. From the foregoing considerations it will appear that the experimental method to be followed is to ascertain whether the velocity of propagation of light through a dielectric is altered by the passing of a displacement-current of electricity in the same direction. If such a change did manifest itself it would probably be a small one, and would only last for the short time in which the displacement was changing and thus the experimental difficulties are two-fold. A small change made in the velocity of light by any influence is best detected and measured by allowing two pencils of light to interfere, and, while observing the fringes, to cause the influence to act upon one of the pencils and to measure the consequent shifting of the fringes. Any form of apparatus for producing interference fringes for this purpose is called an interferential refractometer. The various methods of producing interference-fringes, the methods of measuring the shifting of the fringes, or of bringing them back to their original position by com- pensators of various forms and calculating the shifting from the change of adjustment of the compensator, and the experimental details of the setting up of apparatus for these 174 Mr WILBERFORCE, ON A NEW METHOD OF purposes have all been so fully and ably discussed by Quincke (Pogg. Ann. 132, p. 29, 1867) that there is nothing to be added on the subject. Different experimenters have from time to time employed in their refractometers fringes produced by almost all the methods mentioned in the above paper, but the only instruments of which it is necessary, on account of their connection with experiments of a nature similar to those that I have undertaken, to give a detailed description are those of Arago, Jamin, and Fizeau. In Arago’s refractometer (Zuvres, Vol. x. p. 313, 1858) there is a slit placed in front of the source of light and a convex lens is employed which is placed so that the slit is in its principal focus. Two tubes are arranged so that half the light which has been refracted through the lens passes through each. After emerging from these tubes the two beams pass through two slits a millimetre wide and two millimetres apart which are separated by a small prismatic obstacle whose vertex is towards the tubes. They next traverse two plates of glass of equal thickness which can be adjusted so as to make any angle with the beams and which serve as a compensator, they are then rendered convergent and intersecting by the object-glass of a telescope, and the resulting interference-fringes viewed through its eye-glass. The arrangement is represented in plan in fig. 1. It was used for determining the difference of refractive index of moist and dry air. It will be seen that the method of producing these fringes is in reality Young's, a very primitive one, that the illumination is feeble, as a narrow slit has to be used to admit the light, and that the interfering rays are only separated by a distance of two millimetres, a circumstance which, though indifferent in Arago’s experiment, might be often inconvenient. There is a further disadvantage in that the fringes produced by the method are generally small and close together. They can, however, be made rather larger by inclining the plates of the compensator so that the angles of incidence of the rays upon them are large, The interference arrangement employed by Jamin (Ann. de Chim. et de Phys. 3 Series, Vol. 52, p. 163) is one which could be very advantageously used in all researches on variation of refractive index, for in it all the defects mentioned in connection with Arago’s instru- ment are overcome. The fringes used by Jamin are those which were observed by Brewster and explained by Herschel. Two glass plates with plane parallel surfaces and exactly of the same thickness are required. These are silvered at the back and placed almost parallel to each other, and light is allowed to fall upon one of them so as to be reflected thence in the direction of the other. If the light after reflection at the second mirror be re- ceived by the eye of the observer either directly or through a telescope, curved interference bands are seen which become larger and further apart as the angle between the mirrors is made smaller. The form of the bands has been determined by Ketteler (Beobachtungen iiber die Farbenzerstreuung der Gase, Bonn, 1865) and the subject will be more fully treated at a subsequent period. The general explanation of the formation of the fringes is very simple. If we consider any ray incident on the first mirror, we see that it is divided into two, one which is reflected at the first surface and one which is reflected at the second. The sizes of the mirrors are so arranged that rays which are reflected more than once at the hinder surfaces are cut off. A part of the ray reflected at the second surface OBTAINING INTERFERENCE-FRINGES. 175 of the first mirror emerges from the glass and is then parallel to the ray which had been reflected at the first surface. These two being reflected at the second mirror we have finally four rays, one reflected at two front surfaces, one at two back surfaces, one first at a front surface and then at a back surface, and one first at a back surface and then at a front surface. The first two are cut off by means of suitable diaphragms. The last two will evidently be parallel, and nearly equal in intensity since the mirrors are nearly parallel, and will have traversed nearly the same length of path. Thus they will give well-marked interference phenomena if viewed by the eye. The course of the rays and the position of the diaphragms are shown in fig. & The more obvious advantages of this arrangement are that the rays between the mirrors corresponding to any particular incident ray are parallel, and that the distance between them may be made reasonably large by employing thick mirrors and a suitable angle of incidence, both of which circumstances make it easy to pass the rays through double tubes like those employed by Arago. We may further notice that the size of the fringes will evidently depend upon the angle between the mirrors and can thus be varied at pleasure, and that no slit need be used between the source of light and the first mirror, and thus the illumination will be increased and observation made more easy. We now come to the instrument employed by Fizeau in his experiment upon the velocity of light in moving media to which reference has already been made. The dis- position of its parts will be seen from fig. 2. Sunlight falls on a cylindrical lens A, and converges to f, thence it falls on a plate of glass B and is reflected as if it came from g. The light next passes through a lens C so placed that g is in its principal focus. A part of it next traverses a thick glass plate D, then the two parts pass through tubes containing the fluid to be ex- perimented upon, next they fall upon the lens # whose centre is in the line joining g to the centre of C, and in whose principal focus is a plane mirror F’ perpendicular to this line. From F to g each beam travels over the path that had been taken by the other, they interfere at g and are observed with the eye-piece G. By using the glass D we allow of a considerable separation (9 millimetres in the actual experiment) of the rays within the tubes without making the fringes exceedingly small. Their size may be varied by altering the inclination-of D to the rays. The arrangement may be exhibited in another way by taking the reflection of the whole apparatus in the mirror Ff, as in fig. 3. (The arrowheads show the direction of motion of the fluid.) We obviously may consider a luminous line at g’ to be the source of light. Looked at in this way Fizeau’s instrument becomes a kind of double Arago’s refractometer, but the combination of the plate D and D’ now can only be used to enlarge the fringes and will not serve as a compensator. The peculiarity of Fizean’s method is that he makes each ray retrace the path traversed by the other. One practical outcome of this is that the effective length of his Vor. XIV. Part II. 23 176 Mr WILBERFORCE, ON A NEW METHOD OF tubes is doubled, as may be seen from fig. 3. Thus, by further making the fluid in the tubes move with equal velocities in opposite directions the effect on the velocity of the light is four times as great as would be observed if an ordinary refractometer of the same dimensions were used in one of whose tubes the fluid was at rest and in the other in motion. There is however another and a far more important advantage of the two rays travelling over the same path in opposite directions. In this case if the refractive index of any part of the substance employed be altered by any accidental cause, such as change of pressure or temperature, there will clearly be no effect upon the fringes. The only change in the velocity of light which will shift them will be one which is reversed when the light travels in the opposite direction. The importance of thus being able completely to eliminate the effects of all causes except the one with which we are concerned cannot be overrated. We may indeed be able readily to distinguish between the effects of change of temperature or pressure and those of the motion of the medium, but if the latter are small they may be completely masked by the former, and at least cannot be observed with nearly the same accuracy as if the former did not exist. This difficulty was one under which Roiti laboured to a considerable extent, as will be seen from his account of his experiments, and it is principally due to this that the superior limit which he was able to assign to the velocity of the medium was such a very large one. It may be observed that if Fizeau’s method, or any other in which the rays traverse the same path, be adopted, the size of the fringes and the distance that they shift must be measured and the change in the velocity of light which causes the shifting calculated ; for the circumstance which renders their position independent of change of refractive index of any substance they may pass through will in general preclude the use of a compensator. PART IIL It seemed to me to be necessary to the successful prosecution of a research on the effect of a displacement current of electricity upon the velocity of light that a form of apparatus should be employed for exhibiting interference effects between two rays of light which should allow (i) the rays to traverse the same path (consisting in part of two parallel straight lines) in opposite directions, (ii) the fringes to be large and the field of view bright, and (ii) the parallel parts of the path to be as far apart as possible. -The reasons for the first requisite have been already fully discussed, the second is necessary if a small shifting of the fringes lasting but for a short time is to be ob- served, and the better the third condition is complied with, the more possible it is to arrange condensers in the path of the rays so that a displacement current may be made OBTAINING INTERFERENCE-FRINGES. 177 to travel with one ray at a part of its path without travelling against it at another part. The first of these desiderata has been pointed out to be the peculiar advantage of Fizeau’s method, and the second and third are seen to be to a great extent attained by Jamin’s arrangement. What must be sought after is a form of apparatus which will combine these advantages, and it is further desirable that it should be simpler in its parts and easier to set up than Fizeau’s, in which the cylindrical lens has to be very accurately focussed, and not only the position but also the direction of the small mirror F (fig. 2) most carefully adjusted. It occurred to me that suitable interference-fringes might be obtained by the following method. Two plane reflecting surfaces are fixed so as to be accurately at right angles, and a thick mirror, the surfaces of which are plane and parallel, is arranged so that they are uearly parallel to the line of intersection of the two perpendicular mirrors. Any ray of light incident on the thick mirror has a portion reflected at the front surface, and a portion reflected once at the back surface and then emergent into the air. These are reflected by each of the two mirrors, but in reverse order, and return to the thick mirror where they are again reflected. By properly placing diaphragms we intercept all rays except the two which have undergone a reflection at each of the two surfaces of the thick mirror, and these are in a condition to interfere. The course of the rays and the position of the stop may be seen from fig. 4, where the rays travel over nearly the same path, and it is clear that we can by shifting the position of the perpendicular mirrors make the two portions into which any particular ray is divided traverse paths which would be exactly identical if the thick mirror were accurately parallel to the line of intersection of the two. In fig. 4 M is the thick mirror, A and B the mirrors at right angles, OF the course of the incident ray, and X the stop. It will be seen that by means of this apparatus all the conditions which were above stated as desirable are attained. The form of the fringes obtained when monochromatic light is allowed to fall in all directions upon the thick mirror can be most easily investigated by employing a pre- liminary artifice. We know that when we form an image of any object by successive reflections at two perpendicular mirrors, on whichever of them the light is first incident, the position of the image is the same. Let (fig. 4) M’, O', F’, G’, H’ be the images formed by the mirrors A and B of M, O, F, G, H. Then we see that the rays retumn from A and B to M as if they had had their rise in a ray O'F’ reflected by a mirror M’, where it is clear that the angle between M and its image M’ is twice the angle that M makes with the line of intersection of A and & Thus our fringes will be of exactly the same nature as those obtained by means of Jamin’s mirrors. 23—2 _ ~J io 6) Mr WILBERFORCE, ON A NEW METHOD OF It has been already stated that a discussion of the form of these fringes has been given by Ketteler, but the methods that he has employed for obtaining some of his formulae seem to me to be so needlessly lengthy that it would not be out of place to show how the question may be more simply treated, Let D be the thickness of the glass plate M (fig. 5), w its index of refraction, ¢, ¢' the angles of incidence and refraction of a ray O'F’ falling on M’, y, y' the angles of incidence and refraction of the rays F’h and H’k falling on M. Draw a plane U through H' perpendicular to H’k meeting F'h in n, and draw any plane V perpendicular to the three rays leaving M at k, f, and h, and meeting them in p, q, and r. The difference of path between the two rays from O’ to U is clearly Qu. FG — F'n D (ect Reger tan ¢’ sind = 2uD cos ¢’. The path from H’ to p is known to be the same as that from n to r, Thus the difference of the paths from H’ to p and from n to q is the same at that between the paths from n to q and from x to 7, that is 2uD cosy’. Hence the whole difference of path of the two rays from 0’ to the plane V is 2uD (cos ¢' — cosy’). If now these rays together undergo any number of reflections and refractions and finally meet, the paths from V to the point of intersection will be equal. It will be seen that the difference of path between two rays depends only on the direction and not on the position of the ray which gave rise to them. Thus if all the rays which leave the mirror M are allowed to pass through a lens in the focal plane of which a screen is placed, each point of the screen will be illumi- nated by a set of pairs of rays, the difference of path between the two rays of each pair in the same set being the same, and on the screen we shall see a series of bands alternately dark and bright, if the light be monochromatic, or of bands of white light separated by dark spaces with coloured edges, if ordinary light be employed. In order to find the form of the bands on the screen we must remember that the line joining the centre of the lens to the point on the screen illuminated by any set of rays is parallel to the path of the rays before they reach the lens. Let us for the sake of simplicity consider the two mirrors at right angles to be vertical and the plane of the paper horizontal, and let M (fig. 6) be a section of the thick mirror by the plane of the paper. Let SZ be perpendicular to the sections of the surfaces of M by the plane of the paper, p the angle a ray QS incident on M at S makes with SZ, and @ the angle that the plane QSZ makes with the plane of the paper. OBTAINING INTERFERENCE-FRINGES. 179 Let v be the angle that the thick mirror makes with the vertical. Then we see at once, from fig. 6, that cos = cos y cos p — Siny sinp sin A cos 6 = cosy cosp +siny sinp sin @) © Now, since pw cos f =,/p? — sin’ TET if A be the difference of path, we have = = Rea + (cosy cosp+siny sin p sin 0)? —/w?—1 + (cos v cos p—sin vy sin p sin 6)’. Put ooze. The above equation can be most easily rendered rational by putting it in the form o=Ja4P-Ja=B, whence "8 = Jat B+ Ja-B, and (c + oI =4(a+ £). Thus the equation as rationalized is (c+ 4 cosy cosp siny sin p sin 0)? = 4c* [u?— 1+ (cosy cosp+siny sinp sin @)’]...... (i). Now since, neglecting signs, p and @ are the same for the incident and the reflected rays, by substituting in this o+y ye tan*p = tan@ =7 z where f is the focal length of the lens, and considering c as a parameter, we get the equations in rectangular co-ordinates to the curves of equal illumination on the screen given by light of any particular refractive index. These equations will however only hold for small values of 2 and y, since the focal plane is not the locus of the foci of parallel pencils falling on the lens in different directions, but is only a tangent plane to it. Let us imagine conical surfaces with the centre of the lens for vertex and Q) for the equations to their generating lines to be drawn, and consider their intersections with a sphere of radius (a) having the centre of the lens as centre. These curves will not be unlike those formed on the retina (as seen from behind) if the light were received directly into the eye. Neglecting powers of v above the second, equation (i) becomes c +16 sin*p cos*p sin’@. v* = 4c (u? — 1 + cos*p) + 4c*v* (sin®p sin?@ — cos*p), 180 Mr WILBERFORCE, ON A NEW METHOD OF or, as a further approximation, since both c and y are small, 4,sin*p cos'p sin"@ .2' =a" Gi" SI ps c oasis (ii). The equation to the projections of these curves on a plane parallel to the lens is got by putting 2 2 sin*p = ca L \ tang =2 | a Hence we get C [ata — a? (a? +97) Hy? (@ — a HY) ccc eecevecnecneceseonnsnes (iii). The tracings of these curves for different values of ¢ are given in fig. 7. If the axis of the eye is inclined to the line SZ (fig. 6), as will be in general the case, we shall see only a part of the above system of fringes. For example, if the eye be placed as in fig. 4, remembering that the image on the retina must be inverted to give what is actually seen, the bands will be something like those shown in fig. 8. The distance apart of the rays F’h, H’k (fig. 4), which it is advisable to have as large as possible, is with our notation evidently 2D tan ¢’ cos ¢. If this is a maximum it is easily shown that 2 tan*¢ = ae i For glass «=, therefore ¢ is about 50°. The practical details for the setting up of the apparatus for these fringes are as follows : An ordinary lime-light was used as the illuminating source. Since the rays finally leaving the thick mirror follow the same path as those incident upon it, it was necessary to reflect the light upon it by means of a piece of glass. In order to avoid the trouble eaused by having light reflected at each of two surfaces incident upon the mirror, the front surface of a prism of small angle (about 18°) was used as a reflector. The thick mirror was one of a pair furnished by Elliot Bros. for producing Jamin’s fringes. Its thickness was about 20 mm. and its surfaces on being tested proved to be very accurately plane and parallel. With the two perpendicular mirrors I had some little trouble. I first tried to use a right-angled prism, but could not obtain one whose faces were accurately perpendicular. Ordinary mirrors would be very annoying to use, as the reflections from the front surfaces would render the field of view confused. I next tried mirrors of black glass such as are often used for Fresnel’s interference experiment, but found the intensity of the light reflected from them to be exceedingly small. Finally I used a pair which were silvered on the front surface and which reflected a very large proportion (almost 90 per cent.) of the light that fell on them, The glasses were carefully selected before being silvered and were very accurately plane. I had at first expected that interference-fringes would be produced if the angle beween the mirrors were approximately a right angle, but I found that unless it was accurately OBTAINING INTERFERENCE-FRINGES. 181 so none were to be seen. That this is in accordance with the theory given above is clear, for if the mirrors are not at right angles the two rays to which any ray of light gives rise will be no longer parallel but will include an angle which varies with the angle of incidence of the original ray, and their difference of path will depend upon the position as well as upon the direction of the original ray. Thus the screen cannot be placed so that all the pairs of rays intersect on it, and if at any point of it a pair of rays do happen to intersect the brightness there will not be sensibly altered, since it is illuminated by a large number of rays not in a condition to interfere. The interference-fringes were viewed through a telescope focussed for an infinitely distant object. The most convenient method of supporting at once the telescope and the small prism was to employ a spectrometer from which the collimator had been removed. The thick mirror was mounted on a plate so as to be movable in the direction of its surface, the adjustment being made with a screw. The plate was fastened to a large block of lead with three short legs. This was placed on a board which was supported by three levelling screws. In the block of lead three screws were driven, and a plate of glass rested upon them. The thick mirror being adjusted so as to be accurately vertical, the screws were arranged so that the glass plate was horizontal. Thus by observing a spirit level placed on the glass plate it was easy at any time to bring the mirror back to the vertical position by means of the levelling screws that supported the board. A drawing of the arrangement is given at A in Plate Iv. The thick mirror was in the first instance adjusted to be at right angles to the line of collimation of a good surveyor’s level by focussing it to infinity, illuminating its cross wires, and getting their reflection im the mirror coincident with them. The two mirrors which were to be at right angles were supported in the same way as the mirrors used in Fresnel’s interference experiment described in Glazebrook’s Physical Optics, p. 119, except that the second mirror was fastened in a position approximately perpendicular to the one that it usually occupies. The support of the mirrors was then fastened, so that the fixed mirror was vertical, to the upper part of an alt-azimuth stand, whose base was levelled so that the mirrors as a whole could be turned about a vertical axis. A drawing of the mirrors and stand is given at B in Plate tv. The movable mirror was then by means of successive small motions of the screws against which it was pressed rendered vertical and at right angles to the fixed one. The criterion of the accuracy of this last adjustment was that when a telescope focussed to infinity and its cross wires illuminated was pointed at their line of intersection the two images of the wires coincided. In order that no rays of light should enter the telescope but those that give the interference effects the apparatus is arranged as follows. Between the lime-light and the small prism a lens is placed so that the rays from any point of the lime should emerge as an approximately parallel pencil. By this means the illumination of the field of view will be rendered more intense. A piece of black 182 Mr WILBERFORCE, ON A NEW METHOD OF cardboard with a vertical slit in it about 8 mm. in breadth is placed in front of the lens. The breadth is so chosen that if we look at the thick mirror from a little distance the rays reflected at its first and at its second surface are separated by a narrow interval of darkness. The perpendicular mirrors are then placed so that their line of intersection is in this dark interval. If we now move the thick mirror very slightly from the vertical and look through the prism towards it with the naked eye, we shall see a very bright portion, a fairly bright portion with interference curves, and a much fainter portion. The first is due to light which is twice reflected at the silvered surface of the thick mirror, and can be got rid of by moving the mirror parallel to itself by means of the screw. The third is due to light reflected twice at the first surface of the thick mirror, and can be cut off by placing a diaphragm between the prism and the mirror and moving it until it just encroaches upon the fringed part, which is the portion we require. The course of the rays and the methods of cutting off the superfluous ones can be readily seen from fig. 9. It may seem that, as the diaphragm YX is sufficient to keep the light from falling on the further part of the front surface of the thick mirror, the slit S could be dispensed with, but, by illuminating only as much of the prism as is necessary, we keep out a great deal of scattered light which would otherwise pass into the telescope. With the naked eye I found it quite easy to make out the form of the fringes given in fig. 8. The only rays which are to be employed in the experiments on the velocity of light being those that are almost horizontal in their course between the thick mirror and the perpendicular mirrors, the others were excluded by placing a diaphragm with two small equal holes at a suitable distance apart between the mirrors. When this was adjusted so that the light could be seen through the holes the telescope was pointed in the direction in which they were seen through the prism. On looking into it the field of view was observed to be very bright and crossed by fine large dark bands which were horizontal and had rather narrow fringes of colour at their edges. That the bands should be horizontal is clear since what we see is that part of fig. 7 enclosed in the dotted circle. By altering the inclination of the thick mirror to the vertical we can make the bands move up or down, at the same time becoming larger or smaller, but if we make the mirror so nearly vertical that the bands get very large indeed before vanishing they are no longer suitable to work with, as they become indistinct and much distorted, the latter being due according to Quincke to imperfections in the workmanship of the glass. In order to make it evident that the fringes would not be shifted by any change in the refractive index of the substances through which the rays passed, plates of glass of various thickness were placed before one of the holes in the diaphragm and it was found that the fringes were unchanged if the plate was placed perpendicularly to the rays passing through the hole, but if the plate was inclined the fringes were shifted, as cau evidently be explained by the theory of Jamin’s compensator. OBTAINING INTERFERENCE-FRINGES. 183 It will be noticed that if an inclined plate be put before one of the holes the two rays no longer traverse the same path, the one being raised and the other depressed by passing through the plate if it be inclined to the vertical. The permanence of the fringes in the first experiment is a sufficient guarantee that they would not be affected by any change of refractive index which it is possible could occur, having regard to the symmetrical form which the apparatus will take. I thought that, the arrangement being set up, it might be worth while to show that the fringes can be exhibited on a screen in the focal plane of a lens, for Jamin (Cowrs de Phys. Opt. Phys. Chap. 1.) gives an explanation which makes their formation depend upon the light having passed through some small hole such as the pupil of the eye, and which seems certainly unsound. Placing a screen of ground glass in the focus of the object-glass of the telescope and examining it with the eye-piece I found that the fringes were distinctly visible on it and that therefore Jamin’s explanation breaks down, PART IV. The question as to the most convenient arrangement for allowing a “displacement- current” of electricity to pass along the paths of the interfering rays next arises for decision, First we must notice that the time-variation of the line-integral of the electric “dis- placement” along the path of either ray will constitute the whole displacement current along that ray. Now as the ray returns to the point from which it starts, the line-integral of the electric displacement taken all along its path is always zero if the path is all the time through a substance of constant specific inductive capacity. Thus we shall have to use a condenser containing a dielectric whose specific inductive capacity is different from that of air, The simplest arrangement that can be made is represented diagrammatically in fig. 10. Two plates A and B are placed side by side and two others C and D are placed side by side and opposite to them, being separated from them by a medium of specific inductive capacity K,, that of air being K,. A and D are connected and raised to potential JV, and B and C are connected and put to earth. Small holes in the plates allow the rays of light to pass through, Let us suppose that the mirrors are so far away from the condenser that the potential at each of them is zero. Then, if P be the electromotive force at any point of the path of a ray in the direction of the ray, the line-integral of the displacement will be R 2 ug Pds, M 4c0r _ R C Ay vgs Se Se 2f Fe Pda4.2 | Hee pad Mu 4a A 4cr Vou. XIV. Part II. 24 184 Mr WILBERFORCE, ON A NEW METHOD OF R Now | as —i0) M [, Pas= v.| Hence if +r be the time of charging the condenser the line-integral of the mean displace- ment current will be Ko KG, Vi Wr Tt For the other ray this will of course be the same but with the opposite sign. We see that the result is independent of the thickness of the dielectric between the plates of the condenser. Let the velocity of light in air be (v) and the velocity of the “medium” for a dis- placement-current f be af Thus the velocity of light in the moving medium is OS seers and the time of traversing the space ds is ds 2 ds (1 6 =) since af is small compared with v. Thus the whole time by which one ray is accelerated or retarded by the motion of the “medium” is os 2S KG wie an [fle = 5 : 7 The difference in time taken by the two rays to traverse their course will be twice this, or o K,-K, Vv lp oe eee Thus, their difference of path in air will be a K,-K, V 0) dom 25t. Virekt ie A piece of plate glass was employed as the dielectric of the condenser. Although its thickness does not appear in the formula just investigated, yet it is clear that it must be large compared with the diameter of the holes through which the light passes if the potential over each of these is to be sensibly the same as that of the conducting plate in which they are made. The plate of glass that I used had a thickness of about 26 mm. and its surfaces were very nearly plane and parallel, so that when it was placed in the path of the rays and moved about the fringes were not at all displaced. The conducting plates of the condenser were made of pieces of tin-foil. These might have been fastened directly to the thick plate, but it was thought to be more convenient OBTAINING INTERFERENCE-FRINGES. 185 if different parts of it could from time to time be used if necessary, and another disad- vantage of fixing them was the extreme difficulty that had been found in some preliminary experiments in getting the holes in the tin-foil exactly opposite to each other, so the pieces were fastened with shellac varnish on two thinner glass plates of thickness 3 mm. each which were brought up so as to be in contact with the thick plate. The holes in the tin-foil might have been at first made at any convenient distance apart, so long as it was less than the maximum distance between the two parts of a ray, and the angle of incidence of the light on the thick mirror adjusted so that the rays should pass through them if the plates were properly placed. This would however have been very troublesome, and it was found easier, having set up the interference arrange- ment in a permanent manner, to make the holes at the suitable distance apart. This distance could be most accurately found by placing between the thick mirror and the perpendicular mirrors a screen in which a narrow horizontal slit had been cut so that the height of the slit was the same as that of the middle of the thick mirror, and sliding up and down before it another screen in which two narrow slits inclined at equal small angles to the vertical were cut, until the rays passed through the holes thus formed, The pieces of tin-foil on the glass plates were of the shape shown in fig. 11. The diameter of each of the holes was 5 mm. and their centres were 15 mm. apart. In order that sparks should not pass from one piece to the other a slip of glass was fastened half way between them and at right angles to the plate by means of electrical cement. Directly the condenser was employed, however, the cement was perforated by a spark. A block of paraffin was then melted on to the plate, the surface of the glass being first carefully cleaned and then rendered very slightly greasy with turpentine, and by means of this the pieces of tin-foil were completely insulated from each other. The same was of course done with the other plate. The shape of the paraffin block is shown by the dotted lines in fig. 11. After the paraffin was put on, the rest of the front surface of the glass was given several coats of shellac varnish, so that the insulation of the pieces of tin-foil was rendered very good, The readiest way of setting up the condenser in the path of the rays was found to be to take one of the thin plates and to support it vertically and perpendicular to the light incident upon it, so that the rays went through the two holes in the tin-foil. This adjustment could be made with very little trouble by placing the plate with the holes on the same level and nearly in the right place, and then moving the perpendicular mirrors slightly by turning round the upper part of the alt-azimuth stand. The other plate was then put in contact with it back to back, and its support adjusted so that the holes in the opposite pieces of tin-foil exactly faced each other, which, on account of the small thickness of the plates, was also easy. The second plate was then carefully moved away from the first in such a manner that the rays still passed through the holes. The thick plate mounted on a frame was 24— 2 “ 186 Mr WILBERFORCE, ON A NEW METHOD OF then placed between them and parallel to them, and the two plates moved up to it one after the other, taking care as before that the rays had a path open to them. The plates can be adjusted perpendicular to the rays incident upon them by noticing the direction of the light reflected at their surfaces. The contacts with the tin-foil plates were somewhat difficult to make satisfactorily, as on account of the high potential to which they had to be raised brush discharges into the air were liable to occur. An instrument was designed and made for the purpose which is shown at C in Plate 1v., in which the contacts are made by curved rods which descend from two horizontal cross bars and press by their own weight against the plates. In order that any effect that the displacement-current may have on the fringes should be observable the time during which the current passes must not be less than } second. This condition was satisfied by connecting one of the cross bars of the contact-maker with the inner coating of a Leyden jar of which the outer coating was put to earth, as also the other cross bar. The former bar could be connected by a sort of large switch mounted on a block of paraffin either to the outer coating of the jar or to one electrode of a Holtz machine of which the other was to earth. An earth-connected knob was placed near the knob of the jar so that its potential should not rise above a certain value, depending on the distance of the knobs, which could be adjusted at pleasure. The potential of the jar had to be limited so that the insulation of the condenser should not be broken down. The whole arrangement is shown in plan in Plate m1. and in elevation in Plate Iv. Various sizes of Leyden jar were tried which, together with the condenser, took from } to 1 of a second to charge. On connecting the upper bar of the contact-maker with the electrical machine by means of the switch, the potential of the parts connected with it rose rapidly from zero to a certain limit when they were discharged by a spark passing to the safety knob, and the same variation of potential recurred. A considerable number of experiments was made with this apparatus, but no shifting at all of the fringes could be observed. To be able very accurately to decide upon the question of whether there was any shifting or not, a thin wire was placed in the focus of the object-glass of the telescope so as to be parallel to the fringes as seen through the eye-piece, and the telescope pointed so that there was a very narrow line of light between the wire and one of the black bands which was very well marked. A widening or narrowing of this line corresponding to a shifting of the fringes over , of their breadth could have been detected. Hence the difference of path of the rays is less than ;4, of the mean wave-length of light, that is, less than 6 x 107% centimetres. Hence referring to the value obtained for the difference of path = ies gu <6 x 107 centimetres. v ov T OBTAINING INTERFERENCE-FRINGES, 187 Now if C be the mean value of the displacement-current through the glass while the condenser is being charged, and if 6 be the thickness of the glass, eee 1 ~ 4a 78° Hence gies Ses .6.C<6 x10" centimetres, v Ka or, if 6 be measured in centimetres, KE Vg Binrersy ae aesreee 5 ; and =. C is clearly the velocity of the medium corresponding to a displacement-current C. Now K,=32x K,, 6=26+ 6=32, v=3 x10" in C.G.S. measure ; therefore a. C< = x 10° in G.G5. units, <41 metres per second. If the superior limit thus obtained be compared with that given by Roiti, namely 200 metres per second, the increased accuracy attainable by the use of the interference apparatus which I have employed will be apparent, more especially if it be borne in mind that the vessels containing the electrolyte in his experiment were 45 centimetres long, as against 32 centimetres of glass used by me. The great disadvantage of making observations such as those that I have described upon a displacement-current is that the current is of necessity exceedingly small compared with those which it is possible to have conducted through an electrolyte, on account of the duration which it must necessarily have in order that its effects should be visible, and that therefore, though the velocity of the “medium” oC can be shown to lie within very narrow limits, yet C is so small that we derive little information about the value of a. Thus, in the above experiments, V =about 500,000 volts., =$10* in cas, T=1, SSR Shy K, = ion . Ow Whence we obtain: Re V; 5 C 10m ~ dor 7.8 9X 8" = 22x 10° in ©.G.s. units per square centimetre, = 2:2 x 10” ampéres per square centimetre. IX. On the mutual action of oscillatory twists in an elastic medium, as applied to a vibratory theory of electricity, By A. H. Leany, M.A., Pembroke College. [Read, Nov. 23, 1885.] THE oscillations discussed in this paper are such as may be produced in an elastic medium by the tangential displacement of the surfaces of tubes of small sectional area: the tubes either forming closed curves or extending indefinitely in both directions. The direction and circumstances of the motion are in general analogous to ordinary vortex motions in an incompressible fluid, and it is shewn that, if the period of the oscillation be such that the waves produced are “long”; so that the inverse square of the length of the wave can be neglected compared with the corresponding power of ordinary finite distances; then the displacement due to these tangential displacements is the same as the velocity due to a vortex ring of the same form as the tubular surfaces. There are however sufficient differences between the two theories to prevent the name “vortex” being given to this species of motion, and we shall for the future call the tubular surfaces “ oscillatory twists” or “twists” for the sake of brevity. It is found in the subsequent work, by making use of the ordinary equations of an elastic medium, that two such twists will act upon each other so as to give rise to a force which depends upon the product of the angular displacements of the two surfaces, and which is of course of the second order in these dis- placements. Now the equations of elasticity have been established on the assumption that terms of the second order may be neglected, and hence an objection may be raised at the outset that, if we are to take account of forces depending upon the second order terms, we must investigate the equations of elasticity to the second order, and until this is done that the whole theory is unsound. In answer to this objection it may be remarked that, if the medium in which the motions are supposed to take place is “perfectly elastic” or non-dissipative, as the luminiferous ether must nearly be, if it be capable of trans- mitting vibrations from ordinary stellar distances as it is supposed to do in the undulatory theory of light, then it will appear to be at least probable that im such a medium the energy due to a given extension is equal to that due to an equal and opposite compression. If this be so the terms of the second and all even orders will disappear from the equations of elasticity, and consequently if we keep below the third order of the displacement our equations will be correct and the results arrived at can be relied on. It has been shewn* that if W is the total energy of an elastic body in any state of strain, W is a function of s, s,s,a 8 y where oe ee 1 da 2 dy’ 2 4dz5 _dw dv u , dw dv du eee ae = +— CS aes diet Ode" dz 1 daz dy’ * See Thomson and Tait, Nat. Philosophy, Art. 673 and Green’s Mathematical Papers, p. 249. Mr LEAHY, ON THE MUTUAL ACTION OF OSCILLATORY TWISTS. 189 and since s,, s,, etc. are small W can be expanded in a very convergent series of the form W= W,+ W,+ W,+ ...... 1 2 where W, is a homogeneous function into which these six quantities enter in the n’'th order: wu, v, w being the displacements parallel to the co-ordinate axes. In the ordinary theory W, and W, are shewn to be zero, and the equations of an elastic medium depend upon W,. Following Green notice that the energy must remain unchanged if —z and —w are put for z and w; ie. as much energy is spent in stretching the medium up as in stretching it down. This substitution would change a and £ into —a and —,, and we see that the energy must not contain any odd powers of « and 8. Similarly it must contain no odd powers of y. Thus the 56 terms in W, reduce to 19, 3 3 3 3. 12 Sg» Ss 3 namely : s SGU AS SN, GF, She, Sky, Osea BBS 2 2 2 + 2 2 2 5 2 2 2 ws, BS, 783 %,, BS., Ye; 5, 88,5 9°8,. But, if the energy of compression is equal to that of extension, W must remain unchanged by putting —w for wu, —v for v and —w for w. Thus we see that in this case all the terms in W, vanish and W becomes W,+ W, to the third approximation, whence it follows that the equations, which were established by putting W equal to W,, a will continue to give correct results if terms of the third order are neglected. As practically bearing on the question whether the equations of elasticity can be relied on in the case of the ether [ may mention that Professor Pearson of University College, has, in a recently published* paper, investigated the equations of elasticity to a second and higher orders; and that he finds among several other curious results that, if the second powers of the strain are taken into account, the index of refraction of light will depend upon the intensity. As no trace of such a phenomenon has, I believe, been observed, this fact appears to strengthen the supposition that results obtained from the equations on the ordinary hypothesis will be reliable up to the second order of approximation at least. 2. The results obtained in this paper may be briefly summed up as follows. In the first place a twist along any curve will give a displacement equal in magnitude and direction pb*w sin pt to the magnetic induction due to a current of strength where b is the radius of the twist, wsin pt its angular displacement, and yp the coefficient of magnetic permeability of the medium: the waves being supposed to be “long” asin § 1. This is shewn to be the case if the axis of the twist is either rectilinear or circular, and might be inferred from the corresponding theorems in fluid motion on kinematical considerations, The second result, which does not appear to bear any analogy to the theorems of fluid motion, is that, if the field of vibration is explored by a rectilineal twist of the same period as that of the vibration, the twist will experience a force at right angles to the plane containing the twist and the direction of the displacement which would exist if the twist were removed. This force is in such a direction that a person standing in the twist so that the rotation of the surface of the twist is in the opposite direction to that of the hands of * Proceedings of the Cambridge Philosophical Society, Vol. vy. Part rv. page 296. 190 Mr LEAHY, ON THE MUTUAL ACTION OF a watch, and having the direction of the displacement acting from his right hand to his left will be urged forwards by the force. Displacement produced by a rectilineal twist. 3. Taking cylindrical co-ordinates it is clear that the displacement at any point will be wholly perpendicular to the axis and to the shortest distance from the given point to the axis. Denoting this by v we shall have (by variation of angular momentum) Cra © 2m"), if p is the density of the medium and 7 the tension exerted by the consecutive layers of the medium. p.2a0r Q ID) 0 P! e Q But if O is the projection of the axis and if two points PQ are displaced to P’ and Q’, ne [UU ON 7T=px angle RP'Q =H (GaAs dv 1 dv _v_p dv therefore we get Te Fe po ae Therefore v= Aevtl (hr) + A’e~*L, (hr) 2 is the complete periodic value of w of period ae if os and J, is the solution of Bessel’s equation of the first order, applicable to space outside a cylinder. This is known* to be 1 h?r? — hi'r* h®r® hr\ (hr — hr® I, (hr) =F. {1 - 2 +g} -}- {hr8, 597 8... — (C+ arr + log >) G Toner x!) 6) if T is Euler’s constant and S,=1+4+... = : If squares of hr are neglected, we have, as an approximation, a A sin pt ; r which is the approximate expression for the displacement given by a single oscillatory twist if the wave produced is long. * See Rayleigh, Theory of Sound, Art. 841. OSCILLATORY TWISTS IN AN ELASTIC MEDIUM. 191 Supposing the angular displacement of the twist at time ¢ to be wsinpt and the radius to be b, we have The magnetic induction due to a straight current of strength c is < if w is the ur “magnetic permeability.” Therefore the displacement due to a rectilineal twist is equal to the magnetic induction due to a current whose strength is equal to 4b°o sin pt. 4. From the result obtained for the case of two dimensions, it appears to be probable that a displacement due to any twist along the axis of a closed curve will be the same as the magnetic induction due to a current along the curve if second powers of rh, where h is the inverse wave length, are neglected. The integration of the equations of elasticity is however very difficult when the boundary is not symmetrical; but the required result can be obtained for the case of a twist along the circular axis of a ring of small section. Using “toroidal co-ordinates,” take as orthogonal surfaces a series of rings generated by the revolution of coaxial non-intersecting circles about their common radical axis, and a series of spheres cutting these rings orthogonally, the third set of surfaces being planes through the axis of revolution of the rings. These co-ordinates have been investigated * by Mr Hicks, who has applied them to find solutions of Laplace’s equation at the surface of an anchor ring or tore. If the rings are defined by u=const. and the spheres by v=const. it has been shewn that : sin hu 7? B00 () = 0) coshu — cos v sin v r cos@ = a ———__——_ cos hu — cos v y and @ being polar co-ordinates and @ the co-latitude. Also, if b is the radius of the cross section of a ring, ¢ the radius of circular axis of a ring, R radius of a sphere and a of the critical circle, cos hu = : sin hu= ; : ie AAS CE CAA IrDA COOE ECE ( The ordinary equations of elasticity can be transformed to these co-ordinates and an approximate solution obtained for the case of a long wave by neglecting the square of the inverse power of the wave length. If & be the displacement perpendicular to the surface of one of the rings and 7 that perpendicular to one of the spheres; & being reckoned positive when measured from the * Transactions of the Royal Society, Vol. 172, p. 614, Vor XV. Parr IL 25 192 Mr LEAHY, ON THE MUTUAL ACTION OF centre of a ring; and » being reckoned positive if measured from the axis when the point considered is above the plane of the critical circle; we shall have for the equations of elasticity mer. net sin hu pe dt? cos hu — cos v : de dw dn sin hu 2 ese 0080) Ee eee (A + 2) sin hu om (cos hu — cos v) FF PO Soran de du (X + 2) sin hu — + pu (cos hu — cos v) = Suaiacdoonst: where X and w are the ordinary coefficients of elasticity, p is the density, and* &(1—coshwu cosv) — n sinhu sin v sin hu 1 (dé o) == (aa tae (cos hu — cos v) + ee ee chhee 1 (x2 a ee) ee Esinv sin hu — 7 sin*h “} ; a \ cos hu — cos v These equations give, by differentiation and reduction, — (&sinhu+7 sin a} 5 du du d’e 2 — d’e d’e 5A 1l—coshucosy de sinv de _ OP dé . du** dv * sinhu(coshu—cosv) du coshu—cosv dv (A+ 2x) (cos hu — cos v)* AG); d'o dw l—coshu cosy dw sin v dw oP Oe (6) du’ * di sinhu(coshu—cosv) du coshu—cosv dv p(coshu—cosv If these equations are solved we shall be able to obtain general values of & and 7 which will satisfy any specified surface conditions. I have been unable to obtain solutions of these equations beyond a particular integral, it will however be noticed that when the corresponding equations are obtained in solving for a general displacement in polar or cylindrical co-ordinates, if symmetrical about an axis; the complete solution will, the squares of the inverse wave length being neglected, be the solutions of the equations got by neglecting the effective force. These will not be the general solutions, nor shall we be able to express an arbitrary displacement by means of them; but we shall find values for the displacement that will satisfy several definite surface conditions, and among these will be the conditions resulting from a uniform twist if the ring is of small section. Now equation (5), if the right-hand side be neglected, is Laplace’s ordinary equation, as can be shewn by transformation. This is indeed obvious since it is manifestly the transformed equation of the dilatation The solution of Laplace’s equation in toroidal co-ordinates which is applicable to space without a tore has been shewn by Mr Hicks to be Jeos hu — cosv 3A,P,, cos(nv — a,), * e« and w are connected with the ordinary dilatation and twist by the equations _ 1 d(r*u’) 1__ d(rv'sin 8) . sind /du' d(rv') ‘= 78 dr rsin 0 de oa. \ a0 dr where u', v’ are the displacements along the radius vector and the tangent to the meridian, See Lamé, Elasticity, Art. 84. OSCILLATORY TWISTS IN AN ELASTIC MEDIUM. 193 where P, is one of the solutions of the differential equation n 2 - + cot hu Ses —(v7—})P,=0. Hence we have to this approximation e = ./cos hu — cosv 2A,P,, cos (nv —4,), and it can be shewn that sin hu | aps = Be qa °° (nv — B,) ./cos hu — cos v It might have been expected that any boundary condition could thus be satisfied ; but this is not the case: for, when the values thus obtained for ¢ and are substituted in (4), the constants combine so that A, and B, give only one condition at the boundary. Substituting the value of w and putting —4B,-—{B, cos8, =C,, — 3B, sinf,=D,, 4B, cosB, — 3B, Coa =C,, 4B, sin8, — $B, sin8,= D,, Se i B,cos8,=C,,, = B_ sing," ** B, sing, =D... and making use of the nee relations, which are easily proved to hold from the sequence equations between the P, functions, given in the paper previously referred to, ‘ US A Seri ie ae sinhuP, ibe Weare ) , GP aa dP, 1) mare en _ 1) +4 (2n+ 1) Ta j J coaeP, = a {(2n+1) P.,, + (2n—1) P,_}, SOeOROOICHORCOUC 8); Sj nee Sa Jian) i du or Sn ( nti * n-1/) with sin hu Os +4 coshuP,=34P,, aP,, dP, cos hu aah +4 snhuP,= Fie we finally arrive at SSS UE dP GE dP, dP, ii £ =./coshu—cosv {D, aa + (6, ase C, i) sin v— (2.9.7 —-D,— ai 2) cos v +> (2 nt a — sin nv — (p t: Ll = cos nv du d TA du du (9) hu-— = n= seo ese [-C,P, + (C,P,-8C,P,) cosv + (D,P,—3D,P,) sinv + {(2n-1) C_,P,_,—(2n+1) CP,,,} cos nv + {(2n-1) D,_,P,.. — (21 +1) D,P,,4} sin nv] 25—2 194 Mr LEAHY, ON THE MUTUAL ACTION OF If we had made use of the value of e« we should have simply altered the arbitrary constants: but in the case of a twist there is no dilatation at the boundary, no wave of dilatation is propagated from the boundary, and ¢ is zero throughout, so far as the disturbance caused by a twist affects the surrounding medium. 5. These are the general values for the displacement at any point; it is necessary to see whether it is possible to satisfy the surface conditions over a ring of small sectional area. Taking all the constants to be zero, except C,, we have &=2C, sinv ab a/cos hu — cos », du n = C,(P, cosu—P,) Jcos hu—cos v. We shall shew that these values will satisfy the surface conditions over a ring whose axis is along the critical circle. Resolving along and parallel to-the axis of the system, we have _ &sin hu sinv + (1 — cos hw cos v) cos hu — cos v = C,(P, cosv — P,) J/cos hu — cos v, displacement parallel to the axis — &(1—coshu cosv)+7 sinhu sinv cos hu—cosv_ displacement perpendicular to the axis = . aP, ,~——— =— 20, sinv aa a/cos hu — cos », after reduction by relations (8). But if AA’ is a section of the critical circle displacement along AP =C, (2 ee - i z _ © cos hu — cos 0, displacement perpendicular to AP =—C, { cos ¥ (2 S - P,) +2 = + P, ; Jcos hu — cos v, b being the radius of the circle with centre at A. OSCILLATORY TWISTS IN AN ELASTIC MEDIUM. 195 *But in the neighbourhood of A, P, = 2e-* log (4e"), P, =e; therefore, if bis small, displacement along AP = zero, displacement perpendicular to AP =—-— 4Ce-', - cos hu =-2/20,5- Hence a displacement bw sinpt at every point of the surface of the ring of small section whose axis lies along the critical circle is represented by Coe sna a VIE. n =C,(P, cosv— P,) J/cos hu — cos v, if —2/2Ca=V’o sin pt. The displacement along and perpendicular to the axis of the system has been given to be WS faa (P, cosv — P.) J cos hu — cos v AB OS nn pace, ld ers Pen 11). Ages Bro sinpt dP, , m) = ” fae a dy “nv J/cos hu — cos v : a db ? : Putting P= = 7= | J/cos hu — sin hu cos @ . dd, 9 {eos hu —sin hu cos}? Jo dd — sin hu cos 6” oP ={ * Jo feos hu and, reducing by the relations, r + a 7” wer o cos hu = ae cosy = nae Wa Wotan Bata Mcale SA ae dyer thes ene ncer ek 12), sin hu = 20" S09 sin y = 2” 2989 (12) L 3 ra Ip if L*=(r* + a*)’— 4a’r* sin*@, r being the distance from the origin and @ the co-latitude, dé us Sehor Ee, — et : i n/cos hu — sin hu cos 0 7 P= | Jess hu — sin hu cos 6 dé x Tr — {7 dé 7a 4 = Z ‘ a8 = aia =2VE | Fizmaang ONE By ral J1-Hsinta.do fk’ 2 if =e, ; 196 Mr LEAHY, ON THE MUTUAL ACTION OF we get, in polar co-ordinates, w’' =—baw sin pt | Me aL =a a a 0 (7° + a* — 2ar sin @ cos ¢) i r cos cos Odd 0 (r* + a? — 2ar sin @ cos $)* *y' =+ Wao sin pt These are the components of the magnetic induction due to a circular current of bu sin pt 2 For if @ be the strength of the current, F, G and H the components of the vector potential at any point P, Fev. . (a0: @=2c{" Blo /P+a strength , where yw is the coefficient of magnetic permeability. cos padh ; —2ar sin @ cos¢’ therefore the components of 33 the magnetic induction are Cae r cos 8 cos add Be 2 a a ee? Jo (7? +a? — 2ar sin @ cos d) b=0, C ft r sin @ cosd—a c=-2- Jo (7? +a’ —2ar sin 6 cos ¢) See Maxwell’s Electricity, equations A, Art. 591. As a particular case let us suppose the radius of the circular axis of the ring to be small. In this case we shall get the simplest components by resolving along and per- pendicular to the radius vector from the centre of the circular axis, We get displacement along the radius vector = b’a*w sin pt i ata 3 0 (r* + a* — 2ar sin @ cos ¢) __ ab*a*o sin pt cos 0; * For ‘g »/cos hu —sin hu cos ¢ dp whence, taking P; = f »/ (cos hu sin hu cos ¢ d¢, and re- E 7 do ducing by above relation, we get required result. -|" {cos hu — sin hu cos ¢}#’ fs I" sin hu(1+cos*¢) — 2 cos moos, 9 ie {cos hu —sin hu cos p}# OSCILLATORY TWISTS IN AN ELASTIC MEDIUM. 197 displacement perpendicular to the radius vector and tending from the axis of} =0°a’o sinpt the system T rcosd —asind d 0 (°+a°—2ar sin8 cos$)® pow ( LA) 5 29 ° er OSD 2r* if only the lowest terms of - are taken into account. This is, as might have been expected, the same as the magnetic induction due to a magnetic particle whose axis is along the axis of the ring and whose strength is maa sin pt oe = 6. Before leaving this part of the subject a reference should be made to Mr Pearson’s “Note on twists in an infinite elastic solid,” published ia the Messenger of Mathematics for 1883, to which my attention has been called since writing the above. In this paper the following results are given, P a ee TIT a aw | 1 fn Was ee where V = r= ie dv, dV dU ae emia a a ay | W= ras dv, where u, v, w are the displacements due to any given twists &, 9, €; the system of dis- placements being so chosen that e=0. From these results equations (1) and (13) can readily be deduced as special cases. It should however be pointed out that in these equations it is required that every twist filament should presuppose a definite displacement at any point of the medium and no propagation is considered from a definite boundary. In fact, substituting these values at any point where there is no twist, we get 2 2 2 4 sop rote WES dt dt dt whence it appears that the relations are kinematical in character; and do not give dis- placements which satisfy an oscillatory displacement, or one that in any way depends upon the time. From the way in which equation (1) was obtained it is clear that the effect of a long wave to the first approximation is the same as if the effective force is neglected. Hence we could immediately deduce equation (13) by the help of Mr Pearson’s formule, since they do satisfy the equations of elasticity if these terms are neglected: and similarly it can be seen that the displacement due to twist along a curve of any form is the same as the magnetic induction due to a current along the curve if the squares of the inverse wave length are neglected. I have preferred to leave the work in its original 198 Mr LEAHY, ON THE MUTUAL ACTION OF form, since the displacements given in equations (9) will give the strain at ‘any point produced by several other displacements of the boundary besides a simple twist: in fact all which can be deduced from a dilatation and twist given by e =Jcoshu—cosv &(A,P, cosnv + A,’P,’ sin nv), sinhu P Ole ye o= ee >> (3. cos nu + B, cai sin nv) : J/coshu—cos v du du The general values for the displacement can also by the help of the foregoing be deduced for the case of a long wave, if the displacement is symmetrical about an axis. For in this case the dilatation is known* to be = SL ,r-Z, {1 i eal é to a second approximation if Z, is the zonal harmonic of the order n. Also we have S79 t7 = c/eos TiS ip ig TL r- DZ = Jooshu — cosy S (A,P, cosnv + A,'P, sin nv), since each side of the above equation is the general solution of Laplace’s equation in the case of symmetry about an axis, = LarOvZg, = bee p Hence 2 ny) Jcos hu — cosv 5 (C,P,, cos nv + C,'P,, sin nv), if the C’s satisfy the equations Ag =40, 20,3 A = 26,-4C, —3¢,, AS40730" A,=) $C =40 —56:, A/= 30-40’ -5C’, A,= 50,—40,-T7C,, Ate 50 40270. A, =(2n—1)C,,—40,—(2n+1) 0,4, Ag’=(2n—1)0,,'-40,/ —(Qn+1)C,,/ nei? as can be verified by differentiation with respect to r. Whence we get as the value of e, if fourth powers of hr be neglected, e=J/coshu — cosu &(A,P, cosnv+A,’P,’ sin nv) cos hu + cos v + ka? S(O. cosny + CP.” sii Ne)... ccesevanees (tay; »/cos hu — cos v if the C’s and A’s are connected by the equations just given. The complete value for » to a second approximation might also be determined and values for & and 7 obtained which would satisfy any given conditions at the surface of a ring the w of which is given; but the complicated character of the relations connecting the constants would make the determination of the displacement in any special case a matter of considerable difficulty. * Transactions, Vol. xiv. page 7, equations (8). OSCILLATORY TWISTS IN AN ELASTIC MEDIUM. 199 Leaving this part of the subject let us now see how a rectilineal twist will be acted upon if placed in a field of vibration. For this purpose it will be necessary to find the general expression for a displacement in cylindrical co-ordinates in order to get suitable surface conditions. The case of two parallel rectilineal twists will first be investigated and the general result deduced from this simple case. — 7. To find expressions for a steady periodic disturbance in an elastic medium un- limited in extent: if the displacement is given for all points which are on the surface of an infinite straight cylinder when in their undisturbed positions, the motion being supposed to take place in planes perpendicular to axis of cylinder. The disturbance will reduce to a case of motion in two dimensions, We shall have, by the ordinary theory of elasticity, the disturbance compounded of w and v along and perpendicular to radius vector from the axis where wu and v are given by the equations* de pdy du 2 = OF 2H) Inte db? de ie enmiinneae OS ; 7° a0” dr Poe : _1d(ru) ,1 ad where oS Sie ' ae” 1 du _1 d(vv) Yr dO 7r dr ’ r and @ being co-ordinates of undisturbed positions. Equations (1) reduce to d (~ = a Ide pr de dr\ dr)" r dP X+2py det*| (17) d " “7-2 d*y _ pr d’y al Cp oF G0 pe ae Thus ¢ and ¥y are independent, as can be otherwise shewn; and e¢ is the same arbitrary function of +2 as ¥ is of pw. To solve equation for e in a periodic form, put e= wer, whence w must satisfy the equation de fo da An a Ee ae (" ae) ta age tre =O, if pp = (A+ 2p) h. By Fourier’s theorem w is of the form Xw, sin (nb +a,), : dw, 1 dw, ieee zs : where w, satisfies ene las (x — “) =O". ..ucavaeecssrerereconcnere brs (18). * Lamé, Elasticity, Art. 78. Vout. XIV. Parr II. 26 200 Mr LEAHY, ON THE MUTUAL ACTION OF It is known* that the solution of this equation, which is applicable to space outside the cylinder, is died w, = A, (kr) F oat Wana ctudece dente Sethe sae caesar ate (19), a kr ( ker? Isr ( ker? Isyr* I88 , if w= (P+ im log 3) mor tora} + [ar ~ on a Ss to ae ge Sey devil 1 where B=ltgtgt-..--—, and T is Euler’s constant (= '5772156...). The solution applicable to space within the cylinder is the ordinary Bessel’s function of order n. Hence w, being of the form given above d n => 7 pt (Joyp)” 4 e= 2A, sin (nO +2,) e?* (kr) {3 coal W, with another term obtained by changing the sign of «. Changing the constants and substituting the trigonometrical form of solution for the exponential one, we have =A, sin (nO +4,) (kr)" lawert ae 2 2 pee 2.2 4 if y,=sin (pt + &) (r+ t8'5) (0 ae torpt--) + (Seoaep sees k? r 7 Ir* +5 cos (pt +) {I - Se tgp S| a wietare (20), and similarly, —— = dy B,, cos (n8 +8.) (hr) aaa sf Zo. where pp*=yph? and z, is the same function of hr as y, is of kr. To get u and v we have by equations (16) oe kdr hirdé 21) 5 Lae Lay | et de Wy einen vevmuae ete: eee (21), k’rdé@- hdr which will give general values for the displacement at any point in the medium. We shall require the terms in ¢, y, u and v corresponding to n=0 and n=1. * See Rayleigh, Theory of Sound, Vol. 1. Art. 341. OSCILLATORY TWISTS IN AN ELASTIC MEDIUM. They are ke? r kty* 24a | A e= KA, sin (pt +6){(P +log5) (1 = 9 < (4 Hr. kr* + k°A, cos ( pt + &) a ge tor gett sin eS a) {5 {1 euler 3 +k°A,sin (pt+ &). ky h*r® ee snl@ ir i —K’A,sin (pt+). {(T-+1o s3\(5 3 snl ) w {kh a ae ker a 4 4 —k°A,cos(pt+ &) i -s 4 +B 4 y is the same function of hr as e is of kr. Also for x= 0, (BrP drt ier 76" Tile —— 7 sin (pt + 6) ies oe ae 3)( Poa § é eit sie - ‘ia 2 D2 2 42 2 (pt + €,) ){(r + log iy = , (he a hir* her (H+ 60ST At oa, 2? Aa kr? ee kt et key 6 246" h'r® + — cos Uy sin (pt + &,) hir* —— sin ——* cos any Bae 2? 4 sae oe a Boe 1 “oF Act = j f, being the epoch (in y) corresponding to &, in e. And for n=1, Bh? 2? Thr = Asin (pt + &) sin (044) {1 + = + sin (pt-+ &) sin (0-+4,) {(T-+log e)("5 A, kr? +78 cos (pt + ,) sin (@+4,). Se ue »2 2? 4. ; oe hir* +B sin (t+ &:)sin (0+ 8) {1 - a =z = sin (pt + €) sin (@ + 8,) {(r + log Ls) (‘7 hir* ee =z cos (pt + &/) sin (8 + B,). 3 a(y 34 = +... Wr? Bk*r4 os oe] aw . 4 hiy* 2? . 4 Dae. a ae ae 5 +...) = (‘;- Wy? +...) ra S is 3k'r 2? 4 hir* 92 “s 4 2 + seve +...) 26—2 202 ool & == sin (pt + §) cos (8+ a,) {1 Oe a A + =z cos (pt + £,) cos (9 +4,) 4 ee Mr LEAHY, ON THE MUTUAL ACTION OF ky kety* Fa ae kr\ (er? Ie*r* hers Jer + a sin (pt + &) cos (8 + a,) (r+ log 3) (“ + oa - i - gg S2te as fay? byt eat) 3h'r* ‘Thtr* B,. Tees ss (pt + ¢,°) cos (8 + 8,) {1 reo Sear Bae \ /h2y? 4 22 4 — +2 sin (pt + [,') cos (0 + B,) {(r +108) é 3h vue] = (St a ee her? = 8h'r* ) B, F = a cos (pt + £/) 00s (0 + 8,) .5( 2 oat Mutual action of two rectilineal twists. ) | Let us suppose two cylindrical surfaces to have their centres fixed and to be moving with oscillatory twists about their centres; the angular displacements at any time ¢ being given by a, sin pf, @, sin pt. We shall suppose the displacements to be so small that there is no slipping at the boundary; also that the waves are long, so that the distance c between their centres is small compared with the length of the wave of displacement. be small compared with the distance c. with the wave-length, we have by equation (1) a displacement compounded of and Fe BY perpendicular to AP 1 perpendicular to BP, wb? sin pt r, 2 « and b being the radii of the cylinders. and a component These are equivalent to a component 2 an c sin 6, sin pt along BP, 1 2 2 a ¢ (cos 6,— “1) sin pt+ 22 sin pt perpendicular to BP. 2 1 The radii are supposed to For all points in space, the distance of which from either cylinder is not comparable bo (eS) eo OSCILLATORY TWISTS IN AN ELASTIC MEDIUM. In the immediate neighbourhood of the surface of B these reduce to oa /. 6 —" sin 90 pre eee. : oe (sin ars sin 2 2+ oe sin 30, — sin 40,...) sin pt, 2 2 3 2 a r Uh r : wp” . : (cos 0, — cos 20, + 2 cos 30, — — cos 46,...) sin pt + ——sin pt. c c c c r 2 @ and Hence if the terms due to the reflection at the surface of B are represented by 1,, v,, we have displacement along BP = a (sin 0, - a sin 26,. ..) sinpt-+ Uy ab. tee(20)) 3 displacement perpendicular to BP = : % (cos 6,- . cos 26,. ..)sin pt+r, +% sin pt u, and v, must be of the forms given in equations (24), the constants being determined by the boundary conditions. Suppose for example that there is no slipping at the boundary. The component along the radius from the centre must be zero and the component perpendicular to the radius equal to bw sin pt. Taking the terms dependent on sin @,, cos @, we get for the constants in equation (24) or (h? +k?) Aad Fi oe cer; 7 5 ee we (26), 2 tk log 241) -+itlog (4 3th) 4a’o, C Wb? a= me (h’ + k*) ome (1+ = , 4c0°o, he eee ese whence sin(pt+ 6). > Be Ker? hr 1\ rt ‘ —-z 8nd, Al, 1 + (log ane 5) +B,11 ~ (log ytt-5) tI + cos (pt + &) sin 0,(k?A,—h7B,) = , sin (pt + ) kr hr Vy v= — cos 0,| A, {1 - (log F +T— 5) pte. {1+ (tog 5 ee. Ges ee ] 2 + cos (pt + ¢,) cos 0, (k’A,—h’B,) 7 , 204 Mr LEAHY, ON THE MUTUAL ACTION OF or ao, ( h?-k 7/5 ‘ 2 > 2 \ , , US on a — Pe cos € » sin(pt + &) sin @ ec (r(h me a1e ) R (log P4r)+H(i g 4) 1) 1 2 + = sin &, cos (pt + &) sin @,, ke (tog ee r) +h (log La r z iD cos ch sin (pt + &,) cos8, aaa ao, ( hi-k (0 ” | ( oe: aie (log S42) +2 (log att) - eo lr (+h) \e st “ee sin & cos (pt + €) cos 6,. | | | | | Since the series in equations (24) are convergent it is clear that, at any distance from the twists which is comparable with the length of the wave, the reflected displace- ment will be negligible as compared with the direct twisting action. This will not be the case if we are dealing with distances which are small compared with the length of the wave, but it can be shewn that the effect is of the same kind as that already given, or is of higher orders in : and -. Omitting for the present the terms due to this “induced twist *”, we shall proceed to find the force between the twists—the displacement being taken to be that given in equations (25): u,, v, being given in equations (27). To find the foree on B we must first calculate the normal and tangential forces 12 (Nd a) ; : : : 2 5 which act at any point of its surface, resolve and then integrate for the time = which is a complete period. tm oe sin ¢, a double reflection will give a ae SmrppaPw,w a ll ag - — 15 Sin § cos & {1- 2m cos § +m? cos 24. resolved displacement at the surface of B composed of c(h? +k ate A subsequent reflection will give a force es 1 of fa} ee ] 2 1 i > sin 6, {sin pt — 2msin (pt +) +m sin(pt+26)}. Srrpparw,w, = ¢ (I+ 2) sin G Cos G {1-4m cos § + 6m? cos 26, — 4m cos 36 +m cos 46}, : the series being continued for r reflections at each surface sin 6, and | i¢ (2r+1)c is comparable with the wave length. All these terms will give the same effect as the one investigated in the text since m is small, 2, rs ="! cos 6, {sin pt —2m sin (pt + 6) +m? sin (pt+2¢)}. | aoe Y. This can be treated like the terms = ©*\ cos Gz in equation (25), and will by result (29) give a OSCILLATORY TWISTS IN AN ELASTIC MEDIUM. 205 The force acting along the normal at any point of the surface is yy i =) du N,=(+2u) 7 +2 (=. — and the force along the tangent to the surface is qd. 1du % Pan (te Wz): The only terms to be taken into account are those involving cos@,, sin @,, as all the others disappear on integration over the surface. At the surface u=0, v = bo, sin pt, du dy ‘lea Fa du 4a°w, ik’ ar ombe V+ dv 4a’ | Sto ee : an => = a sin 4 sin ( pt + a) cos 0; sin € sin ( pt + €) sin @,, therefore, since (X + 2p) k? = ph’ = pp’, 4a°pp*w, : : ie eile = hehe sin € sin (pt + €) sin @, 2S abe (h? + k*) sin € sin (pt + €) cos 8, In calculating the force on B it must be remembered that P is in its displaced position: hence the angle which the normal to the surface makes with AB is no longer 0, but 6,4 @, sin pt. Therefore, Force on AB revolving along AB in the direction AB Ss [om cos (8, + @, sin pt) — T' sin (@, + , sin pt)} bdé@ 0 hota Aaippio, » xi ; Pu Ie BY = Ae sin € sin (pt + €) i, sin (w, sin pt) dé _ _ 8a'pp*o,0, .. : ty (8k?) sin € sin pt (pt + €), since w, is small; Force perpendicular to AB 2 2 = fee cos (w, sin pt) sin (pt + €) sin €. The only admissible part of this is Spp'a’o, c(h? +k’) the term ,o2 being negligible as being of a higher order of small quantities. sin (pt + €) sin &, 206 Mr LEAHY, ON THE MUTUAL ACTION OF 2 To find the whole force on B we must integrate for the time a and we get the 3 : eee force perpendicular to AB to be zero; and the force in direction AB during the period & 8 aww, . es ele +k) sin € cos t. minis ivieie.sia’elsiateleie:alnielp/aiaie/atalvinloieyeialetwinievere (29). This is an attraction if the twists are in the same sense, and a repulsion if they are in opposite senses at the same time. 9. More generally, let a tube of twist be placed in an elastic medium which is undergoing any periodic disturbance F'sinpt, the angular displacement of the surface of y the tube at time ¢ being w sin pt, and reckoned to be positive in the direction indicated by the arrow heads in the figure. Let Ox be the direction of the component of F resolved perpendicularly to the axis of the tube, the trace of which on the paper is O. The component which is parallel to the axis will clearly have no action on the surface 20 during a complete oscillation, for its action during that period will be 2zb | ae sinptdt, 0 which is zero, Z being the component parallel to the axis. The section of the tube being small, X may be taken to have the same value all over the surface: and even if the small variations of X are taken into account it can be seen that they will disappear on integration over the surface like the higher terms in equation (25) above. As before we shall have to add terms similar to those given in equation (27), namely, along OP fee I (log War)+e (tog +r) : ( - cos £,} sin (pt + €) sin@ r I? (tox 5 + r) +h? (tox + r) + X sin & cos(pt + £) sin8; OSCILLATORY TWISTS IN AN ELASTIC MEDIUM. 207 and perpendicular to OP ars ;: RP log +1) +1 (log +1 —-X Mena ae (G1) sing, + = z ee # (log +1) + (logy +1) + X sin & cos(pt + &,) cos8, cos ‘ sin (pt + ¢,) cos 0 X being the radius and ¢ given in equation (26). Hence we shall, as before, find the force on the twist to be in the direction Oy and of magnitude a Suppo Be and we have the following results. sin€, cos€,, If the field of vibration be explored by a twist which is not supposed to have inductive action on the surfaces giving rise to the displacement, and if its axis is along “) Oz, the direction of the twist bearing to the positive direction of OZ the same relation as the thread does to the axis of a right-handed screw, then if the displacement in the medium at any point is represented by Fsinpt; a force will be experienced by the twist in the direction Oy, where Oy is at right angles to the plane containing the axis of the twist and the direction of F, the magnitude of the force being equal to kFo sin FOz, where & is a constant, depending on the nature of the medium and the period of the twist. Also, if Ox is the normal to the plane of yOz drawn in the same direction as F; Oz, Oy, Oz will be in cyclical order. This is the same as the relation between the magnetic induction, the electromotive force, and the current given in Maxwell’s Electricity, Vol. u, Chap. 8, where it is stated that if the primary circuit is fixed and the primary current kept constant, and if the magnetic field be explored by a linear circuit conveying a current it will experience a force similar in magnitude and direction to that given above if F represents the magnetic induction and the current. 10. It remains to add a note on the possible application of these results in the formation of a theory of magnetic action. It has for some time been inferred from Vou; XIV. Parr IL. 27 208 Mr LEAHY, ON THE MUTUAL ACTION OF several experimental results* that the particles of a magnetic body are polarised, and that magnetization simply means the inclination of their axes in the same direction. If we suppose these particles to be small oscillatory circular twists, we see by result (15) that the displacement produced by each of them will be the same as the magnetic induction 7)'a*@ sin pt 2 surface, b the radius of the cross section, and a that of the circular axis of the twist. due to magnets of strength , where @ is the angular displacement at the Also, it has been shewn that a force will act upon a rectilineal twist similar to that which would act on a current in the magnetic field, and it can be proved that the force on another small ring would be that experienced by a small magnet at the point. If magnetization of a body means the inclination of the axes in a particular direction, the summation of the effect of these rings will be the summation of the supposed magnetic effect of the particles of the body. The general condition of a bar magnet on this theory will be, that its surface is vibrating tangentially, so that the direction of vibration is parallel to the axis of the body. A cylindrical body carrying a current will, on a similar hypothesis, have its surface also vibrating tangentially; but in this case the direction of vibration will be perpendicular to the axis of the cylinder: and, since it has been shewn in a former paper} that pulsating spheres in an elastic medium will act upon each other like electrified spheres, those in like phases repelling and those in unlike phases attracting one another; a statically electrified body may be considered to have its surface vibrating normally. Thus, if the surface of a body is vibrating in any manner, the components of the vibration will correspond to the magnetic, electrodynamic, and electrostatical condition of the vibrating substance. The chief objection that I see to the theory is that it is difficult to imagine how the alternate polarizations and depolarizations of particles ranged along a wire could give rise to oscillatory twists. Several suggestions might be made as to shape, mode of motion, etc. of the particles, in order to explain this, but they would be mere speculations while they have no experimental basis to rest upon. This is at present an impediment, but does not appear to be a fatal objection to the theory. That magnetic and electrical action is not propagated through an intervening medium is almost inconceivable after Faraday’s experiments and Maxwell’s investigations; and that there must be tangential stress: ie. that the medium is not a fluid appears from the con- dition of the magnetic field due to a current. Moreover it is difficult to think that the remarkable results given in Chap. xx. of Maxwell’s Llectricity are a mere accident, and that the media through which light and electricity are propagated should not be related. There remain the possibilities of transmission by means of statical stress, finite displace- ment, and vibration. This last has always been objected to on the ground that the flow of electricity takes place in all directions, and not in a straight line at right angles to the wave front, as is the case with light. But this property of the transmission of light * See Proceedings of the Royal Society, Vol. xxxv. p. 19 + Transactions of the Cambridge Philosophical Society, and p. 178, Vol. x1v. Part 1. OSCILLATORY TWISTS IN AN ELASTIC MEDIUM. 209 in straight lines depends upon the shortness of the wave lengths; and if we consider long waves the transmission in a straight line will no longer follow. It may be noticed that, if the medium of propagation is identical with the luminiferous ether, a long wave will not imply a slow vibration; for example, a wave length of 200 miles will correspond to about 1000 vibrations in a second, taking the velocity of light to be 200,000 miles a second. It may at least be said for a vibratory theory that it gives results which correspond with observed facts, namely, that like normally vibrating bodies repel and unlike attract: also that like twisting bodies attract and unlike repel; and that in the case of simple forms of the bodies the law of force is the same as that of observed electrical and magnetic actions. Moreover, by a small change in our hypothesis, an explanation of gravitation can be deduced from what has been proved in the case of pulsating spheres; which is in accordance with what has always been expected to be the case, if the ex- planation of electrical actions should be found. In the result worked out for pulsating spheres the radius at any time ¢ is supposed to be a(1+4, sinpt), which is the natural supposition; a being the radius of the sphere in a state of equilibrium. But if we vary this supposition by supposing the radius to be a(1 + hk, sin pt —k, sin’pt), [which is not very unnatural since the work required to compress a sphere into one of radius a(1—k) is greater than would be required to extend it to a radius a(1+h)], then, in addition to our previous result that two spheres repel if the amplitudes k,, k, have the same sign and attract if they have opposite signs, we should get a statical stress, the coefficient of which would be —4k,, and which would give an attraction varying according to the law of the inverse square, as may easily be seen by geometrical considerations, The possibility of electrical action by means of statical stress or of finite displacement still remains untouched. It may be shewn that in the former case a twisting stress would give rise to a force of the kind required between two cylindrical surfaces, but there would in this case be a much more important term urging them along the lines of magnetic induction, which is not consistent with observation. As several experiments point to the idea that an electric current is accompanied by a twisting action, this looks like an objection to a theory of statical stress.) Moreover the mathematical theories of finite displacements in an elastic medium or of stress in it are far more difficult than that of vibrations, and I would venture to submit a remark of Lamé on this question which had better be reproduced in his own words,* “En général, sauf quelques cas simples, les problémes “relatifs & I’équilibre d’élasticité sont incomparablement plus difficiles & traiter par l’analyse “mathématique que les problemes relatifs aux vibrations...... Une si grande différence dans “Ja facilité d’aborder les deux genres de problemes pourrait étre une indication naturelle. “Parmi les questions de Physique mathématique qui résistent aux efforts des géométres, “ou quiils traitent péniblement par des formules loagues et compliquées, il en est beaucoup “dont l'importance est fort douteuse. Au contraire, un grand nombre de questions qui se “yésolvent par des calculs et des formules simples sont d'une importance incontestable. “Gerait-ce done que l’équilibre d’élasticité joue dans la nature un role moins important “que les vibrations ?” * Lamé, Legons sur Vélasticité, Art. 69. ra a, » thie goes Lr) ane. vi » nO te eee Peer ” i oi "ral ae ‘ o Mob potas « poor ogo ~ lie ba ( - ~~ ni edi deity tie , ga: y in, Pal i Hiyol #74 ore , a « A gf? Ls “Orr? re i + y - . a a aap 4 ' e ¢ mi@- iF, pehlarig @ , a4 @Jl, eso ba hee wh Vera Feed tite WS eee m1 ; a. Nia Lihoat ie tne. 2 ee } a uD ha wt a ra : ; : a asa Chameleon o = iwae! sai tL papcubegd IT OP i ul > ys hore ads 7 = © va > 7 bh y - ¥ - sb, : Vi i" : S tiem 5 id - dati » - * ' 8 - ' ree we ie pa ates “4 t 4aye 4 yar Y oe Ts Wain 9 a kort ite» his 4 *huede a ee i as Sshe : hits rs (leche oat“) Seige «e @nf ik limes & 6%). 40 ‘ cite eobke, ets ig) ee) Dee. 49+4s (bo Tlie a apiet eee eh ; y LW. wee oe 6d pall i4 eine i i tlie: ake eT rupted) elim aU qx » inet eit yap TS UL tk ee dy dl eel, A aoe Os mang aid 8 Pend ef Ay eee? » j )Avomid pay ‘ oe DPeorignw bea wt.pnssitew a) peek boa tp ened Wag - walt @b. asf) Pp diee atid believe © b-44 age wes Ue ~ on oo hime” Gm oe ‘they 'ly won lids pd 7 i. ee ae > sl er eo INSTRUMENT COMPANY CAMBRIDGE SCIENTIFIC THE 8. THE CAMBRIDGE SCIENTIFIC. (INSTRUMENT COMPANY. Vol XIV. . Trans X. Ona class of Spherical Harmonics of complex degree with application to physical problems. By E. W. Hosson, M.A. [Read Feb, 28, 1887.] THE object of the present communication is to present an investigation of the properties of Spherical Harmonics of degree —$+p,./—1, and to give some examples of the appli- cation of these functions to express the solution of certain questions in the theory of the potential. In the Appendix on Spherical Harmonics in the first volume of Thomson and Tait’s Natural Philosophy, the importance of these Harmonics in the determination of the steady motion of heat in spaces bounded by concentric spheres and co-axial cones is pointed out, and some general remarks are made about Harmonies of fractional and imaginary degrees. These remarks are the basis of the investigations in the present paper. The functions with which this paper is concerned were introduced as independent functions, defined by certain definite integrals, by Herr Mehler (see Crelle’s Journal, Vol. LXvItL, and a memoir entitled “Ueber eine mit den Kugel- und Cylinderfunctionen verwandte Function,” Elbing, 1870), and are called by him “Kegelfunctionen,” which we should render as ‘“Conal Harmonics.” An account of these functions, with Mehler’s application of them to potential problems connected with the infinite half-cone and the space bounded by a rotating circular segment, will be found in Heine’s Kugelfunctionen. I have treated these functions entirely from the point of view of Thomson and Tait, that is to say as a particular case of ordinary Spherical Harmonics. The only writing besides the above-mentioned Appendix of Thomson and Tait, with which I am acquainted, in which the problem of conduction of beat or electricity in the space bounded by two concentric spheres and a cone with the vertex at the centre is touched upon, is a note by Mr W. Burnside in the Mathematical Messenger for 1885. In this note two problems are solved, in which the potential function is symmetrical with respect to the axis of the figure. In the first part of the present communication the zonal and tesseral harmonics of degree —}+p,/—1 are exhibited as series in various forms; in order to make the requisite transformations free use is made of the general theorems for hypergeometric series in Kummer’s paper on the subject, in Crelle’s Journal, Vol. xv. Vos: XIV. Parr III. 28 212 Mr BE. W. HOBSON, ON A CLASS OF SPHERICAL HARMONICS The addition formula for the zonal harmonic of the first kind is then obtained. In the second part these functions are applied to the following problems : (1) The determination of the permanent temperature at any point in a uniform con- duetor bounded by portions of two concentric spherical surfaces and by a right circular cone with its vertex at the centre of the spheres, when the temperature is given at every point of the conal boundary and is zero over the spherical boundary. (2) The same problem for the space bounded by two such spheres and two such cones, the temperature being given over each of the conal boundaries. (3) The case in which, besides the boundaries in (1), there are also two plane boundaries each passing through the axis of the cones, and which are maintained at temperature zero, (4) The problems corresponding to the above, when there is only one spherical boundary. (5) The case, given by Heine (Kugelfunctionen), of the space bounded by an infinite half-cone. (6) The case in which the boundaries are those in (1) but in which the temperature is also given arbitrarily over the spherical boundaries. (7) The determination of the electricity induced on the surface of an infinite con- ductor with a hollow of the shape in (1) due to an electrified point inside the hollow. (8) The corresponding problem for the space in (4). (9) By inversion, the determination of the potential at any point external to a con- ductor bounded by two non-intersecting spheres and the surface formed by the revolution round its chord, of a segment of a circle joining the two common inverse points of the two spheres due to a free charge on the insulated conductor. A direct method of solving this problem has been indicated by C. Neumann (Math. Ann. Vol. 18, Ueber die Meh- lerschen functionen). (10) The determination of the potential of the electricity induced on a conductor of the shape in (9) by an external electrified point. (11) The determination of the current function of the motion in an infinite perfect incompressible fluid, due to the motion of a solid body whose surface is of the form generated by the rotation of a circular segment, in the direction of the axis of rotation. In conclusion a remark is made as to the two dimensional harmonics of imaginary degree. It should be observed that the arbitrary functions in these problems are supposed to be such that the integrals and series are convergent and that the latter have not in general (i.e. except at particular points or along particular lines) infinitely slow con- vergency. This will be the case for all functions which need be considered in physical problems. I hope on a future occasion to consider some approximations made from the results obtained, especially in the problems (9) and (10). These last problems are interesting from OF COMPLEX DEGREE WITH APPLICATION TO PHYSICAL PROBLEMS. 213 the fact that when the rotating segment is only a small portion of a circle the body becomes two spherical conductors connected by a conducting wire. It may be interesting to know how the charge is divided between the spheres and the wire, and thus to know how great the effect of such a connection is upon the distribution of electricity on spheres. PARI Investigation of the form and properties of Spherical Harmonics of degree —+ +pN—1 1. The equation satisfied by the zonal harmonic P,,() is d du} (te) gat te +) u =(()2 if for n we put —}+p 5 ale this equation becomes d », du he = fir) ale (ler sete =) cagoaeaponndsene sseseudouocu0cD0N il). Fe {O— #9) Seb (oD w= 0 (1) If we assume that the series a, +a,u+4,u"+...... satisfies this equation, we obtain by substituting and equating the coefficients of the various powers of mu to zero, per (22 = Magy © a Qn (Qn —=1) =. 1)( 2n) Ao,» p+ (2n—3)° Fonts = (2n st 1) (2n) Bon—1> hence, writing A, B for z,, a, respectively, we obtain the solution of (1) in the form (= wer Sees eo roe oe oon geen Po If we denote by F(a, b, c, x) the hypergeometric series @.b6 a(a+1).b(6+1) , ich uaa ec(ecei) I+; this solution may be written v= AF (4+4p1, }—4pu, 4, w)+ Buk (2+4pe, — bps, 3 py?) occ. (2). 2. Each of the hypergeometrical series in (2) is convergent for values of ~ between +1, but is divergent when w= +1, for we find m+1 a, 4n?—2n?+... an : — 3 2 ’ ie ass 4n° — 2n aa 4n? + Qnt +... 4 a ) ipa 4n® + 2n? 214 Mr E. W. HOBSON, ON A CLASS OF SPHERICAL HARMONICS which shews that the terms of each series alternately diminishes at a rate comparable with the rate of diminution of the terms of the series 1+4+4 }....... It is however possible so to choose the ratio of A to B that the value of v in (2) may be finite when w=1. The series F(a, b, c, z) may be written II (e—1) — M(a+r—1)1(b+r-1),, ae +i@_-1l 6-1 TWH n@** II (r) We +r —1) hence, if TI (— 3) IT (3) “Toil Te aetna the value of v will be finite if we suppose that an equal number of terms of the two series is always taken. Strictly, we ought to write the terms of the two series alter- nately, but for convenience we retain the form (2); thus, if A I (= $+ pe) —}- tpt) B 2U1(—4+44pu) Te $—}pu)’ the value of v in (2), under the supposition just mentioned, remains finite when p=1. If the ratio = have the above value with the sign changed, v will remain finite when w=-1. We can now divide the value of v into two parts, one of which is finite when w=1 and infinite when « =—1, whilst the other is finite when ~=—1 and infinite when ae 3. It is convenient to choose absolute values of A and B such that v=1 when y=1. This may be done most conveniently by means of a theorem given by Kummer (Crelle, Vol. xv. p. 82), . a a+b—-1 ge Se ni 3° 3 *) / 2, Jatt (* — *) “4 Ss ‘( e When w=1 the left-hand side of this equation becomes unity; if we put a= }+ pu, b=4-—pr, r=p’ =c0s'6, we obtain the equation Ja F(t+4pu, = ae we) Zale i+ hp, $— spe 3 FH’) F(44+p.4- pe 1, sin’ = te —444p0) I (—4—43p%) my eee pnie ae, ' If we denote the right-hand side of this aoe by K, i the complete solution of the equation (2) is v= CK, (mu) + DK, (— fp) cceeeeeeeecereeeeeesreeneteeeenneneees (3), where CU and D denote arbitrary constants. OF COMPLEX DEGREE WITH APPLICATION TO PHYSICAL PROBLEMS. 215 The expression ,(“) is equal to unity when ~#=1 and is infinite when p»=—1, and K,(—) is unity when «= —1 and infinite when p=+1, Moreover K, (+) =F(A + pe, $—ps, 1, sin’) =] eas 4 sin®? ies a Se 8) sin'g +... e 0) A? +1 4 K,(—p) =F (3 + ps 4 — pt, 1, cos’ be eel ee costs + pe satin (4). The first of these equations shews that K,(u)=P-3+,.(u). Heine has shewn (Handbuch der Kugelfunctionen, Vol. ul. p. 225) that K,(— pw) = 4 {Q-1+p.(4) + Q-3-» (u)}, where Q(x) denotes the zonal harmonic of the second kind, 4. If in the equation (1) we put 2?=1—y* and change to z as independent variable, the equation will be found to become du (1—22*\ dv = see [Ae G ) die =| Zz ee (Dea): Assuming v= >A,2" as a solution of this equation, we find A,,, (n+4)+p n+2 Za. (n + 2)? a. plz Al MID\ fate Gee thus eae {1 Peal ny eae i a1) = +4 Vinee = 2 3 4 7/2 +A,sin@ j1 pp BEaealk tal n’6+ (p +l) = oa sintO+...}. Since K,(cos@) does not change sign with @ we have 2 2 Kx, (cos @) =1 ee oH al sin’@ De +e el ip sd +4) ) si Benoa sepenaienee ki (5), a result which is valid so long as @ lies between + oe 5. The function K,(cos@) may, by means of two theorems given by Kummer in his above-mentioned memoir on hypergeometric series, be expanded in cosines of multiples of 6. These theorems are 2ab 2a (a+ 2) b (b+ 2) ig ine os gy 008 28+ ee otf, a—b es b+1 3 | -) ms} 2 2 2}. = =a 5 ee le) (a+VO+)) , (a+1)(a+3)(b+1) (6+ 3) (a—3) (6-3) © G=G= 3) ChO25 ic |eo 36 +- 0s 30 + 216 Mr E. W. HOBSON, ON A CLASS OF SPHERICAL HARMONICS If we substitute in these the values a=}+pi, b=}$—ps, we have, referring to the value of A,(cos @) in equation (2), Sy ele ae 1 Ge: SR ee (4p? + 1) (4p° + 5°) K,, (cos Od)=a | 4p ans 03 20+ (4p? + 3") (p+ 7?) cos 40 +... rg ARS ve ee (4p? + 3°) (4p°+7°) 8 | eoso+ Zt zg oO 830+ Gira 5 dpha OS DOE sam herawsee swiss (6), 2Qer Bea? T= $+ ip) WE Spor’ 2ar 8= TF 4 Spo MSI Sp) WE + 3p) WG — 4p This agrees with the formula obtained by aoe (Kugelfunctionen, Vol. 1. p. 241) by means of the addition theorem for the HK function. The above series for K,(cos@) may be ob- tained directly by substituting saa 1S in the equation (1), which then takes the form 2*(2°—1) o = ae + (p" eres ed If we assume v= >A,2" as a solution we get A,{p’+(n—4)}=A,.,[p'+(n—$)'}, and we thus get a series of the form above, but the determination of the constants is troublesome. 6. In order to express K,(cos @) as a definite integral we shall require to express 4 o 1 tet a oe F(a, b, i da , x) as a definite integral. The general term of the series is n(2*5- ‘\ta+r 251) Wb eg 8 7 rl Ul a4 0—* +r) 1-1) N@-1) / ¢ TI(2a—1) 2” ai and since Tl(e—1) an ), this may be written a+b-1? a+b .\ CE El ee eer per (Sa 2 -1) Il(a +k —1)1(b+k-1) pee a at —_ LS = eee ee ee WH Tq Wal) Wo —1) I(@+b +2k—1) ate ant ty Pl! e n (248-7) 1(24* 1) Lisle ay ol a. pe VERE I eh Pe ae 2 (4a | ja Em (1 Ww) du Vor Il (a—1) (6-1) a a+b_,, ar ! the sum of the series is therefore ee ihe! a+b n (75 ‘a (*3 =-1) : gare 1 a-l i= B-1 = ane eee e eee ee senses Wah eee Cae {1 = 4aru (1 = w)} * u which is the required expression for F(a, b, eee. ip bo — “I OF COMPLEX DEGREE WITH APPLICATION TO PHYSICAL PROBLEMS. - ew hak coeeclt: If we put a=}$+py b=}—ps, e=sin'g, we get CL y—d+pe (1 — y) 2-4 y-3-P (1 — u) “PP oa {1 — 4au (1 — u)}? ; Zz 1 E, (00s 8) = srr (= + ps) I(—3— Pd) Jo now II (—4+ pe) I (—$— pe) = ee ; if we make the substitution u =a , which gives lsech*}4, this integral becomes u(l—-u)= cos pa cosh pwr i cosh $4 1 > aa ere : sin? 46 ue | ii — ea ge a es 7 o (cosh a + cos @)? Also, changing @ into 7 — 8@, Fieke » K,(— cos 8) = 2 cosh pir [ cos px du. 7 Jo (cosh a — cos @)? The functions K,(cos @), K,(—cos@) introduced by Mehler were defined by means of these definite integrals and their properties were deduced from this definition I shall now investigate the form of the tesseral harmonics of degree — 4+ pu where wu, satisfies the differential equation - (. sin Those of the order s are of the form eas sh .U,, d ( may eer Sys du | Sisal) Pe See tat If we put u,=(1—y4")*w,, the differential equation for w, is d’ (—w) qa 2+) m a {p> +(s + 4)} w, = 0. The value of w, obtained from this differential equation is found as m the former case (s=0) to be (+d) ep’ sy ete + ph {(s+5)* +p} A|1+ 2 1.2.3.4 tee epee 7 Lerwtrl (es ae 5+ pi} pit..|, or AF (hs+4+}u st}—hys $, 2?) + BuF (s+ 3+4 ms, 4s+3—4 ms, 3, o’). Employing the theorem of Kummer’s used in Art. 3, we find that the expression bo pam io 8) Mr E. W. HOBSON, ON A CLASS OF SPHERICAL HARMONICS J/7 Il (s) Fi Gs sd tipy ll Gender ee eee Nos Ob sere rk F(js+i4+ t ts+2-—4 3 ?) (9) ~ s— + dp) ds—F- spy) ¥ a Phra ee is convergent when ~=1, and is equal to unity when ~=0; and that it is in fact equal to F(s+4+pr, s+$4—ps, s +1, sin’ $6). If in the above expression the two series are added instead of being subtracted, the resulting expression is convergent for ~=—1, and is equal to F(s+}+pt s+4—pu, s+1, cos’ $6). It may be shewn as before that, provided @ lies between +47, the expression considered is equal to F(js+4t+hp, $s+4—4 ps, s+, sin’ d). If we denote this expression (9) by K,*(u), the most general value of w, is sin’ @. {C'K,* (cos 0) + D'K,’ (— cos @)}, where KX,‘ (cos @) is finite except when @=7, and K,‘(cos @) is finite except when 0 =0. 8. If we apply the theorem F(a, By, #) =(1—a)r** F(y—4, y-B, », @) to the expressions just obtained for K,*(), we find that they become Jr I (s) ; ey te <4 Wigs= fn) Gp (l—p)*F(4—ts—}p, —48+} ps t, p’) 2 Jr UT (s) - : ae = the oe 4m) I jes ai —p)"wE (}—48—4pe, $-}s+4 pu, 8, pw’), cos “10. F($—pu, $+pe, s+1, sin?46), and cos F (is+3—4 pu, s+ $+4 pe, s +1, sin’), respectively. 9. If in the theorems quoted in Art. 5 we put a=s+}3+p., b=s+4-u, we obtain for K,'(+ cos @) the expression (Qs+1)+p* {(2s + 1)* + p*} {28 +5)? + p’} = E * (2a—3) +p" eee (CPT) +p} ((2s—7)' +p} mes | 44 (28+3¥ +p {(28 + 3)’ +p*} {(2s +7) +p} as) ep [eos 04 Cet AE co 330+ C= 5+ pt (2s— 9)" +p a a8 50+... aieans (10), where a= — J7 I (s (s) I (—s—4) © eh fl (4s—4+4 pr) (4s — 4-3 pe) I (—$8—44+4 ps) I (—$8—4-—4 pr)’ B= 2 Ja II (s) 1 (—s — 4) I (Js—}+ hp) ds—]—F py) UM (—-4s+4+4py) U(—he+h—ypy) OF COMPLEX DEGREE WITH APPLICATION TO PHYSICAL PROBLEMS. 219 10. I shall now investigate the addition theorem for the function K,(u). Let cos y = cos 8 cos 8, + sin 8. sin 0, cos (p — ¢,). g=0 Assume K, (cos y) = Xu, cos s ($ — g,) s=0 substituting in the equation d ( dz 1d FTN — 1) Gh Gta SS =O; which 4’, (cosy) satisfies, we have by equating to zero the coefficient of cos s(¢— ¢,) the equation d a Se oF x dp (l-—p Wa = |p ++ 73 «=0, u,=4, K,'(u) + B, K,' (—p). In this case we must have 8/=0 for K,(cosy) remains finite when cos @=1, whereas K,'(-—1) is infinite, also since cosy contains ~ and mw, symmetrically «/ must contain K,; (w,) as a factor; therefore K, (cosy) = yy, K," (u) K;' (u,) cos s ($ — 4,), to determine y, put @=0,=47 and ¢,=0; this equation becomes then K, (cos &) = Sy, K,' (0) K,’ (0) cos sd, the solution of which is comparing this with equation (6), we have {4p + ie} eee: ‘{4p° ate (2s = 3)'} Ys {K," (0)}?=a "{4p" + 37}. BA Te {4p + (28 — 1)} ’ = 26 {4p* + 37} ...... {4p* + (2s — 3)*} {4p? + 5"} ...... {4p + (2s —1)*}’ according as s is even or odd. Now equation (9) shews that 0) = Il(4s—3i+ ao. -jf- $ pe) i vr II (s) eeaueeE Tier Qa Also (Art. 5) a= = - Wreonee fat} (G) +St (GQ) +3 8B = ot soddac OPO ae a as na (Ap + 1°) (4p as 5 —— — + (2s — 1)*} Vou. XIV. Parr III. 29 220 Mr E. W. HOBSON, ON A CLASS OF SPHERICAL HARMONICS Hence we obtain the addition theorem K, {cos @ cos @, + sin @ sin 0, cos _ — ¢,)} =, (cos @) K, (cos @,) =< ad ? _— Me =(-1)'.2. od oak —— sl ales DA ee (cos @) A,” (cos @,) cos m (p—¢,)...(11). pied PART II. Applications to physical problems. I shall now give some examples of the application of the functions of which the properties are investigated above, to the solution of problems in electricity, the conduc- tion of heat and hydrodynamics. 11. If we determine n so that the harmonic (ars 2.) S, may vanish for both the values a and b of r, we obtain from the equations “ Aa" + 7=0 AU + a= 0| the values n= ESE =, where & denotes any integer; thus spherical harmonics of degree “log, = a ! =, are adapted to the solution of potential problems in which the boundary of “log, = a the space considered consists partially of portions of the two spheres r=a and r=b. 12. Suppose it required to determine the permanent temperature at any point in the space bounded by the two concentric spherical surfaces r=a, r=b and the right circular cone @=a, when the two spherical boundaries are maintained at temperature zero and the temperature is given as a function of r and @ at every point of the conical boundary; 7, 0, @ denote the ordinary polar coordinates of any point. Suppose b>a, and let r=ae?, b=ae’, then if S;,,, ; denote a surface harmonic of o 2 Tbe al ‘ é 1 k ; : ; degree =5 to the solid harmonic ale =). Sim 1 Vanishes over each of the 2 r ok spherical boundaries. The surface harmonic must not be infinite when @=0 and _ the potential function V must be unaltered when @+27 is substituted for @¢, therefore a suitable form for V is Kin (cos 0) ahs . (karp\"=” V= > —-= sin (= i) X (By cos mp + C;., sin mp) ——— fac ; ad Ki. ker (COS a) g OF COMPLEX DEGREE WITH APPLICATION TO PHYSICAL PROBLEMS. 221 where B,,,, C,, are determined from the condition that when @=a, V may be equal to a given function of p and ¢ for all values of p between 0 and oc. Suppose the value of V over the conical boundary given in the form BEAD, ); assume Jr J (ps &) = by (p) + = {bn (p) cos mp +f, (p) sin mo}, then we have b()=s-[ Fm 6) ag Pa(p) cos md + My (p) sin md = =| "Ff (p, $') cos m (¢ -- d’) d¢', and for values of p between 0 and o we have Freee. Hemi eas “emp, je AO SI (p, Sg |, sin. (p, $) dp’, We). kro fo (2. kemp = SS; ae [ i Le ' ’ ’ , hence PO ae agape eee eI ap ap 1 #=> m=” sin karp (7 [2" . karp’ , ee ee ton > = maze | (: sin — cosm(p—¢') f (p', 6’) dp’ dd’ ; thus we have K jx (cos 8) r We EEN ais = esin (=) oC — 27° Jrk=1 m=0 . kip’ , , , , , Te cay os sin ——— cos m (b— $') f(p’, $') dp’ dd’, og when e=1, except when m=O and then e=}. This value of V satisfies the potential equation, is finite throughout the space con- : : ‘ 1 3 sidered, vanishes for r=a and r=6 and is equal to Te J (pe, @) over the conical boundary ; Ps it expresses therefore the temperature at any point within the space considered. In the particular case in which the temperature over the conical boundary is constant and equal to V,, the value of V is Ky, (0088) eas sin ut aaa alae ( Sa ee o ° Kin (cos a) Jo ee i! K,,, (cos 0 ; Vie ei eye ; tx (C08 8) > Nr 4h? + 0° ( Gey Kjn (cos 4)" oc 13. In the case in which the space throughout which V is to be found, is bounded by the two spheres r=a, r=b and the two cones 0=a, 0=£, we require both the functions K (cos @) and K(—cos@) to express the value of V; these are now both finite throughout the space considered, 29—2 222 Mr E. W. HOBSON, ON A CLASS OF SPHERICAL HARMONICS An 2 of the form V== s sin (ee e) {(A,,, cos mp + B.,, sin mg) Kien (cos @) + (C1, cos mp + D mk mk sin mp) Kj, (— cos @)} G satisfies the boundary conditions V=0 for r=a, r=b and is finite and continuous throughout the space, and the four sets of coefficients may be determined so that Vass, (ep, ¢) for @=a, and Vash (p, ) for @=8, for values of p between 0 and oa. The above expression becomes for 6=a V=>> — = sin ts (eel. Kin (cosa) +C), Kn (— cos a)} cos mp + B,, Kj, (cosa) + D,, Ky, (— cos.) sin mg], now we have as above =f (pole Son a ik [sin a ces rip ede des i =a hence we obtain the equations mk o flr heer oy! A,,,K™ (cos a) + C,,K™ (— cosa) = aa] | sin = cos mp’. f,(p', &') dp'dg’, 2m Jo Jo | ein ae sin mo’. f (p', ¢') dp'de’, 0 B_,K™ (cosa) + DK” (— cosa) = eal 0 and by putting @= 8 in the expression for V, we obtain in like manner A,,K" (cos 8) +C_,K™(—cos 8) = ma \ sin = cosmo’. f,(p’, ¢’) dp'd¢’ BK" (cos 8) + D,,,K" (— cos B) = on? =n [os in SH sin mo’. f,(p', $’) dp'dd’. Determining the values of the constants A, B, C, D from the equations, we obtain the following expression for the temperature ._ ker, 3s : eine a Qn? |p K fm (0S ) Ke (— 008 8) — Kien (cos 8) Kin (— 08 2) o gv fo fin S i a = a x | ; ¥ sin a cos m ($6— ¢) A (p', -) {Kee (cos 8) Kix (— cos B) — Kien (c08 8) Kien (cos a} + f(p’, #)| Ki (—cos 0) K fm (cos 4) — Kin (cos 8) K ir (-- 008 a) | dp'd¢’, where e=1, except when m=0 and then e= }. OF COMPLEX DEGREE WITH APPLICATION TO PHYSICAL PROBLEMS. 223 In the particular case in which V is constant over each conical boundary, say V = V, for @=a and V=V, for =f, we must put f,(p, ¢) =Vie", tile, $) = Vie" in the ex- just obtained; it then becomes L sin ine ie sin Lae ie dp' Or a ial (cos a) K (— cos 8) — K (cos R) K (— cos a) a Jo ae ee * [K (cos @) {VK (— cos 8) —V,K (— cosa)} +K (—cos 0) {VK (cos a) —V,K (cos 8)}, which is equal to 2aV, Fe ih ae ler Ks (cos 0) Ken (— cos 8) — Kew (— cos 6) Kew (cos 2) Vr pa 4? +o og ° Ky (cos a) mae cos 8) — Kin (—cos a) Ky, (cos B) in (OS 8) Ken (— cos 2) — Ky.,(— cos 0) Ky, (cos «) Bes bs i sin iit = ra io er * 4kear’ + 0? on™ Kee (cos a) Ks (— cos 8) — Kirn (— cs 4) Kin (cos 8)” This is the expression for the permanent temperature throughout the space considered when the spherical boundaries are maintained at temperature zero and the conical boundaries at constant temperatures V, and V,, 14. In the case in which there are besides the two spherical and the one or two conal boundaries, also two plane boundaries ¢=0 and ¢=¥y, over which V is to vanish, the value of V will consist of terms of the type = sin (=?) sin (TE) x2 KT alee cos 0) for this vanishes over the four boundaries r=a, r=b, 6=0,6=y. We find that for values of p between 0 and o and of ¢ between 0 and y, r il . (kr, p . mrp Fe? f(e, ¢) =>— 2 ain a(* ze i pmae Se sine -S (p', $') dp'dd’. Hence for the case in which there is one conal boundary @=4 maintained at temperature = t (p, ¢), the permanent temperature at any point is r Ke (cos @) 1 . krp . = f 1 me ee saa a i: sy: mm K ¥ (cos he ker Cc The case in which there are two conal boundaries 6=2, 0=8 may be treated in a similar manner, but it seems unnecessary to write down the result in this case. 15. The solution of the problems corresponding to the foregoing ones, when there is only one spherical boundary, may be deduced from the above solutions by making a 224 Mr E. W. HOBSON, ON A CLASS OF SPHERICAL HARMONICS indefinitely small and replacing the summations with respect to k by corresponding de- finite integrals, but the solutions may easily be obtained directly as follows. Terms of the type or b\ sin ae a sin (p log, | as mp .K,” (+ cos 6), are suitable for this case, since they vanish for r=6 and are finite and continuous for all points in the space considered (except infinitely near the origin which is the vertex of the cone, and may be supposed excluded by surrounding it by an indefinitely small sphere). Let r=be-*, then p has values between 0 and o for the space considered. If there is one conal boundary over which 1 a Jad $), we have Sp, $= = Xe [re ¢') cosm(p— $) dd’, and fle, $)==] [- sinppsin pp’ .fe', $') dp dp’ TJolo for values of p between 0 and ; hence the value of V is =e Fa 3 pal af cosm (@— $’) sin pp sin pp’. f (p’; $) R™ (cos q) mone d¢' dp dp’, for this expression is equal to + f (p, ¢’) when 0=a. If V is constant r= V, over the cone this expression becomes Db tiaite, ° K, (cos @) 7 — va5el sin pp sin pp’ . p2 Toa dp’ age * psin pp K, (cos) , >» L+4p° K, (cos a) 16. If there are two conal boundaries =a, @=£, over which 1 1 v= Te > F] y= ae 5) ? Wak (p, ) Wak (p, $) we obtain by proceeding in a similar manner r 2 1 cial Nae ’ a , , , J = [ Ry cos m (p — ¢’) sin pp sin pp’ f, (p’, $’) K,” (cos @) K,” (— cos 8) — K,” (— cos 8) K,," (cos B) Bag K," (cos a) K,” (— cos 8) — K,” (— cos a) K,” (cos 8) ne ep copa ral oe gee ? i 7S =o 7 xe [/ is [ cos m (b — ¢’) sin pp sin pp’ f, (p’, ¢')- K,," (cos 0) K,." (— cos a) — K,” (— cos @) K,” (cos a) K,"" (cos a) K,” (— cos B) — K,”(— cos a) K,” (cos 8) dg’ dp' dp’. OF COMPLEX DEGREE WITH APPLICATION TO PHYSICAL PROBLEMS. 225 The cases in which there are also plane boundaries 6=0, ¢=y may be treated as before. 17. If in the preceding cases we supposed ) to become infinite we should obtain the solution of the case in which the sole boundary is the infinite half cone, the value of V being zero at infinity; in this case it would be necessary to alter the meaning of p, we should put r =e and integrate for p between the values +; for f(p, $) we should use the formula 5 ri 1 o (oe , , , , fe. ¢)==| | cosplp—p) Se, #) dp ap’ instead of the formula used above; we then obtain for V in the case of a single conal boundary, the value K,,” (cos 0) ra Se[ [| cosm(—$') cosp (p— pF) $) gimp.) UP dp dp’ Im? Vr m=010 J -o) -@ 7” (cos a) which is given by Heine (Kugelfunctionen, Vol. U. p. 243) and is obtained by him in a direct manner. 18. In all the problems discussed above, the value of V has been supposed to be zero over the spherical boundaries, and to haye arbitrary given values over the conal boundaries. If the values of V over the spherical boundaries are not zero but given functions of @ and ¢, we shall have to add to the values of V throughout the internal spaces, which we have obtained, values of V which satisfy the conditions of vanishing over the conal boundaries and having given values over the spherical boundaries; we should then have the complete values of V, which satisfied the conditions of having arbitrarily given values over the whole of the boundaries. The natural way to solve the problem of deter- mining V so that it may vanish over the conal boundaries and have prescribed values over the spherical ones, would be to assume an expression for V in a series of spherical harmonics of degrees and orders so chosen that they vanish for the one or the two values of @ cor- responding to the conal boundaries. However the determination of the degrees and orders of the harmonics which satisfy this condition presents such difficulties that I have evaded it by using harmonics of integral degree which express the given temperatures over the whole of the spherical surfaces of which the boundary is a part. If the temperature over the portion of a spherical surface exist by a cone =a is given in the form F'(@, ¢), I suppose this function so extended as to apply to the whole of the spherical surface; the value of F(@, }) outside the cone is entirely at choice, and may be taken either as zero or as an analytical function of the same form as that which expresses the given temperature over the portion of the spherical surface cut off by the cone; this arbitrarily assumed form for the value of V over those portions of the spherical surfaces which do not form part of the boundaries of the space considered cannot affect the result, since the value of V inside the space only depends on the boundary values of V. The extension of the function F'(6, ¢) to the whole spherical surface is merely an analytical artifice. 19. Consider the case of the space bounded by two spheres r=a, r=b and a cone @=4a; suppose the temperatures of the portions of the spherical surfaces maintained at 226 Mr E. W. HOBSON, ON A CLASS OF SPHERICAL HARMONICS given values F, (@, $), F,(@, $) respectively; we have to determine a value of V which shall have these given values over these boundaries and shall vanish over the conal boundary @=a. As explained above, I suppose /\(6, ¢), F,(6, ¢) functions given over the whole spherical surfaces r=a, r=b, their values for @>a being chosen so as to make the functions most simply expansible in surface harmonics of positive integral degree. Suppose F, (0, ¢) = degree n. Assume (0, 6) ==Z, VY, and Z, denoting harmonics of integral ») n ah 1 ; V= =r" Y n = gt Z, , then from the conditions i F = Z, = qn n a"Y' + BY.’ 4+ Ae Zr, we obtain Y;,’, Z,’, and thence the following value of V, ve eee (n+) )(o=p) 5 = aes s sinh (n + 3) p sinh (n + 4) o sinh (n+4)o this value of V which I shall denote by Wane ¢, @) has the prescribed values over the spherical boundaries but does not vanish for 6=a, so we must add to it a value of V : : ; 3 : 1 which vanishes over the spherical boundaries and is equal to ace E-(p, b, a) over the conal 4 boundary. This is (Art. 12) 2 K’.,(cos 6) ; 1 > ATP ol o fln cmp’ ; che Se, sin? = | sin ——cos m(¢— , , a) dp’ dd’. Sa mee: ot Kix (Os &) sik DE es (p—¢$) u(p', $, 4) dp’ dd o Hence the complete value of V which has the prescribed values over the spherical boundaries and vanishes over the conal one is y= gee (n+ 4) (c—p Feta 2 oe sinh ( (n+4)o ™~ sinh ( (n+4)o = P Kj (cos 8) aay ae Dene it =e ; Ke (cosa) Tv YT k= tae i) kn (COS & opie | kip 5 sinh ( n+4)(o—p') » A [y sin | cos m (p — $) % ~ sinh (n+ 4). yi Si(a, $’) dp’ dd F p Kix (cos 8) eS Sean ee 2a? Vir k=1 m=0 7 Kix (cos a) s sinh (n+ ey ok [i [sin Ct AP — sb) % 3 ays Z, (4, $') dp’ do’. OF COMPLEX DEGREE WITH APPLICATION TO PHYSICAL PROBLEMS. 227 We have ie sin Ze sinh (n + 4) p’dp' =— = ip [eos ies —(n+ bipe /-1 1} 6 0 ker th alles = 008 f+ (0+ 9) v= 1 6 | dp : _v=I. fem {ka —(n+h)oJ—1 1} | Tate] 2 kr —(n+h)0J— 1 ka + (n+4)o0/-1 “a _N=1.¢ — 2hkm cos km sin (n+ 4)oV-1 2 ken? + (n + 3)? +L ] Ais aay = re Dr aes Migs 2) ¢ , and similarly | sin ee sinh (n + 4) (o — p’) dp’ Keo + (n+ re Te karo sinh (n+ 3) 0 len? + (m +4)’ 0° ? thus the integration with respect to p’ in each term has been performed. A further simplification of the expression above is obtained by substituting for Y,,, Z, their expansions in tesseral harmonics m,=n > ; (A jm, CoS m,h' + By» Sin m,¢’) T',"" (cos «), m= 2Qr : cos ; } Me: ; then since | ee mp cosm (p—¢) dd’ is zero unless m, =m, and is then equal to 0 cos 3 : : é T md we have the following expression for the temperature at any internal point, under the given surface conditions v=,/ DS) Sn: Bilas) VR = sinh (n + 3) p Z, r sinh (n+4)o r sinh(n+3)o ” Ky, (cos ess ys! Jac SS Cay sen lemipe aac jp Ln” £08 2(Anm cos mp + By m sin mp) 2 Jr k= k=1 m= =o o Pee (cos ae n=m ken + (n 4 ic oc n Kir(cos0) ; : nihergs Jbo Sys harp a n=” kT.” (cos 4) (A’y Cos mb + Bey m Sin mp) ral a = ar ae n=0 o K jx (COS 2) n=m = k T+ (n+ 4)? o oc te ita a | io YT” cos mo dudd where Anm= SSS (see Ferrers’ Spherical Harmonics, p. 86) n nL, a7 n—m! w+ 2 ib | ‘ YT” sin mopdudd B, a a n—m! 2n+1 A',m= an expression similar to Ax,» with Z, for Y,, Biggnicte eestor tacts carta ei tout Bigt, Mae Saesty soak ot Wor, SO. IPArie 1DUE 30 228 Mr E. W. HOBSON, ON A CLASS OF SPHERICAL HARMONICS This value of V added to the value in Art. 12 gives the complete values of V throughout the space when the values over the whole boundary are given. In the particular case in which the temperatures are given constant over the spherical boundaries V=V, and V=V,, we have Ken (cos @) yall. —4V, a vars ray k = r (b= Jr k=1 p "King (6082) o + 4kn?* (V,- Vy: 20. An important application of the preceding formulae is the determination of the potential of the electricity on the surface of an infinite conductor with a hollow of the shape just considered, due to an electrified point placed inside the hollow space. If we suppose a negative unit of electricity placed at the point, the problem is equivalent to the deter- mination of the value of V throughout the space considered when it is equal over the bounding surface to the reciprocal of the distance from an internal point. This value of V is the Green’s function for the space, and when found enables us to determine the potential at any point of the space considered, due to any arbitrarily given surface dis- tribution of matter; for if G@(P) denotes the Green’s function of any point P, the potential at P of any surface distribution in which o’ is the density at the element dS’ of surface is {@(P)o'dS, the integration being taken over the whole boundary. In this case the value of V over the bounding surface is 1 il a xe + e% — 2e°* {cos 8 cos 0, +sin @ sin 8, cos (b — ,)}’ if p,, 9, ¢, are the coordinates of the internal point. Over the conal boundary the value of V is then to be 1 1 ae +e — 2ePt {cos a cos 6, + sin asin 8, cos (> — ¢,)} and we must substitute this value for the function =f ¢) in Art. 12. r wT + 1 a habe If p>p, this expression is equal to a > amie P,, {cos a cos 6, + sin a sin 0, cos (f — ¢,)}, and this is equal to 1 e"o ep aa n—-m! = ar rat oe: (cos a) 7’ (cos 8,) cos m (fp — ¢,). If p eet =r (cos a) T” (cos 8,) cos m (f — ¢,). We must therefore in accordance with the method of Art. 12 expand a function which is e”Po : e(n—te equal to athe for values of p between p, and a, and is equal to Seti for values of p OF COMPLEX DEGREE WITH APPLICATION TO PHYSICAL PROBLEMS, 229 ; the required series will be karp between 0 and p, in a series of sines of multiples of 7? SAag sin —- ’ o Pp by where Ankt= z | "en (n+) pot (n-3) sin al dp += i eneo~(n+3)P sin — dp, o 7 Po carrying out the integrations, we get 2 2 hen? + (n— 4)? a ie _2 en (nt) 7 —o(n+4h)e-™t sin =| ker (e- (+3) ¢ cos kar — e~ ("+3) > cos ee — 20 e”Po 7 ken? + (n+4)o° The part of V due to this distribution over the cone is then Kix (cos @) 2 —M! Frm a T” (cos a) T” (cos 8,) cos m (b — $,) Kee (e084) Dn g Pee a = = SS 2S. etm A, n+m! 1 ar Tk=1 m=0 n=m Over the sphere pat the value of V is to be = = e- (+1) P cos 8 cos 8, + sin @ sin 8, cos (¢ — ,)} and over the sphere p=ca it is to be = ¥ er-(n+1)¢ P {eos A cos 6, + sin O sin 8, cos ( — $,)} 1 DS An» cosmp + By» sin mp = a (n+1) po Q>, am cos m (fp — ,), 05 m(h— g,): hence using the formula in Art. 19 we have, since in this case 1 A’ nn os Mh + By, sin mp = — eens (mtt)o 9s” eee T™ (cos@) T\” (cos,) cos m (f — ¢, The value of V, v= we y sinh (n+4)(o- p) 1 ee "Sos | n—m! 3 r snh(n+4)o0 “a mao n+m! h ek F m=n ee + 1S > a ethe enpo- (n+1)o 2 >a a -T'™ (cos 0) T."’ (cos8,) cos m (p — ¢,) m= Kin (cos Os Peles ay san oa 2 Jr k=1 m=0 @ He@ae) n=m 3.8 m! kT” (cos a) cos m ios d,) kn + (n+4)’o° : 1 a Ne (Bes (n+1)o /b) 9 = 30—2 230 Mr E. W. HOBSON, ON A CLASS OF SPHERICAL HARMONICS The sum of V, and V, gives the complete value of V, that is the Green’s function of the point p,4,4,. The corresponding problem for the case of the space bounded by two cones and two spheres may be solved in the same manner, but it is unnecessary to work out this case in detail as the formulae do not essentially differ from those just obtained. 21. I shall now obtain Green’s function for the space bounded by a cone and a single spherical surface, this might be deduced from the last case but is as easily obtained directly by means of the formula in Art. 15, Using the notation in that article, suppose Py» 8, , the coordinates of the point for which the Green’s function is to be obtained. The reciprocal of the distance between this point and (p, @, $) is 1 b Je-2* + e- 2» — 2e-P-»» {cos O cos 8, + sin @ sin 8, cos (fb — ,)} : It is well known that the value of the function over the sphere is that due to a mass e? at the inverse point whose coordinates are be, 0, , and the value of this at any internal point is ePo b eo + e- 2? — er? {cos 8 cos 8, + sin O sin 8, cos (b — ,)} We must add to this the value obtained from the formula in Art. 15, where we put Jb e"® 2 + @~ 2» — 2e-P-P {cos x cos O, + sin a sin O, cos (f — ¢,)} tT (p, = 7, Pom 4 Jb eo” 4P ~ Be — e-% — QerrP {eos a cos 8, + sin a sin 8, cos (f — ¢,)} | The value of Green’s function is therefore 1 b /1 —e- 2-2» — 2e~P-P {cos 8 cos 8, + sin 8 sin 8, cos ($ — ¢,)} i paces (Pe nae P > m ae Jb 7 = Ps i | cos m (p— ¢’) sin pp sin pp’ SF gl") Al Jo. Jo IX," (cos a) e- We + e-*e — 2e-P-P {cos acos @, + sin a sin 8, cos (¢ — ¢,)} ePo-*p \ dg’ dp dp’. Je — e-% — ero [cus a cos 0, + sin a sin O, cos ($ — ¢,)} This is the potential of the electricity induced by a negative unit placed at the point p,, 9, @ The integration with respect to ¢' may be carried out by expanding the denominators of the fractions in Tesseral harmonics as in the last case, distinguishing the two cases p= py. 22. I shall now apply the method of inversion to obtain from the preceding cases the solution of the corresponding problems for the spaces bounded by the surfaces which OF COMPLEX DEGREE WITH APPLICATION TO PHYSICAL PROBLEMS. 231 are the inverses with respect to a point on the axis, of the spherical and conal surfaces. The inverse of a spherical surface with respect to any point not on the surface itself is a spherical surface, and that of a cone is the surface generated by the revolution of a circular segment round its chord. i : A hea mht | a Fig. 1. Let O be the centres of two spheres and the vertex of a cone as in Art. 12; take a point C on the axis in the space between the spheres and invert the system with respect to C, taking CO=k for the constant of inversion, so that O is its own inverse point. The two spheres of which AA’, BB’ are diameters invert into two spheres on aa’, bb' as diameters and the cone inyerts into the surface formed by the revolution of a segment of a circle of which OC is chord, and which contains an angle equal to the semi-vertical angle of the cone. Pr J6 yi Ds wa le Fig. 2. Let P’ be the inverse of any point P, then we have, if p, 0, @ are the coordinates ta OF C— - POG = OF also Oana es thus the locus of P’ for which p is constant is a series of spheres for which O and C are common inverse points. Corre- sponding to p=0, p=a we have the two spheres for which = is equal to t and ae and x ; kra k*ae? the radii of these spheres are easily found to be Rog? =e k 2» In the inverse figure 232 Mr E. W. HOBSON, ON A CLASS OF SPHERICAL HARMONICS we regard p, 6, ¢ as the coordinates of any point P’; we have CP* = a? (e* + e — 2er** cos 8) = a*. e+" {2 cosh (p — r) — 2 cos 8} where k= ae’. According to the method of inversion if V is the potential at P due to any distribution of matter, the potential V’ at P’ due to the inverse distribution is given by k V=V.ap» pt . W=V.e2 ,/2cosh(p+7) — 2 cos 8. 23. To the problem of determining the potential throughout the space bounded by two spheres and a cone corresponds that of finding the potential at any point in the space external to two spheres aa’, bb’ connected by the figure OC formed by the rotation of a segment of a circle. In the case in which the two spheres aa’, bb’ are equal and the segment is flattened down we have a body approximately the shape of a dumb-bell slightly thickened in the middle. ; eS een : Fig. 3. Suppose that it is required to find a potential V’ throughout the external ‘space so that at the surface of the body just described V =F (p, 08): We have vo— te ez »/2 cosh (p + T) — 2 cos 8, where V is determined by the formula in Art. 12, substituting for f(p, 0, $) the value F (p, 6, $) -e* ./2cosh (p+ 7)—2cos0 24. First suppose it required to find the potential at any point due to the dis- tribution of electricity on a conductor of this form charged to potential C. For the spheres p=0, p=o the values of V are —r Cet oo% 2 cosh t —2c0s 0” J/2 cosh (¢ +7) — 2030” OF COMPLEX DEGREE WITH APPLICATION TO PHYSICAL PROBLEMS. 233 and for the cone 6=a V is Ja Ce Jr /2 cosh (p + 7) —2.cos a Hence the part of V due to the value of V over the cone is z Ky, (cos 6 . karp’ Cer Ja . krp ix (008 ) o SU a — } sm — Sf ———— dor 7 k=l o Ky, (cos a) » V2 cosh (p’ + 7) — 2 cosa To find the parts of V due to the values of V over the spheres we must in accordance with Art. 19 expand these values in harmonics of integral degree, we have at once Ce n= = (' & e-™P, (cos 8), /2 cosh 7 — 2 cos 8 n=0 ( ) ‘ei(t-2) Gs = Ce-*Ze-™("+9) P. (cos 8). ./2 cosh (+ + ¢) —2 cos 8 Hence the corresponding part of V is a. sinh (n +4) (o —p) ae hee ete ae erie Js Gece a NG eminent nn ER Kin (cos 0 Nae 3 hog a ix ( Mime k.e-™ P,,(cos a) ar aA o ° Kyn(COS 4) nao K'm* + 0°(n + 3)" Kee 0 Joao Sigh kp ee ) na ke-™—(™*Ve P (cosa) Ar kat o Ky, (CoSa)nz9 =a" +0°(n+4) oc Karp’ < sin The integral | 5’ can be expanded in a series, thus: = 0/2 cosh (p’ + 7) —2 cosa a E karp! St oc fon Che, ko if — SSS ey, | =) sin ~“P Se-(m+)(6'+7) P_ (cosa) dp’ I; /2 cosh (p’ + 7) — 2 cos a ee o (essa) de ki de P_ (cos 2) -—(n+4)7 _ cos hor Om CTBT o)))) 234 Mr E. W. HOBSON, ON A CLASS OF SPHERICAL HARMONICS Hence the value of V’ is Des ae a ., sinh (n +4) (o—p) Be ; C /2 cosh (p + tT) — 2 cos 8 l4/2 See Ty CEST e-*" P,, (cos 8) Bes sinh (n ate 3)P, -ar-(ntl)e ; We 2 ih the” P,, (cos @) ee 9) ko P, (cos a) C =5 Dual Scns —————— : a 2 cosh (p + 7) wos 4/ Se = 7 Relea phat (nt hyo g {e-™* (1 —cos kar) — e~"7-*D-e (1 + coskz)}. This is the potential at any point in external space of the electricity on the conductor when it is at potential C. 25. Next suppose it required to find the electricity induced on a conductor of the same shape induced by a negative unit of electricity at any point in the external space. We shall obtain the solution of this problem by inverting the case solved in Art. 21, supposing that a mass = is placed at the point P, the inverse of the given point (p,8,6,). In order to find V we must therefore multiply the value of V in Art. 21 by e'l™-7) ,/2 cosh (p, +7) —2cos ,, hence the value of V’ is elle+)-7 ,/2 cosh (p, + T) — 2 cos , »/2 cosh (p + 7) — 2 cos 0 (V,+V,), where V, and V, are the expressions in Art. 19. 26. For the case of the infinite half cone with no bounding spheres, if »=e? denote the distance of any point from the origin, it is known (Heine, Vol. m. p. 243) that the value of V at any point in the cone corresponding to V= = J (p, &) over the surface is ES | ee Lae (ere 8) ioe mn C08 (p—p') cos m ( — 4"). dip du Re m= = a This may easily be deduced from Art. 15, by making the radius 6 of the bounding sphere infinite and altering the meaning of p. By inversion we find the potential V’ of the space external to a figure bounded by the surface of revolution generated by the revolution of a segment of a circle round its chord, the value of V’ which is such that the potential at the bounding surface is F(p, ) is J/2 cosh p — 2 cos 0 ™5” Fp’, ¢’ »K." ‘ : ; ieee = oak | aia Sams o05 cosy (p—p’)oosm (p—¢") ddp'd. 27. As an example of the use of this formula I shall find the current function (x) in an infinite fluid when a solid of this shape is moving with a velocity V in the direction of its axis. OF COMPLEX DEGREE WITH APPLICATION TO PHYSICAL PROBLEMS. 235 Let y be distance of any point P from the axis and « the distance from C’ measured along the axis, then yw satisfies the equation dy ay 1 yd_ da dy a dy P y CX B Fig. 4. In a paper on the potentials of the surfaces generated by the revolution of Limagons recently read before the Philosophical Society, Mr A. B. Basset has remarked that this equation is of such a form that Vin is a potential; in fact the equation may be written cp GP il dh 1k GE \ (Ab io (ie tat y ay ty ag) | ae and since at the surface y~=4Vy’, _ is a potential of the external space and is equal at the surface to $Vysin ¢. Now referring to the figure in Art. 22, we see that I? sin 0 : a ky =sin @. 2 1 9 cosh p — 2 cos 0” ee ko ® * @ert {2 cosh (p + 7) — 2 cos 6} putting k=1, a=0 to suit the new notation. Hence the value of vane over the surface sin 6sin ——_+__. Thus referring to the formula in the last article we have for 2. cosh p—2 cos 0 is } the value of the current function V sin 6 sin a K,, (cos 6) At a ———— ie =A cos 4“ (p — p’) dp'dy. 4er? 2 cosh p — 2 cos 0 —@ (2 cosh a —2cos a) Ke (cos a) 28. It may seem worthy of remark that corresponding problems in two dimensions may be solved by means of two dimensional harmonics of imaginary degree. ~ , da\ Sila ‘ If ( Ar" + — a n@ vanishes for two values r=a, r=b of r, we must have r"] cos Aa” + zi = A b” + = => OQ, a b Vou. XIV. Parr III. 31 236 Mr E. W. HOBSON, ON A CLASS OF SPHERICAL HARMONICS therefore =o OL att log” —=2./—1 kr, kor krO _kné If b=ae", r=ae*, then sin? P (4. e7+Be P)gatnhesiocy 0 and p=a. Fig. 5. For example to determine the steady motion of heat in the space bounded by r=a, =b, 6=0, @=x when the temperature is f,(p) for @=0, f,(p) for @=a and vanishes for =a, r=b, we have to determine 4, and B, from the equations (4,4 BY =|" fie) sin “ZF ap, Chee ce a )g=[" Ft. (p') sin PP tp, hence the temperature is p = al inh #9 (9!) + sinb = @-) k=1 9 4.(9) sin "2a XI. Table of the Exponential Function é to twelve places of Decimals. By F. W. Newman, Emeritus Professor of University College, London. [Communicated by Prof. Adams. Read October 31, 1887.] Tus Table gives the value of e* for values of « from «=0-001 to c=2-000, taken at intervals of 0-001. The mode of forming it is exactly similar to that described at length by the author in the introduction to his Table of the Descending Exponential, which was published in Part mi. of Vol. x1. of the Society's Yransactions. The object in both cases was to secure a systematic verification of the results. First from e, we may find e* by the formula saben irae 1 ‘ e=e.e mefl + ratragt et. 1 1 1 S = yt _ _ “4 also ree ef iti 5 reg te}. Hence, if MJ denote the sum of the odd terms and NV that of the even terms of the above series for e’, we have e=M+N, also 1=MU-N, which sufficiently tests the accuracy of both M and WN and therefore that of e’. Similarly e*® may be found with verification from ¢’. 1 1 il Si 8 =e 3 For e=e.€ é(14+p+7 5ty 9-3 tke), if! lt il =e en a — A also Ca— ene é(1 i*L2 rath): And if M’ denote the sum of the odd terms, and NW’ the sum of the even terms of the series for e*, we have e=M’+N', and e=M'-N’; and since e is already known, this tests the accuracy of both M’ and N’ and therefore that of M’+N’ or é&. After this the intervals are halved, and e* is obtained when « is taken successively = 2:5, 15, and 0°5 by means of 31—2 bo oo oO Pror. F. W. NEWMAN, TABLE OF THE EXPONENTIAL FUNCTION e : Teer Ipeee | e e t-(li5+sagteagt®)> since deta Std Set=e, c=", e.et=e’, by which all are doubly verified. Further, by means of a ye ig 1 . : "=(I +79 + 10.20 * 10.20.30" &e.) still working downwards from e*, we may obtain in succession, with verification at every fifth result, ea Ana pas) Sy eas, PTR PAG PAS), goo 24) WA RSS Son tly OHS OSH one OAL eyavel 10). The table thus formed is called by the author his skeleton table, in which the calculations are carried to 16 places of decimals, but from #=3 to #=2 it is left as a mere skeleton. Then between «=2 and w=0 the entries were filled in, making x vary at each, step by ‘01. First, all the intervals were halved by means of lhe 1 05 __ cats FS + 50 * 20.40 = 20.40.60 * &e.) T ore (pe De eae ae e 207 95° 0.40 20.40.60")? : 1 and again gta _" (1 + as = 40 * 90.40.60* &e.) ; 1 and tale (1-5 29 7 5. 40 20.40.60 + we.) ; so that if M”’ denote the sum of the odd terms of either of these latter series, and N” the sum of the even terms, then MeN ere and WAL IN ERC CB and since e'” has been already calculated independently, this tests the accuracy of both M’ and WN”, and therefore that of M’—WN” or of e'*. In this way may be obtained with perpetual verification, the value of e” from r=2 to £=1°95, 1:9, 1:85, &.... down to ‘1, ‘05, 0. After this, by means of gale io0 + x00.500 * 100;200.800* ©): the process is continued from #=2, so that # may descend at every step by ‘01, and verification is obtained after every fifth entry. TO TWELVE PLACES OF DECIMALS. 239 Finally the table was enlarged, without any change in the process, so as to make aw descend by only ‘001 at each step. Thus, the intervals are first halved by means of 1 1 1 =) 00p) aes — eo" =1+4 500 * 200.400 * 200. 400.600 * & which gives go oN + i etaes : + &e. 200 * 200.400 ~ 200.400. 600 , from which may be derived in succession the values of ¢ for w=1:985, 1975, &c... down to ‘005, with verification at every step. After thus halving the intervals, the table is completed by employing in a similar manner the formula goods

w nr _ oo on toe] co io} Ne} un {1:000—1'199] TO TWELVE PLACES OF DECIMALS. 245 2 e” xr e” | x er xr ef I'000 | 2°7182 8182 8459 | 1:050| 2°8576 5111 8063 I'0oI | 2°7210 0146 9882 || 1'051| 2°8605 ro1g 8483 L'002 | 2°7237 2383 2306 ||1:052| 2°8633 7213 9414 1°003 | 2°7264 4891 8454 || 1°053| 2°8662 3694 3716 1'004 | 2°7291 7673 1052 ||1°054| 2°8691 0461 4256 I'005 | 2°7319 0727 2826 |/1'055| 2°8719 7515 3901 1'006 | 2°7346 4054 6508 || 1'056| 2°8748 4856 5522 1007 | 2°7373 7655 4830 |1'057| 2°8777 2485 1991 1'008 | 2°7401 1530 0530 | 17058) 2°8806 ogor 6185 T'009 | 2°7428 5678 6345 || 1°059| 2°8834 8606 0983 100] 3°0041 6602 3946 "IOI| 3°0071 7169 2554 370101 8036 8334 "103| 3°013I 9205 4294 "104| 3°0162 0675 3447 "105 | 3°0192 2446 8807 *106| 3'0222 4520 3391 3°0252 6896 o221 *108| 3°0282 9574 2320 "EOQ) 30313 2555 2715 "I10| 370343 5839 4436 “ILI | 3°0373 9427 0515 3°0404 3318 3989 "113| 3°0434 7513 7897 "114| 3°0465 2013 5279 "I15| 3°0495 6817 9183 "I16| 3°0526 1927 2654 30556 7341 8746 "118| 3°0587 3062 0510 119} 3:0617 9088 1006 "120| 3°0648 5420 3293 "121| 3°0679 2059 0434 3°9799 9904 5497 "123| 3°0740 6257 1549 "124| 3°0771 3817 1664 "125| 3°0802 1684 8918 "126| 3'0832 9860 6389 3°0863 8344 7159 128| 3°0894 7137 4312 "129 | 3°0925 6239 0937 '130| 3°0956 5650 0125 "131| 3°0987 537° 4969 371018 5400 8568 ELS S| NonO4 Oa 425 4020 "134| 3°1080 6392 4431 135] SILII 7354 2995 136] 31142 8627 2553 3°1174 O211 6488 138) 371205 2107 7826 "139| 31236 4315 9684 '140| 3°1267 6836 5186 "141| 3°1298 9669 7457 371330 2815 9624 "143| 3°1361 6275 4820 '144| 3°1393 0048 6179 "145| 3°1424 4135 6839 "146| 371455 8536 9941 3°1487 3252 8628 "148| 3°1518 8283 6047 "149 | 3°1550 3629 5350 "150| 371581 9290 9690 "151 | 3°1613 5268 2222 3°1645 1561 6108 153} 3°1676 8171 4509 "154| 371708 5098 0593 "155 | 31740 2341 7528 156] 3°1771 9902 8486 3°1803 7781 6644 158} 31835 5978 5179 "159| 31867 4493 7275 ‘160| 3°1899 3327 6116 161] 3°1931 2480 4891 31963 1952 6790 163) 3°1995 1744 5010 "164| 3°2027 1856 2747 165 | 3°2059 2288 3203 "166 | 3°2091 3040 9582 3°2123 4114 5092 168) 3°2155 5509 2943 "169 ) 3°2187 7225 6349 "170| 372219 9263 8529 "171 | 3°2252 1624 2700 3°2284 4307 2089 173| 3°2316 7312 9921 174] 3°2349 0641 9426 "175| 3°2381 4294 3838 176| 32413 8270 6393 ‘177| 3°2446 2571 0331 "178| 372478 7195 8895 "179| 3°2511 2145 5332 "180| 3°2543 7420 2890 "181| 3°2576 3020 4822 3°2608 8946 4385 "183| 3°2641 5198 4838 184] 3°2674 1776 9443 "185 | 3°2706 8682 1466 186} 3°2739 5914 4176 32772 3474 0846 "188 | 3°2805 1361 4750 "189 | 3°2837 9576 9169 "190| 3°2870 8120 7383 "E91 | 3°2903 6993 2679 "192 3°2936 6194 8345 "193| 3°2969 5725 7674 "194| 3°3002 5586 3960 F195) NSO 5e oii O5S2 "196| 373068 6298 0602 "197 | 3°3101 7149 7565 "198| 3°3134 8332 4700 "199| 3°3167 9846 5319 32—2 Hee ee i) e} nN Hee ee 4 on N Se = = Se = o} x Le oe Ae ee oD Lal on a Toro | 2°7456 o101 5017 || 1:060| 2°8863 7098 9268 Torr | 2°7483 4798 9290 || 1061 | 2°8892 5880 3924 I’012 | 2°7510 9771 1912 || 1062] 2°8921 4950 7839 T'013 | 2°7538 5018 5631 || 1'063| 2°8950 4310 3904 1'014 | 2°7566 0541 3201 || 1'064| 2°8979 3959 5012 Tots | 2°7593 6339 7376 || 1:065 | 29008 3898 4060 I'016 | 2°7621 2414 og15 || 1:066| 2:°9037 4127 3947 1017 | 2°7648 8764 6579 || 1°067| 2°9066 4646 7575 1018 | 2°7676 5391 7130 || 1°008| 2:9095 5456 7851 I'019 | 2°7704 2295 5336 ||1':069| 2°9124 6557 7681 020 | 2°7731 9476 3964 || 1°070]| 2°9153 7949 9977 Vo2t | 2°7759 6934 5788 || 1071 | 2°9182 9633 7653 T'022 | 2°7787 4670 3581 || 1°072| 2°9212 1609 3625 1'023 | 2°7815 2684 o121 ||1'073| 2°9241 3877 0814 17024 | 2°7843 0975 8189 || 1'074| 2°9270 6437 2141 1'025 | 2°7870 9546 0566 || 1'075| 2°9299 9290 0534 1'026 | 2°7898 8395 0039 || 1:076| 2°9329 2435 8919 17027 | 2°7926 7522 9396 ||1'077| 2°9358 5875 0229 1°028 | 2°7954 6930 1428 || 1'078| 2°9387 g607 7398 1'029 | 2°7982 6616 8931 || 1'°079| 2°9417 3634 3364 1030 | 2°8010 6583 4699 || 1:080| 2°9446 7955 1066 | 17031 | 2°8038 6830 1534 || 1081 | 2°9476 2570 3447 1'032 | 2°8066 7357 2237 || 1'082| 2°9505 7480 3455 1°033 | 2°8094 8164 9614 || 1083] 2°9535 2685 4038 1°034| 2°8122 9253 6472 || 1'084| 2°9564 8185 8148 1'035 | 2°8151 0623 5624 || 1085] 2°9594 3981 8739 1'036 | 2°8179 2274 9882 || 1°086| 2°9624 0073 8772 1'037 | 2°8207 4208 2063 || 1:087] 2°9653 6462 1205 1°038 | 2°8235 6423 4986 || 1°088)| 2°9683 3146 goo2 1'039 | 2°8263 8921 1474 || 1'089| 2°9713 0128 5132 1040 | 2°8292 1701 4352 || 1°090| 2°9742 7407 2563 T'041 | 2°8320 4764 6446 || t'091| 2'9772 4983 4268 1042 | 2°8348 8111 0588 | 2°9802 2857 3224 1°043 | 2°8377 1740 9612 | 1'093 | 2°9832 1029 2408 11044 | 2°8405 5654 6354 | 1'094| 2°9861 9499 4803 1045 | 2°8433 9852 3652 | 1095) 2°9891 8268 3393 1°046 | 2°8462 4334 4349 || 1°096| 2°9921 7336 1166 1°047 | 2°8490 gIoI 2°9951 6703 I113 1'048 | 2°8519 4152 7321 || 1°098| 2°9981 6369 6227 1049 | 2°8547 9489 5295 | 1'099| 370011 6335 9505 He ee A 4 i] nN ie eel cal oO tN HAR RH H H ~ HH eA 4 lon ~ See See _ N N = SSS Leal I nN Lo oe oe ee - nv ~ Hee ee HR ee H io) nN ee | H oo to ee! K w ~ = Se eS 4 co x 4 0) \o rs) Ss = = Se ay nS iS ln A oe A oe oe i iS} (oe) ‘oO 4 fo} ‘Oo ~ =x Ss = Se - a Nn |e oe A on oe 246 ror. F. W. NEWMAN, TABLE OF THE EXPONENTIAL FUNCTION ec” [1:200—1°399] x | er | x er x e x e | 1°200 | 3°3201 1692 2737 | 1'250| 3°4903 4295 7462 | 1°300] 3°6692 9666 7619 |1°350| 3°8574 2553 0697 1201 | 3°3234 3870 0271 | 1251 | 3°4938 3504 6173 || 1301 | 3°6729 6779 9547 | 1°351 38612 8488 5584 | 1202 | 373267 6380 1245 || 1°252| 3°4973 3062 8719 || 1°302 | 3°6766 4260 4443 || 1°352| 3°8651 4810 1755 | 1°203 | 3°3300 9222 8983 || 1°253 | 3°5008 2970 8596 || 1°303 | 3°6803 2108 5981 || 1°353 | 3°8690 1518 3075 | 1°204 | 3°3334 2398 6813 || 1°254| 375043 3228 9303 || 1304 | 3°6840 0324 7841 | 1°354| 3°8728 8613 3411 1'205 | 3°3367 5907 8067 || 1255 | 3°5078 3837 4343 | 1°305 | 3°6876 8909 3705 | 1°355 | 3°8767 6095 6633 1°206 | 3°3400 9750 6081 || 1°256| 3°5113 4796 7221 | 1°306] 36913 7862 7258 |1°356| 3°8806 3965 6617 1°207 | 3°3434 3927 4193 | 1257 | 3°5148 6107 1447 | 1307 | 3°6950 7185 2190 | 1°357 | 38845 2223 7240 | 1208 | 3°3467 8438 5744 | 1258 | 3°5183 7769 0535 || 1308 | 3°6987 6877 2194 | 1358 | 3°8884 0870 2386 ‘adie 373501 3284 4080 | 1259 | 375218 9782 Soor || 1°309 | 3°7024 6939 0967 |1°359 | 3°8922 9905 5941 1210 | 3°3534 8465 2549 || 1:260| 3°5254 2148 7365 || 1310] 3°7061 7371 2210 || 1°360| 38961 9330 1795 r2ur | 3°3568 3981 4503 | 1°261 | 375289 4867 2151 | 1°311 | 3°7098 8173 9627 1°361 | 3°9g000 9144 3843 1212 | 3°3601 9833 3297 | 1262 | 3°5324 7938 5886 || 17312 | 3°7135 9347 6926 | 1°362 | 3°9039 9348 5983 | 1°213 | 3°3635 6021 2290 || 1'263 | 3°5360 1363 2100 || 1°313 | 3°7173 0892 7819 | 1°363 | 3°9078 9943 2116 | 1214] 3°3669 2545 4843 || 1°264| 3°5395 5141 4329 || 1°314| 3°7210 2809 Goz2r || 1°364 | 3°9118 0928 6150 | 1215 | 3°3702 9406 4322 || 1-265 | 3°5430 9273 6109 || 17315 | 3°7247 5098 5251 | 1365 | 3°9157 2305 1993 | 1216 | 3°3736 6604 4095 || 1°266 | 3°5466 3760 o982 || 1°316 | 3°7284 7759 9233 | 1'366 3°9196 4073 3559 | 1217 | 3°3779 4139 7534 || 1'267| 3°5501 8601 2493 || 1°317 | 3°7322 0794 1693 | 1°367 | 3°9235 6233 4766 | 1:218| 3°3804 2012 8016 || 1268] 3°5537 3797 4191 || 1318] 3°7359 4201 6360 | 1°368| 3:°9274 8785 9536 1'219| 3°3838 0223 8917 || 1-269 | 3°5572 9348 9626 | 1°319 | 3°7396 7982 6971 | 1°369 | 3°9314 1731 1795 "270 | 3°5608 5256 2356 | 1°320| 3°7434 2137 7261 | 1°370| 3°9353 5069 5470 ‘271 | 3°5644 1519 5938 | 1321 | 3°7471 6667 0973 | 1°371| 3°9392 8801 4497 j 8139 3936 | 1322) 3°7509 1571 1852 | 1°372| 3°9432 2927 2813 273.) 3°5715 5115 9915 | 1°323 | 3°7546 6850 3647 |1°373 | 3°9471 7447 4357 "274 | 3°5751 2449 7446 | 1°324 | 3°7584 2505 O11 | 1'374 | 3°9511 2362 3077 275 | 3°5787 of41 o102 | 1°325 | 3'7621 8535 5000 | 1°375 | 3°9550 7672 2921 "276 | 35822 8190 1459 | 1°326| 3°7659 4942 2075 |1°376 | 3°9590 3377 7841 6597 5099 | 1°327 | 3°7697 1725 5099 |1°377 | 3°9629 9479 1796 ‘278 | 3°5894 5363 4604 | 1°328 | 3°7734 8885 7842 | 1°378 | 3°9669 5976 8746 ‘279 | 3°5930 4488 3564 | 1°329 | 3°7772 6423 4073 |1°379 | 3°9709 2871 2656 "280 | 375966 3972 5569 | 1°330| 3°7810 4338 7569 | 1380 | 3°9749 0162 7495 281 | 3°6002 3816 4214 | 1°331 | 3°7848 2632 2108 | 1°381 | 3°9788 7851 7236 4020 3098 | 1°332 | 3°7886 1304 1475 | 1°382 | 3°9828 5938 5856 "283 | 3°6074 4584 5822 | 1°333| 3°7924 0354 9454 | 17383 | 3°9868 4423 7335 "284 | 3°6110 5509 5992 | 1°334| 3°7961 9784 9838 | 1384 | 3°9908 3307 5659 285 | 3'6146 6795 7217 | 1°335| 3°7999 9594 6419 | 1°385 | 3°9948 2590 4817 "286 | 3°6182 8443 3111 || 1°336| 3°8037 9784 2997 | 1°386 | 3°9988 2272 8800 0452 7290 || 1°337 | 3°8076 0354 3373 | 1°387 | 470028 2355 1607 288 | 3°6255 2824 3373 || 1°338| 3°8114 1305 1353 || 1388 4°0068 2837 7238 "289 | 3°6291 5558 4985 || 1°339 | 3°8152 2637 0746 | 1°389 | 40108 3720 9697 "290 | 3°6327 8655 5753 || 1°340| 3°8190 4350 5366 || 1°390| 4°0148 5005 2994 ‘291 | 3°6364 2115 9307 || 1°341 | 3°8228 6445 g030]|1°391 | 40188 6691 1142 5939 9284 | 1342 | 3°8266 8923 5559 || 1'392 | 4'0228 8778 8156 "293 | 3°6437 0127 9319 || 1343 | 3°8305 1783 8777 || 1393 | 4°0269 1268 8059 294 | 3°6473 4680 3057 || 1°344| 3°8343 5027 2513 ||1°394| 4°0309 4161 4875 "245 3°4729 3479 9337 | 1295 | 3°6509 9597 4141 || 1°345 | 3°8381 8654 0600 || 1°395 | 4°0349 7457 2632 246 3°4764 0947 1183 | 1'296| 3°6546 4879 6222 || 1°346| 3°8420 2664 6874 ||1°396 | 4:0390 1156 5365 3°4798 8761 9438 || 1°297 | 3°6583 0527 2952 || 1°347 | 3°8458 7059 5174 || 1°397 | 4°0430 5259 7109 I I | Ny nN NN w Les) Oo ios} .2) a ao Co ies} ~I oO o WwW | N ~ N w UL an ~s Ko} nN tN o Ww > > | on A | - Oo Wo oo OH on = un CO mr “10 nM ew Me ty ~ ~s ies) ui o on oo ~~ ty rs) Ww te iS) oo °O ~ i.) oo > N w oo on = SS eS eR Ny ice) N Ww loa ° Ww ao N w fa) w > Bs - o oo) oo ima) ~ ao =) i?) oo “a ee N w x w ESS fb n iS) nN n = mn Keo} fo] oO ° Ce rs) oo mY Ww a nN a Ko} ~~ nN P= N w > ia) nN wW ~ ia) ° ~ wat N ho) ee eR tN © N w aN + ° ° 248 3°4833 6924 7581 || 1°298| 3°6619 6540 7987 || 1°348| 3°8497 1838 9346 ||1°398| 4:0470 9767 1907 "249 ©3°4868 5435 9094 | 1'299 | 3°6656 2920 4989 | 1°349 | 3°8535 7003 3236 ||1°399| 470511 4679 3802 ~~ e e N te ~ [1°400—1°599] TO TWELVE PLACES OF DECIMALS. er 247 ev 470551 470592 470633 4°0673 4°0714 40755 4°0796 4°0836 40877 40918 4°0959 4°1000 41041 4°1082 Actin 22 41164 4°1206 4°1247 41288 4°1329 41371 41412 4°1454 471495 4°1537 41578 4°1620 41661 4°1703 41745 4°1786 41828 41870 41912 42954 41996 4°2038 4°2080 4°2122 4°2164 4°2206 4°2249 4°2291 4°2333 42376 42418 4°2460 4°2503 4°2545 4/2588 9996 5719 1848 8383 5325 2674 0430 8595 7167 6149 5540 5340 5551 6172 7203 8646 o5olr 2767 5446 8538 2044 5962 0296 5043 0206 5784 1778 8187 5014 2258 9919 7998 6495 5411 4746 4500 4675 5270 6286 7723 9581 1862 4565 7691 1241 5214 9611 4433 9680 5353 6845 5088 2588 34°97 1611 1268 6452 1240 9715 5962 4071 8136 2254 0528 7065 5973 1369 737° 8098 7682 0251 9942 0892 7245 3149 2756 0221 97°04 5369 1385 1923 T161 3280 2464 2902 8788 4319 3698 1129 0824 6997 3865 5653 6586 0897 2820 6596 6469 6686 1501 5169 1952 6115 1928 3664 5602 2024 7216 5469 1079 8345 1571 5065 3239 Orit 0301 8034 7641 3455 9815 1063 1546 5617 7630 1946 2930 4949 2378 9594 C079 ogI9 3805 4032 6000 4113 2779 6410 9425 6244 L293 9004 3810 O152 2472 5220 2848 9812 57/5 9603 1366 4°4816 4°4861 4°4906 4°4951 4°4996 4°5041 475086 45131 45176 4°5222 4°5267 45312 45357 4°5403 475448 4°5494 4°5539 4°5585 4°5630 4°5676 4°5722 4°5767 475813 475859 4°5905 45951 4°5997 4°6043 46089 46135 4°6181 46227 4°6274 4°6320 4°6366 46413 4°6459 4°6506 4°6552 46599 4°6645 4°6692 46739 4°6786 4°6832 46879 4°6926 46973 4°7020 4°7067 8907 130° 6141 5432 5172 5363 6003 7995 8638 0632 3279 5978 9331 SUSY) 7397 2112 7282 2907 8988 D025 2519 997° 7879 6246 5072 4356 4101 43°95 4969 6095 7682 973° 2241 5215 8652 2552 6917 1746 7041 2801 9027 5719 2878 0505 8599 7162 6194 5695 5665 6106 Pror. F. W. NEWMAN, TABLE OF THE EXPONENTIAL FUNCTION e* [1°600—1"799] e | £ e 2 e v ev | 4°953° 3242 4395 || 1650] 5°2069 7982 7180 || 1°709} 5 4739 4739 1727 | 0267 6006 49579 8793 4161 |) 1651 | 5°2121 Sg4r 1365 || 1°701 | 5°4794 2407 7005 || 1°751 | 5°7603 6015 6942 4°9629 4840 1916 || 1°652| 5°2174 0420 7740 || 1°702 | 5°4849 0624 1708 || 1°752| 5°7661 2339 8240 4°9679 1383 2620 || 1°653| 5°2226 2422 1520 || 1-703] 5°4923 9389 1317 || 1°753] 5°7718 9240 5661 4°9728 8423 1237 || 1654] 5°2278 4945 7924 || 1°704| 5°4958 8703 1320 ||1°754| 57776 6718 4975 49778 5960 2740 || 1°655 | 5°2330 7992 2178 | 1°705 | 55013 8566 7211 || 1°755 | 5°7834 4774 1957 49828 3995 2102 || 1656] 5°2383 1561 9513 || 17706] 5°5068 8989 4489 || 1°756| 5°7892 3408 2387 | 499878 2528 4305 || 1657] 5°2435 5655 5163 || 1°707] 5°5123 9944 8656 ||1°757| 5°7950 2621 2051 479928 1560 4333 || 1658] 5°2488 0273 4371 5°5179 1460 5223 || 1°758| 58008 2413 6743 49978 rogt 7178 || 1°659| 5°2549 5416 2382 || 1°709| 5°5234 3527 9706 | 1°759| 5°8066 2786 2258 50028 1122 7834 || 17660] 5°2593 1084 4447 || 1°710] 5°5289 6147 7624 || 1°760| 5°8124 3739 4403 | 570078 1654 1301 || 1°661 | 5°2645 7278 5824 ||1°711 | 5°5344 9320 4504 || 1°761 | 5°8182 5273 8985 0128 2686 2585 || 1662] 5°2698 3999 1773 || 17712 | 5°5400 3046 5878 || 1°762| 5°8240 7390 1820 0178 4219 6697 || 1°663 | 5°2751 1246 7564 || 1°713 | 5°5455 7326 7283 || 1°763] 5°8299 0088 8730 0228 6254 8651 || 1664] 5°2803 go2r 8467 |) 1-714] 5°5511 2161 4261 || 1-764] 5°8357 337° 5541 0278 8792 3469 || 1665 | 5°2856 7324 9761 || 1°715| 5°5566 7551 2361 || 17765 | 5°8415 7235 8086 0329 1832 6174 || 1°666] 5:2909 6156 6728 ||1°716| 5°5622 3496 7138 || 1°766| 5°8474 1685 2204 "0379 5376 1799 || 1°667 | 52962 5517 4658 | 1-717 | 5°5677 9998 4150 || 1°767 | 5°8532 6719 3740 "0429 9423 5377 || 1668} 5°3015 5407 8843 || 1-718 | 5°5733 7056 8962 || 1768) 58591 2338 8543 "0489 3975 1950 || 1°669| 5:3068 5828 4583 || 1°719| 5°5789 4672 7145 || 1°769| 5°8649 8544 2470 "0539 9031 6564 ||1°670| 5°3121 6779 7181 ||1'720| 5°5845 2846 4276 || 1-770] 5°8708 5336 1383 0581 4593 4268 || 1°671 | 5°3174 8262 1948 || 1-721 | 55901 1578 5936 ||1°771| 5°8767 2715 1149 0632 0661 o118 |) 1°672 | 5°3228 0276 4198 || 1°722 | 5°5957 0869 7711 ||1°772| 5°8826 0681 7644 0682 7234 9176 || 1°673| 5°3281 2822 g250 || 1°723| 5°6013 0720 5196 || 1°773 | 5°8884 9236 6746 0733 4315 6506 || 1°674 | 5°3334 5902 2432 ||1°724| 5°6069 1131 3989 || 1°774| 5°8943 8389 4340 0784 1903 7189 || 1°675 | 5°3387 9514 9073 || 1°725| 5°6125 2102 9693 || 1-775 | 5°9002 8113 6319 0834 9999 6273 || 1°676 | 5°3441 3661 4510 || 1°726| 56181 3635 7919 || 1°776| 59061 8436 8580 0885 8603 8867 || 1°677| 5°3494 8342 4084 || 1°727| 5°6237 5730 4282 | 9350 7025 "0936 7717 0047 || 1678 | 5°3548 3558 3141 | 1°728| 5°6293 8387 4402 || 1°778| 5°9180 0855 7564 0987 7339 4995 || 1°679 | 5:3601 9309 7035 || 1°729| 5°6350 1607 3907 ||1°779 | 5°9239 2952 6113 8537 | 1°680| 5°3655 5597 1122 || 1°730| 56406 5390 8428 || 1-780] 5:9298 5641 8591 ‘1089 8114 6043 || 1°681 | 5-3709 2421 0766 || 17731 | 5°6462 9738 3604 || 1°781 | 59357 8924 0927 "1140 9268 2532 || 1°682 | 5°3762 9782 1334 || 1°732| 5°6519 4650 5078 || 1°782 | 5°9417 2799 9052 ‘1192 0933 3113 || 1°683 | 5°3816 7680 8200 || 1°733 | 5°6576 0127 8498 || 1783 | 5°9476 7269 8905 "1243 3110 2994 || 1°684| 5°3870 6117 6744 || 1°734| 5°6632 6170 9520 || 1°784 | 5°9536 2334 6432 "1294 5799 7027 || 1685 | 5-3924 5093 2350 | 1°735 | 5°6689 2780 3805 || 1°785 | 5°9595 7994 7583 ‘1345 g002 0608 || 1686 | 5°3978 4608 0406 || 1°736| 5°6745 9956 7018 || 1786 | 59655 4250 8314 1397 2717 8780 || 1°687 | 5:4032 4662 6310 || 1-737 | 5°6802 7700 4831 || 1°787 | 5°9715 1103 4588 "1448 6947 6679 || 1688 | 5°4086 5257 5460 | 1°738| 5°6859 Gor2 2921 | 1°788 | 5°9774 8553 2373 "1500 1691 9448 || 1°689 | 5:4140 6393 3263 | 1°739| 5°6916 4892 6972 || 1°789 | 59834 6600 7645 "1551 6951 2235 || 1°690| 5°4194 8070 5131 || 1°740| 5°6973 4342 2672 || 1°790| 5°9894 5246 6383 "1603 2726 o1g1 || 1°691 | 5°4249 0289 6480 || 1°741 | 5°7030 4361 5716]|1°791 | 5°9954 4491 4574 "1654 9016 8475 || 1692 | 5°4303 3051 2732 | 1°742 5°7087 4951 1804 || 17792 | 6'0014 4335 S2r1 "1706 5824 2250 || 1°693| 5°4357 6355 9316] 1°743| 5°7144 6111 6643 || 1°793 | 6°0074 4780 3291 1758 3148 6683 | 1°694| 5°4412 0204 1663 | 1°744| 5°7201 7843 5943 || 1°794 | 60134 5825 5820 1810 0990 6949 || 1°695 | 5°4466 4596 5213 | 1°745 | 5°7259 0147 5421 || 1°795 | 60194 7472 1807 1861 9350 8224 || 1°696| 5°4520 9533 5409 | 1°746| 5°7316 3024 0802 || 1°796 | 6:0254 9720 7270 "1913 8229 5694 | 1°697 | 5°4575 5015 7701 | 1°747 | 5°7373 6473 7813 || 1°797 | 6'0315 2571 8230 1965 7627 4546 | 1°698| 5°4630 1043 7544] 1°748| 5°7431 0497 2190] 1°798 | 6°0375 6026 0717 "2017 7544 9975 | 1699 | 5°4684 7618 0397 | 1°749 | 5°7488 5094 9672 | 1'799 | 6°0436 0084 0764 Lan! xi wn ° on ~i mn + a = I ° oo cs) ~ ~ _ on oo = N ° AMUN UMN AM Annnn Annnn NAOMI OnN Monon WUT “ =) ° Ww oo ~I > ~I ~ [1°800—2'000] TO TWELVE PLACES OF DECIMALS. 249 ev av Ee” 1800 | 6:0496 4746 1801 | 6°0557 0013 1°802 | 60617 5886 1°803 | 6°0678 2365 1°804 | 670738 9451 1805 | 60799 7144 1°806 | 6:0860 5446 1°807 | 60921 4355 1°808 | 60982 3875 1809 | 6°1043 4003 T°810} 6°1104 4743 1811 | 61165 6093 1812 | 6°1226 8055 1°813 | 6:1288 0629 1814 | 61349 3817 1815 | 6:1410 7617 1816 | 61472 2032 1817 | 6°1533 7062 1818 | 61595 2706 1819 | 671656 8967 1°820] 61718 5844 1821 | 6°1780 3339 1822 | 6°1842 1451 1°823 | 6°1904 0182 1824 | 6°1965 9532 1°825 | 6:2027 g5o1 1°826 | 6°2090 oogt 1°827 | 6:2152 1302 1°828 | 6:2214 3134 1°829 | 6°2276 5588 1830] 6°2338 8665 1°831 | 6:2401 2366 1°832 | 6°2463 6690 1°833 | 6°2526 1639 1°834 | 6°2588 7214 1°835 | 6°2651 3414 1°836 | 6°2714 0241 1°837 | 6°2776 7695 1°838 | 6°2839 5776 1°839 | 6°2902 4486 1°840 | 6°2965 3826 1841 | 6°3028 3794 1842 | 673091 4393 1°843 | 673154 5623 1°844 | 6°3217 7485 1°845 | 673280 9979 1°846 | 6°3344 3105 1°847 | 6°3407 6865 1°848 | 6°3471 1259 1849 | 6°3534 6288 4413 7710 6707 7464 6044 8520 0968 9471 o118 9004 2231 59°5 6141 9058 0782 7445 5184 o144 8476 6335 9884 3292 8733 6390 4450 EOS 6556 3008 4673 7771 8525 3166 7931 9064 2813 5436 3193 2353 g1go 9986 1027 8606 go25 8588 3607 0402 5297 4624 4720 1929 63598 6°3661 6°3725 6°3789 63853 6°3916 673980 6°4044 6°4109 64173 6°4237 6°4301 64365 6°4430 64494 6°4559 6°4623 6°4688 6°4753 6°4818 6°4882 6°4947 6°5012 6°5077 6°5143 6°5208 6°5273 6°5338 6°5404 6°5469 6°5535 6°5600 65666 6°5731 6°5797 675863 6°5929 6°5995 66061 6°6127 6°6193 6°6259 66326 6°6392 6°6458 6°6525 6°6592 6°6658 6°6725 6°6792 1952 8252 5188 2762 COTA: 9825 9314 9443 0213 1624 3677 6372 9710 3691 8317 3588 9505 6067 3277 1134 9639 8794 8597 go5t o156 Igi2 4320 7380 1095 5463 0486 6164 2498 9489 7137 5444 4409 4033 4317 5262 6868 9136 2066 5660 9918 4840 0428 6681 3601 1188 2602 3°94 9772 8998 7154 0621 5786 9°45 6799 5456 1429 1140 1015 7487 6997 5991 og21 8247 4434 5955 9287 og16 7333 5038 0533 03 3° 0947 8908 9743 2989 2191 4898 7668 7263 9653 2015 0732 2394 3597 0943 1043 O512 5973 4055 1394 4633 0421 5413 6274 9670 1°900 1901 I°g02 1°903 T'904 1°905 1°906 1°907 1908 1909 I°9Q10 I°QII I°9t2 I°913 1°94 Tgt5 1-916 I°Q17 17918 Igtg 1°920 1921 1°922 1°923 1924 1925 1°926 1°9Q27 1928 1929 1°930 1931 17932 1°933 1°934 1°935 1°936 1°937 1°938 1°939 1'940 1941 1'942 1943 1°944 1945 1'946 EQOA 1°948 1949 9444 8368 7961 822 9157 0762 3039) 5989 9611 3908 8879 4526 0849 7848 5925 3879 NS 2625 3018 4092 9 5846 8283 1403 5206 9694 4866 0724 7268 4499 2417 1024 0319 0305 0980 2347 4405 7156 0600 4737 9569 5°97 1320 8239 5856 4171 3185 2898 3311 4425 6240 2°000 | 7°3890 5609 8930 6502 2279 0783 1872 2240 8592 7635 6087 0669 8112 SHS 8531 5000 1315 4238 0541 6999 0397 7525 5179 o164 9291 9376 7245 9729 3665 5899 3282 2673 M37] 4947 1581 7727 0276 6128 2191 5378 2610 0814 6924 7883 0637 2143 9363 9266 8829 5°33 4870 5336 3436 6181 0589 3685 2502 4078 5460 3701 5862 goro 0219 6572 5157 3069 7412 5295 3836 or58 1394 4682 7167 6002 8347 | 1369 2242 8148 6276 3820 7986 5981 5024 2341 5161 0725 6279 9°77 6379 5453 3576 8029 6103 5°95 2310 5058 0661 6443 9739 7891 8246 8160 4998 6129 XII. The Equations of an Isotropic Elastic Solid in Polar and Cylindrical Co-ordinates, their Solution and Application. By C. Cures, M.A., Fellow of King’s College, Cambridge. [Read October 31, 1887.] CONTENTS. SECT. SECT. I. §§ 1—6. Fundamental Equations in Polar Co- IX. §§ 63—69. Fundamental Equations in Cylindrical ordinates. Co-ordinates. Il. ,, 7—19. Equilibrium of Solid Sphere or Spher- X. ,, 70—77. Equilibrium of Cylinder, solution in ical Shell. ascending powers of r and z. Til. ,, 20—22. Tendency to Rupture. XI. ,, 783—83. Equilibrium of Infinite Cylinder or IV. ,, 23—31. Gravitating nearly Spherical Mass. Cylindrical Shell. V. ,, 32—39. Solid Rotating Sphere. XII. ,, 8490. Vibrations of Infinite Solid Cylinder VI. ,, 40—47. Rotating Spherical Shell. or Cylindrical Shell. VII. ,, 48—50. Sphere or Spherical Shell with given XII. ,, 91—99. Equilibrium of a Finite Cylinder under surface displacements. purely surface forces. VI. ,, 51—62. Vibrations of Sphere or Spherical Shell. SECTION I. FUNDAMENTAL EQUATIONS IN POLAR CO-ORDINATES. § 1. In considering the form taken in polar co-ordinates by the equations for the equilibrium or state of vibration of an isotropic elastic solid, it is convenient to deduce the result from the formula for the energy expressed in Cartesian co-ordinates. Using Thomson and Tait’s notation, let a, 8B, y denote displacements parallel to three fixed rectangular axes, and let da _ dB. dy _ da re d dy = 4, dp is dy da mt oe a (1) dy da dp SRT teat ie eS - weer . dz dy * da while es ap + dy = da dy dz Mr C. CHREE, ON THE EQUATIONS OF AN ISOTROPIC ELASTIC SOLID. 251 If m and n are the elastic constants of a homogeneous isotropic material, the potential energy W, per unit volume answering to the displacement a, 8, y is given by 2W,=(m+n) & +n {a> + 0? +0? — 4 (ef + eg tfa)} ..cccceceecsereeee (2)*. We must transform this expression to polar co-ordinates. Moreover it is most convenient to take the axes not in fixed directions but specially for each point considered. Thus if r, 0, 6 denote the co-ordinates of any point P in a sphere referred to the centre O of the sphere as origin, to a fixed plane from which to measure ¢, and to a fixed axis from which to measure @, it is best to consider the displacements of P as u along OP, v along a perpendicular to OP in that plane which contains OP and the fixed diameter from which @ is measured [this plane we shall call the meridian plane at P], w in a direction perpendicular to the former two directions, and so perpendicular to the meridian plane. u, v, and w are of course functions of r, 6, and ¢, and we have to express e, f, 9, a, b, ¢, 5, W, in terms of 1, 0, ¢, u, v, w. § 2. To do so we require certain elementary geometrical relations between the directions of the fundamental axes at P and at adjacent points. The following figure shows these at once. Let P be the point 7, 0, ¢, (@ sagopaehcocouAacaar r, 0400, d, T 1 Rees Oa eee r+or, 6, ¢, : ISP Bea ancentestcne r, 0, 6+0¢. Let PN, SN be perpendiculars on the axis from which @ is measured, and PY, S7' tangents to the meridian sec- tions at P and S; i.e. PY’ is perpendicular to OP in the meridian plane at P, and similarly for S7. Then A PNS = 0¢, ee Se) eae POS = PNS J, = sin 009, [Nee Nae SING ie V6 peat ee PTS = PN: gr = 8 00d. At R the fundamental axes are parallel to those at P. the axis of w makes angle 0@ with axis of wu at P, / Noe Dine (senor enc aan OR cashes nineties GO! etree siesta Ue INE SIS: es Baan onneReeeion DO -coogsocosondeds COS)OO Diet «= cnicisaict- Os sesouts NOCHE ETO Sane QU Soaieiioierlvalsisneittis et RtO Mme c settee) osteo * Cf. Thomson and Tait’s Nat. Phil., Part 1., Equation (7), p. 232, putting k=m—-4n. Vor eV eARD ITT: 35 252 Mr C. CHREE, ON THE EQUATIONS OF AN ISOTROPIC ELASTIC : du nee : é , If now we understand exactly what is meant by >, and similar differential coefficients, dé we can deduce at once what we want. : haw AY , d - If u, denote the displacement at Q along the direction OQ, then pee fy 0 ao when Q moves up to P. Suppose however we took the axes of a, y, z, which are fixed in space, to coincide in direction with the axes of u, v, w at P; then u=a, but ie is not = olay But dy rdé da Lt, Component of displacement at @ parallel to OP — u = lit. = Se , dy PQ when Q moves up to P; fie (u+ = 00) cos@ — (v+ 2 28) sin 00 ~w i.e. ar Lt. a0 , when 0@ indefinitely small, _du_o a0 +r The other differentials are to be found exactly in the same way. This method is really that employed in Rigid Dynamics in the case of moving axes, and is practically the same as that given by Mr Webb in the Messenger of Mathematics, February, 1882. We thus easily find e = = = = sinteec ageas e e e e (3), pao 22 I PO rn ee ck (4), g= a 4 cot a a Te Sthncsdpine) ual eos ee (5), ==, 7 “pts 22 GOO nsh tae ee ee (6), pe ey nS ee ee (7), dz dz dr r rsind dd _dB da_ldu,d _»v Cle a dé ap eer Se cro pels fue he nea Od aes eal a (ys It is to be noticed that most writers on Elastic Solids measure 6 from the equator so that they take v opposite in direction to the above, also @ of our notation is equivalent to 7 ¢ of Lamé’s. Taking this into account the above values agree with those of Mr Webb. 3 du 2u 1ldv_ veoté 1 dw F > — = — - RS slelaie vial vip cle eiefecce 9 F We thus find dé=et+ft+y ayes ey tt ae Ais (9) or, as is often more convenient, 1 (d. ur 1 d.ur sin 6 fies up, d.wr sin + r\ dr ‘sind dé sin’ db 6= SOLID IN POLAR AND CYLINDRICAL CO-ORDINATES. 253 fl d.w* 1 d.vsin@ 1 dw)” (7 dr +rsind dd 'rsind dg) ita, du /2u 1 dy vcot 8 ees “= u 4) wu v cot 8 1 oo) ae” (naa r + rsind dd +( G? r + rsinddd 2W,=(m+n) dr r i r dé 1 dw _ weot 7 Tp a dw _w Ls aN dv pea ih du\? e dé r rsin 6d Ce r vi rsin® dp i= r % Tr 76) § 3. It seems on the whole the best way of finding the equations of motion or of equilibrium to employ the principle of the conservation of energy along with D’Alembert’s principle. Thus the increase of the potential energy of the system must be equal to the work done by the forces, other than elastic, which act throughout the mass or over the surface and by the reversed effective forces. Of course this is merely another way of saying that the sum of the increases in the potential and in the kinetic energies is equal to the work done by the external forces. Let us denote the non-elastic forces which act throughout the interior of the mass, ie. “bodily forces,” by R, ©, ® parallel to wu, v, w respectively, per unit mass; and denote the surface forces similarly resolved by F, G, H per unit area. We have then to find the increase in the potential energy of the entire mass due to an arbitrary displacement ou, dv, dw, and also the work done by the systems R, ©, ®, F, G, H and the reversed effective forces, and equate the two. Thus (a _ tw e = =o) / f If pr’ sin Odrdédp [(z 7B) au+(@ dé du + (® dE dw | + [ds (Fou-+ Gov+ How) =change in potential energy of solid due to displacements Qu, 6v, dw, for all possible syaluessotethemratios ci: NOUN OW) ste--eeteetaseeeeceereae steamer anaes (12). The triple integral applies to the whole of the interior of the solid, the double to the whole surface. Owing to the length of the expression it seems best to write out separately the changes in the potential energy due to du, dv and dw. If then W denote the potential energy of the entire solid, fies ur 1 d.vsi w\* 2W= [[fresin Odrdé dd (mem (5 =e ray g - 2 +f ao 73) du ;2u 1dv_ vcoté 1 dw u ldv\/w. veoté 1 dw +n\a 4 (Bet ay" teaapde) jets wiles 7 aoaa ldw weot@ 1 d\* (dw w 1 du /dv v il1du =i) - fae Bh O% a a ie » eae =) a (G-ats do) ] 3): Denote the variation in W due to displacement du by OW,, ieapeBhabuded bhanbbeartebaqs SaseodanSeM ee AAO SEDER Ryn RSE eee to dv by OW,, tes ie a, S and to éw by dW,. Then we shall find 0W,, 0W,, dW, each to consist of two series of terms; one for the interior, the other for the surface. To find these, only elementary knowledge of the Calculus of Variations, or rather integration by parts, is required. 33—2 254 Mr C. CHREE, ON THE EQUATIONS OF AN ISOTROPIC ELASTIC To show clearly the method employed we shall find the variation of the term in 6 due to du fully. It is : 1 d.ur? 1 d.vsin#@ 1 dw\/1 d.r*du [[frsin earaeag [2im+m (J dr *rsn@ dO b aaO aa) Pe )): ie. [[]s Odrdédd 2 Gn ems 2 ae Integrating by parts, this is equal to 2 [[m +n) 5.7°.dud0dd — 2 [[Jom +n) - dur’ sin Odrdédd¢. To obtain the entire change dW in the potential energy, we have only to add together the changes dGW,, 0W, and dW, which are to be obtained exactly in the way illustrated above. This involves no absolute difficulty but algebra of some length. Introducing the values so obtained for OW in (12), supposing the normal at any point of the surface to make with the directions of u, v, w at that point angles cos*A, cos“, cosy respectively, so that 17 SIN AED AO= NAG coo ocee cies he aces dsecst shee ccs e nee (14), misini OAD d= WASe.ees. tee Aceon Oost eee Rae (15), and POO Or = OS or. sas ce oedsanod to nbs cee eetee (16), we find |fas [au \F- x ((m +n) — 2n (3 — =) — pn (2 = = + : = — vn (= = ~ + = = 9 aa) + dv \e- an (5 - 2) —# (m+n) 3 — on(3—%— 29%) _ ex w cot 0 all 4 \r ddr r sin 6 dd/) dw ow 1 du 1 dw weoté 1 d + dw {H— an Ge = ees ee i =aind di) —» (m+n) 8—2n (3-3 -2ete - a) ioe a? dé d d 1 du - || |rsin @draeag [eu iP (Fe -B)-(m+m +e ag: o(G +o -- 5) n 4 ( 1 du dw ¥\ ~rsinOdd\rsinddd dr r { (fv_e@\_m+ndé n ad (dw 1 dv\_nd/ dv 0) +0010 (a S i i Fame as ao ee nb aa) a a 78) ow § (qe ) eh gb 8 Ee (= se +ou\p(Ga-®) rsind dé rdr\snddp ” dr w) dw 1 du — 53a (ae + woot 0 = | Sinan (17). SOLID IN POLAR AND CYLINDRICAL CO-ORDINATES. 255 § 4. The displacements du, dv, dw are arbitrary and independent, thus we must have the coefficients of du, of dv and of @w in (17) in the interior, and at the surface, all zero. We thus get three equations at each surface (if more than one), and three internal equations. The latter are o(Se-2)= Pace e Bi d sind (+ v -7%)+ n ad ( 1 du dw ~*), (18), dé dr rsin@d@’ r rdé) rsin@d¢d\rsin@ db dr dy _ m+n db n ad /dw 1 dv nd du e (ae )- r db r ain dde (aa tw orO— eae eae ("5 ete za) ae a (19), dw _m+nds5 nd/ il du _ ow n d dw 1 du (ae | eeu aera aa dr = 0) + Aa8 (‘a9 + cot O— = 9 a5) Bee. ey These equations may be put in a very neat form. Write 1 d.wrsin@ d.ovr y waa at is )- MD oes i (21) 1 /du d.wrsin@ and da I) ARR con eeecceseenteececsneeseenee (22), d d.vr du sin 6 ( = =) (LRA ees nn: See eae See (23). Multiply both sides of the first internal equation by r’sin 8, -Od dS C TOO REODEE 6 COREE COE aERS SCONCE dee eeececetesasseee oT SINC, Rees eo be. erates and ofetheyhird:seeeets see sceeeeee eT and they become respectively dd | as dw au - (m+ n) r* sin 6 ae gtr dé = pr’ sin 6 (G-2) aa sect oraas cane (24), dé dA ad d’v 5 (m + n) sin OF — Eee ee sin 6(5 2-9) SPADE REA Roa pone (25), dé 08 dA (d’w (m +n) cosec 0 a= rea gL 7a 2h (Ga -) obec tunlentshe tbat ie (26); while the surface conditions are ie 78 du\ dv_v,idu dw w i vdu : fi {(m-+ n) d—2n (3 = al +pn fe A i 7a) +n (+ =— ie and a SRRDOLEGEES (27), dv v 1 du 1 dv uUu ) 1 dw w 1 dv’ c= rn (Fe - pie =) +m 4(m+n) d— 2n(3- SG —*) + vn as — pot O + ag) (28)> ? dw w 1 du ldw w 1 dv Hann (FF + sino ag) tH" (> ag ~ 708 + rane aa) uv 1 dw’ + f(m-+n) 8 2n (8-E-Fcot 0 at susees (29). 256 Mr C. CHREE, ON THE EQUATIONS OF AN ISOTROPIC ELASTIC Usually the surface is that of a sphere and so ~=0=v while X=1. In this case we get d F=(m+n)5—2n (s - ir) sls stand iaaSTS Tan Suhre eRe (30), a r ldu onal, ai 1a) BSH O OL CORE OE COCODIRDO TOC BONGO] DaTcane (31), a , 1 du won( 1 t aime iaceiaivinys {etnjels p)aioieveloinlelpievelerele/a\aleiaeleretets (82). The equations both internal and at the surface are really identical with those given in Lamé, page 200, ete. He however gives merely the results and we have nowhere seen them actually worked out for the general case. The form above is slightly different from Lamé’s, because he uses as his angles of reference Ww and ¢, answering to our @ and 5 —@ respectively, and his v is measured in the opposite direction to ours. § 5. The reason that has led to the use of a different notation here is that it seems objectionable to introduce in Elastic Solids angles different from those ‘generally employed in solid geometry, at least in English books. Also since the development of these equations depends largely on the properties of spherical harmonics, it is awkward to employ a notation different from that which occurs in the standard English works on that subject. This awkwardness is due to the two facts that it would make reference to the ordinary spherical harmonic properties more difficult, and that it would make it harder to recognise these properties when they turned up. § 6. The surface conditions may be employed in finding the elastic forces or “stresses” at any point in the interior of the solid. For we can draw an imaginary surface through the point considered so as to include a limited quantity of the elastic material whose con- ditions of equilibrium can be separately studied. It is in equilibrium under the action of the system R, ©, ®, of the reversed effective forces if motion exist, and of the surface tractions. These surface tractions are as before given by the values found above for F, G, H, but they are now really the stresses existing at this boundary. We can suppose the boundary moved about so that its normal lies in any required direction and so can find the whole series of stresses at the point considered. Suppose we denote these stresses by letters indicating their direction. Thus let R,=Normal traction parallel to w on the face r*sin 0 d¢ dé, R= Tancential”27.escsee eee CRORE SE RA, rsin 6 dd dr, Ry = Tangential’..,..........2 2.0504. Ce A cas te rs r dé dr, 6;= Normal ie eee Pimentel fae wcindeey r sin 6 d¢ dr, © = Tangential’, cee res dene ME daa stand see r* sin 6 dd dé, @, = Tangential... we oaete ene ee ie Sem rile ri r dé dr, ®, = Normal. . £02 ..cce ee Wliwododeaneresar r dé dr, ®, = Tangential)... 39.034, «60. ites AUD Rastetiece tote 7 sin Odd dé, @,'="Tangential’; ov castcaressoroteas Discs aeccicsee soe rsin Odd dr. SOLID IN POLAR AND CYLINDRICAL CO-ORDINATES. 257 Putting A=1 in the surface equations (27), (28), (29) we get the tractions on the face r? sin Odd dé. They are F, G, H parallel to wu, v, w in the one notation, se Ole, SDE ars cars ioclyseeee is foe oy Waves OWING home doseoe 3 * R.=(m+n)6—22n (8 — a) Sede Dace MeO RGAE Cee EASE (33), “ d. el aay ao! SC ep OGa ChE DECOM aER dome ar acct ene (34), ap” r I Gin # spell “dr ‘sind dd CAP Re GononncHodncodcoabooenccanarc (35). Putting ~=1, we get the tractions on the face rsin 6 d¢ dr, v d.- r ldu cele 21d) Ao sc adie aowontonsedeanDiusare doedooadebenoses (36), u ldv O, = (m+n) 6 — 2n (s— ss = SGhO mono een sober peaCe too (37), ldw w 1 dv’ Dy =n fe dé = a cot 6 ar aa) de) Ngan COTAOO oO atobO Ope dec 40.6 ob (38). So putting v=1, we get the tractions on the face rdé dr, de | "iE I dar oe Sie 3 Seria i BA OCR COD Oe COC OUT CIGTIn S COC DONE (39), ldw w_ 1 d O, =n (= dé = A cot 0 + waa) = apadaleateteleletetaefeletelalstaletctelststvinielclctelelutetetstatste (40), ©, = n)3— an (8—*— "oot @— 2, 4) 41 = (m+n) o— n( = emer CO eG ee ee (41), From these values we get at once the well-known relations (O}e — Re 5 Ry = @. 3 O, = D, Sfafe viel elevevs/elerslw'cvelalainietelat inie.cieie (42). Of course we might deduce these at once from the form of W, putting dW dw eee’ Po= 7,7 ote We have thus deduced all the circumstances whether of equilibrium or of motion from the expression in Cartesians for the potential energy. This expression requires only the knowledge of the expression for the relative displacements as given by the equation to the “strain ellipsoid,” the invariants of which, viz. et+f+g and eft+eg+fyg—ftia+b'+c’), are all that is required. ix or D Mr C. CHREE, ON THE EQUATIONS OF AN ISOTROPIC ELASTIC SECTION II. EQUILIBRIUM OF SOLID SPHERE OR SPHERICAL SHELL. § 7. We shall now apply these polar equations to the following problem. A solid sphere, or a shell bounded by concentric spherical surfaces when undisturbed, is acted on in its substance by a system of forces for which a potential exists, the potential being expressed by a series of solid spherical harmonics, and over its surface or surfaces forces are applied which are expressed as a series of surface spherical harmonics. It is required to find the displacement at every point of its substance. This problem has been solved by Sir W. Thomson in Cartesian co-ordinates, see Thomson and Tait, Vol. 1. Part m., beginning at § 737. Lamé also has given a solution in polars in Liouville’s Journal for 1854; but it is extremely complex, and is entirely different from what follows. Professor Pearson in the Quarterly Journal for 1879 has solved the case of normal surface forces on a solid sphere expressed as Legendre’s co- efficients. Professor Darwin has converted Sir W. Thomson’s solution into polars for the case ~=0, see Phil. Transactions, 1879. The following method is entirely different from any of the above, it solves the most general form of the problem, and seems less diffi- cult than Sir W. Thomson’s method, especially in the case when the surface forces are purely normal. The same or closely analogous methods are applied hereafter to the case of equilibrium or vibration of a cylinder, and to the vibrations of a sphere. § 8. If V denote the potential of the non-elastic or bodily forces, then z= ~ 2 O= 5 a’ oad a We have, see (9) and (10) Section L, jd Bus Vda ceo lp Tuy (1) ares rade. r rsin@ dp al akon 1 d.vrsin@g 1 d.wrsin @ =a ape ENO 7 ain? 9 dd ) snbbnansaabdauuvee (1 a) The surface conditions are (30), (31), (82) Section 1, viz. (m—n)6 + 2n a MOLMAal LLAChION PEL WN, ACA) Jac ace cade come mdeesinsirilele siaesisat (2), ies 1 i) : ee nh em dean = tangential surface traction along tangent to meridian ......... (3), w rd 1 iw) ; ; : ar Obi Hit 0 dd = tangential surface traction along perpendicular to meridian...(4), putting =a for a solid sphere, and =a and a’ for a shell. SOLID IN POLAR AND CYLINDRICAL CO-ORDINATES. 259 The internal equations are (24), (25), (26), Section I. viz. (m+n) 7° sin 6 ei — os +n p= — pr’ sin 0 ay amano tea hae cients te (5), (m + n) sin poe oe 0D Pre saioens, tisineeiassrites (6), (m +n) cosec ign a = —— +n am = — p cosec 0, niaaires samie a anieitaleaetal wishin (i); erere ——— E LS oa = OO 8, aR len 20h (8), 22% ( = me we | heey See ee (9), © =sin 6 (“i - =) Tee. 5). | 20 a en (10) We shall first consider a solid sphere, and afterwards find the additional terms to be introduced in the solution for the case of a shell. Differentiate (5) with respect to r, (6) with respect to 6, and (7) with respect to ¢; adding we get (Go BION SS aioe Vocootedat aucosecnereeessee vnbeeh eon sdbeh (ali). \ 2 Nae dic eat, 20s J il Go. 70s ee dé re 5 ai where as usual Wo= FEI Sau ra qe. sin 0 dot sin? Tay ee (12). aan 364 According to the data of the problem V=(r'V,), where V, is a surface spherical <>v). Ape: tlée harmonic of degree 7, and so r'V, a solid harmonic of the same degree. At present we shall retain only one term 7'V, as a type of the solution. The solution of (11) is obviously a AVA Vane fers Ness Coty eae pecs (13), m+n r where Y, is an arbitrary surface harmonic of degree i, retained as a type. Substituting this value of 6 in (5), (6) and (7) we get d& dB m+n re Le ieee a sin gee 1 Hep ecen ae Bonehe CRTSREHCOCET (14) ee a VE Maen ars Gis). dg dA m+n i1 ad Lg ‘ Fis ar | aa Gneekyn 'a/0/0\e,0 [0 olafnivlelaisiejeleieleleialeisiulsjatarelovatete (16). Substituting in (14) the values of © and 48 given in (10) and (9), we get @ d.vr du 1 /du d d.wrsin@\ m+n 2 d = ees a dr ~ 8) ~ sin 8 (age dr’ dd js ) ae ge ges Vou. XIV. Parr III. 34 260 Mr C. CHREE, ON THE EQUATIONS OF AN ISOTROPIC ELASTIC In this substitute from (1 a), d.wr sin Gy d. ur 1 d.vrsin é dé sin SU ee +a eT. ) +83 sin’ 0, and we get sin 6— — = + 7g: sind— — —sin 0 — ak gece dé = d du Li Gwe ah eae depres d (/d.uwr 1 d.vrsin @ de" dé sin @ d¢ dr al ) ne +n n +sino ©, 76 = iin One ee dr : Substituting for 6 from (13) and reducing, the terms in v cut out. Bringing over the term in & to the second side, performing the differentiations with respect to r, and dividing both sides by sin @, we get 7 du Ll dtu ur aay sin 0— = Wy, (42 dr = 3: do + sm™6 dg SP a ee m+n ) ——- 4 : m = n n The left-hand side=0 is the general equation in w; the right-hand side gives the typical terms, treating the 7* surface harmonics as typical. We shall find that part of the solution of (17) depending on r‘** by assuming this gives to determine wu, the equation ih sa! du i akin Mi o\+ (6+2)p snd go 9 Gg + cara age t G8) +2) m= -(™-2) ¥,- meV, «(18). Since the equation satisfied by a Laplace’s function X, of the 2 degree is na dX, aXe 7 sin 6 dé" sin OF dé ee 6 d¢* eran SU the particular solution of (19) is (= = 2 _\n 2 (¢+2) pl’, 2(27+ 3) 2 (20+ ae because (¢+ 3) (@+ 2) —7 (+ 1) =2 (274 3). an arbitrary surface harmonic of degree 1+ 2; me 9 (i+ 2)pV, le =e an pl (= "2 (2i+3) (m+n) 2 (2043) The general solution is Z,,,, VT. eine (20) itl is the coefficient of r‘** in the complete value of u. This shows that the part of the solution of (17) which involves surface harmonics of degree 1, is ma = — 7H p(+2)V, n i=t 21 u 7 2(2+3)(m+n) +2 QrTs) © AGS RR, shocidrinsd 0000 (21), where Z, is an arbitrary surface harmonic of degree 7, which however must be zero if 7=0. SOLID IN POLAR AND CYLINDRICAL CO-ORDINATES. 261 § 9. Again in (15) substitute their values for A and € from (8) and (10), and for d.wr sin @ 1 deur* il GLP op S|) 8 7 dr rsnéd dé ) ae? 2 Bs EC ee apy «ap deur. ad.ursin 0 1.e. = ah r™ sin? @V,+ 7? sin? 6 Y,— sin?@ aR sin 0 7 : retaining typical terms. We thus get 1 d{[-p P an dur . ,d.ursind SSaRT lata *sin’OV, + 7° sin’ OY, — sin?@ — a — sin | 4 ur _ in Gur du ) Te a OY. r’ sind dd? ae ( ai. dede) me de® whence @ersn@, 1 d Be g der sin @ 1 @.orsin®@ _ US Cie eee 29 d.ur* Ge sin 0da0° >" dé z 7 sin? @ dé ——sr* sin 0d" Suey ae Pp nippy DER an oe DY +sind 4 Siam 5 (sin' OV) + =F 9 ag (ein Oy Jig — fr sin @ A Substitute for w the value given in (18) and (20) so as to obtain that part of v which has 7" for its coefficient. This is the same end as we obtained above in our choice of the particular part of 6 which we put in. Reducing now the second side of the last equation by performing the differentiations and putting the terms together we get 2 2 -ursin @ Ld d.vrsin@ 1 d.vrsin | — —pi+1) (sine 7 ~iV,c0s8) dr + rsinddd sis) do sin?d d¢* ~ (m+n) (2i+3) Mm ,. = 8) +2 mae ye ree aly oe i+1 i) —2 (sin 0% 6 a, g + +3) cos 02...) welseseee( es)! To obtain the coefficient of r** in v, put vsiné=7*"y,, and (22) gives ; ; iL al 1 dv, —p i+l dV, ee GE 2) ct sin 6 dé" ae 0% = sin*dd¢® m+n +3 (sine HGH 6v,) ne as Y. ad ate = a (sin — 70s 6Y,) - 2 (sind de + +3) cos 02.) ... (28). To find the particular solution of (23), substitute for v, in the first side sin go , and we find as the result of the operations on the first side ay, Rete Peal Sb ak a Seen sae ae) ae do ae Substitute in this age = {i+ 1) sin*@Y,+ sin 1S sin 0 ae and it immediately reduces to 2(¢+1) (sin 0G: — a eos eY,). 34—2 262 Mr C. CHREE, ON THE EQUATIONS OF AN ISOTROPIC ELASTIC Thus the part of the particular solution of (23) that depends on Y, is Mine > “ pees sin gue 2(¢+ 1)(20 +3) dé’ psin 0 dV, and the part depending on V, is similarly — Fim+njQi+s) dd- In exactly the same way we find that the part of the particular solution depending on Ie eer OEA: is ——z sin 9 — , 4, 3s i+2 dé The general solution of (23) is v,=X,,,, an arbitrary surface harmonic of degree 7+1, which our surface conditions will show to be zero if the surface forces be purely normal. The complete solution of (23) is thus —(¢+3)+2 sini hel Tae give Sal OZ ee pees %=—~Fim+n)(2i+3) dd 2atljats) dd i+2 do But 5 is the coefficient of r** in the complete solution for v, therefore that part of the solution of » which depends on those surface harmonics of the i degree occurring in (21), adding the term in X, to remind us of the existence of a series of that nature given by (24), is TU eee BODES eee cea ee 25) "=~ 92143) dd |mt+n § i+] 40 SiO aes 29) The complete solution for v would require us to write a = before each of the typical terms in (25), because these in themselves are not surface harmonics. We use this form of solution for its convenience as it shows at once what terms in the solution depend on the same surface harmonics V,, Y,, Z,. This is especially the case when the surface har- monies are purely normal, and so X,=0 as we shall see shortly. §10. We might determine w in lke manner from (16), but it is obviously simpler to deduce it from (1 a). Substituting in (1a) the values already obtained for w and v in (21) and (25), we get dors ging (=P v4) + 62) so (# (243) " lmtn (+204 (— m+n n Br 2) vr Tee eee sin 68) Sp. dV eS a ane vam —— : dX of n ssseed |e tH; 2(21+3) (m+n dd i+] dé aed A sin@7 a ns ia, ye! — (i+1)sin?Or'Z+ sin 8 76 | as the equation giving those terms in w which depend on V,, Y,, Z,, X,. SOLID IN POLAR AND CYLINDRICAL CO-ORDINATES. 263 On reduction this at once becomes d.wrsin@___psin’@r'** ane eee Ly, dp 2(m+n)(2i+3) iG +04 9 qo-8in 4 states et 8) +2 Wy IO Scheer Se + 7 sin OsG+DOUrS) \iG+D¥+ a0 Fee a Sees \ G+) 44-5 : ==, sin oe resin Xe, Employing the well-known differential equation for surface harmonics of degree 7, and dividing out by 7, we get = (6+8)+2 d.wsin@ _ — pr? CV, ay, rn? &Z, ee ao = .. (26), dp 2(m+n)(2i+3) dg? 2(¢+1) (+3) dg? t i d¢ ae ype = 3) +2 Mohd THs od p ma Eo OZ < gence 7k 2(21+ 3) sin @ dd (m+n Ons cc in Tee isin 6 dd Te a (27), Ga dw, dX. ‘y where w, is given by Chia aden ogee en (27°). The form of w, is given at once by that of XY, Thus for the Tesseral* harmonic expression, if O77 cosa¢+C'T7 sinog denote the part of X, depending on cosc@ and sin og, we have Jae isotee = : o sin og — C’ cosa), where > denotes summation with respect to co. (26) would also admit of a solution w=/(r, 9)+ because in the expression for 6 given by (1) w occurs only as varying with ¢. § 11. To determine Y,, Z,, X,... we have the surface conditions (2), (8), (4). Let us first consider the case when the surface forces are purely normal, as this is much the most manageable case and seems most useful. It also will best bring out the parallelism of the typical solutions (21), (25) and (27). If a be the radius of the sphere, we get from (2), =p m(20+3)+n+i—1) _ ee DT Se aN a 5 ame gi W483 B43 + 2n(i—1) a’ °Z,=S,... (28), where S, represents the normal tractions depending on surface harmonics of degree 7. From (3) in like manner, leaving out the common factor n, Sten ect ee ets tei ee oS Gal) a 5 53 men 2+3 ' maeI@ae) 1 Ae a sin @ Se) * Ferrers’ Spherical Harmonics, p. 80. + See Ibbetson’s Mathematical Theory of Elasticity, Example (18), p. 380, noticing misprint = for (3): 264 Mr GC. CHREE, ON THE EQUATIONS OF AN ISOTROPIC ELASTIC So obviously from (4), 1 d(-—p itl y, m(t+2)—n ~ ,, 20-1) 5 ; ae ae ae fee RES ACEC) oe Oo ee where w, has the value given by (27”). Since (29) and (30) are to hold for all values of @ and ¢, i.e. all over the surface, they are satisfied by X,=0, and therefore w,=0, and by mt Si a+1 ie my +2)— 1 a m+n u+3" ~ &+1)(2i+ 3) For this satisfies the surface conditions (29) and (30); also (31) and (28) enable us to determine the two unknown quantities Y, and Z, in terms of V, and 8S, by two simple equations. X, thus occurs, as will be shown more satisfactorily presently, only when the surface forces are other than purely normal. In the case of purely normal surface forces (3) and (4), besides indicating that X,=0, give only one equation. Hereafter when V4 “= Te ee (31). we speak of these giving only one independent equation this is what is meant. Multiply (31) by 7 and subtract from (28) and so eliminating Z, we get, after easy reduction, y= t+1 m (ze + 4+ 3)—n (204+ 1) whence, and from (31), z= S.a {mi (i + 2) —n} + pa’ Vi {m (i + 2) — n} a 2n(t—1) {m (21° + 40+ 3) —n (20+ 1); m+n j@i+a +e — V,{m @i+3)—)] ogee (32), § 12. The coefficient of V,r" in the i harmonic term of w is then from (21), substituting for Y, from (32) and reducing, — pi {m(i+1)—n} Qn jm (27° + 44+ 3) —n (21+ 1)} The coefficient of — in v is from (25), substituting for Y, from (32) and reducing, 2 m(i+3)—n 35 Pon it es emi a (35). The same expressions occur in the equation for w, therefore the complete solution, depending on V, and S,, is 2 2 is ipV, {m(i+1)—n} + S,a“ (4+ 1) (mi — 2n) iar In {m (27+ 41+ 3) —n(2i+D} gt [ O% *Vilm(it+2)—n} + Sa {mi (i+ 2) —n} (36 On (i —1) fm (G2 + 46+ 8) —n(F1)} sds t tae aes 36), Ps. a ee {m(i+3)—n}+S.a~ {m (i+ 3) + 2n} Qn {m (27° + 44 + 3) —n (27 + 1)} 41 d [a'pV,i{m(i+2)—n} +S,a {mi (i + 2) —n} dé 2n (i—1) {m (277 + 414+ 3) —n (204 1)} SOLID IN POLAR AND CYLINDRICAL CO-ORDINATES. 265 rtd [ev {m (i+ 3)— n} + S.a* {m (@+3)+ at ~~ sin Odd In {m (2? + 4¢ + 3) —n (2+ D)} te rd Fate Va {m (i+ 2)—n} + Sa {mi (a + 2) — 0} (38) sinOdb | 2n(i—1) {m+ 4i4+3)—n(i+)}o Jo” raed suse ipV,+ (¢ +1) (224+3) S,a* | m (20 + 40+ 3) — n (204+ 1) § 13. One very curious relation between the displacements due to S, and V, when ane pes : : SYS ay ; “ is negligible is obvious from these equations. Neglecting — the coefficients of pV, and m m : of Sat in u, v, w are everywhere identical. Thus a surface pressure —S, has exactly the same effect in producing displacements as bodily forces whose potential, per unit volume, is S, (“) acting throughout the mass of the sphere. This is noticed by Professor Darwin in a particular case which he treats in his paper “On the Bodily Tides of Viscous and Semi-elastic Spheroids,” Phil. Transactions, 1879, Part I. He considers there a gravitating nearly spherical body whose surface is r=a+o,, and considers the effect of the surface pressure gpo, and of the gravitation potential due eee 2 OOM ATEN. hich is 52 (F) a. to the protuberance which 1s 5-4 (7) % Calling a’ the displacement in any direction due to the surface pressure gpo,, a” the displacement in the same direction due to the bodily forces he finds «” = — oie 1” This agrees with the relation stated above, because in this case 3 = (S18 iar og il 2 ee cia 2+ I find that the equations (10) in the same paper, noticing that p=u, T=%, V=Y, ete., and putting 7,=Sa%, and “= 0, give identically the same values as (36), (37), (38). § 14. Suppose now that the surface forces are no longer purely normal. The condition (28) remains unchanged; but on the right hand of (29) we shall have terms =i, where 7’, is a surface harmonic of degree 7, and on the right-hand side of (30) 7 SS, where U, is a surface harmonic of degree 7. Also on the left-hand sides of (29) n and (30) we should write a >. Call (29) and (30) when so modified (29 a) and (30 a). Multiply both sides of (29a) by sin@, then differentiate with respect to @, and add it to (30 a) differentiated with respect to ¢. Then, noticing that Gh a ACHE devs qo 2? ag eaea ten Gk, sin OV,, 266 Mr C. CHREE, ON THE EQUATIONS OF AN ISOTROPIC ELASTIC and similarly for Y, and Z,, we get aye ; pa sel mee a ee Bs @+1) sing | PE 43 taaal) ies wo : SIGE cia) il inne dU, Saal aes a [ees PSE A AY peo lc +i (¢-lj)a | qa | 1 | go-sin OL + | dh v dh |’ but, by (27”), = + ORT 0, : t+1 ., mi(t+2)—n py 201) 3 . St(¢ + 1) Ez 443°" le +7 G41)(esy 7 a Z, 1 Li die: I @ilor = SS Mien Seok y ae i \ ne Ee r) qa nOli+ =o 7 | Scio (31 a). Again differentiate both sides of (29a) with respect to ¢, and subtract (30 a) first. multiplied by sin@ and then differentiated with respect to 6. We thus get i-1 7 7 Syren a aX, id Git, a) = = (¢-1) E& ‘dé a: sin au, | =2 = |Te- da: sin 0U, |. : ae : : : a dw, aX, Differentiate both sides of this with respect to ¢; then noticing that prea 1b” this becomes ae pee an ee TX, ax) ol ed dU, Y(i-lja siete de * .sin @ abla a aaeanaes Bin Orel os nae esi dU, @T, ) Le. zat (?—1) X,= nan ~ E . sin 6 dd © ge (re (40). YT, and U, are known, and thus the right-hand sides of (40) and of (31a) can be expanded in a known series of surface spherical harmonics of the various degrees. Thus X, is given by (40), while Y, and Z, are given by (28) and (81a). It follows from the parallelism between (31a) and (31) that the terms in Y, and Z, which depend on S, are the same as before, and so are given by (32) and (33). Also X, is inde- pendent of S,. (40) shows clearly that X,=0 if U,=0=T,, i.e. if the surface forces be purely normal; also that X,=0 unless U, and 7, are functions of ¢, so for a series of forces expressed as Legendre’s coefficients X,=0. § 15. Let us now find the additional terms to be introduced when we have no longer a solid sphere but the material contained between two concentric spherical surfaces of radii a, a. The forces are supposed similar in kind to those of the last case. Let us number the equations so as to show their connection with the preceding. Thus (13’) means that this is the additional term or terms to be added to the right-hand side of the equation (13) of a solid sphere. Taking the 7 harmonic as typical, suppose the body forces have a potential SV/r-™; then Se er Verse en ee ee (13’), m+n where V/, Y,’ are surface harmonics of degree 1. SOLID IN POLAR AND CYLINDRICAL CO-ORDINATES. 267 d& dB min,. Ge aa ; Also Tee a r’ sin 8 aR (CARED) ara can cane tee poorar acces (14"). Substitute d.wrsin @ p Ayesha ce rap Cai Chee d.vr sin 6 ie Pca Or“ Vi + sin? Or Y,’ — sin One S sin 6 ——ap ; and get ae ts i @ du il akan Ee! inn. vena (WD We ae oo de eo Viggen ie +1) +2}. JERI) =U aocdocooooae (18’), and get : : d du, A UST p , (Mm | } le 8) + Sin SE ae Oe entogae 7 meen: c+, iF Cac ese) The solution ee is (ee Sart 2 MAO S= 1) 2p : U; ean 2 (2i—1) == we en@aM,) 2 Lie BOUOODEODOCH OOO dS (20 ys where Z’,_, is an arbitrary surface harmonic of degree 7—2. Thus the additional part of wu, not included in (21), which involves a surface harmonic of the 7 degree is Cat) oe ee sti api) Sarton a ony Ue 5 ay iro he fa CD HEEL: Oseeseaeee (21’), where Z, is an arbitrary surface harmonic of degree 7. Treating in precisely the same way as we treated (15) the equation, which is here the equivalent of (15), viz. dA d& mtn. d dat an a sin 076 SMS EP eset eS oe seeds (154); we get dvr sin @ ad me d.vursin @ 1 d@.ovrsind de® ‘snddo"" dO ‘smd dg = War aV, ; | m (t—2)—2n bf Gh fos ore hee = OO dé la ee ae (42-In dé ; a7 +74 {2 (i — 2) cos 0Z', , — 2 sin 8 or Soe eRe riceatc dee (22%. Assume v sin Sas and we get ‘ eee fl dv, IL Ga, — pt (omer ecg. ae) sin sin og ae sin’ 6 dd? ~ (m +n) (21 — iia Cas dé 5 + (i+1) cos 0! eee {sin r) = + (i +1) cos OY, | +2 \@- 2) cos OZ’ _, — sin 0 soe . .(23'); and we find, precisely in the same way as we did in the case of (23), that the solution is (X’,, being an arbitrary surface harmonic of degree 7 — 1) ‘pe bey adVi mG—2)—2n. ,dY; sm@dZ,, yx, ple, "=m +n) 2 (20 — i) ear 2ni (2i — 1) sin? 0, waa edge coe Vou. XIV. Parr III. 35 268 Mr C. CHREE, ON THE EQUATIONS OF AN ISOTROPIC ELASTIC Therefore the additional part of v, not included in (25), which contains the surface harmonics of degree ¢ which appear in wu, adding the typical term X/’, is oars | ee ee _m G22 | az; eras 25 dO | m+n 2(2i— 1) Qne(2a—1) t+1 dé S10 42% where X, is an arbitrary surface harmonic of degree 7 Similarly, to find the additional part of w proceed exactly as before, i.e. substitute in d.wrsin@ , 55 NOt me. MICO? SING aca Saal sin 95 — sin’@ —7——sin 8 de the values of 6, uw, v as given by (13’), (21’) and (25’), and employ the surface harmonic differential equation. We thus get a differential equation similar to (26) whose solution presents no difficulty. This solution is uel a dl p Pole Mi (t— 2) 20 Gh aes hte owe 0 sin 0 dp E= 2(21-—1) —- 2nt(22 — 1) nok ~t+1 Z| + wir = ee a (27), where dw; ie dX dd dé ° It must be understood that the values of v and w given by (25') and (27’) do not profess to be solid harmonics of the 7* or any degree, or putting =a constant to be as they stand surface harmonics. The terms merely represent part of a series, a } being understood. This form is adopted as showing conveniently the parts of the solution for u,v and w depending on certain surface harmonics of the i** degree, i.e. on V/, YJ, Z/, X/. é If we deal with purely normal! forces or with Legendre’s coefficients, Y/=0=w,. In the complete solution of a spherical shell the value of 6 is given by (13) and (13’), that of uw by (21) and (21’), that of v by (25) and (25’), and that of w by (27) and (27’). Thus a type of the complete solution for a spherical shell is oe eS ee p(t+2)V, _mi—2n- ae 1 eee Z, _ (ep G@—-)V, , mG+1)+2n_, ai : 7 os Set) | Smee VA ea Liwnceeeise (36 a); be earn) A aie m(t+3)+2n,) dé, r "= 2(2i + 3) dé lag te Diag ao. amos r'?* dZ, nag, Pt ad { p v’ m (1 — 2) — 2n us ve| dz; +227 1)d0 \m+n ' ni Yi mel de Pane Aon GH/E) YP Le A apy a eee mt dd, , = 2(2i + 3) sin 8 dd lm +n i n(i+l1)— i LOST: Gore Sp SE a oth fy BOGE et v;| re? a2; 2(2i—1) sn dd |m+n ' ni ‘\ 441 sinOdd The expression for wu gives all the terms which, when r is constant, are surface spherical harmonics of degree 7, and in v and w are exhibited those terms which depend on the +r lw/...(38a). SOLID IN POLAR AND CYLINDRICAL CO-ORDINATES. 269 above surface harmonics of degree i in w, with the addition of the typical term X,. For the complete general solution we must write a = before each of the above expressions. § 16. In the most general case when the surface forces are not purely normal there are six series of quantities Y,, Z,, X,, Y,', Z,’, X,’ to be determined, and there are six series of equations to determine them, three at each surface. If the forces be purely normal exactly the same method as for a solid sphere shows that X¥,=0=X,. In this case, as in a solid sphere, the two surface Ect (3) and (4), besides showing that X,=0=X,, are of the form J (SL) =0 and = q $5 S(L)= 0 , where ZL, stands for all the terms of the 2" degree in surface harmonics. These relations hold all over the spherical surface, and thus the solution is Z,=0. Thus (3) and (4) furnish at each surface only one com- plete equation for the surface harmonics of any one given degree. The surface condition (2) gives one similar equation at each surface. Thus we can regard Y,, Z,, Y;, Z/ as four independent variables given by four simple equations, and so we can find them all in terms of V,, V/, S, S/; S/ being the normal traction on the interior boundary. § 17. Let us then consider more fully the case when the surface pressures are purely normal. The additional terms for w, v, w in (21’), (25) and (27’), putting X/=0= supply towards the boundary condition (2) the terms _ pil al xT [ m(2i—1) —n(* +7—1)} = ee + {m(@+3i—1)+n} ral — 2n(t + 2)r**Z!...(41). Towards the expression (3) they supply d[—r*( np ..,,m(e—1)—n_,) 2 (i +2) np Fl lot ea 5 Gees Fail n1 Zo Sieleierese'sleiciejele and so obviously towards (4), ad Tie a oer ee (et A 2 (v +2) an ve in dg | HoT eat i a foie. By | oboe (43), writing r=a at the external boundary, and r=a’ at the internal. It will be as well here to write out these surface conditions in full. They look cumbrous but in any special case are much simplified. The surface equations are, from (28) and (41), (29) and (42), (30) and (43), een = ~ a V, {m (204+3) +n (+7%-1)}4+ Vi {m (—71—-3) + “| +2n (i-—1) a?*Z, ee e SPY n(n (Ss 7" Son (P+ 3i— —2n (i+2)a7*?7/'=S, Pet lan {m(2¢—-1)—n(?4+71—-1)} + Y/ {fm (+32 1) +n) | n(i+2)a**Z’=S....(44), = V,n(it+1)+ Y, UE Gan a se alls *] +2n U = y) a*Z, 214+3|m+n v+1 U Ojai i -+ m(v?—1)—n (v +2) ao 2) 0 wed ge. as ‘2 7 CON ASS al Eel Wie a1 er V, w+, i |+ + 2n Ze — ee (45); 4+1 30—2 270 Mr C. CHREE, ON THE EQUATIONS OF AN ISOTROPIC ELASTIC = Ez = V, {m (2643) +n (P+%i—-—1)} + Y, {m(?—-t—3)+ 0) | + 2n (t—1)a**Z, re ME PS Be toe ase Pe = Ee re ree ey tele {m (2i—1) —n(P@+%-1)} + YZ {m(?+32—1) +n} | — 2n (0+ 2) a“ Z =S/...(46), —a"[ p ; ,mi(it+2)—n| 5.0 —1) may 2+3 Ee shal aaa nial t+1 pad ee (-i-1 "2 iad *, ae = Wee mEaD=") 420 Creag 0. ae (47). 24 —1 [m+ v i+] These equations, supposing i>1*, determine Y,, Y/, Z, 4 without ambiguity of any kind. § 18. The following method gives the values of u, v and w very shortly. It is suggested by the form of (25) and (27). Supposing we have got (1a), (13) and the solution (21), and have worked out the first side only of (22), we then get as before for the general solution of (22), selecting the term im X, as a type, rX, a sin @ From (1 a), since this solution is independent of w or 6, we have dws d.wsin@.. aX, dp da mae Cie * w=wy', where dua ee. Fi ap ae We then want particular solutions for v and w answering to mi—2n +. oe u=— 1 i+ 8) - Ai\ } sin 0 dd Then from (1 a) 5 Kamer dy, Tye ees 1 ni — 2 f(r se gin @ te = ¢ A pny enn d tt He ath Ve sin 6 de dé * sin? @ dd rdr' 2(2+3)n A special case occurs when i=1 whether for solid | For explanation, see paragraph r of § 737 of Thomson and sphere or shell. In the surface conditions Z, occurs mul- | Tait’s Nat. Phil. The same remarks apply to the case of a tiplied by i-1, and so its value would become infinite. | cylinder. SOLID IN POLAR AND CYLINDRICAL CO-ORDINATES. G3) 2 : le Va verity & i.e. i(t+1)f, (7) Y= Va J =o roa +3) 42 mpiVNe = ta Se) = eeu) For the second assume w= flr) Gy and precisely as before we get from (1 a) ole pos SDS PE 1 JEN) cr 2 (m+n) (204+ 3) For the third assume 5 Oe v=f,(r) Ga : a7 UO) eT GER and from (1 a) we get f@) aot ey a LV WA pak. But v=f, (r) = + f, (rn) 7 +f, (7) a oo a @° which, on substituting for the functions of 7, we see is identical with (25). Similarly (27) is given by the above values for w. The same method applies equally shortly in the case of a shell. The forms assumed for v and w in this method would scarcely suggest themselves unless we had already found them by some direct method, so that the process is from one point of view unsatisfactory though practically serviceable. § 19. The case i=0 seems slightly different from that of the general solution, but is really included in it. It differs mainly in the number of arbitrary constants, but, as we shall see presently, there are enough to satisfy the surface conditions. To save space let us consider first a shell the radii of whose surfaces are a and w’. From this we can pass at once to a solid sphere. From (13) we have = Gomevenmi= 0% consanenoy c@uasacquoareccusoncnconadnonanc bo “I bo Mr C. CHREE, ON THE EQUATIONS OF AN ISOTROPIC ELASTIC The surface forces are purely normal, call them F and F’. Supposing no bodily forces to act, we get from (21) and (21’), leaving out V, and V,’, U=>5 Z, is zero, for as it is a constant there are no corresponding terms in v or w, and 0 ata ae Bie ; : ‘ Z. ste in (1) it is impossible that § should vanish while b= Also Y,’ is zero, for Osa is inconsistent with (5). ul d.ur [Of course the above value of w follows directly from b=5 ats YS but sitias desirable to show that it is contained in the general solution.] = ok du u Noticing that now (m—n)S+2n5 =(m+n)6—4n-, 5 dr r it will be seen from (30), Sect. 1, that the surface conditions are (m+n) 6 — dn“ =A whee = Usaaator se ee noieneceaeeeeee (50), = Ww Otic 110 nasa nscecener nen eee nee (51). 3 ‘3 We get thence i a (m- 5) (a — a’) Z _— ava” (F— lt) . ° 4m (a — a*) ? i ak — a°F’ ava’ (F- =i . srefor = == —, || = 0 a in ascccereeerecencens 52), therefore u zm | Saas ae (52) 8 — 7/3 BV 8 =< is MMAR en Re, (53). (m - 5) (a° — a’) For a solid sphere put a’ =0; F”’ goes out as it should, and Fy ‘ SE ESR OSH CR ECR Eee oRE Ep cere ocSaandecoon 4 u anon (54) $= Bath yyy 46. eaten Vaes baht Ae (55) m—-. 3 Suppose next that the shell is very thin, having a’ =a(1—e), where e° is negligible. Then, PER ees ai ei if u=( 3e +f) tO PG ae Tiaiiieveie'e oie ces etepeniele (56), Ti eV ge Oe , 8 = ("5 +P) —— ieee ee. tl, eine nei (57). Thus unless F—F”’ be very small, u is very large. SOLID IN POLAR AND CYLINDRICAL CO-ORDINATES. 273 If F=F", a 3-—.. This is exactly the same expression as for a solid Ui Ss 3 : : : é 2 UW sphere, as is obvious, since in the solid sphere (m +n) 8 — 4n — is constant throughout. April 20, 1888. If the tangential surface forces instead of themselves being surface harmonics P ? i=} 5 : : : - dQ 1 dQ; be derived from a potential expressed in harmonics, so that T= 6 and U.=s ads’ represents the i‘ harmonic terms, then by (40) we have X;=0; also the = can be dropped i where Q; on both sides of (31a), the right-hand side reducing to —7i(i+1)Q;/n. In this important case the solution is thus obtainable explicitly. The additional terms to be added to (36), (37) and (38) being respectively w= Mi (i+ 1) O,[((@-1) (mi —2n) 7a —{m (@-71- 3) + nb ra), Shor _ MN, dQ, iM Nie pe hae where for shortness 1/M,= 2n (i —1) {m (27 + 44+ 3) —n (27 + 1), NV, = 7 (i-1) {m (6+ 8) + Q2nkbri a - (6+: 1) {mW -i- 3) 4 nh ra. With similar surface forces the case of a shell can also be solved explicitly. | SECTION III. TENDENCY TO RUPTURE. § 20. One of the most interesting questions in the problems we are to be engaged in is the tendency of the body to rupture which we shall now consider. It is fairly obvious that the tendency to rupture at a point inside a solid will depend only on the state of stress or of strain in the material surrounding that point, and not at all on the nature of the external forces which produce the strain, nor on the form of the external surface of the solid. Various measures of the tendency to rupture have been advanced. Lamé considers that a body will rupture when the greatest stress rises to a certain value, obtained experimentally for each substance. On this hypothesis the tendency to rupture would be measured by the greatest stress. By others, e.g. Professor Darwin, the difference between the greatest and least principal stresses, called the “maximum stress-difference,’ has been taken as giving the tendency to rupture. On this hypothesis the solid would rupture when the maximum stress-difference at any point reached a critical value determined experimentally for each substance. By others, e.g. Saint-Venant, the greatest strain, pro- vided it be positive, has been taken as the measure. On this hypothesis if the strain exceed a certain value, obtained experimentally, the solid will rupture. In all three methods the actual critical value will be modified according to the different ends aimed 274 Mr ©. CHREE, ON THE EQUATIONS OF AN ISOTROPIC ELASTIC at. If the stress is to act only for a short time a higher limit may be taken, but if the stress is to be long continued it must never reach a point at which it may begin to cause deterioration of the substance of the solid. In the first and third methods the limit might be obtained by experiments with the traction of a bar of uniform cross section. In the second method the limit would be best obtained by experiments on shearing force. In some cases, for instance in the simple case of longitudinal traction of a bar where there is only one stress, the different methods coalesce. I consider the last two views much more probable than the first, but do not venture to assign the palm to either. In fact I am doubtful if certain cases do not throw doubt on the universal application of any such measure. § 21. It may seem premature to consider the question at all while holding these views, but so long as the physical basis is clearly stated no possible harm can ensue from the mathematical developments. The results we obtain may possibly even be of use in showing how to discriminate experimentally between the different theories. Further any one of these different methods requires the greatest principal stress to be known, or is most easily applied when a knowledge of this stress is obtained, so our results will be at once applicable to any of the methods. To show this for the third method, suppose e, f, g the three principal strains, P, Q, R the three principal stresses, and 8=e+/f+g. Then if P be the algebraically greatest stress, e is the algebraically greatest strain, and the relation between them is P=(m—n) 6+ 2ne, : _P-(m—n)d F or Z2= on ee RE ty es LR ae eee! BAe (1). If then we know 6 and P we at once know e. For our future applications we require a knowledge of the stress-quadric. For the properties of this quadric see Thomson and Tait’s Natural Philosophy, Vol. 1. Part U. S$ 662—665. The greatest and least principal stresses act along the least and greatest principal axes of this quadric, the properties of which are of course independent of the system of axes employed. In the cases we are to consider one of the principal stresses @ is perpendicular to the meridian plane, the other principal stresses R, and ©, lie in the meridian plane. If the equation of the stress quadric referred to our usual system of axes be Rare Oy Diigryt De: = Ali ana scssasesareaenspccsereereeaett (2), then referred to the principal stress axes it is Ri + Ong? + BA HL vies seis satan dauen se betieeenidmetater (3); SOLID IN POLAR AND CYLINDRICAL CO-ORDINATES. 275 and by ordinary geometry py sue ae sO Gabe aeaeGUditeanecnoro te accor escnodentar (4), R+0-S 8,= 5 where S ya CPO) eA shea taster tate c denem eettotie baat says’ (5). We shall suppose this square root always taken positively, so that R, is the algebraically greater principal stress in the meridian plane. It is easily seen from (33), (36) and (37) of Section 1. that a 1 dv ay 1 du 2 / 2 = , as S* = 4n Cae aa aP\\ G aeitee aay) |." seacdeeonoacsenoan5 dl) If a be the angle which the axis of R, makes with the radius vector drawn outwards, cosa 2sina Ro S+0-R' The greatest stress algebraically is thus either R, or ®, and the maximum stress- difference is S, R,—©®, or ®—@,, according as ® is the mean, the algebraically least, or the algebraically greatest principal stress. We have ® = (m—n) d+ 2n ( +" cot 6) wchedeenaiane ana eeconn ses sciuaats (7), R, + O, = B+ O = (Bm =n) 6 = @ oo... eseecreccescenscsecensees (8), and Jig = 2) Su) scoptogscacgbecnocosd9anbcHoboSronesaaEosa28 (9); whence we get 2(® — R,) =2n {3 = +° cot é) - a| -S eet Md ined (10). 2(b- ©,)=2n {3 (442 cot) ~ 5} +8 We can divide our results into three cases. 5 U4 ane ‘ uv z Case (i). 3 (= aa cot 0) —6 positive, and at the same time ;3 (Z se cot 6) — 6} num- erically greater than (=)- In this case @—R, is positive, thus @ is the algebraically greatest stress and the maximum stress-difference is uv ies, > -0,=n|3 (245 cotd) — 3} +5 mlatete(ela(alalelelolplelaialeja}puis\olelarslvistere Gal): ie P : WE Case (ii). 3 (“ + cot 0) —6 negative, and at the same time {3 é + = cot @) — 6} ) 2 numerically greater than (=). In this case ® is algebraically the least principal stress, Vou. XIV. Parr III. 36 276 Mr C. CHREE, ON THE EQUATIONS OF AN ISOTROPIC ELASTIC and the maximum stress-difference is R,-d=3—n{3 (242 cote) — 3) sia teeta eaesseaeeeees seen (12). cy , 2 2 Case (ii). 3 [e+e cot @) - 3 numerically less than (5): In this case ® is the =< mean principal stress and the maximum stress-difference is S. In what we have called the third, or Saint-Venant’s, method the tendency to rupture is in Case (i) measured by oe if positive, i.e. by . ie cok Ot. fae VISE eae Oe ee (13) ; —— 7 Hinks in Cases (ii) and (iii) by 7 , if positive, i.e. by Set u,v r= 4. 2 {5 = (= + = cot 6) ureiataitarses tatetereletere is atcteiatota rata ctelenatetetetetenes (14). It will be found that in general there is a surface or series of surfaces over which ®=R, or =0,. These surfaces are of importance on any of the three theories, as they indicate the limits within which the several cases mentioned above apply. The surfaces themselves may be regarded as loci where Case (iii) applies. § 22. As an easy illustration of the hypotheses as to the rupture of a solid we may consider the solutions (52) and (54) of Section 1. In the case of (54), giving the displacement for a solid homogeneous sphere, = obviously =<, and so R=@=®; thus no stress-difference exists, and according to the second hypothesis no amount of uniform normal traction or pressure would have the least tendency to cause rupture anywhere. Pon se ge u cd : Te , The greatest strain is in either case ee ae indifferently, and is positive or negative according as F is a traction or a pressure. Thus on the third hypothesis no amount of uniform normal pressure tends to produce rupture, but uniform normal traction on the other hand will eventually produce rupture if = exceed a certain experimental limit. Thus both theories agree in stating that no amount of uniform normal pressure will rupture a solid sphere; but they differ completely as to the effects of uniform normal traction. There is no very obvious experimental method of testing the results of a uniform traction, but it certainly seems probable that rupture would follow from a very severe traction. Greater interest attaches to the result (52), Sect. u., which is for the case of a shell, and is 3m —n 4nr* 1 EE —a°F’ aa? (F— =) —>—— r+ — > |. bo sI ~I SOLID IN POLAR AND CYLINDRICAL CO-ORDINATES. Here @=b=(m—n)d+2n-, lu as Spe 5 and R=(m—n) 6+ 2n ee therefore R is algebraically greater or less than © according as F'—F" is negative or positive. The greatest value of R~@, ie. the maximum stress-difference, occurs when r=a’, or at the inner surface of the shell, and is taken always positive. As being of greater practical interest we shall suppose #=—p, F’=—p’, so that p and p’ represent uniform pressures, e.g. of a gas, on the outside and inside boundaries. Fag aL : F ; Then the greatest strain is Fe or - according as p is greater or less than p’. If p>p' the greatest value of = occurs when r=qa’, and is 3 (m—n) a& (p—p') —2np' (@ — a”) Woes Ko On the third hypothesis no tendency to rupture will exist unless this be positive, i.e. unless p-p 2n a =) SG (1 ale te LM (17). Thus no rupture can ensue however great p may be unless the ratio of p : p’ exceed a certain value depending on the ratio of the limiting radii of the shell. ; ee eet : ; : If p and equating separately to zero the coefficients of the harmonics of different degrees, taking the 7‘ as a type, we find os payer . pee m(2i+3)+n(?+7—1) Bn Vae m(v—-t—3)+n “ 2 fe 4: e (m+n) (21 +3) i 243 a'Y¥,—2n(i—1) a‘? Z,— gp (Sm+m) _ (8). 5 (m+n) bo io 6) 0 Mr C. CHREE, ON THE EQUATIONS OF AN ISOTROPIC ELASTIC — a, In the second surface condition, (28) Sect. 1., all the terms are of the order = at least, thus we may write r=a over the free surface. Substituting for Y, from (7), and making the terms depending on the differential with respect to @ of each surface harmonic separately vanish we get, with the aid of the left-hand side of (31) Sect. 1, equations of which a type is mn +1 _, mi(i+2)—n ,, 2-1) ,, 2gpn =a nN+3 3 Po V, SE @+1)(i+3)° 1D (@i+3) ay,— ne Dae weer Z, ~5 (m+n) One eeeen (9). Treating the third surface condition, (29) Sect. 1., similarly we find that the coefficient sab ae is simply the left-hand side of (9). Thus by making the expressions of which this is a type vanish we satisfy the third surface condition as well as the second, and this is the only way we can satisfy both conditions. Putting for shortness 5 (m+n) {m (27 + 4043) — 1 (204+ L)} HD, 0. ccc eeeeeeeeeseee ees (10), we find from (8) and (9), ——— eo) [5 {m (24-48) — n} V,— (2648) (5m —n (i —1)} ga] «sas. (1D), he beat [5 (m+n) t{m (i +2)—n} a’V, — {5m (¢ + 2) — mn (2° —7 — 101-1) — n* (27 + 38)} ga] ... (12). For the case we are more specially interested in when V,= ml 7 a", these expressions become , Vii TD {5m (21+) —n (4?-+10i+9)}, -t+2 Z,= ety pi p, {Lom*i (é + 2) = mn (4° + 4a? — 24 + 1) — n? (44 — 3)}. Substituting these values in (3), (4), (5) and (6) we obtain for the strains in the nearly spherical body (1) acted on only by its gravitational forces 4 gpr {2_ a’ (5m + n) ~ 10a(m+n) | 3m —n a,c + gp on Qi + 1) D, [r'"a~‘ {10m (® — 1) — mn (40° + 40 + 340 + 297 + 10) +1? (81? + 8? + 137 —2)} ~ r1g-4%5 (10m (6+ 2) — mm (40? + de? — 26-4 1) — 1? (46 8)}] oes (13), 22 aie dé (14) SOLID IN POLAR AND CYLINDRICAL CO-ORDINATES. 281 gpa. Hg Z F " Rs Y v= (2 +1) D, [ra {10m? (i — 1) (i + 8) — mn (40°? + 122 — 6 +17) — n? (8:2 — 17)} — ra? (10m (i + 2) — mn (40? + 4 — 24 +1) — n? (41 — 38)})......(15), na gp 572 — 38a7 (5m + } ~ 10a (m+n) 3m —n aor — 93 gD [5m (47 + 127 + 87 + 3) —-n (8i* + 207° + 102? + 101 — 3)] sGcodd (16). In every case { denotes summation with respect to 7, and o, represents all the surface harmonics of the 7'* degree that occur in the equation to the free surface. The fact that the magnitude of the strain depends only on the degree of the surface harmonic and not on its special form seems worthy of notice. § 26. This problem has been treated by an entirely different method by Professor Darwin*, mainly for the special case “=0. He has applied his results to the deter- mination of the tendency to rupture, on the stress-difference theory, in the earth regarded as an isotropic body. Except in this case and on this theory, with an exception presently to be noticed, the tendency to rupture will depend only to a trifling extent on the divergence of the earth from a truly spherical form. It will usually depend mainly on ger {n-ne} ve ae, at AR ae (17), ~ 10a (m +n) 3m —n the strain which is very large compared to the other terms. Neglecting for a moment the other terms in the strain, we should get for our principal stresses Rae omtn) =a) 10 (m+n) a e gp {(5m — 3n) 7? — (5m + n) a? SOEs 10(m+n) a ? Thus the maximum stress-difference is Rex 27eMree (18) 5 (m at n) ea sae aosene, 2 slots going s\zicieieigieiee A and the greatest strain 29m +n) 3m — nf ainin/nYo\d/p}aieleis (ois, e(eleipiaiais\ele\eislevejelejats Both these expressions have their greatest values at the surface. These values are respectively du___—gp fone _ dr 10(m+n)a | 2gpna ; 3 (m+n) ala(a\eipials}4cetele?ata|ele/elsyaialejs\=laia''a)alayals\alp/a\s\e/elale/s)avalele (18 a), and » oy ; SINR heh says AY aes (19a). 5 (m+n) (3m — 1) * Philosophical Transactions, 1882, pp. 187—230. 282 Mr C. CHREE, ON THE EQUATIONS OF AN ISOTROPIC ELASTIC . oy : n deuya ; § 27. In the special case a it is thus correct to neglect the stress-difference arising from these terms, but in an ordinary elastic solid this stress-difference would have in the circumstances of the earth an enormous value. For instance, if following most foreign elasticians we suppose m=2n, these greatest values of the maximum stress-difference and greatest strain would be the same as in wires of the earth’s mean structure exposed to tractions equal to the weights at the earth’s surface of lengths of their material exceeding 500 and 250 miles respectively. In fact on neither theory could an earth of ordinary elastic solid structure exist for an instant. On the stress-difference theory the earth regarded as isotropic could remain a solid only if m were extremely small. From (17) we see that = is everywhere negative, and from (19) = is positive only at distances from the centre exceeding r,, where 5m+n )3 r, =a ts Got n}ojo (05 wiinivtore la[e.cisiafajarele ¢/olare)ie vista nlenal stein ioce fate (20). So on the maximum strain theory it is only at distances from the centre exceeding r, that there is any tendency to rupture. As mm increases from 0 to 1, r, increases from ‘74a to a, while for the mean value m=2n we have r,= ‘86a. So on this theory there would be in the earth, regarded as isotropic, no tendency to rupture except at a distance from the centre exceeding a certain fraction of the radius depending on the material. For the mean case m=2n there would be no tendency to rupture at greater depths than 600 miles from the surface. For the limiting case m=n there would, but for the deviation from a truly spherical form, be no tendency to rupture anywhere, and the smaller m=" the smaller the region exposed to rupture and the less ; v 5 : : the tendency to rupture. Since a measures the ratio of lateral contraction to longi- tudinal expansion for longitudinal traction, it can scarcely vanish much less be negative; but it is at least conceivable that there are materials for which it is small. It is thus on the maximum strain theory by no means impossible that a gravitating isotropic sphere of the same size and density of the earth might remain solid throughout. It would however if composed of such a material as iron be incapable of remaining as a connected solid within distances not exceeding a few hundred miles from the surface. § 28. From the previous remarks it follows that there is no use in considering the tendency to rupture due to the terms represented by those in o, except on the stress- difference theory for the case ~ very small and on the maximum strain theory for the case i — TN a ©Yery small. In thus further considering the matter it will be as well to limit ourselves to the most important special case when o,=P,. SOLID IN POLAR AND CYLINDRICAL CO-ORDINATES. 283 Supposing then 7=2 and neglecting terms in n compared to terms in m, disregarding fur reasons already stated terms independent of P,, we get from (13)—(16), 6 3 1 2 al U= gpa lk, 6r" — l6a'r D5na* yee AP RSC Oia) adn. deep Laon Nat vetisterra. dest (21). ~ IPO TG 95na2 w=6=0 av 1 du _ 24gpa pete egeeee Se pa, 2 A) ain 9 Whence we find ria ne 95 na2 (a — r*) sin 20, wu oe r 1 dy _ 39pa,,. vee AN ire ean gee — § (a’—7") cos 26}. From (6) of Sect. mL it easily follows that /95a28\2 (Gan = (Ga = OP $5 SPx (Gr = (igs) coooeeacoseon ooo scnoeone (22). 2 Thus for a given value of r, S is a maximum in the equator and a minimum in the polar axis. Further, as depending on the sum of two squares, S can vanish only at the two points in the polar axis at the distance from the centre MTOM yan ss Toiciesteite soa datnusitandacecwadeochonssbeoacee (23). Again from (21) we find, noticing that 6 vanishes, 3 (2 ae cot @) —6= 39 pt, vial OS5nat tea Oy Sb Gi ata Os -opaopanoscescnecae (24). This expression is finite and continuous, being of the same sign as a, inside and of the reverse sign outside the spheroid Sap Ore On SiN IO), saeeaaceecemeeentacoret ete (25). The polar, or minor, semi-axis of this spheroid is 7, and the equatorial is a (8)?, The spheroid cuts the sphere, r=a, in points whose polar distance is = sine (1)%. Vv Since 7, is nearly equal to a and @, is small, 3 (e+e cot 6) — 6 is positive throughout the whole of the sphere excepting a small superficial portion round the poles. We have next to find the locus \3 (= +" cot @) =: | (5) = 0, which determines the surfaces over which @ is equal either to R, or to @,. From (22) and (24) the locus is easily found to be Beer sini (Gar — NO ra-F Or sin O} SOs 2 «ian sade seebciee LT = hee oS ; : (5) _ (a'—1")? (4m — 1)? _ ont (a* —r*) cos 20 (4m — n) (5m? — 22mn + 5n?) “Oy (1 9m—5n)* 10 (m+n) (19m — 5n)* (5m? — 22mn + 5n’)? r delle aim ae "Too (aslo =a ae (11). A convenient form of (11) is Ne (5m* — 22mn + 5n*))* = AG — — a 2 {5a 9m BUM ={@ 7°) (4m —n) — 7 1O.Gnen) | spa ees (4m — n)(5m* — 22mn + 5n*) + 4r° (a — 7°) sin’ @ Osi en (11a) Again S can vanish only when both (9) and (10) vanish simultaneously. Unless in the special case 5m*—22mn+5n*=0, when S vanishes all over the surface of the sphere, this happens only when sin 20=0, fais 5m* — 22mn + Sn 10 (m+n) — (4m —n) (a* — 7°) cos 20 = 0. The first gives 6=(0, 7) or ae i.e. S vanishes along the polar axis, or in the equator. Putting 6=0 in the second equation we get Wp xa 1 ii 1 5m* — 22mn + 5n*® 10 (m+n) (4m — 2) This gives r, less or greater than a, according as 5m*—22mn+ 5n’ is positive or negative. SOLID IN POLAR AND CYLINDRICAL CO-ORDINATES. 289 Putting @== in the second equation we get _ : 13 am SGI ODHAEIE EG orev erste tesee sense eens (13). 10 (m+n) (4m — n) This gives r greater or less than a, according as 5m?—22mn+5n’ is positive or negative. In both these cases we have assumed that 10(m+n)(4m—1n) — (5m? —22mn + 5n’) is positive, which it certainly is if “> “09. If then 5m*—22mn+5n* be positive, S vanishes inside the sphere at two points in the polar axis at a distance from the centre given by to tio 1 12 = Soin OHS, SR AUR (12). 10 (m+n) (4m — n) While if 5m*—22mn+5n’? be negative, S vanishes inside the sphere in a ring in the equator, whose radius is given by at SP Simp Bah oe reereerneeceeeeressee (13) 10 (m+n) (4m — n) For 5m? —22mn-+ 5n’=0, S vanishes all over the surface of the sphere. uv § 34. We have next to find 3 G a5 cot @) —6. z 4 Pie GN ON Secon : From (5) and (6), noticing that cot @ , i 3(sin’@ —1), we obtain wv mr sin®@ (a*—71") 5m+n 4m —n r ny r CoE n (19m — 5n) zi 3 13 (m+n) (38m — n) = n (19m —5n)) _ 2 __(5m*— 22mn + 5n’) —" T0n (3m — n) (19m — 5n) ae: Thence and from (7a) we get {s (: + = cot 6) - 3} n (19m — 5n) =r? sin’ @ (3m — n) + (a* — 7°) (4m — n) - (5m? — 22mn + 5n’*) aa) 10 (m +n) Thus, supposing of course 3m—n positive, if 5m*—22mn+5n® be negative 3 & + = cot é) —6 is positive throughout the whole sphere. If however 5m?—22mn-+5n’ be positive there is a spheroidal surface » (5m® — 22mn + 5n*) _ 2 ot? rs Bien ye a ees ees r* sin’ @ (38m —n) + (a* — r*) (4m — n) —7 10(m+n) * over which 3 ( a: = cot @) —d=0. 290 Mr 0. CHREE, ON THE EQUATIONS OF AN ISOTROPIC ELASTIC It is easily seen that 3 (K+ 2 cot @)—8 is positive inside this surface and negative outside. The polar axis of the spheroid is given from (16) by : : 1 y= OTe EO on sltefe'vieisininfoieininie claletnfeieiuarere » see (12); eT (m +n) (4m — n) and so the ends of the polar axis of this spheroid are the curious points where S = 0. These lie inside the sphere only if 5m*—22mn+ 5n* be positive, and at them R,=0,=®, and the stress quadric is a sphere. The equatorial semi-axis is given by r= : (17) an See ey ee 10(m+n)(4m—n) 4m—n n 3 . be positive while 25m*+42mn—15n’ is positive, and if we assume m>n this is certainly the This gives r>a if 5m*—22mn+5n® be negative while m> Also r>a if 5m?— 22mn + 5n* case. We shall see afterwards that 5m?—22mn-+ 5n? is positive if = > 4159. Thus in all practical cases the equatorial semi-axis of the above spheroid is >a. The spheroid entirely encloses the sphere provided 5m? —22mn+5n® be negative; while if 5m*—22mn+5n* be positive the spheroid cuts the surface of the sphere at a polar distance given by ite < La ) } 5 2 ees a 5m — 22mn+ 5n 18). : GME GINS (18) aa : : : m1 This value is certainly real supposing — > 3: n : uw ov ie SNe § 35. We have next to find the surface {a(¢ + 7 cot 6) a - (5) 10): = From (11a) and (15) the surface becomes after reduction 2 on pata os r* sin’ 6 | om —n)’r* sin’ @ + 4 (a? — r*) (4m — n) cee 5m? —22mn + 5n? -r e o 77 = VU ucceee . r’ (3m —n) 5 Gen) | 0 (19) So the surface consists of the polar axis r’sin?@=0, and the spheroid (2m —n)* 72 sin? cde __ Gm? + 16mn — 5n*) 3m —n)’ 7° sin’ 6 + 4 (a? — r*) (4m —n) BEND Pet YM) Bnet <9 (in —n) ee eso) 5 (m+n) SOLID IN POLAR AND CYLINDRICAL CO-ORDINATES. 291 The polar semi-axis is given by 4 = (21 1 3n—n 5m7—22mn + 5n* mae 1+ 4° 4m —n 5m* + LOmn — dn? n : A Bs ae cat For m>s, r, is >, =, or 4, greater than a. This is very easily seen however from the fact that the spheroid cuts the surface of the sphere at a polar distance Oe 5a? — 22m Ayn é,=sin™ ee eetllel Be ede eeea eae ais (22); 5(m +n) (3m —n) thus 0, is imaginary if 5m*— 22mn+5n* be negative. In this case the polar semi-axis is >a, thus the spheroid, as it never cuts the sphere, lies entirely outside it. If 5m’—22mn+5n? be positive @, is real, because 5(m+n)(8m—n) is certainly > 5m? — 22mn + 5n® for any value of m:n>4159, thus the spheroid cuts the sphere. Also in this case the polar semi-axis is a. 5 5 : uo : S\? If 5m’? —22mn + 5n* be negative, \s (f+2 cot 6) - st - (=) cannot pass through the ) ? value zero except along the polar axis inside the sphere, and so being finite and continuous must have the same sign throughout. This sign, being the same as that of the left side of (20), is obviously positive for any value of Ce 2 Vi? If 5mm? —22mn+5n’ de positive, the surface {3 G += cot 0) = a} = (5) = 0, besides the polar axis, consists of a spheroid part of which lies inside the sphere. Inside this spheroid the sign is the same as that of the left side of (20) and so is positive. ’ 2 2 Lastly for 5m?—22mn+5n?=0, the surface Sy Z + oth = 3} — = =0 consists of yi | ip j r 2n the polar axis, and of a spheroid whose least (polar) semi-axis is a. Thus the whole surface so far as concerns us is the polar axis alone. The problem resolves itself into three parts, according as 5m’—22mn + 5n’* is (1) negative, I > 8 8 (ii) positive, (iii) zero. The equation 5m?—22mn+5n’=0 is a reciprocal equation whose 1 : -. m roots we shall denote by a and a where a= 41596 approximately. In case (i) must : Ls ys a, 1 Fe as lie between a and ae im case (11) = must be greater than a or else less than a? while in sont LD, : 1 : . case (111) 7 equals either @ or ae In what follows we shall suppose m>n, which is : 5 - : Hie certainly true of all ordinary materials. Thus we shall have in case (1), Fk and 1. ju v é te ‘ , Throughout the whole sphere 3 (page Ct @)—8 is positive; and the same is true of ) = S\? : 4 A 5 : : 3 (. + cot a) — a} _ (5 i; excluding the polar axis along which this last expression is ( P r / an zero. Along the whole polar axis ®= R, so that either may be looked on as the greatest stress, and the maximum stress-difference is correctly given, writing in w’p again, by vy 20'p_ | ~ 92 ~ 29 : 5m? — 22mn + dn 2 2 2 = ——"_ 1 (q@?— 1°) (4m — n) — YF? — J 19m —5n 1S ( ) 10 (m+n) and the greatest strain by vr 5m +n tm Ole 2 5m* — 22mn + 5n? 3 13 (m+n)(3Bm—n) n(19m— 5n) 10n(38m —n) (19m — 5n) ey. “cot 0=a'p | aaa = Throughout the rest of the sphere, the maximum stress-difference is n {3 (i+ Zeot@)-8+ 5}, : 2n : wi 9 and the greatest strain 7 ty cot 0. These may be got at once from (lla), (14) and (15) writing in the factor w’p. It is easily seen also that 3 (Gre cot 6) —8, S, and “+ “ cot @ are all algebraically greatest when 7=0, and diminish continuously as r increases. Thus on either of the two last theories the tendency to rupture is greatest at the centre. On the one theory it is measured by ae S&S ga 20'pa (4m —n) =—TOm io Bn TT tteteeeeeeeeenernee ees (25), and on the other by uv ow pa” 4m—n 5m +n =i A eek ak bya rR. : 3 tr (19m—5n) 5(m+n)(3m— “st ape 28) § 37. Case (ii) oe n 3(“ + "cot 8) —6 is positive inside and negative outside the spheroid (16) ; ( 2 / 2 3 pt 7 cot 8) — 8} - (=) is positive inside and negative outside the spheroid (20), excluding the polar axis along which it vanishes. The first of these spheroids is that whose polar semi-axis is r,, and which cuts the surface of the sphere at a polar distance 6,; the second has for its polar semi-axis r,, and cuts the surface of the sphere at a polar distance 6,. From (18) and (22) sin"@, = Fist 8. ane acter ice ane (27), SOLID IN POLAR AND CYLINDRICAL CO-ORDINATES. 293 also from (12) and (21) a ( 1 1 ) _ 5m* — 22mn + 5n* 3m —n 1 ) ONS, eae 4in — 10 5m? + 1l6mn—5n* 10(m +2)) m . *. 6,<@,, and for —>a obviously 7, >1,. n 2 x : : uv \ So far then as concerns the sphere, the region throughout which 3 (- +/, cot 8) —6 is es ) positive is greater than and includes the whole region throughout which " (u v \ i ils N Ss 2 {3 ie + 7 cot g) ai (5) is positive. Thus inside the spheroid generated by (20) Case (i) of Sect. Ill. apples, ® is the - : . u v algebraically greatest stress, ®—©, the maximum stress-difference, and — + —cot@ the greatest UP if strain, The extremities of the polar axis of the spheroid (16) are the points where R=O=®; along the polar axis between these points ®=R,, and beyond them @=0,. Between the surface of the spheroid (20) and that of the sphere Case (iil) of Sect. Mm. applies, S being : 5 S se) : the maximum stress-difference and eae = 2 - 5 + = cot a) the greatest strain. : _ Aa sees 1 Since the least value of r, 1s Fig and the greatest value of @, 1s sin /1, these values occurring for “=, the region throughout which Case (iii) of Sect. 11. applies is limited to a comparatively small superficial portion round the poles. In this region a*—7° is very small, and it is easily seen that the maximum stress-difference and the greatest strain are both very small compared to their values near the centre. At a given distance T 9° from the centre the maximum stress-difference and strain occur when Car ie. in the equator; they both diminish as r increases. Their absolutely greatest values occur at the centre being as in the former case given by (25) and (26). eee ae ae v nae : § 38. Case (ili), 5m*— 22mn+t 5n'=0, or ™ = 41596... All the expressions become very simple ; ge 2 (a? — 1°) (4m — n) a 19m — dn and so vanishes over the surface of the sphere, at every point of which the stress quadric is thus a surface of revolution. Along the whole polar axis P= R,, the extremities being the points R=@=®. Throughout the whole of the sphere correct measures are for the maximum stress-difference 2 (a? — r*) (4m — n) +r’ sin’ @ (3m —N) (30 oe RR a oR 5U), & —@,= 38-9 294 Mr C. CHREE, ON THE EQUATIONS OF AN ISOTROPIC ELASTIC for the greatest strain eae mr* sin?@ 4 pS r eas: ~ n(19m — 5n) 3 Sm +n fi 4m —n 5(m+n)(8m—n) n(19m—5n) These have their absolutely greatest values given as in the previous cases by (25) and (26), writing in op. The above value of ~ gives ~—” =-3793.... Thomson and Tait, Nat. Phil. Part 11. n 8 2m a 2m § 684, mention that for brass Kirchhoff found ” =-387. Thus Case (iii) may represent very closely the phenomena attending the rotation of a solid sphere of brass. § 89. The value (25) for the greatest value of the maximum stress-difference may be put in the form 1 Dy 5 ms ha scan oe Cac ee oa neiseee ETS A S = 2o'pa niaieat © (32); 4 4m —n ; : m. : : thus it obviously decreases as , «iucreases from } to 2. This decrease is extremely small however, the values being m n | 8 ll He br po ar &, , >" e a Thus for any ordinary substance, }4w’pa* is a very close approximation to the maximum value of the stress-difference. If we suppose the sphere to have the same radius, mean g density, and angular velocity as the earth, we know that wd = 525 very approximately, g being the acceleration due to gravity at the earth’s surface. This sphere would thus have a maximum stress-difference given by S = at This would represent the pressure, per unit area, on the base of a structure on the earth’s surface and of the earth’s mean density whose height was , or about 5-9 miles. pee: 680 The value (26) for the greatest strain is capable of greater though still very limited . Ct i 2 variation with the ratio m:n. For m=2n it is very approximately a If E=&n . 2 2 denote Young’s modulus, this is very nearly = Taking the earth again as an illustration, this would represent the strain at the end of a wire of the same mean density as the earth and of a length slightly exceeding 4 miles; supposing the point of suspension the required height above the ground. On either theory the effect of the rotation of the earth in producing rupture is very considerable, which seriously affects any conclusions drawn from the results of Sect. Iv. alone. SOLID IN POLAR AND CYLINDRICAL CO-ORDINATES. 295 SECTION VI. ROTATING SPHERICAL SHELL. § 40. As an illustration of the general formule for shells let us consider, what is in itself an interesting problem, a spherical shell rotating with uniform angular velocity @ about a “polar” axis, having special reference to the case when the shell is thin. The potential is V= sor - = PPR eas soasio tein clidacineiacnaasionciasynome atc As in the case of the solid sphere consider the parts of the solution separately. First , V= 30" Se adatom hata sectnc Mua teae ates (2). As in the solid sphere, this gives el ay, ewer ES Ho Olen to nO ruaer MROCHGACP rn Acree neReRceer (3), a apr Or . j Da ace fag FL Siefeletelaatclstalaletelstets/=[olelalalsislslstotaletaietetals (4); where C and # are constants, determined from the surface conditions (m—n) 8+ 2nS* =0 where r=a or r=a. 1 op a’ — a” 5m +n Fe Thus Mma ee (5), op p30 —a* 5m+n . Teak ) oe? Ibn woe (6). a ED Next Fe 1S, cegpcteene ace Hacesot econ) cc ecneeacnn chee: (7). Employing the general solution (36a), (87a) Sect. 1, writing Y,=A.P,, Z,=B.P,, Y’,=A’.P,, Z',=B'.P,; we have from the general surface conditions, (44)—(47), Sect. 11. —m+n _, 9m+n re —p wa im+5n —@A = 2) 8 = oe } a ; +2nB+ A’a er 8na° B 8 ; (8), = 28 12 ee eA ee ear Boe Oe oe) (9), 3 m+n 3 ia 8m—n 3 0m — nN Mee —p wa 3n —a B-A'a? Bie Pp an wa Tl +n A’'a 5 i= min 3 7 Uc (10), 8m — 3m—n 8n ,5y —p wa 3n ed / 1-3 mie pes = sek gent ee aA ar v4 nB—A'a Sap tg B min 3 Tce (11). 296 Mr C. CHREE, ON THE EQUATIONS OF AN ISOTROPIC ELASTIC Eliminate R from (8) and (10), and from (9) and (11); thus 19m —5dn 5 40 ep Tm—n Al (gat Yee ald een ee EET wAa 21 +4mA’a 3 nBa mene 00 (12), 4 L9m—5n pa AO etl : a a +4mA’a 3 nBa mtn 21 STelmsiein(asts:vioialelasecel (13) Eliminate B’ from (8) and (10), and from (9) and (11); thus 25 ,, 9m ow =p m+ 2n Bae 5 3 At = 2 tee a a aAm+5nB+a°A 3 “mane ae (14), -3,,9m+5n —p m+ 2n — q’* i 1-8 A ee a” Am+5nB+a’*A 6 ato Sorceress (15). Eliminate B’ from (12) and (13), and B from (14) and 15); thus 7» 19m—5n —p 7m—n hes eta Nyt eae = 2 i, A (a'—a’") 21 + A’(a’ — a”) 4m ALi (a’ —a’”) Dp (16), a ,a@—a"°9m+5n_ po” ,. .m+2n = A(a’—a”)m+A (aay G = aaein | i aes 5 ascatnameneaeeeeee (17) We thus find 2 1 ; Se {168m(m + 2n) (aa’)’(a?— a)’ + (Tm — n) (Im + 5n) (a*’— a*)(a'— a)}...(18), m+n D A'= aa {20 (4an — n) aa’)? (a? =a) (a = @)} sain os. ccceecacesasennsancteeccmseecieeevoay (19), where for shortness D = 504m? (aa’)’ (a? — a”)? — (19m — 5n) (9m 4+ Sn) (a® — a*) (A — A") vee eee eee eeee (20). Multiplying (14) by a’, (15) by a* and subtracting we get po Case i ee r 30n = (aa’)* (a? —a”)’” o, 9m +5n 2(4m—n) (a — a’) (a'—a") .”., from (19), B= an Aidt esc gsGuacconer ss (21). Similarly from (12), (13) and (19) we find B= »m 6 (4m — n) (aa’)* (a? — a”) (a° —a*) — po* — : ue The expressions for A’, B, B’ are comparatively short; and in a given case, when m : 1 is given and also a: a’, they would be quite simple. SOLID IN POLAR AND CYLINDRICAL CO-ORDINATES, 297 From the general solution, (86a) Sect. 11, the coefficient of r°P, in w is 20'p m—n ¥ 21(m+n) Tn : and so is nD As here all we have to do is to write the values (18)—(22) for Y,, Y/, Z, Z’ in the general solution (36a), (87 a) Sect. 1. - | 24m (m — 2n) (aa’)’ (a? — a”)? + (9m + 5n) (m = 5) (@—a’*)(a'— a”) ssa (23). § 41. The expressions for u, v, and 6 are of course long, and a determination of the stresses and tendency to rupture would be a tedious process, so we shall confine ourselves to the case when the shell is thin, say GE JOM (Uiete) a acto nrcne teen clases cide ce etees ne se sesso ssa (24). The method we shall follow is this. First determine the parts of A, B, A’, B’ that do not contain e in their expression. The physical side of this step is that we assume the shell so thin that the displacements at any point are the same as at the point on the surface which lies on the same radius vector. We thus trace merely differences at different points of the surface both for the displacements and stresses, and so do not attempt to find how the stress rises inside the surface of the envelope. We next shall find the additional terms introduced in the values of A, B, A’, B’ when we keep terms in ¢ neglecting those in e, and thence find the variations in the displacements and stresses along the same radius vector. § 42. So first neglecting terms in ¢, and dropping wp for the present, we get from (18) and (20) after arithmetical reduction, 1 19m —n atc RSet sn alee conde eles eee bane 25 m+n 5 (3m—n) 2) Sue 8 a 4m—n 5 ‘= 96 Similarly A TB main Bm teen neeee (26), 2 a@ (4m—n)(9m + 5n) 5 B=- Ta 77a (27), , 4 ad m(4e—n) By = men n= (28) DG es 29) =F ie CDG ane Sao cee 29), oe ‘Ya’ 5m +n\ : p>: = iB si Tee a a eames (30). - The coefficient of 7°P, m u is, by (23), 57? — 2(); 2 1 57m’? — 30mn — Tn BD. 298 Mr C. CHREE, ON THE EQUATIONS OF AN ISOTROPIC ELASTIC Substituting the values (25)—(80) in the general expressions for 8, uw, v, (18 and 13’, (36 a), (37 a) Sect. m1, and in (3) and (4), we get we 8a° — 27° (21m + n) + Ge (4m — ) a® (5m +) — 1° (3m — n) = 2 Vo ‘ 2 15 (m+n) (8m —n) i. 3(m+n)(8m—n) Co id eet hs be 3 5Tm* — 30mn — Tn’ on (9m + 5n) (4m —n) ~ 5 (m+n) n (3m — n) |" S21 = 9 _ 4a’ (8m+2n) (4m—n) _ 4a" m(4m—n) fe 9 ars i 1 + ng Dmtn Ya 5m+n ————<$_ 3{— 89° + Barr HN vee cece ect e cee ee een ee ‘ +55 (m + n) ara 3am —n er" n (33), Manne We cal 7 pis, 3 3 95m* + 48mn—Tr* _ a*r 2” (Om ae eal ag, n(4m — n) nee m (4m — n) _ dP, 14 3 r 7 (34) = 0 15n (m+n) (8m — - “Sg § 43. From the assumption we have made as to neglecting terms in € we might v C= ie, ik (oh — mean =(), for it is so at the surface, this being in fact the assert at once that r dr second surface condition. But as the expression will be useful afterwards when we retain ¢, and is a very strict test of the accuracy of the results (33) and (34), we shall find its value at present. eee P, : Noticing that ao 3 sin 20, we have, after addition and reduction, ie eae sin 20 (4m —n) 2 - 32 a’ ee a aa Lone E r (19m +7n) — ; ea (9m + bn) ~ 3 # (@m—n)+ em Now put r=a, and the expression inside the square bracket in (35) vanishes. The principal axes at any point are thus of course the radius vector and the perpendiculars to it in the meridian and perpendicular to the meridian. Calling the three principal stresses R, ©, ®, in this case SOLID IN POLAR AND CYLINDRICAL CO-ORDINATES. 299 : é 1 é From (33) and (34) putting 3 a for P,, and reducing, we get U es 1 dv 15x (m + n) (38m — n) \r “dr +r do 3 cos 20 E (152m* + 18mn — 14n*) — (4m — n) {5 (9m + 5n) — 5 “(9m + 2n) +a te = mn} 22 (5 eT, ~ a (57m? — 30mm — Tn?) + (4m — n) las (3m + 2n) — o = ii ( Ta’ Smee — 2n (3m — n) oar a ee “I are se Lamu ee eT erah Ua rae. vf (37). Putting r=a we easily find that the right-hand side of (37) becomes identically zero, and therefore Again from (32) and (33), bar gee that du = HAH Dyn tee =(m—n)64+2 20. we find, after reduction, 15 (m+n)(3m—n)R bo Q, af bo =4P, E (12m* + 25mn —7Tn*) — (4m — n) 5 (9m + 5n) — 3 3 “(9m +7) oe nt | sb (Ge =70) (Gib) Weak 3S P| Gansaas-onnne- aon. Sccne-en05 cos pode aodoa ceceednaddedc (39). Putting r=a we find that the right-hand side of (39) becomes identically zero, and thus k=0. This is in fact the first surface condition, and might have been assumed without proof, but (39) will be found useful when we cease to neglect the thickness of the shell. From (38) it follows that @©=0; and so the only stress which does not vanish is ®, which we proceed to calculate. § 44. From (33) and (34) we obtain, on reduction, 15n (m+n) (38m—n) (¢ _ = cot 0) = 3m cos’ 6 \-= = (19m + 39n) — 2 (ann —n) bea o (4m - nh iia 3a 6 a’ aiid (57m? + 12mn — 21n”) + 12a7m (m +2) + a (7m? + 2mn — n*) — 7 pe (4m —n)...(40). Put now r=a, and write 1—sin’@ for cos’@; then, denoting Young’s modulus by £, and reintroducing the factor wp, we find after an easy reduction ~ +" cot 0 = a ibaa earache ne ann cncnennae (41). Wier, oul... Parr Ty, 39 300 Mr C. CHREE, ON THE EQUATIONS OF AN ISOTROPIC ELASTIC From (32), putting r=a, and reintroducing @’p, we easily find - — Sue our cree (42), therefore from (41) and (42), remembering that ®=(m—n)d+2n (= +2 cot), we obtain OD Seo 0, ALIN 0 ocd dra dele aang Se Sains ge tere (43). Since ® is the only stress which does not vanish, the maximum stress-difference is simply the right-hand side of (43). The maximum stress-difference thus vanishes at the poles, and has in the equator its maximum value Comparing this with the results (32) and (33) of Sect. v., we see that according to the stress-difference theory the tendeney to rupture in a thin shell bears to that in a solid sphere, of the same radius and rotating with the same velocity, a ratio which in any ordinary isotropic substance is very approximately 40/17. Since ®, the only existent stress, is by (43) necessarily positive, it follows that the 3 ; v : ae corresponding strain, viz. ~ +" cot @, is everywhere the greatest strain. Thus the greatest strain is given by (41), and so has its maximum value in the equator. Comparing (45) with (26) Sect. v., we see that according to the greatest strain theory the tendency to rupture in a thin shell bears to that in a solid sphere, of the same radius and rotating with the same velocity, a ratio which in all ordinary isotropic materials lies between 3:1 and 5:1. It should be noticed that the longitudinal tractions in a standard bar which the two theories would indicate as producing the same tendency to rupture as rotation in a thin shell are absolutely identical. § 45. We shall now consider what change is introduced when we retain terms in e, neglecting terms in e. We have first to find the change introduced in the values of A, A’, ete. Returning to (18) and (20), and retaining those powers of e which are required, we get Woes 168 m (m + 2n) (1 —3e) 4e* (1 — €) + (7m —n) (9m + 5n) 3(1 — €) &*.7 (1 — 3e) Sic ~ 4 x 504m? (1 — Be) (1 —e) & — (19m — 5n) (9m + 5n) 3(1 —e) e?. 7 (1—3e) We see that e& cuts out above and below, and so also does 1—4e, neglecting terms in e& The value of A is thus unchanged if we retain only terms in «. From (19), 20 (4m —n) (m+n) (2e — €*) (1 — Be) Te (1 — 2) A (m+n)=-— - . same denominator as for A SOLID IN POLAR AND CYLINDRICAL CO-ORDINATES. 301 Dividing out by ¢’, we have (1 -5) (2132) Be) new value of A’=old value x 1 —4e 5 Seta 4m —n therefore Ab a= ‘ = (lee 15 m+n 3m—n =a 3 So from (21), new value of B=old value x eesti) 1 —4e 2 2 = F Paiore fp see pt (4am i) (9m + 5n) ater: 45 m+n n (5m — n) (58) (2 A 5 (1 — 2e) So from (22), new value of B’ =old value x z als, 4 a m(4in—n) eeretore ~ 35 m+n n(3m— n) eae e . 1—2e From (5) new value of C=old value x 5 =SE a” dm +n the , C =——_ 5 ,— —€). profore m+n 3(8m—n) Oe) Gas) = 5) From (6)*, new value of #=old value x —, ao 5m+n therefore v= oad 400i (1 — Se). Thus we get, reintroducing the factor w’p, apP, | ,57m* —30mn—7Tn* 5 (m+n) 21n (8m —n) (9m + 5n) (1 —e) In (8m — n) — (4m — n) {o'r 4a’ (3m + 2n)(1—$e) 4a" m(1—Fe) i] rr 9n (3m — n) r* Tn (8m —n) +45 oF 5) | Br* + bar Su —«) = aS == i ph tt (46), "= T5n (m a (3m —n) “e eee me ane = ON aa) Le z. VAS) + S m= ro} eileen 28 (47). * The values given for EZ in (6) and (30) are each 4 times too great. + The value given the coefficient of the last term in (33) is 4 times too great. 39—2 302 Mr C. CHREE, ON THE EQUATIONS OF AN ISOTROPIC ELASTIC It will be found useful to notice that im (46) and (47) we invariably find (1 —.) associated with a’r, (1— $e) with a’r*, and (1—{e) with ar“, while ¢ does not appear in the terms in >”. § 46. Let us now find the maximum stress-difference and greatest strain at points at a distance from the centre r=a(1—.), where we neglect 2”. First let us find the value of S. We can do this at once from the expressions (35) and (37), by noticing from what terms in (46) and (47) the terms of these expressions are derived, and modifying the coefficients by the introduction of ¢ after the scheme indi- cated above. Doing this and writing r=a(1—2), we get, from (35), » ce b d Y s i dul 10n (m + n) (38m —n) | dr 3 dé) wpa’ sin 20 (4m — n) 2(19m + Tn) 32m \| - (1 — 2x) — 2 (9m + 5n) (1 — e) — 4 (8m — n) (1 — $e) (14+ 3x) + “wy (1 — $e) (1+ 52). When terms in e, 2°, and ex are neglected this vanishes identically; and thus, even when terms in e and « are retained, the three principal axes of stress at any point coincide with the radius vector, the perpendicular to it in the meridian, and the perpendicular to the meridian. Similarly, from (37), we find, after reduction, hea < - Ldvu>m+n — 22 ; a : |r — = an ea = Gn 5 [(8m —n) (7m —n) — 8m (4m — n) sin* @] ... (48). Remembering that u d.- R-O=+S=2n ra * = : we accordingly have 2 2/¢ O-—R ud Ue EF 3m — n) (7m — n) — 8m (4m — n)sin*9} ......... (49). ~2(m+ n)(3m—n) | By noticing from what terms in (46) and (47) the several terms of (39) are derived, and introducing the modified coefficients in ¢, we in like manner find 1 — 22 15 (m +n) (3m —n) R/w’pa’ = 4P, | (12m? + 25mn — Tn’) dee tus: +4 (3m —n) (5m +n) {5 (1 — e) — 3 (1 — 2a) — 2 (1 — $e) (1 + 3a)}. — (4m — n) £(9m+ 5n) — 3 (1 — $e) (1 + 3x) (9m +n) + 9A (1 — Fe) 1+ 5x) mt | When terms in e, z*, and ex are neglected this vanishes identically. Thus R vanishes throughout the entire thickness of the shell, and so (49) is simply an equation giving ©. The only other stress which does not vanish is ©, which we proceed to calculate. SOLID IN POLAR AND CYLINDRICAL CO-ORDINATES. 3038 § 47. Returning to the expression (40), and introducing the modifications necessary in consequence of the retention of e and the coefficient w’p, we get 15n (m + n) (8m — n) (= aa - cot 6) /o%p = € 9/5 7 3 cos? |- Sn — B9n = (1 — Se) m (4m —n) + = 5 (1 — $e) m (4m — | 2 dim + 12mn — 21n? 2 — f 14 + 12a’ (1 —€)m(m+n) 5 Ki + 3 = (1 — Se) (7m? + 2mn — nn”) 6 ee (ze im: (Aan — 1) a ccncsossesens note oes (50). 27 3 ue 4 Write now r=a(1—~2) and, neglecting #’, the right-hand side becomes m(19m + 39n) =o 3a*cos’@ - (1 - 22) 2(1 — $e) (1+ 3x) m (4m — n) +4 (1 — Ze) (1 + 5a) m (4m — n)| 57m? + 12mn — 217n? 14 — $a? (1 — Ze) (1+ 5x) m (4m —n). —a’(1—2z) + 12a’ (1 —€) m (m+n) + 3a (1 — Be) (1 + 3x) (7m? + 2mn — 2’) Reducing, we easily find arpa: yal ae Gl Gun , B (2 27) sin’ @ + (2x — e€) ae Ba seaeeaacenetes (51). D se et rT Similarly from (32) modified we obtain 5(m + n) (8m — n) 6/@*pa? = — cos’ 6 {(1 — 2x) (21m + n) — 4 (1 — $e) (1+ 32) (4m —n)} +2 (1 — 22) (m+n) +3(1—€) (5m +n) — 4(1 — $e) (14+ 32) (4m—n); reducing, we find o pu é= eA) Gm =n) [(m +n) sin? 6 + (22 — €)(7m —n) + 2 {e(4m —n) — x (9m —n)} sin’ O]...(52). Thus PD /w*pa* = Tm—n ie 2 sin* 6 2(m+n) (m+n) (38m —7n) ‘sinto + (2x7 —.) {e(m—n) (4m —n) — & (11m? — 8mn+ “| so-((553}): Of the three principal stresses we thus know ® from (53) and © from (49), while R is zero, From (53) and (49) we have e®—O= wpa? sin? 6 {i a 9% (5m — —— - } 304 Mr C. CHREE, ON THE EQUATIONS OF AN ISOTROPIC ELASTIC If @, be the angle given by . » (3m —n) (7m — n) ae sin 0, = of tnGm=n 6 ae ee (55), then at polar distances not exceeding @, the maximum stress-difference is given by (54) when a is less than e/2, and by (53) when 2 is greater than ¢/2; at polar distances exceeding @, the reverse relation holds. The absolutely greatest value of the maximum stress-difference occurs on the inner surface in the equator, and has the value © — © = «'pa° (1 PS ee eee (56). Since R and ©, two principal stresses, vanish the corresponding principal strains are each equal to — 8(m—n)/2n, where 6 is given by (52). These two strains are clearly negative except in the immediate neighbourhood of the polar axis where sin’@ is of the order «. Since in this extremely limited region all the strains are of the order e, we may for all practical purposes, in treating of the tendency to rupture on the greatest strain theory, confine our attention to the third principal strain, given by (51). This increases with 6, and has its absolutely greatest value on the inner surface in the equator, where the measure of the tendency to rupture is given by uo _ @ pa® m+n E = comes B (1-<" 2") hse sath BIE 8 (57). Thus on either theory the tendency to rupture in a thin rotating spherical shell is, for all isotropic substances, greatest in the equatorial regions of the inner surface. Since e is supposed small the remarks made at the end of § 44 apply with very little modification. SECTION VII. Sphere or spherical shell with given surface displacements. § 48. Let us briefly consider the application of the general solution (36 a), (37 a), (38 a), Section mL, to the case of given surface displacements, say w= 2U,, vx==V,, w==W, at the surface r=a; w= 2U/, v==V,/, w==W, at the surface r=a’; where U,, V,, W,, U’, V/, WY are surface harmonics of degree 7 As in the case of given surface forces this form of solution is especially suitable for purely normal effects, so we shall consider this case specially; afterwards we shall briefly consider the change to be made when v and w are not zero at the surface. Using the solutions mentioned above, our surface conditions shew us that X,=w,=X/=w/=0. Thus the equations v=0, w=0, when r=a, give us one complete equation between Y,, Z,, Y/, Z/. SOLID IN POLAR AND CYLINDRICAL CO-ORDINATES. 305 For v=0 gives an equation ae ) =0, and w=0 the equation aD CAAT )=0 an v= q SiGe , p) =9, all over the surface ; therefore the complete equation is f(0, ¢) =0. Thus the surface conditions are mag are ™(i+1) +2 ay ae > eae Y,+a Z.— 2 aye ye +a* *L, = sae tech sete sis eceineL)s = (i+ 8) +2 i-1 = (i—2)— 2 —i-2 SSS Z,+ Se Se Zh = Goooo 6aciced (2), me 2 Aas ~@+1)+2 eet) Te _ YORI) he Y,+a' aI es 2 +a‘ Z, = (Uf ABO bodticis tas Seem (3) m(i+3)+2 get G-2)-2 pee = aye. o be et ee SIT A, =e ncasacosae 2)) Multiply (2) by 7+1 and add it to (1), also multiply (4) by +1 and add it to (3); then we cet a ie oy eet 1 ag _ m+) +n(2t+1) — (Tia Ue pea mat 5), 2n 0 : nt (21—1) : : -) Meer pees a,S m(i+tl)+n(204+1) iy, = quay, gL gag MEDERMA payee oe 2n u nt (27 —1) : : whence, eliminating Z,, Teves PA Geto) on (23a) a — ee sey ae a = (#@—a") Y,—- — 7 ,=-Ua UG Nat aen dette t Qn, es net (24 — 1) (Gi 7 ies wy Again, multiply (2) by ¢ and subtract it from (1), also multiply (4) by 7 and subtract it from (3), then we get aa Bere 55 mi +n (21+1) ai, — = a'Y! + 21 2c il Say OL be 7 Le = nn es A = [ape A Me a 3), n(i+1) (+3) eT) ace (8) Da CeO) = nC nn ane = ‘ DCN Y.-5,% Yit7y 4 The SEER Re aR ae (9); whence, eliminating Z/, mi+n(2i+1) _ ,, pc aye We ah ey teats ae rie eae sa )Y,-5, @—a*)Y, = Ua" — Ua Direyeicreraye (10). (7) and (10) give Y, and Y/, whence we may obtain Z, from (6) or (5), and Z/ from (8) or (9). 306 Mr C. CHREE, ON THE EQUATIONS OF AN ISOTROPIC ELASTIC If for shortness mi +n(2t+1) | sss _ 7iaits mM 7/3 pla pa| eth Gia) csi ood oa ) ra Ld Re uk (i+ Ijt+n (21+ 1) 1-21 -2i41 2n ae ni (2% — 1) ia re nit then Ua — Ua’? a @—o) Diye— : ; 11) ; ‘ae G31) Oe Peter (11), Usa Ua ea) (a -a™) | H2_ T7142 mi+n(2+1) | sis _ [tits | ; (eis OEE et DY =| / eta ee PE me eee 2 A 0 ee Cece (12). * m Titi WT 7-i UL | Ufa = Ua oa (a? — a”) | § 49. If the sphere were solid Y/=Z, =0, and a’=0; therefore from (1) and (2), which are now the only conditions, areal) 43) | Y,= Ua nak Ce ee (13), Rai ae Te (14). 2 {mi +n (27+ 1)! Substituting these values in the general equations we get _ str 1 (t+ 1) (21 +3) ; 8 = Uz'a ipo eee ee snltgiectsracaanrelbaes eee (15), u= ee LG) “i {m +8) + Bn} — (=) (i+1) (mi — 2n)] u, lle ee (i +3) +2n dU, Ir\? r\ih ne {mi +n(2i+1)} dé \(2) = (“) } Oe Eee ramaiochacegl slelocistar Gaiaicta ee ote Detaeeeiee eater zor _ m(t+3)+2n 1 dU, ((r\ rv Y= Timi t+ n(2i+1)} snd do {() - (5) pel epee rane nd. > (18). § 50. The method to be adopted when the surface displacements are not purely normal is exactly the same as in the case of surface forces which are not purely normal, so we shall merely glance at the case of a solid sphere under surface displacements u=ZU,; v=2V,; w= zW.. SOLID IN POLAR AND CYLINDRICAL CO-ORDINATES. 307 Instead of (1) and (2) we get from the general equations of Section I. mi» uta) tv iy mag ( oe 5) a ode ak WG, = Onpocnsadesoogon nose cugnaboondcoogsad (19), ee Ts — 3)+2 - d | n Ca Gils aX, Bn eee Si in ce a S J Ss Ve 9 ao~ | FG +1)\(u+3) Y,+ B Aye eee =V, rabolefetelenaiomisieltistetes Helerase(eisyeie ( 0), IL. Gl ait 3)+2 Gi | —, ,:|\-—_—_—— "7,4 -—Z Mian Wide naceereeerectiocte tt 21), snd d~“| 2@+1yu+s)- Ue gases ) : dw, dX, 55) where aes a Sr [eee (22). Multiply both sides of (20) by sin 6, then differentiate with respect to @, and dif- ferentiate both sides of (21) with respect to @ Adding these two results we eliminate X, and w, by means of (22) and get, using the surface harmonic equation, S ES aes aiaen +: See ps ove | ae oro ee dW, . Qn (i +3) iw"Y,-(@+1)a@ Z| = sin V+ ||... ono es (23). sin @~ | dd" dd Multiply both sides of (21) by sin@ and differentiate with respect to 0, then differ- entiate (20) with respect to ¢ and subtract from (21) so modified. This eliminates Y, and Z,. Finally using (22) and the surface harmonic differential equation, and then differ- entiating with respect to $, we get SHG lane = 1 dies ACW; a) > eS sin 0 ~ a ssune db dd¢* [September 18, 1888, If over the surface r=a@ of a solid sphere the displacements be derivable from a potential expressible in spherical harmonics—i.e, if w= 31 = (Sea), i dr d a eg Hers ak cel wade (Sele i: and Sear) FE (sre), when r=a, representing by S, the typical i-1 surface harmonic—then throughout the sphere w=38,(r/a)', v= 2% ol (<) » and a @ w =>, TL. GIS} (: isin 6 dd shell be similarly derivable from potentials involving typical surface harmonics S, and / i-1 | . If the displacements over the surfaces r=a and r=a' of a spherical respectively, then X,, X/, w,, w, ete. vanish, and an explicit solution in terms of S, and S/ is in like manner easily obtainable. Since in these cases 6=0 the solutions are applicable to the case of bodies bounded by spherical surfaces which are immersed in or contain incompressible fluid. | Vou XV. Parr IL. 40 308 Mr C. CHREE, ON THE EQUATIONS OF AN ISOTROPIC ELASTIC SECTION VIII. Vibrations of a sphere or spherical shell. § 51. Supposing no internal non-elastie forces to act, the equations of motion are, see (24), (25) and (26) Section I, (m+n) 7* sin nee nig tn ge = pr sin oS = desks See Spanos cate tment QQ (m+n) sin 0 a Tt +n _ =pr sin @ a Beas cande Gendt saree (2), (m +n) cosec 6 a7 ne +n a pr = Sa baSactos Sean aoa eee (3); where u, v, w, 8, A, 3, © have the same signification as before, viz. ad Ce 1 d.vrsiné 1 d.wrsin °) (4) =ialmae ape ae co a ee (4), &e. Differentiate (1) with respect to 7, (CA eacosasncae ces mane 6, and" (3) eee): arnpanebosceed: ¢; adding we get 2 a6 : (m+n) V7d=p 7 Ta (5), where y’ has its usual meaning. In (5) we may at once assume 6 x coskt and then determine it as a function of r, @ and @. This gives PS 2d8 1 od 4H 1 a kipd _ det r dr? 73n0 do" ” db* 7 sin® dp m+n Regarding 6 as a function of @ and ¢, we may suppose it expanded in a series of sur- face spherical harmonics of different degrees. Then to determine the function of r forming the coefficient of Y,, the surface spherical harmonic of degree 7 in this expansion, we have Gd 2dd o (ioe t(t+1)) _ ats da ee og wig Yajulnt ufoihfateietararamia ditaliebs etuieiotade etetete (7), where for shortness a=) nl a est NR Aaa IR DLR Fats Sa (8). m+n Put also =? 552 ttitet asides Sea ae ae (9). We may write (7) in the form PAS 1d.rl8 (ee (+3) de +7 et ee -ED) SOLID IN POLAR AND CYLINDRICAL CO-ORDINATES. 309 Equation (10) is the well-known Bessel’s equation having two solutions J;+; (kar) and Y;4; (kar) in the usual notation. Only the former occurs if the sphere be solid, because Yi4,(kar) is infinite when 7=0. But for a spherical shell both occur. Of course the Y here has no connection whatever with the Y used above in the expression for a spherical harmonic. For the present confining ourselves to a solid sphere we have SiSteos WEY Gre Seeg (aP ies Aid gov tyasdcoots De cae Deacon oceans (11). § 52. The method to be followed in determining wu, v, and w is exactly the same as in the case of equilibrium. Thus in (1) substitute the values of 43 and &, confining our- selves to a single surface spherical harmonic, we thus have ah 8 d.vr = 1 (4 d d.wr ame) — Ps ee noes M+N » . gf a, ee ~ d0) sind \d¢®? dr’ dd n dt? io de Pp Lac N Oy. aa t (12) Substitute in (12) from (4), d.wrsin@ . 9, /d.ur’ 1. d.ursin@ ,.\ . de eka a(* Ga 1-ae 3) we thus get Gh Gh 1 dvu 2 Chop ik Gl d.u* | 1 d.vrsin6 : ee em anelag do ae dae | eildr” sin dr ‘sind dO 5 Ds Pees du mn Boe oe a ae sin 8 dé ind The terms in v cut out leaving a differential equation containing wu and 6. Divide out by sin@ and, confining ourselves at present to the spherical harmonics of the 7 de- gree, substitute the value (11) for 6. Assuming where U, represents spherical surface harmonics of degree 7, we get to determine w as a function of r the equation 2 Se — +i ua E A Ji, (bar) — ae. fa Wie ar) | Y; coskt...(14). ‘| We may write (14) in the more convenient form a a ure + ii78* — i te al dr*’ r dr d 7 : = E i Are Ji+3 (kar) = (5 = 2) Tix a) | } ; coskt ee eeeenee (15). The general solution of (15) is obviously ur = BS pa (GBI) ee cea ssee en ceiects sasietes sn acaeevieemaret (16), 3510 Mr C. CHREE, ON THE EQUATIONS OF AN ISOTROPIC ELASTIC where B is independent of 7 and is, in accordance with (13), of the form Z, coskt, where Z, is an undetermined surface harmonic of degree 7. Thus for the general solution of (15) we have u= Z; coskt rh Tia, (EBT) Sash he. as de RUE a oe (17). The particular solution is clearly of the form w= Y,coskt f(r), where f(r) is a function of r. Now from the Bessel’s seve it is easy to show that E ae +4 ' re — — tes r — Jia; (kar)=h? (8° — aad ale af) S543 (kar) — 2h’? J ;.4 (kar), dr* r dr dr i.e. by (8) and ( RP m a i = = ae Ee 1S = itd (kar) — QS +4 (ar | alaepieted pis kletdae bieve tite ehevoticety (18). Also FE 5 tie es 5+ ee @ = 4) ead Ji; (kar) = kh? (B? — a?) Ji, (kar) = Kp m m+n n Six (har). Thus the particular solution of (15) is obviously jhe | Sere Mian oe en) | LAE Ww = | 7 app ea ean) kee Klean) Namco, Thus the complete value of uw, so far as depends on coskt and surface spherical harmonics of degree 7, is i ; u = coskt EE 3 V; \r Jix4 (kar) —r~ = Tiny (tar) te rea, (ke | Repro re (1,2) § 53. In the case of v let us follow the same plan as in the case of equilibrium, and find the portion of the solution for v which depends on Y, and Z, and on the general solution. The method is the same exactly. E F : ; d.wr sin@ . In (2) substitute their values for @ and © and for ae in terms of wu, v, and 6, also put in its value for 6. We thus get 1 @.or 1 d Ay as eS 1 d.vrsin@ sind dg? +7 sind dd’ rie ee ead tec hee ny | GRE Ts | d*u |p do m+n. ,d¥; 4 a Oy es + sind ate sin Or - wate a sin@ da" J.) (kar) coshkt. Here v is assumed « coskt, so put onze, and the above becomes 1 d.vr sind il @ pil ICRIAO Oe SUNG aeons fate. a tHaah doe ie ae. 1 a 7 ee m+n dy = A os a esto ot Jes. (on =7 | aie qe. ie 6Y;,- =i nd hal kt J;.4 (kar) Mu 1 d sin? ‘oe ur* +sind a9 and do" SOLID IN POLAR AND CYLINDRICAL CO-ORDINATES. dll Substitute for w from (19), and this gives 1 (ours 1 d . ,d.vrsiné 1 d?.vr sin® 222» sind aie sin@ +h?’ sl dr sind do’ dor sin®0 a = (sino +2 cos) : tS gy (kar) — ane i i Si44 (kar) —r ue Ji 44 (kar | dé ‘pr? dr dy; m+n = 1 = d ad = |- a Tig (hear) + aie Fury (har) 1? © Sis (iar) | 7 = reds (k8r) —— (inet +2 cosb%) © —.r 2 Tz4 (kr) ahs (20). To simplify this we must employ Bessel’s equation, and carry out the differentiations when required. We thus find d (r* ad i+1 = i Tey (kar) — 18 © cas (hart = 9 {i —— M Ses ER es (21). Thus, noticing that a =a, we reduce the coefficient of sin@ = +2 cos@Y, in the first line of the second side of (20) to = Me Ea et ae ee (22), é : ‘ Vee Santee ‘ So again the coefficient of sino) “in the second line is reducible to 0 2nd (m i(@+1)+1 ae Se a io Vane ONG) suet 23 Pe op a: Ji+3 (kar) ip ae (kar) as Per ; soccauod¢oonconnn} (23). Thus equation (20) becomes 1 [d’.ursiné 1 ad). 7d aunisind Ld; onisim@ : coskt | Pt and g0-2° ag at age TR sine meioneTs t(¢+1) dy; jar4 oe Mm 4 | peat Pa a S543 (kar) + sind —~ We Nears S S544 (kar) — sl + pa? J; +1 (kar) —2 cos 7 = 1 Biren (k8r) + sin 8 — a {—2) a Bsr (7.517) ppogemneee eve ae aoCe i (24). From the form of the first side of (24), for the general solution assume vrsin@ x X,, a surface spherical harmonic of degree 7; whence, to determine vr as a function of 7, = iae JME eu or = 0, : @ vr? d P = Sele ahd EES, ir Fra. 7+ {ee - aS Cw = (0; or ort x Jes (kBr). we have CHREE, ON THE EQUATIONS OF AN ISOTROPIC ELASTIC S12 Mr C. So, writing in the coefficients cos kt and X,, we get 4 exy (KBr) v=cos kt —— The analogy of the solution = is a typical term of this part of the solution for v We have next to find the particular solution of (24) as the function of @ i dé ”. Thus sub- obtained for the case of equilibrium suggests the trial of sin@ ind ¢ which will give the part of the particular solution depending on } stituting cos kt f (7) sin 9 oa in the left-hand side of (24) for vr sin @, we get FQ AD Bad A: a eG) Cs eae 6% sin? — .sin6 qo * sin®d is sind > a t+ RP (r) sin ECR AG) nO 70 dr? * sind do"? aa x La ¥; ie 1d dy; After using ain?d ae a sind d- sin@ — "a6 ° this gives for the first side of (24) the expression . ae nao! 3 oe ") y +( ee = as => F6 jt EG COS tO Va rsces ster (26). Y,. Equating is really the particular solution of (24), then the expression (26) a If cos kt f(r) sin 6 q must be identical with that part of the second side of (24) which contains ‘in (26) and (24) we get two equations which coefficients of cos@Y, and of sin @ the e ought to be identities. The former agit f(r)=- ae se Heat) on eoroaios cise sc sa toss hah le see OSE (27); and thus in accordance with the second we must have Dawa tC +1) rt sar) = : : Were la + k*p* — 2 (= jagaediens (har) = coefficient of sin @ do 8 (24). Substituting for i+3 (kar) from the Bessel’s equation we reduce this expression to rail 2a (Gee | ~ faa |— 3 geen ar) + |B 2) +5) Jers ar) ~~ = it is obvious that this is actually identical with the co- Noticing that Thus our hypothesis was correct, and the particular solu- efficient of sing Ui in (24), ' (24) depending on Y; is cos kt Jo MA Ne ut fits acinus gay ose neR eee trea (28) tion of YS en Teaan z SOLID IN POLAR AND CYLINDRICAL CO-ORDINATES. 313 x) as the form of particular solu- dé tion of (24) giving the terms Tossing on Z,. Writing in Z, for Y, and y(r) for f(r) in (26), we see from (24) that if this hypothesis be correct y(r) must satisfy both the equations 2¢+1 Z oO PRT Gai Recee teBerneedee sess (29), and o x (r) + Bx (r) -— a (7) == rk Tipsy (HBT) oceececsceeec ees (30). aa, respectively. dé These come from equating the coefficients of cos @7, and of sin @—— Now (29) gives {rd So ae es) 31) Me) = aE (" dr 5) Jiiy( i api slaieheiprela cls iaesisteeinis ( )s and using the Bessel’s equation it will be found that this satisfies (30) as it ought. d ith (kBr) atte atat states atelaisletalelelststeisvaterelateisietetstast= (32), Ue x)= ae 1L dr” and the particular solution of (24) depending on Z, is coskt dZ; pid 1 Nou Qe ~iG+)) de ee (kBr) SSOOOOOOOODODOOCOONOnOCoCorG (33). Thus the complete value of v so far as depends on Y,, Z, and the additional typical term Xe. as 1 dY; 4 1 d%Z, 4.4 Xx; 3 v= c0s ht | =a Juss (ler) + ay = i TES: (kBr) + =a Fay Br)} (34). § 54. Just as in the case of equilibrium* we need not form the differential equa- tion in w given from (3); for we know now uw, v, and 6, and so get w from the expression (4) for 6. The plan followed before applies here in its minutest detail, as it was independent of the functions of r occurring in w and v, and depended only on the differential equation satisfied ee Vee Ze eande xem nus coskt d WG; = ae 4 Qn Sand a6 |- Fat r 2543 (kar) + op ae = =. Ted 544 (esr) | + cos kt w,r2J 4+ (kBr) ...(35), dw; _ dX; 5yz where Fg ea ogee ane (36), (34) and (35) give that part of the solution of v and w which depends on the typical term for w in (19) containing surface harmonics of degree 7, and also the typical independent term Y,. § 55. The foregoing proof is rather long, but the result, now that its form is known, can be deduced quite shortly as follows. Proceed as before till equation (19) is obtained ; then, to find the corresponding terms for v and w, assume ur = cos kt ae a + (r) A el alee Ms (37), * See § 10, Section II. 514 Mr C. CHREE, ON THE EQUATIONS OF AN ISOTROPIC ELASTIC _ cos ki ad - and w? AVE aaa e582 C97 ta aon docboagac ncsodeadosscoBac (38). First to determine f(r) use (4), substituting from (37) and (88) for v and w. This : d dy; CRN ae Ghote gives cos kf) 9 ap: sin 0 5 + sare age POH FOE. Substitute for 5 from (11) and for uw from (19); we then get ose \_ y 3 d rey i d —t(t+1) f(r) Yi= VY; | ri diay (har) - = 2 dp Ji+4 (kar) — ah Ji+3 (kar)} |...(89). But the coefficient of Y, on the second side of (39) is equal to the coefficient of (sin @2F +2 c0s 6Y,) in (20) multiplied by 7°; thus from the form we got for this coefficient in (22), it follows at once that : rt : yf (r) =— aa vitt (kar) BOODOCOUOOOOOOCOOOROOAOOCOOOCOOUSOOOCNO (40), and the corresponding term in ¥v is a ae 4 Ji43 (kar), The term in Z may be got in the same way with even less difficulty. Then to obtain the typical term X, we require only the first side of (24), which it is com- paratively simple to obtain. § 56. We have now to determine X,, Y,, Z, &c. from the surface conditions. Suppose the surface forces depending on coskt are =S,coskt normally, 7 cos kt tangen- tially in the meridian, and £U,coskt perpendicular to the meridian; where S,, 7, and U, are spherical surface harmonics of degree 7. Then so far as depends on cos kt the surface conditions require (m—n) 6+ 2n uo DEAS yuh 4 Ren anneaerie FS noceaaeenceeantaticoc (41), a.= ie. Ake ahyi nN : Bias + ie Z => >T, COs kt aialuialdinio relat eisietatsiaistalsjnsretelale eralatarntetetate (42), oe “7 1 du A FEES th stent blainrevele(arsinlelniaipiarwnip elmiatalele(viaieielaieteleiete (43). We shall find, as in the case of equilibrium, that if the surface forces be purely normal X, and w, are zero so far as depends on the surface forces, and so exist only as “free” vibrations. In this case we can drop the = in (41), (42) and (43); and in every case we can drop it in (41). Thus from (41), (11), and (19), we have, putting r=a after differentiation, SOLID IN POLAR AND CYLINDRICAL CO-ORDINATES. 315 (m — n) V,a4 J; (kaa) + We = (3? v4 J,4, (kar) — v3 = it} (kar)} + ant, = {ret Si+4 (ven) = Sooosee (44). Further, we see that ae d eG grt c+ (har) — 978 & Ties (kar) = 20 AT Tox (har) +18 Tiny (har) {tet EDEN Using the notation = ie (kar) =J’;.; (kar) ete, we thus reduce (44) to a* Y; [Fe (kaa) {m+n eo Ghs is =} pees ‘44 (aa) | Rea kaa + Anka? Z, {ou “s43 (kBa) — Lan Sinn (ka)} = 8, Woes (45). So from (42) it is easy to find d Ya* 3 Za? 2kB , yeh £: J; (kBa) ~ 16 ee J5.4(kaa)— 2424 (haa)} == ceom {2Jes(kBa) + (6%a — 98 e+ 141) sare kBa De 3 Seay (l > on 9 ba 3 wr i+3 (kBa) — ea ae = =o eae (46). So from (43) mea | Ya*( 3 ; eee Laks aes +1 (kBa) [Al - {hea Tes(ba)— 27. (kaa) —: = os Jia) + (8%? 2.44141) 2! ae I] = 7 i+3 k + SwkBa Jen (kBa) — 5 2 ie - *3U, deen, ns (47), h dw, dX, where db =A) . As in the case of equilibrium we can from (46) and (47) get two more convenient equations, Multiply both sides of (46) by sin@ and differentiate with respect to 0, differentiate both sides of (47) with respect to ¢, then add together (46) and (47) thus modified ; we obviously eliminate XY, and w., and noticing he d 1 te We ae ‘age W. | a0 88 Sate ap 7 An sin @ d¢” we get , Ya Za kp a Jixs(kBa) Yi (i+1)n EE ae Tery( kaa) — 25 (kaa)} — 76 at ’,44(kBa) + (2 B'a?—27.7+1 +41) bBa =| 1 d dU, =- 55 =| 5 sin 87, + 3 | dueaee eeicteeetee (46 a). In the special case 7;=0=U,, we may drop the = on the first side of (46 a). Vou, XIV. Parr LIE 4] 516 Mr C. CHREE, ON THE EQUATIONS OF AN ISOTROPIC ELASTIC Next multiply both sides of (47) by sin@, then differentiate with respect to @, and subtract (47) so modified from (46) modified by differentiating both sides with respect to d. We thus eliminate Y,, Z, ete, and get Jp. ~ ds, a i sir ' z ee) on ee S lane i a0" sin Ow,- nkBa-® i i+, (KBa) —5 Ba — de waar sinOU,|, all over the surface. Differentiate both sides of this last expression with respect to ¢; then, noticing that a= T': we find that the first factor on the left-hand side becomes ( q CX, : an 9 ix: XxX. sin 0 age t i dd ; ie. is equal to —1 mae sin 0X,. Thus nkBa-* Si (i+ 1)X, Went kBa) — : fe CO} - SS a ee sin OU, — Al (AT a)*. From (45), (46 a), and (47 a) we can determine all the quantities of which Y,, Z,, X, are representatives. The coefficient of X, in (47 a) equated to zero, gives the frequency of those free vibrations in which there is no radial motion. o § 57. Let us now consider more particularly that part of the solution depending on purely normal forces. Since in this case we can separate the spherical harmonics of different degrees in the surface equations, we may treat them separately. Thus we have (45) and (46 a), dropping the = and putting 7,=0=U,, as two simple equations to determine the two panes unknowns =|, «- Seas We thus get Dey, = — Eg (kBa) + {k®BPa® — 21 (i+ 1) +1} Zep 20) ceo (48), D.Z,= os [Mn O — 25's, (koa) 0 Ee eee (49) where D is the value of the determinant Sos (kaa) \ meen) — 72 GEE DY 4 TT's (ha, Bhar | Jury (WB) ggg ~ 2’ (vey | eee RI Ne (kaa) ie : ae (kBa) + (6a? — 2i (6-41) +1] fee 80) Re eh I AM ce ice i si (50). * If instead of being themselves surface harmonics, | monic of the i degree, the right-hand side becomes a _ as"; ait ire 1 ds", i(i+1)S”; and the = may be dropped from both sides. rT and ap? where S”; is a_har- SOLID IN POLAR AND CYLINDRICAL CO-ORDINATES. 317 The values of Y, and Z shew that for a given frequency of vibration the ratios Y,:S,, and Z,:8, are, for the same sphere, independent of the form of the spherical harmonic of the degree 7. Thus we have the following important law. The ratio of the amplitude of the forced vibrations to the amplitude of the con- straining normal surface force of a given period depends only on the degree of the surface harmonic determining the law of distribution of the force, and not at all on the form of the surface harmonic. Again the periods of the free vibrations, answering to S,=0, are given by D=0, and so are independent of the form of the surface harmonic, depending only on a and 1. Thus for a given sphere all free vibrations whose expression for wu contains surface spherical harmonics of the same degree are of the same frequency, the frequency for degree 7 being given by (50)=0. Thus we could obtain the law of frequency from a consideration of the simplest form of surface harmonic of given degree, say a Legendre’s function. We have thus a large variety of possible forms of vibration of the same frequency, and so the frequency alone is not sufficient to distinguish the vibration. From the form of (50) it is obvious that the equation D=0 may be regarded as an equation in ka, ie. & and a@ occur always as a product. Thus for a given material and for a given form of vibration, the frequency varies inversely as the radius of the sphere. This remark applies to all the possible forms of free vibration, cf. (47 a). § 58. The case i=0 is very easily treated, and is found in most books on elastic solids. Let us here however follow the Bessel’s Function method. Proceeding as in the general case we get equation (10) with 7=0; whence, in place of (11), 6=cos ktA’r? J, (kar), where A' is a constant, = ae SIM, (7) IC OSIE ater acc esc crore aaee atari suche aacliaey vasasrt anes (51), ka 7 if dee Ay) Ee Tv 6 is independent of @ or ¢, and so — Ht SEN (52 ) dp top np dp ene (52) Oar Ho thus a a (kar) cos kt, 5 el ee sin (kar) Pee and so ur = [a = cos (har) |. No constant is required as this gives u=0 when r=0, sin (kar) kar A > COS kt * 1 a tia) — cos (bar)| arene hencennn isa se (53). 41—2 318 Mr C. CHREE, ON THE EQUATIONS OF AN ISOTROPIC ELASTIC a Wek wu : ’ : The surface condition is (m—n) 6+ 2n 7, = normal traction, or, as 1s more convenient, (m+n) 8S—4n ~=normal traction. If then this traction be Fcoskt on the surface r=a, 7 sin (kaa) 4n (sin (kaa) iss sath . F=A jim + n) Fan Fea { lag (ay Seen issatteeed (ai) For free vibrations F=0, and so tan (kaa) _ 1 he a Eger rc as a. siicolbjeis aft ardisiela lia caalentewiaamieeeeteeaee (55). 1- ka 4n Possibly a simpler method than the preceding is to substitute Spe at once for 8 in (1). Then assuming ux coskt we easily put the resulting equation into the form 3)? P 1d : (5) on Pr (urd) + ae (urd) + \Pa? — = QPy=0) Fike (56), whence ur =A" J; 3 (kar) cos kt, where A” is a constant. § 59. The general solution is equally applicable to the vibrations of a spherical shell. There is required only the introduction of the second solution of the Bessel’s equation. Since 7+4 is neither an integer nor zero, the second solution is simply J ~ (+5 (kar). Thus in place of (11), in the general case we should have 5 = cos ktr® [VJy4, (kar) + VT 44) (Ory) «occ cceceececccsesssees (11), where Y/ is a surface harmonic of degree 7, and (11’) is used to signify that the equa- tion takes the place of (11). It is much easier to introduce the additional terms in wu, v and w than it was when dealing with the equilibrium of a shell, for in that case the coefficients of Y, and Y’, of Z, and Z’, were affected differently by the necessary differentiations with respect to r. This difficulty does not exist here. Thus for instance the coefficients of Y, and Y/ satisfy the same differential equation, and are operated on by exactly the same differentiations with respect to r. Further the resulting expressions are simplified only by a use of this same differential equation, and as i+4 occurs only as a square its sign has no influence. Thus all that is necessary is to introduce in (19), (34) and (35) wherever Y, and J;,, (kar) occur a new term in Y/ and J_(+,) (kar), and similarly in addition to the terms in Z, X,, w, and Jj+,(k@r), new terms in Z/, Xj, w, and J+, (kBr), where the dashed letters are similar in signification to the undashed. Thus the complete solution for a shell is u= cos kt E ME \ rt J,4; (kar) — pit Jros(kar)) Lele ead = joes Kea z, \ a ae y (kar) —r am -+» (kar)| +Zr 8s, (kBr)+ Z, r ten (br) | 19, SOLID IN POLAR AND CYLINDRICAL CO-ORDINATES. 319 »=cos kt + do 0 [i Fe HY, J;4,(kar) + Y/J_@; (har) ie l u i r fia! + aD Tai Ph Teay (KBr) + Ze. FAT 80) | eoske [x JeshBr+X Fah} ie «Oe Wh 1 OG ea De. . e ee (34), _coskt d es rr ~ sin @ ists “AY, Jia (har) + Y; (F-csy (kar) +GED aa a 3 ,.3(kBr \ he -. PD —G4y (KBr) | seosktr hh kr) +m Fern (h87} Pak a’ puked Relea prachbecaet eee em (35’), ao dw, _ aX, oe dw, dx; or dp do dp de” and 8= cos ktr? {V Ji+4 (kar) + Y/J-G+y (ietR)| Wiateonue to onan eebooesedce (et): Supposing surface forces to exist over the bounding surfaces r=a, r=a’, we get as usual 3 series of equations at each surface, and thence, exactly as in the case of equi- librium of a shell, determine the 6 series of unknowns. If the surface forces on the shell be purely normal the equations get much simplified. ; d Th. tf ues There occur two expressions aot ) and and Sele, ¢) vanishing all over the surface, whence there results the one equation (0, ¢) =0.- We thus get for a shell 4 simple equations to determine Y,, Vi, ZZ, im) terms of the normal forces S,coskt and S/coskt. The coefficients of these quantities are as before constants, so there is no theoretical difficulty in the process. § 60. We may by this means find for example the periods of the free “transversal” vibrations. We get exactly as we got (47 a), putting U,=7,=0 over both the surfaces X,[ Js (ha)— 5 5 OBO) 4 x7 |S —asy (Ba) — 5 Fey CPO | _o, , r=d,Tr=a, kBa kBa x,|y “44 (kBa’) — 2 5 2 OO) + x aes sn hBa’)—5 Zeus Ce) _ 0: whence the periods are given by \y ‘ity (kBax) — : pe} pep | (ka’) — 3 ee y — [o'-rn (8a) ~ 5 gS en CEO! bry 80) —5 He ae (47 a) If the shell be not very thin probably it is best to transform (47’a) by means of the relations between consecutive Bessel’s functions into {i —1) Ji-y (8a) — 6+ 2) Seog (RBa)} (6-1) T—-» (KB!) — G+ 2) J+ 9 (KB0)} ~ (G1) Ji-5 Ba’) — (6 +2) Feng OBa)} (6-1) T—@-» (480) — +2) Jr) (KB0)} =0...ATD). 320 Mr C. CHREE, ON THE EQUATIONS OF AN ISOTROPIC ELASTIC For the special case i=1 this easily leads to tan (kK8 (a—a’)) _ 3 (kB)*aa’ +9 kB(a—a@) ~—s (kB)*a’'a* — 3 (kBY' (a? — 8aa’+ a7) +9 00 (57) If the shell be very thin the use of the relations * Ji+4 (kBa’) = Jin, (hBa) — (a — @') kB 45 (kB) ay LN (i+ >) -on(58) kBa' J’ j44(k8a’) = kBaJ' s+, (kBa) +(a—a') a )k*B? — (m+n)1 a ae de 7 Pr (Fe -Z) OT OGHOOHOORIO So SO ae etG (28). These equations are of course really the same as those given by Lamé. § 67. The surface conditions may be much simplified if the cylinder be a right circular cylinder with flat ends. On the curved surface X=1, and so a ay (Hg ae , dew F=(m+n)6—2n le era 7s) Be ess eee otis eet ine machine (24), ae ieee or eliedas ae ean | nogdnnoaseacndonaogonadsoUonudod opacusaabae (25), du dw See ee 26) ; H=n (5 7) ee Rte arse Ge ne ee (26); where F is the normal traction, G the tangential traction perpendicular to the axis, and H the tangential traction parallel to the axis. For the flat ends =i, LNEOS/H, F—n (¢ + ae ) =tangential traction along radius vector ...................... (27), dv 1 dw : ‘ : : G=n (a eT )=tangential traction perpendicular to radius vector .....(28), dz a lil= 6-2 eas othe i) =normal traction (29) =(m+n)o— nla. ‘ Gh ab Ses Nia Te OAT EN ee 29). In the case of an infinite right circular cylinder, i.e. the case where w is not con- sidered and the forces on the flat ends are neglected, we have to consider only the curved surface, over which now § 68. By means of the surface equations we can also determine the elastic forces or “stresses” at any point in the interior of the solid. For suppose an imaginary 42—2 326 Mr C. CHREE, ON THE EQUATIONS OF AN ISOTROPIC ELASTIC surface to pass through the point considered and to enclose a limited portion of the solid. Then our equations must apply to this limited solid, and our surface equations to its bounding surface. Thus by taking the normal to this imaginary surface at the point considered in any given direction we get the stresses exerted across a plane perpendicular to that direction. To indicate the nature of the stresses at the point call them Ian Ro, ive @,, Oz, ©., Z., Zo, Z.; R, being the normal stress on a plane perpendicular to the radius vector, R, ... tangential stress parallel to the radius on a plane perpendicular to 6, Ke. Taking then X=1 we get the surface perpendicular to the radius, and so R,=(m+n)6—2n C : = + = ey ee (32), ee r 1 du ‘ eae rae dé, Eyapeleiava'c eloie hel aaseintertleielaisieiatacie cisveieie stale cee a meee ete (33), Z,=n (+) Shino dead conte aulvc sieoeuie eee onntees sac Gee ee Gee eaeeeee (34) So taking p»=1, 7 ee - 1 du 1 = He Ai se | els (Oh ateivictarecaly min ia\einin close! «impel alninia}s nin incin|=in(acclotelnfadstelaiannts (35), Op, = (m aa n) 8 _ an (= oa 2) ecw eee eee eee e resets eee canto eaee see (36), dv 1 dw fan (+e S| cot eiteelea sae ce Re (37). So taking v=1, du dw ae es (z + =) AS VE a 9 OY fe re (38), dv 1 dw @,=n(E+, a) ae Core! iat ate ao serio ile bale (39), Z,=(m-+n)8—2n (See +2 _) Riinw, Se ee (40), Whence R,=0,, R,=Z,, @,=Z,, which are the well-known relations between the tangential stresses. Hereafter we shall denote R, by R, @, by ©, and Z by Z simply. No confusion with the bodily forces will ensue, because these will be expressed in terms of a potential. SOLID IN POLAR AND CYLINDRICAL CO-ORDINATES. 327 § 69. We shall now consider the tendency to rupture in a cylinder in which the displacement w is zero. Regarding the radius at any point as axis of z, a parallel to the axis of the cylinder as axis of z, and the perpendicular to these two directions as axis of y, the equation to the stress-quadric is Tae AD Bgaenpiats Oy et ize Mer 3 eee e sg acr akan esas ees oie (41). When transformed to its principal axes, the equation will become g ERT ONS gi 97 aN Loa eee oe RT ee (41 a), where #,, ©,, Z are the principal stresses. By solid geometry 2R, =R+0+S, and 20,=R+0-S, where (SN (C=C) PoRL de. Saree meee eee (42) taken positively, so that R, is the algebraically greatest stress in the cross section. If e, and f, denote the principal strains in the cross section, we have R, =(m —n) 6 + 2ne, ©, =(m—n) 6+ 2nf, f BGA Selce iapictela Ben ARecUS aneteaoO Nee (43), Z=(m—n)6 } e, being algebraically not less than /,, and ¢,+f,=6. We see that the greatest strain is e,, if positive, otherwise it is zero. R,-—m—nd 6 § Also = on Qt gy ees (44), since R+0=(R, + O,) =2mé, and ov. R= ms + e a etea Tener acta --2 i} r Of , dz (1) The surface conditions are, see (24), (26), (27) and (29), Section 1x., d ; on the curved surface (m—n) 6+2n = = Normal) brachion.ce scence sseeeee eee eee (2), du d i : 4 and n (Fe + ic] = tangential traction parallel to axis.... .................(8)} dz) dr : (du dw : : ; on the flat ends n {a d Ta diale bane enulal Urachlonr ss. esti seer es nee (4), \dz = dw : S and (m—n)6+2n aS normal traction......... std ated-op, cae Ng OR RDC RCE OC EE (5). If the bodily forces have a potential V, the internal equations are, see (21) and (23), Section IX., supposing V independent of 0, 4, ds dv ; (m+n)7 ae 7 aie Oa aR (6), (m +n) pon _ or sue ty ans A particular s S ea 3 pie (OF onan ch ondonbsouusenodoniabso ode sanceo. \ particular solution is w on B ane (17), and the general solution may be found by trial to be w= a2z+B, (9 — 2z*) + €, (22° — 327")... eeeee tet neeteeeees (18) ; -. the complete solution is Myo ™m : ; i ‘ w=— 5 Re ee +a2+ 8, (7? — 22") +, (22° — zr") ..... cece (19). We have certain relations between the constants in 6, w and w in order that dw 1d. ur dz r ar = This gives in fact a,—~ Be—— Cs — 48.2 + 66, (2 7) 42 or 4 204 B(Gr—) 4 22 ae +e (= - 6zr) +n (82° — 47°) =A+Bz+C (2 - 5) identically...... (20); whence a,+2a=A, y= 48, = 257 m+n matlaechert Y ah lid toa eee eee (21) 6} 0 These relations leave us six independent constants. Select A, B,C, a, B,, and ». Then a,= A —2a, | B/m+n =9 ( i )+26,, haere 2 ee (22), C(m+n)_, te Ghia — 8% and the complete solution is 7 ie b=A+Bz+0(2-5) ives oe A ee eee (23), . 5 m+n ) ™ 2 3 u=ar+\5 +28 re Po Oe ae 9) SE a de tt PP, eine re OE (24), n+ n m >, a Orhan! w= (A —2a)z— 5 Bz* + B, (7? — 22’) — an C2 + (22* — 3zr*) ; a dradtnene | (2s SOLID IN POLAR AND CYLINDRICAL CO-ORDINATES. 331 Substituting these values in the surface conditions (2), (3), (4) and (5) we get, if a be the radius of the cylinder, (m=ny{a + Bz +0(2 - 5h {2 +2(5 "4 Mtn = normal traction on curved surface +2,) + at oe 24 9 (42° — 3a? I ie ae (26), n {a (57 Meta + 28 :) + 8nza + 2B.a— za (== - Sn) } = tangential traction parallel to axis of cylinder on curved surface......... (27) r m m OS 3 ~ = 8 (m—n) {4+ Bz +0(2- 5) + +2n)A —2a— ms 4B 2 — = C2? + (22? — r’) 5 = normal traction on flat end......... (28) n |r (5 aie + 28,) + 8n2r + 26,r — zr io - 81)| = tangential traction in a radial direction on flat end......... (29) In the two last equations we substitute for z the values it has at the two ends of the cylinder. § 71. Let us first consider the terms of the lowest powers, viz O sep Aenea a cmon pire antrlraas: (30), B=CIP pocosooscnspacoq909n0np0pcC (31), NSLS csoostonsoascaar (32) These give (m—n) A + 2na = normal traction on the curved surface = — p, say............ (33) and (m—n)A+2n(A—2z)= normal traction on the flat ends = == Pais geiastateist ote Matsa es (34). This gives the case of a cylinder under the action of different, or the same, uniform tractions or pressures on its curved and flat boundaries. From (33) and (34) we get _ Pp, (m+n) —p,(m—n) 2n (8m — 1) i p, (m — n) — pm page n (8m — 2) : Pe (m+ n)—p,(m—n) } ppacnee _ 2n (3m — n) ; (35) seeeia sisslaseac oiaseeroealen aaentlae 5). iene 2) (m—n)— p,m | oF n(3m—n) ~ J Several useful applications of this solution present themselves Vor -—_ 2n (3m —n)’ surface of a cylinder whose ends are free. rule Gascglicn al ae ana ie (Bm—n) 3 Finally if we wish to produce no radial displacement, we get for the ratio of the pressures on the curved and flat boundaries i TEE) A ARE asp ne ee (41); ia TS alae ae aca ; while if we wish no longitudinal extension we must have Py gE I eke ee (42) Pp, m—n These results are all, it will be observed, independent of the length or section of the cylinder. SOLID IN POLAR AND CYLINDRICAL CO-ORDINATES. 333 Similar problems are considered by Prof. Clerk-Maxwell in the Transactions of the Royal Society of Edinburgh for 1850, but the method employed is entirely different from the preceding. § 75. Let us consider the solution given by (28), (24) and (25), more specially in the case when the surface forces are symmetrical with respect to the plane z=0, which we shall suppose to bisect the axis of the cylinder. Then from symmetry uw must be the same at corresponding points on opposite sides of the plane z=0, while at two such points the values of w must be opposite in sign while equal in magnitude. It follows at once from (19) that B and @, must both vanish. Suppose the length of the cylinder to be 2/, then we get from the surface con- ditions (26), (27), (28), and (29), after a little reduction, (m—n) A +2na+ a Ca? — 6nya® + 2° (Cm — n+ 8nn) =normal traction on curved surface ............... (26,), za {16nn—C (m+ n)} = tangential traction on curved surface............ (2a): (m+n) A — 4na+ CP? (m+n) — 16n7l? + 7° {sun - ole c| —MOrmMal inaction: On! Hat) eNndsijse.ss-eeccene cose: (28,), lr {16nn —C (m+n)! = tangential traction on flat ends ............... (29,). We can get rid of the tangential surface forces by taking SECC OUR SCOR CROC EE: Maoncicncsoeenmbcnnd (43) 16n eee this leaves on the curved surface a normal traction -— = M—N >» Ea | TepAL 2 nage 5 Je | AG | and on the flat ends a normal traction (m+n) A —4na— mC?* ) We can get rid of the constant terms by suitably determining A and «@ in terms of C; but we can get rid of neither the traction «z* on the curved surface, nor the traction xr? on the flat ends unless we put C=0 when both vanish. C=0 returns to the case of uniform surface pressures already considered. This solution would apply to the case of pressures or tractions on the flat ends proportional to 7°, and gives that law of force which applied to the curved surface would maintain equilibrium. By means of the two constants A and a we can make the surface forces on the flat ends have a zero resultant, or vanish at any assigned distance from the centre, and the forces on the curved surface vanish over any transverse section at a given distance 2 from the centre of the axis. 43—2 334 Mr C. CHREE, ON THE EQUATIONS OF AN ISOTROPIC ELASTIC § 76. This solution may be applied to the case of a cylinder rotating with angular velocity about its axis. We may regard the cylinder as at rest under the action of a 2,,2 *p » ¢ . or 2 “centrifugal” force of potential V= a whence, from (9), = ee 5 j= TT QT ee rteseaeee norte caeetinaens (45) is the additional term to be added to 6 in (23), putting B=0. This is obviously a case of symmetry about the central transverse section of the cylinder, so equations (23), (24), (25), with B=0=8,, apply. A particular solution of (6) and (7), which also satisfies (1), is w'pr° Uu ~~ 8 m+n) fexu nyo\ujojaieuj=/e/=ainivle ainfeinvaya(n/ajalo\u (eis atadsiainjatnialsialein (46), w = 0, We may use the surface conditions (26,)—(29,) with the addition of p = _ 3n wpa” rea (- m+n 2 4 m+n’ 2 29 5 @pa zn +n . Le — “— aca? to the normal traction on the curved surface ; m nr m—n w'pr® and m+n 2 to the normal traction on the flat ends. We can thus solve the problem of a rotating cylinder whose surface is exposed to no forces save normal pressures or tractions on the curved surface. (m+n) To do so, from (27,) and (29,), 7=C ién? see (43). From (28,), noticing the addi- Wi — 1 pr tional term — cere ae) Tah get m—nwp Bm +N eae Sn - 2 OC SO wae, ne Be eee (47); A m—n m—n C=-—- =——_. op; =- BO eter signe peace’ 48). 2m (m + n) al a 32mn (48) Also from the constant term in the same equation we have (=F iT) PAA =O. recs sete Ae tres nenenmteneamitacade (49). This leaves on the curved surface a normal pressure 2 Et ly ae anes 1 ee (50). Sai 16m 4m (m+n) id SOLID IN POLAR AND CYLINDRICAL CO-ORDINATES. 335 We can get rid of the constant term by taking UUW ane ene = Bin Gas DHfAGr secoqsoacnbadddoobbebptosnes ooqaedee (il) re or we can make the pressure vanish at the rim of the cylinder by taking (7m—n)a m—n,,) w’p = A= {i Cap ee L Pe (52). This solution would suit the case of a cylinder rotating with its surface in contact with a smooth solid, in a cylindrical groove of which the motion is taking place, or the free rotation of a plate the square of whose thickness is negligible. We can also get a solution for a rotating cylinder with only tangential surface forces, by taking m— x n=—C = Sorsecddo ch -aacoosenOatEee cen aoneancener (53), from the coefficient of 2° on the curved surface ; call - Wir Mah a e — 8nn + C+ Ue CAT Glade SEARCH RA aa Rea (54), from the coefficient of r? on the flat Ame. _ —(m—n)o'p (m—ny op La (m+n) (5m—n)’? 7” 8n(m+ 1) (OT — 1) aan i 2); and suitably determining A and «a. This leaves the following tangential stresses ree on the curved surface parallel to the fo Cae) iebiee Ws Bis ei diste A ie es Ler enter onl (56). (8m — n) (m — n) on the flat ends radially ere (m +n) (5m —n) J A third case occurs when no forces exist on the curved surface, and only normal forces on the flat ends, and comes at once. Put (OPE XV Sia ase baodecaando meee eaGan hace tev saco te nenee soc (57), (m—n) A +2nz—w'p “ae = Oy ina semnke tus doled temas ees (58). This leaves on the flat ends a normal pressure 2m +n wpa’ mtn 2 ee m+n = (Bin = 1) Al ea hive evncsiccios concusisnenien (59). This would suit a ae with its ends pressed against two yielding smooth surfaces. er 3@*pa°m a If rim just free, = 5 CRC ae i (60); Se wpar(7m—n) __ wpr ) 8 (m+n) (8m — n) 8 (m+n) ’ | (61) a ee) | aoe ae ae ee 4(m +n) (3m—n) J 336 Mr 0. CHREE, ON THE EQUATIONS OF AN ISOTROPIC ELASTIC The normal traction on the flat ends is, substituting for A in (59), wp (a® — 7°) (m —n) 2 (m+n) Or we might secure that no alteration would take place in the length of the cylinder by taking Pea) al | aes Aaa Ae ae ee RPL eee eae, As (63); 2 2 58 syst penis whence from (58), ae is Peery. 2, 2M+n 5 ; and so iE ee in {a Pr rt Te a MT res cake stata (G4), w=0; and the normal traction on the flat ends _@p(m—n){ .2m+n , ns = 4 (m+n) \a a —2r'| alatatpie aie cvelal tants e[u'atalalatefs/aiaratole sie atn( sar aeterae (65). This last hypothesis is really the same as that of an infinite cylinder and is con- sidered fully presently. § 77. We pass to the case of an infinite right circular cylinder rotating with uniform angular velocity about its axis. By the term infinite we mean that the surface conditions at the flat ends need not be taken into account. We shall first con- sider the more general case of a shell the radii of whose bounding surfaces are a and a’. By putting a’ =0, we get results applicable to a solid cylinder. We have and’ (m + n) = St en rt = OFS esa See age tals eee (67). Substituting for 6 in (67) from (66), we get (m+n) = : 4 ne aging) x 2.5 (68). By straightforward integration this leads to 2 8 PPE +4r+2 Sidhe oe ee (69), ~~ 8 (m+n) where A and B are arbitrary constants, the latter however vanishing in the case of a solid cylinder. From (69) we find opr ge 2(m+n) + 2A ee ee ee rd (70). SOLID IN POLAR AND OYLINDRICAL CO-ORDINATES. 337 To determine A and B, if we suppose the surface free of forces, we must apply at both the boundaries r=a and r=a’ the surface equation (m+n) 3 —2n==0 eth eee a ce (71). Z < 2 2 2. This gives invA = one eran) i 4 (m -+ n) 2, 2/9 _, wpa? (2n+n IMA — 2nBa’ = a'pa” (2m +n) : 4A(m+n) ? whence A= ap = ae 11)" a eee eT et ee (72), Be a ee ek, TA) chk rads ae (73). Thus S=5 oe = pt Fie +6) — 2r*} He ioduversdl cada. 8 (74), “= ee ee r(a@+a*)+ ae = -r| deems zhi (aid) w= 0. The principal stresses act along the directions of u, v and w at every point, and so are R, ©, and Z, and their values are respectively du dr (m—n)d+2n—, (m—n)d+2n , and (m—n)6. Since r is intermediate in value between a and a’ it is obvious from (74) and (75) that 6 and = are always positive. Thus, supposing m greater than n, © and Z are certainly positive, or are tractions ; : ‘ d and © is greater than Z The sign of ir depends on the value of r and on the ratio m:n. It is easily seen however that au is always algebraically less than , thus © is dr always the greatest stress, and = the greatest strain. The absolutely greatest value of = which in accordance with Saint-Venant’s theory measures the tendency to rupture, occurs at the inner surface r =a’ and is 2 — Gk (CATAL MARTHA) bacasssecnnansedooooooconoubusoUSOCE (76). 338 Mr C. CHREE, ON THE EQUATIONS OF AN ISOTROPIC ELASTIC It is unnecessary to determine the regions within which Z or # is the greater if we merely wish to find the greatest value of the maximum stress-difference, which in accord- ance with the stress-difference theory measures the tendency to rupture. It is in fact easily , u F ; seen that the greatest value of 0 —Z, which equals 2n occurs when r= a’ and is w’pn ( , 2mt+n 2) 4 ta \@ = +a f BRS concn od Sabbath icie O's SERRE GAR while the greatest value of @-—R, which equals 2n (:-3), also occurs when r=a’ and is 2 ita 28) fat 2ia EM) RE") ose san techs eon se Peoreeon (ie) Now (78) bears to (77) the ratio 2m: m+n, and thus is the greater, and so in accordance with the second theory measures the tendency to rupture. On either hypothesis the tendency to rupture is greatest at the inner surface of the cylinder. For a very thin shell (76) and (78) are approximately equal to ow pa” (m + n) 4mn and @ Pirie. ssckheadgis cues Coats Boeeierteee ce anaes (80), respectively. If on the other hand a’ be very small, or the cylinder solid nearly to its centre, (76) and (78) become approximately wpa’ (2m + n) Bea ee etacsoneegtestencascaseosernenzessnenee (81), a wpa? (2m + n) 82) an ~ Sens en ee (82), respectively. For a solid cylinder we get from (74) and (75) 2 ep 2n+n 4 r 6 = ae a me 4 —2) } sodtdaarcucisene wie eerste Roe eEEe (83), — opr “(2mm ae os “= ma = art slip bidekinlece ae cooaaentioncee ne areeeee (84), du_ —§s wp) =(2m+n , ; £ and so Ae VCE FR ACC a — ar} albspiceridewtnwe es cleanses cee (85). Here, as for a hollow cylinder, 6 and uw are always positive; while © and Z are always tractions, of which the former is the greater. Inside the coaxial cylinder r=a 2m +n Ba ACen renee ope saccnc (86), SOLID IN POLAR AND CYLINDRICAL CO-ORDINATES. 339 ub. es : ; : dr 18 Positive, and so R greater than Z; outside this surface the reverse is the case. In both cases it is obvious that : is algebraically greater than _ thus throughout the whole cylinder © is the greatest stress. Inside the surface (86) the maximum stress- difference is Sp a C) ee erect eee (87), whose greatest value occurs at the axis, and is o pan (2m +n) ~ 4a (m+n) eee eee eee eee (88). Outside the surface (86) the maximum stress-difference is @— R=2n ( EES eee (89), which has its greatest value, viz. w pan Weaaeee ea (90), at the surface. This is, however, less than (88), which in accordance with the stress- difference theory will thus measure the tendency to rupture in a solid cylinder. The greatest strain is obviously ~, which is always greater than ie except at the di axis, where they are equal. It is here, however, that it has its greatest value, viz. ow pa’ (2m +n) 8m (m+n) In accordance with the greatest strain theory (91) is thus the measure of the tendency to rupture. It should be noticed that on either theory the tendency to rupture is greatest in the axis of the cylinder. Comparing (78), (80), (82) and (88), or again (76), (79), (81) and (91), we see that the tendency to rupture in a solid cylinder is very much less than in a hollow cylinder of the same external diameter however small be the diameter of the core removed. It is also noticeable that the absolute amount bored out of the solid cylinder is of comparatively small importance. The absolute ratios of the tendencies to rupture in a solid cylinder, in a cylinder from which an extremely small core has been removed, and in a cylinder of extreme thinness, all of the same external diameter, are from the theory of maximum stress-difference n: 2m: Bi i) 5 2m+n and from the theory of maximum strain Gsm) n:m+n: (m+n) Seen Vor S1V. Parr II. 44 340 Mr C. CHREE, ON THE EQUATIONS OF AN ISOTROPIC ELASTIC SECTION XI. Equilibrium of inginite cylinder or cylindrical shell. § 78. Let us now consider the general case in two dimensions of an infinite cylinder at rest acted on by a series of bodily forces which have a potential V,—satisfying V?V =0, and which may be expanded in a series of sines and cosines of O6—, and by a series of surface forces expressible in a series of sines and cosines of 0. We then have, see (7) Section IXx., duu, ildv_1/d.ur , dv’ b= tte aur arta) Mogan She Rios. See (1). The surface conditions are, see (24) and (25) Section Ix., (m—n)§ + 2n 2 ==ormal fOrcer socessseerssessecensenseeeee (2), a2 n (, = + = a) =tangential fOLCE) ice scece- ern seene ee senses (8). The internal equations are, see (21) and (22) Section Ix., dé d€ dV (m+n) ro —n ag =— pra one osvs TE SaassaNEcoaDoDoonEccA A (4), m+ndé d&_ pdvV pi = jo: ™ ae = =e PT ae als (5); l/d.vr du where C= = (c - a) STi A A ee (6). Differentiate (4) with respect to r and (5) with respect to 0; adding we eliminate € and get (Ria FENG Oo — BF wera sat nats vais wanteselte sel eee Gin: th 2 — g 1d i cs ai ok Vr dP rd Pde Take as a particular solution ae Ui aa ae « Hie «alan ch game: AnvseTaD ate (8) Let V be expanded in a series satisfying y’V=0, viz. —V = cos i0 (art + a'r“) + & sin 10 (Brit B/7*) cccccssenereeeeeeeeees (9), where i is a positive integer and a, a/, 8, 8, constants. The case i1=0 is considered specially afterwards. SOLID IN POLAR AND CYLINDRICAL CO-ORDINATES. 341 The general solution of (7) is 8= 2 cos 10 (Ag + Are) + sin 20 (Birt Bir) oes ennsownenesenns (10), where A,, B,, A;, B/ are arbitrary constants to be determined by the surface conditions. If the cylinder be solid A; =0=B,. It is unnecessary to retain the series for V or 6 in sin7@ in the work, as the corre- sponding series for w and v can be at once deduced from the parts of their values depend- ing on the series in cosi@ in V and 6. So leave them out at present and take = Vy COS EO OES tiers ds acaistip saisicn'sca cnc Re daen eS (11), A : pa; =i pe, 8 = 3 003 10 {1 (4 Th, a) te (4/ +) See oe ae see ae (12). Substituting in (4) the value of @ given by (6), we get ldu 1d ad m+n aoe ae ape Preostd,..6 (Art — A eh leads, Sites. eed (18). , du 1d.ur Substitute Apa : me mt) , and for 6 the value (12) and we get, multiplying by 1, " dr m+n ear as {pac - jij est =| pti acts: (14). m+n n ee fu + Tem 5 008 | font?) -A me Let wu, denote the coefficient of r* in w; then, equating coefficients of r‘! on the two sides of (14), we get du, ag t +2" U; = eos i | pa; (¢ + 2) _ 4 mi—2n mtn rn The complete solution is y= 00829 [om @ + 2) mi —2n : “a+ mt m+n 4," +C,,. 08 (t+ 2) 6, where C,,, is an arbitrary constant. Again, let w, denote the coefficient of r“*t in wu. Then similarly the complete solution is eT ae cosi@ (pa/(t—2) , ,mi+2n U; => C’_, cos (4 2) Oia (ea PA a ais\eanleecelelacislaleenia (16), where C’,, is an arbitrary constant. ie 342 Mr C. CHREE, ON THE EQUATIONS OF AN ISOTROPIC ELASTIC Thus that part of uw which contains cos 7@ is — con io | tt fom +2) yg mi= 2M) ges u=cos v0 lzarn{ rar A, n + C7 iH "G9 mi+2 a ae ‘eee Me ak He “| HOS hil iu. Gl where C,, C;,' are arbitrary constants. The corresponding terms in v come from (1), which gives, putting together the co- efficients and integrating, in sly mG) 2 v=sin 6 | A, Lid ast Marte SW 2 | — Cr 4(¢+1) n m+n Tis »m(i—2)—2n — zpa/’ = ~ 4-1) n m+ A = gral +f(r) ...(18), where f(r) is a function of r only, which we shall here neglect. The expressions for wu and vy depending on those terms in 6 and V which contain sin 7 can be at once deduced from (17) and (18). Thus in (17) write sin 70 for cos 70, B, for A,, B/ for Aj, 8, for a, B/ for a7, D, for C, and D, for Cj, where D, and D, are two new arbitrary constants, and we get the new part of u. While in (18) write —cosi@ for sini@, B, for A, etc, and we get the new part of v. Calling (17) and (18) with these additions (17,) and (18,) respectively, then (17,) is the complete value of u, and (18,) of v depending on cos 7@ and sin 20. § 79. To determine the constants we must have recourse to the surface conditions (2) and (3). So far as depends on the cosine series in 6 and V, and clearly the cosine and sine series must separately satisfy the surface conditions, we find for the value of du | VO Oe (m—n) 6+ 2n 77 a series in cosines, which after reduction gives du : m (t — 2) 2m+ ni ps, 2 ee at 2 u (m n) 8 + 2n 7 = % cos 10 [ { 3 A,+ 3 res i 2, -(m(i+2),,, 2m—m pa, i-2 t i +2n(i-1) Cri? +r } 3 Aj+ 2 or Similarly, po-¥ Veh es | aa licg (Uk ni pa, : eee (Te 2a = S sina | {5 A, 5 Pe} an) sn F we , = | —2n(i +1) one] ar (20). The complete value of (19), depending on both cosine and sine series for 6 and V, is got by adding to (19) another series in sines, the term in sini@ being got from that in cos7@ in 19) by writing B, for A,, B/ for A,’ ete. SOLID IN POLAR AND CYLINDRICAL CO-ORDINATES. 3438 Similarly for the additional term in (20) write —cos7@ for sini6, B, for A,, B/ A; ete. Call these complete values (19,) and (20,). Then we have (192),=a = Normal force on outer surface, (19,),=a = Normal force on inner surface, (20,);=a = Tangential force on outer surface, (20,),=a = Tangential force on inner surface. Since at the surface the cosine and sine terms must be identically satisfied separately, we can consider separately the terms depending on the cosine and sine series in 8 and V. Thus suppose those parts of the surface forces depending on cosi@ and sinié to be S, cos7@ + V, sin 7@ normally on outer surface, S/cosi0+ V, sinz@ normally on inner surface, T, sini — W, cos7@ tangentially on outer surface, T/sini@— W, cosi@ tangentially on inner surface. Then A,, Aj, C, and C/ are determined in terms of a, a@;/, S,, S/, 7, and Is while B,, Bj, D, and D; are determined in terms of £,, 8,, V5 10, Wand We. Also the value of B; can be deduced from that of A, by writing 8, for a,, @/ for a/. V, for S,, V/ for S/, W, for 7,, and W, for 7/; and the same substitution deduces Be é from Aj, D, from C,, and Dj from C{. We need thus consider only one set of surface conditions, say that depending on S,, 7, S, and 77. Doing so we get _™m ee 2) A ab ae nt ee : gh on Gea) Cans m me 2s) are . a oe - CAM (Git a ON it Sg csensne syste (21), Ln -aS= Ag+ a Pag! + Bn (i = 1) Cal? + a A/a’ ee =P a =P s> 1) Che PP Ii sncoccuoncont & (22), a Aai— ie ae a’ — 2n (¢—1) C,a*? + = A/a‘ — fe One 1 Oat? aT os caste. (23), mt Aa = a’ —2n (i -1) Ca’? + = Atal — a tome 1) Cra ee se (24), S44 Mr C. CHREE, ON THE EQUATIONS OF AN ISOTROPIC ELASTIC From (21), (22), (23), and (24) we determine the four unknowns A,, 4,’, C,, C/ in terms of the knowns S,, S/, 7,, Tj, @, @/. The values are determinate in every case except when i=1, which makes the co- efficients of C, throughout zero. Thus if the values were expressed in determinants one whole column of the deno- minator would vanish when 7=1. To avoid a bodily displacement we must put C,=0, and to ensure statical equi- librium we find two relations (one in a solid cylinder) between S,, S/, 7, 7y, a, and a,’, In a solid cylinder Aj=C/=a/=0, and we get two equations (21) and (23) to determine A; and C, in terms of S,, 7, and a, We get at once in that case ah S,+ i -i p% rR A. = +i m+n © 1c e/0lue ejeicie vvicieieie'seeelinecs cin» uawes (25), _ 8, + (i — 2) T, i442 piaa® ‘ 0, gt OR ccstntntnes wnt (26) ; whence typical terms in the solution for a solid cylinder are =the “al ot ipa, “mt — An Se ea a 98, G2) i= cos i6| yf i } Gam lip See Gee pee 2) 2 mn a i+1 ae r 2 v= sin 10 Fea {- (i+ 2) pa,+ a a (, 846-27 + pany fina, —St eat i (28). meee m i Supposing = negligible, then the parts of w, and also of v, which depend on pa, and on = are absolutely identical. This exactly corresponds to the result we ob- tained for the case of a sphere in § 13, and seems rather curious. In words, this signifies that the displacements at any point of an infinite solid cylinder due to a surface pressure —S,cosi@ and to a bodily force whose potential per unit volume is *8S, () cos 10 are, if = be negligible, absolutely identical. §80. The general equations (21), (22), (23) and (24), though slightly long, are very convenient for the purpose of eliminating the various constants. Proceed as follows. Multiply the sum of (21) and (23) by a, and from the result subtract the sum of (22) and (24) multiplied by a’‘*; we thus eliminate C, and C/, the result being mA, (a — a’) +m (i +1) A/ (a? — a) = (8, + T, a? — (8) + T,) a’ _ PINE, 7 ata _ aie _ mn Li Nes ne meng Qa) EE pa, (a*— a”)...(29). * Notice that potential V= —Zcosi6 (ay), &. SOLID IN POLAR AND CYLINDRICAL CO-ORDINATES. 345 Again, subtract (21) from (23) and multiply the result by a’, and from this subtract the result obtained by subtracting (22) from (24) each multiplied by a’. We thus get m G— 1) A: (a? ca! a?) —mA{ (ie eee) = (ey aes S,) qi? Cy 83 8’) qi? LOL SE a ra Tag ee Oe Bgmararh Oe ene ces (30). (29) and (380) give A, and A; at once. Then (21) added to (23) gives C/ in terms of A,, Aj, and known quantities. Similarly (21) subtracted from (23) gives C,. Thus 1 Ie -i-2 i o Pra} pma; 4nO; (i+ 1) a? =mA,a+m(i+1) A/at—(S,+ 7) + = _ mam al at Ta pn POE attteceetetecnecesrncceennnee vessescesscceennee (31), ; i A ear eee gree on _ py _m+m : 4nC;, (i — 1) a**? =m (t —1) Ajai — mAja*+(S,— 7,) Fe — pe, Sy ars ne are (32). After eliminating A, from (29) and (30) and simplifying, we get ma [Pte ote (2) (EI wena [540 (2) =a(2 apn e408)" e018) a] serena foan(l)-i(S)"fenfeensy'—een)-f Es Pe - [m {i (a — a”)? + aa” ( = ay} + ni (i +1) (a — a] Baye POR when sent - wai [steer ow (8) nova [5f(8)"—0-m(8) —1}— nfm (8)-6-2 8)" a] recnra[afe($) 6-0 )-1}—mfo-n(8)- eon YH] — ipa, (a? ma a”) (a? = ane) ce ay = ni (i —1) (a - a") | Sere: (34). Similarly Ee [m {i (a? — a”)? — aa” ig a 346 Mr C. CHREE, ON THE EQUATIONS OF AN ISOTROPIC ELASTIC n=l Since a=1 is a root of (¢+1) forts ee —1=0 for all values of 7 and of x, it is obvious that a*—a” divides both sides of both (33) and (34). This gives Aq’ and EE = a 0° when a=a. If < be very nearly unity, ie. if the shell be very thin, putting a =a(l—e) and neglecting ¢ when terms in e exist, the resulting values of A; and A, are much simplified. We thus get, after some simple reduction, CAS SA el Ue) da 3p a; a” a,’ . Ne eS ae (; + *) op bet ee (35). Similarly ’ 3 (Li- T/) —i(S,-S, ies 3p at me aka, ) es r(-4- Se ) wikia tote see (36) It is quite simple to get OC, and C/ now from (31) and (82). If e« be very small the parts of A,, Aj, C, and C/ which depend on the surface- forces are much more important than those depending on the bodily forces. The same remark of course applies to the displacements and to the stresses. Substituting the values obtained for A,, A;, C,, C; im (17) and (18) we obtain the general solution for a thin shell. § 81. Let us now consider specially the case of a solid uniform cylinder exposed only to surface forces, viz. a normal traction S, cos 10 + U, sin 70, and a tangential traction T, sin 10 — W, cos 70. Then from (27) and (28), paying attention to the remarks already made as to how to get the terms depending on U, and W, from those depending on S, and T,, we get i _ acos 16 (iS, + (¢— 2) B, (1 _ mi a 4n | a—1 (Ey ean +2) (= ) asin 70 (i1U,+ (¢—2) W,/r\1 mi—2n 2 + oa ; wo (=) Sem ( if ones oe (37), _ asin 18 m (i +2) + 2n my (7. Se ta) LES a; 4n | m (t+ 1) (8,47) (*) i-1 (=) acosi@ (tU,+(i—2) W,/r\*1 m (i+ 2) + 2n r a 7 ie ees 2) ~ am (i+ 1) Cae & pana (38), m 6 = cos 70 sant (“) +8 n71@ ——— ee (2) ind on diewean Veedmtneecaeos (39). SOLID IN POLAR AND CYLINDRICAL CO-ORDINATES. The elastic forces are, see (32), (36), (35), and (40) Section Ix., du = (m — n) 8 + 2n 7 ; du @= — 2n — (m+n) 6 2n 7. : ‘ v mean (read), 5 dr 'rdé Z=(m—n) 6. We thus get R=cos 10 Bas a = Ss (S,+ T,) (=) 6 + sin2 are @ = cos 10 57 (8,4 7) (Z) - EPEC) + sin 70 + (U, + W,) (ny) + (i—2) W, ee Fe cos'i0 ean (“) =. wy (5 “ + sini6 {i (S, + T;) (") ms = JES (e) weet AU... aR i ae (Ey Gosia (Se) Pains (OC: WY Sete mv It is easy to find the principal stresses from these values. The value of S or /(R— ©)?+48,’ is very simple. From above a [s+ G27) og -i(S +7? )] + sin i6 ke (i —2) W} (E)- ~i(U,+ W) ¢) | Hype tion Noticing the similarity of this to the value of 2R,, we get gt [fis +- 2) T} (=)~ eat te () | + [iu + 6 -2)w}(t) -#.+™) (:) | heyntinis. Vou. XIV. Parr Iii. UGE oe SE ye 348 Mr C. CHREE, ON THE EQUATIONS OF AN ISOTROPIC ELASTIC This is independent of @ which is very curious. S can vanish in the following cases: i=2, r=0, when S,=0=J,, } : oer ee [ocerereeteeeeereneeesaeees (46) ; Se , fw @=2) 7, l aes 0 — Wi and | => a(S, + tT) | oh rP_40,+ @-2)W, . S = 0= EF; and a = a(U, ni W,) ) SSponosochedvaoracdendconees (47). iS,+(¢—2)7, iU,+(i-2) W, ae (ty | oe Pease UW et keh e This last requires SS Ts i Soe strstr cinereus te seven tecragarsel CRORES (47 a) This condition includes the preceding cases S.=0=0,, U—0—W,, and= S,=0)— 7... S =0 implies that the stress quadric is one of revolution about a parallel to the axis of the cylinder, and also that there is no stress-difference in the transverse section. We must of course have a 1 in order that the solution may apply to our problem. Clearly this is the case in (47) if §S, and 7, have the same sign, and if U, and W, have the same sign, 7 being < 2. § 82. We shall consider more fully the two most interesting cases, (i) no tangential, (ii) no normal surface-forces. Case (i) i — 0 —= We v2 {. ei 5) ; y2 Tu se = = — a i Ay ear) fe = i (Z) (1-S)} wet uy) (48). Thus S=0 only when r=a, or when r=O and 71>2. The surface S*—(2né6)’=0 which separates the regions in which S is and is not the maximum stress-difference may be at once determined from (39) and (48). For our present purpose however it will be sufficient to determine the maximum values of S S x ? ; and of 5 +n6 and the points where these exist. If i1=2, S has its maximum value 2/8, noe dla 6bi0 oa piel eines ehofselaie'ssinieias ol\sielspin sis'oir/ate)¢ (49), at the axis. = S ; 2 The greatest value of 5 +76 also occurs at the axis, but is only half the above SOLID IN POLAR AND CYLINDRICAL OO-ORDINATES. 349 as 6=0 when r=0. By § 69, taking 6 positive we have aae for the greatest strain. Thus if 7=2 the tendency to rupture is certainly greatest at the axis on either theory. If 1>2, S has its maximum value 2 =o yz 2 * 2 ( ; 5 SE aL a heen MS Re Ae Re (50) over the surface of the cylinder r=ay/ GET T ET rt ereeeeeeeeeet ete cceeeees (51) The + or — being so taken as to make the term in 8 positive, oo nd has its maximum value poe oa 2 2 ( 3 SG ok. ne ace a (52)* m when tan 70 =+ = SEUEESISE 34 Dat aR cane aia aces eee area meee (53), ar Ben. are == ie, Sait ee BAIN ened oy (54). 4 eS Which of the two expressions (50) or (52) is the greater must be determined specially for each value of 7, greater than 2, and of =. In any given case the de- termination would be easy, and the correct measure of the tendency to rupture and the locus of its occurrence on the stress-difference theory would follow at once. On the theory of greatest strain, dividing (52) by 2n we have a measure of the tendency to rupture, cf. § 69; and the locus where the tendency is greatest is given by (53) and (54), taking in the former the + or — sign as is required to make 6 positive. § 83. Case (i) S,=0=U,, Thus S vanishes when ION fee a AS OEE ee RCE RSPR SS SoA (56). When r=0, R,, ©,, Z all vanish if 12; the same is true of the strains, thus there is never any tendency to rupture at the axis. When i= or >2, S has its greatest maximum value DR Waa temaen, awake. od, Jy. weeuedh leatens (57), when 7 =a; here also occurs, for Wi TA wr Maa pe oa DPA * Treating these as indeterminate forms we can deduce the results for the case i=2. tan 10 = + 45—2 350 Mr C. CHREE, ON THE EQUATIONS OF AN ISOTROPIC ELASTIC 1% the greatest value of > +8, which is ~ b (1 Se IS EE tere tae cart caseanennyerrt ray eed (59). The former is always the greater supposing m>n. Referring to § 69, we see that on either theory the tendency to rupture is greatest at the surface. If i>2, S has a second maximum value —2V 2 ( =) VETER ot Geek he en (60), 1—2 over the cylinder lel acer ee (61). A [he sign being taken so as to make 6 positive, 5 + nd has also a maximum value SM EEEE Want tet “22 54c ates eee (62), whens... ~~. «© ® 2 SOtsitaG ae i ciccanches cot acnoaerne eee (63), AT TRE ee PERE SEIT NS ig) ane oes wale ree nae AT On ele Ree Seas (64). On the theory of greatest strain (62) divided by 2n may be looked on as a second maximum of the measure of the tendency to rupture, the locus where this maximum occurs being given by (63) and (64), taking in the former the + or — sign so as to make 6 positive. It is obvious however that (57) is always greater than (60) or (62), and likewise (59) greater than (62), so these second maxima need not be further considered. To compare the effects of normal and of tangential forces in producing rupture, suppose S?+U?=T77+ W?. Then from (49) and (57) for 7=2, on the stress-difference theory, the tendencies to rupture are the same; while from the remarks after (49) and 59), on the greatest strain theory, it follows that the tangential forces are most apt to cause rupture. If7>2, it is easily seen that (57) is greater than (50) or (52), and (59) greater than (52). Thus on either theory if 7>2, tangential forces are more apt to rupture the cylinder than normal forces of the same magnitude. SOLID IN POLAR AND CYLINDRICAL CO-ORDINATES. SECTION XII. Vibrations of infinite solid cylinder or cylindrical shell,—including the case of a cylinder of finite length whose length is maintained constant. § 84. We shall first consider the case of a solid cylinder. : l/d.vur du If (ay = an — = a) sislosacalhierslaialetsislarsfoisla/aletstntete el inta otal ela (utsieielel sisla=tarsf« (i). ae 1 d. ur dv anc ae ( ae ar 2) BROODS BLOORE MaeHdEE Sic dosoorse BOUT Go056 (2); then, supposing there to be no bodily forces, the equations of motion are, in accordance with the results of Section IX., dé da du 3 Cat Mar at dé oad 7 (3), 1 dé dt d’ (m+) FT a Fon a A Mosaensissehleraites eaioe nase (4). Differentiate (3) with respect to r, and (4) with respect to 6; then adding, we get as 1dé 1 d’s ds . (m+n) = aie ara ? ae) so OF | ie (5). We may assume by Fourier’s theorem that 6 can be expanded in a series of sines Suppose then that the term in cos7@ in 6, when a and cosines of multiple angles. vibration of frequency & is taking place, is given by 6 = A, cos kt cos 16 ¢(r), where A, is an arbitrary constant, and 7 is an integer. Substituting this expression in (5) we find that @(r) must satisfy the equation d*p(r) , 1dd(r) kip _ 5] Ras : arene - + Ze — fife + (= amy Pe p(7 a 0 Mvalaia‘ete later sieve resels/afe/e's (erann\shelicajoXo (6). If we denote a2 hye m+n SS SRISC OT BIAS RR BE Ero BE OSM IET CORRE (7), ant Pp 2 and - by B J then the solution of (6), rejecting the second Bessel’s function as applicable only to a hollow cylinder, is Bi ee eee (8). Thus the typical solution is Oe Al oust Cas Odd (He rooocedanopecacecedoonssc. poonoaREsco56 (9). The corresponding term for the series in sines of multiple angles is similarly 6 = B,cos kt sin 10J, (kar). 352 Mr C. CHREE, ON THE EQUATIONS OF AN ISOTROPIC ELASTIC This at present we shall neglect, as it can easily be introduced subsequently when required. To determine u and v, we proceed exactly as in the case of equilibrium. Substitute in (3) the value of @ from (1), and substitute from (2) the result and finally use the value (9) for 6. Carrying out the necessary differentiations in due order, we find du 38du u lLdw pdu Nee (ares m da ’ a + = adr ss + 7 ae = dé = A, cos kt cos 20 ‘- Ji (kar) => Ri an J; (bar) Dasaiepataiatand (10), as the differential equation giving that part of w depending on the typical term in cos kt cos 76. Assume Ris, CA CORIGE) COS COP acces irisiaguctes soneh seer ue Nee ee ATOR (11), and we get du , 3du Ce a ae Be ea) m d ‘ a +u les = — "| = A,c0s kt cos 70 fF (bear = m & J, (kar)} alejsreeie (12), as the equation giving wu as a function of r. (12) may be written in the form @.ur 1d.ur ae Moe oe ; ie SOE a St ol : a eee = + (KB -+) uw = A, cos lt cos i |2J, (lar) =r 5 J, (kar)... (03) It is comparatively easy to get a particular solution of (13). For, using Bessel’s equation, it is easy to prove that r dr d* ld Pe d : d ae et; ant (#8 -5)| r dp Us (bar) = (8 —@) r 7 J (kar) — 2k’a*J, (kar) .. (14). apt ee ee But Bn a mp (oa ee eeeeteette essen (15); therefore the right-hand side of (14) is Ket ee) © = cos 2kt + Se, a, cos (hk, +k) t + Zo,o, cos (kh, — hj) t ..--- (49), where CN Try! he P20) yess eee das are Meee ie cemeccemecusenCe act: (50), and j is any integer other than 7, including zero when k, is supposed to vanish. The solution of (48) will be of the form U = Uy + Su, cos 2kt + Yu,,, cos (hk, + hk) t + Zu,_, cos (k,— k) t+ U cos kt...... (51); where u, is the solution of 358 Mr ©. CHREE, ON THE EQUATIONS OF AN ISOTROPIC ELASTIC u,cos 2k,t the particular solution of du 1ldu w , pr cos (2kt) pau 3 ees -— —— : _——= SAUCE, ss ctarclajniaravarelstaretelede (alone (era 3 F dr* thar a (m+ n) m+n dt* Pa ka) and Ucosxt the solution of du 1 du eee d*u d® rdr r m+n de’ which is suitable for a solid cylinder. In fact U is a type of the free vibrations. (or* J = — Ea 87; WY eR aT aco RAC erence Ener COOOL 54); We have Uy Sin) + A,r (54) AT. (ear) ticace. teehee nemetes «a5 Seen teetee nae eee eae (55), where « is given by alee) = J, Ra PL i nal (56), nm m where as usual tPF m+n To determine u,, we get @u, 1ldu, u, _ 4k2pu,, pr 3 —i+-—t-= ae | La an ase inn ohUnGan MORENO ct: 57). dr ordr r m+n 2(m+n) . ae) - at be or 5 a The solution is U,=— ora Seabed (CARCI) etes cotensader asoncsbosbooddannbdecése (58). We get similar results for the u,,, and w,, terms. Thus the complete solution is pr? — Ae “ 0 8 (m + 7) + SA cos («t) J, (Kar) sy b Mr RE LA BD AW ee + cos (2k;t) | Aol, (2k,ar) sit + cos (k,+k,) t {Aa (k, + kar) a + Scos (k,—&)t1A, J, (k,—k, ar) ——! “Fh ais Si. ase eae (59); : ; (k,— ky) where A,, A,, A,,,, A,, are constants to be determined by the surface-conditions. The only surface-condition is, see (24), Sect. Ix., (m+n)d— 2n~ =0, when r=a This must be satisfied independently by the expressions involving the different cosines, and by the term independent of the time. “ Opa? (2m + n) ' From the last we get oo “8m(m+n) wiahe oiwfulelatale. 67 ha'aia' dip 0/aainivisisjsie)b\« sib \ciaa' «Ie (61). The type Acos «tJ, (kar) of the terms derived from the free vibrations gives of course no term towards the surface force. As an example of the others consider fully 2 cos 2ht {Ad (2h,ar) — Bs r : SOLID IN POLAR AND CYLINDRICAL CO-ORDINATES. 359 7 - ' J, 2h,aa o, This gives (m+n) { 2h,aJd,' (2kaa) + 2k,a — oe ; A,- a | 2h 2 —2n [Pind —— = Sis} =(); Ca I. therefore by an elementary property of consecutive Bessel’s functions, @,m 6 A,= PGC Cna eo (62). Similarly we should find Aw= = ote Behe ike Geren e RCL NY (kh, + k,)® a (mJ, (k, + kaa) — nd, (k, + kj2a)} A,,= soem mesh Taye’ Auer, GH), (k,—k,)° a (mJ, (k, — kaa) — nJ, (k, — kaa} Thus the complete solution is Op ar (2m +n) “= 8 (m+n) m 7 at ee) 5 098 (2k,t) @, mJ, (2kar) ie (2k)? 2 |kaz\mJ, (2k,aa)—nJ, (2kga)} yy 008 (k, + k;) t Bi | : 2mdJ, (k, + kar) eee | (k, + ky) (k, +h) a (mJ, (ke, + kaa) — nJ, (k, + k,aa)} =: cos (k, — k;)t =~ QmJ., (k, — ka) @O.@ G=ky *” Ee —k,)a{mJ, (k, — kaa) - nJ., (ke, — k,aa)} Z "| i If any of the denominators vanish the corresponding vibration becomes infinite. This happens if any one of the following series of equations is satisfied, viz. : mJ, (2k,aa) —nJ, (2k,aa) = 9, mJ, (k, + kaa) — nd, (h; 1 k,aa) = 0, mJ, (k, —k, aa) —nJ, (k,—k,aa) =0. But the equation determining the frequency of the free normal vibrations is mJ, (kaa) — mJ, («2a) = 0. If then «=2k,, or =k,+h, or =k,—k jp? and the consequent stresses and stress-difference also become infinite. the corresponding vibration becomes infinite ; Now we included 0 as one of the values of j, answering to the terms 2, @, cos kt, ete. in @”, and k, then stands for zero. Thus our last result signifies finally that if «=/, the vibration becomes infinite; and the conclusions we have come to lead to the following law. In a rotating cylinder if the angular velocity be analysed into its component har- monics, and if the frequency of one of those harmonics or double its frequency, or if the sum or the difference of the frequencies of two of the harmonics coincide with the frequency of one of the radial vibrations natural to the cylinder, then the corresponding 360 Mr C. CHREF, ON THE EQUATIONS OF AN ISOTROPIC ELASTIC radial vibration will become infinite, and the tendency to rupture will also become in- finte. A near approach to coincidence will similarly cause a large radial vibration and a great tendency to rupture. Che forced vibration of frequency on k, is proportional to @,, while those of the trequencies : 2k : (k,+k,) and 7G. —k,), where j is now other than zero, are Se See se. Bae Ne ed oT J : ‘ quite independent of @,. Thus when the cylinder is being very rapidly rotated the first is in general much the most important. These results seem not unlikely to throw light on the dangers attending the working of some kinds of machinery. SECTION XIII. Equilibrium of a finite cylinder under purely surface forces. $91. In now proceeding to consider the equilibrium of a right circular cylinder finite in all dimensions, let us as a preliminary briefly regard the differential equation d°y l1dy a OA Sa = = is || (0) da? * x Tet : 7 This differs from the ordinary Bessel’s equation only in the sign of k’, and conse- quently the two solutions may be conveniently written under the forms J, (kx. fJ—1) and Y, (ka J —1). The second solution is infinite when #=0 and would be required only for a hollow cylinder. We shall here consider only solid cylinders and so may confine our attention to the former solution. This may be expanded in a series differmg from J, (kw) only in the fact that all the terms are of the same sign. It contains only consecutive odd or even powers of ke,/—1, and thus by the selection of a proper constant multiplier may be regarded as altogether real. In our subsequent work we may suppose this to be mates a (kr J—1) by J/ (kr J/—1), so that J—1 J/ (kr /=1) k/—1dr may be regarded as wholly real if J,(kr./—1) is so. It will also be useful to notice that if J,(kr./—1) be defined as the form taken by J,(kr) when k,/—1 is written for k, so that it is now no longer necessarily real, we get relations among these consecutive functions exactly similar to the well-known relations for the Bessel’s, viz. to the case. Let us denote ke d= i ea Gir) dao) ath Np: sate G aill cena (1) Ji (ker) =4 {J (ler) — Jy, (ler)} SOLID IN POLAR AND CYLINDRICAL CO-ORDINATES. 361 § 92. Supposing no bodily forces to act, the internal equations of equilibrium are, see Sect. IX., dé dt dB (m+n) 7 ae ue q+” aay Rea ite aint cictaisisish sisislsialave'elo.o Sdcicee sonic (2), 1 dé dA de (m+n) ~ ay—” FEE oR iy be epee Mensa ae oniecielac ejsclesisissietcneieise (3), (m+n)r gf =n S +n ss = (U) sonseasn0de don.ccanercpopadespoanconccans (4); ld.ur ldv dw bs where 5 => a ara + - dé SE FEB ConacdeaanstaMaDonHons agMOnodbodr Coors orn coos (5), 1/dw d.ovor A=— & < <7) OTR ay. Sah Oi ace at es: (6), du dw = Bb =7 (= = a SHaAOAOOS 7 DOM OD IO? 460 door HOD pon sa pe AapDopOCenCOD ( ( ) l/d.vur du C oe ee = a) faieluleia}elateletelaielphsie/=;eisfotecesivi aia (viele ipieloin(ninje)« piela aiu(vle\«isioiaia/eia/eialelete (8). Differentiate (2) with respect to r, (3) with respect to 6, and (4) with respect to z; then adding we get d’§ 1d5 10% d’s_ de rdr' Pde da § 93. Assuming 5 = 2 c0s 70 f, (r) f, (2), where f,(r), f,(z) are respectively functions of r only and of z only, we easily find as a - type of the solution 5=cos 20 [.J, (kr) (A cosh ke+ Bsinh kz) + J, (kr J=1) (C'cos kz + D sin kz)]...... (10), where A, B, C, D are arbitrary constants. An exactly similar series in sines of multiples of @ of course exists. From (4), (5), (6) and (7) we get = —— : = 2 =." « _ 5 10 (J, (or) (A sinh ke + Boosh kz) + J, (kr J—1)(Deoskz —C sin kz)}......(11). By treating (11) in different ways we get two different particular solutions which seem adapted for application to different problems and will be considered separately. lw In either case we must suppose w to depend on cosi@ and so may replace +s by —z*w. Thus let us assume ——, = w' (A sinh kz + B cosh kz) + w” (D cos kz — C sin kz) ; cos 10 362 Mr C. CHREE, ON THE EQUATIONS OF AN ISOTROPIC ELASTIC then w’, w’ are determined as functions of r by dw’ 1dw’ Ned mk a dr > dr +(#-)w See Cw" 1dw” a Sty en ae at oe (eta) “a Je(ersf—1). Using the Bessel’s equation, it is easily seen that t= er Ze); w= — r— 2Qnk dr similarly oe —5? fy, (kr J —1). Thus form (i) of the particular solution of (11) is m w= 57 C08 v0 E = J, (kr) (A sinh kz + B cosh kz) +r = (kr /—1) (C sin kz — D cos he) Bee Scuehouc (12). For that part of the general solution of (11) depending on cos i@ we have obviously, referring to (9), terms similar to (10). Thus as a type of the complete value of w we get ee ie = \r = J, (br) (A sinh ke B cosh kz) +r : J, (ker /=1) (C sin kz — D cos ka} + J, (kr) (A’ sinh kz + B’ cosh kz) + J, (kr /—1) (C' sin kz + D’ cos k2)| ana (133): where A’, B’, C’, D' are constants. Substituting from (6), (7) and (8) for @, 33 and © in (2) and (8), and then for a from (5), and lastly introducing the typical value (10) for 8, we obtain two differential equations of the second order in w and v. In the first, v appears only as de and in dé’ the second wu appears only as ae For a typical solution answering to (10) and (12) dé assuming u%*cos7@ and ye sin7i@, and eliminating v between the two equations, we get a differential equation of the fourth order to determine w as a function of 7 and z. So far as z is concerned the solution will consist, as in the case of w, partly of exponential and partly of circular functions. In obtaining the corresponding functions of r we may replace by +* in the first case, and by —* in the second. The conse- quent differential equations of the fourth order in r may be regarded as resulting from the successive application of two operators*, in the one case such as occur in Bessel’s equations of orders 7-1 and i+1, in the other case such as occur in the corresponding sworn, [eZ ardor ane] [Serban]. SOLID IN POLAR AND CYLINDRICAL CO-ORDINATES. 363 equations where & is imaginary. It may thus be deduced that the solution giving the typical term is u = cos 10 |- x {v4 (kr) (A cosh kz + B sinh kz) + rJ; (kr J/-1)(C coskz + Dsin kz} + : J, (kr) (A, cosh kz + B, sinh kz) + a (kr) (A,' cosh kz + B,' sinh kz) - : J, (kr J —1)(C, coskz + D, sin kz) + a J, (kr ./—1) (C/ cos kz + Dy sin k2)| ...(14), where A,, B,, A,, B/, C,, D,, C/, Dy are constants. A similar treatment leads to a corresponding differential equation of the fourth order in v, which is however much simpler than that in w as the right-hand side vanishes. The typical solution is »y =sin 70 E J, (kr) (A, cosh kz + BY sinh kz) + EB J, (kr) (A, cosh kz + B, sinh kz) 7; +1 J, (kr J) (CO; cos kz + D,' sin ke) + © J, (br JT) (C008 kz + Dy sin k2)| Pi (15), where A,’, B,’, A,, B,, C,, D,’, C,, D, are constants. The constants we have introduced in (10), (13), (14) and (15) are not all arbitrary, as the existence of the identity (5) necessitates the following relations: = ee = {A (m+n) — nk A} | Pr = B= = {Bm +n) — nkB? | iA,=-A, zed re a eee (16) apoaing C, =e {C (m + n) —nkC’} = = ID) == {D (m + n) + nkD’} iW, =—-C, uD). == ID), Thus the solutions in w and v are u = cos 10 lu. (kr) e ue r(A cosh kz + Bsinh kz) + : (A, cosh kz + B, sinh kz)l eel iS, (kr) CA m+n—nkA’) cosh kz +(Bm+n—nkB’ sivh ie kin dr~* { ) n + J, (kr at) = = r(C cos kz + D sin kz) + - (C, cos kz + D, sin ke)} + = =i (kr /—1) \c m+n—nkC’) coskz +(Dm+n+nkD’) sin ie} | pee llits): Vot. XIV. Parr III. 47 364 Mr C. CHREE, ON THE EQUATIONS OF AN ISOTROPIC ELASTIC Seed = a) v = sin 70 (4 m+n—nkA’) cosh kz + (Bm-+n—nkB’) sinh kz ld eas J, (kr) (A, cosh kz + B, sinh kz) Ob os ay ab epee tS. — _wW, = 1) Ke m+n—nkO") cos kz +(D m+n nkD’) sin key 1 = J,(kr JS) (C, cos kz + D, sin is)| bate chaaeonpcueseneataaee (18). A type of the complete solution of the internal equations is thus supplied by (17), (18) and (13). § 94. For the special case ¢=0 the equations are much simplified. The expression for v is quite independent of those for w and w and is to be considered separately. The typical solution may be easily seen to be u=-— J, (kr) r (A cosh kz + B sinh kz) z a J, (kr) [4 (m+n) —nkA’} cosh kz + {B (m + n)—nkB’} sinh ks| = J, (ers f=) os r(C coskz+ Dsin kz) Le er: ee kr /=1) [ {7 (m+n) —nkO"} cos kz + {D (m+n) + nkD sin ks] wich (19), Eh (kr) (A sinh kz + Bcosh kz) + J, (kr) (B' cosh kz + A’ sinh kz) = et UT + on! g J, (kr /—1) (Csinkz — D cos kz) + J, (kr /— 1) (D' cos kz + CO’ sin kz) ....(20), 8=ZJ, (kr) (A cosh kz + B sinh kz) + J, (kr J=1) (Ccos ke+ D sinkz)............ (21). For v there is the independent typical solution, deducible from (3) directly by supposing u and w to vanish and v to be independent of 6, v = J, (kr) (K’ cosh kz + E sinh kz) + J, (kr /—1) (N’ cos kz + Nsin kz) ......... (22), where LZ, 2’, N, N’ are arbitrary constants. If there be symmetry about the central section z=0 we must take in these equations B, D, B’, D', E, N all zero; for w must change sign with z, and w and v must not. In this case the expressions are thus much simplified. The terms in the solution involving exponential and circular functions occur separately, and seem suitable for application to distinct problems. SOLID IN POLAR AND CYLINDRICAL CO-ORDINATES. 365 § 95. The surface-conditions are given in equations (24)—(29), Sect. rx. The solution obtained above is not in general capable of satisfying these when the surface-forces are given arbitrarily. For instance, supposing z=0 a plane of symmetry which includes the case when it is held so that w only vanishes with z, it is possible from the solution to satisfy the condition that the curved surface of the cylinder shall be exposed to any system of purely normal forces, But the arbitraries left limit us to a choice of two alternatives, either an accom- panying perfectly determinate system of tangential forces on the curved surface, or a determinate system of normal tractions on the flat ends. § 96. The most manageable part of the solution seems to be that given in equation (22). Confining ourselves to the case when v vanishes over the plane z=0, and taking first the circular function, we have as a type v=N sinkz J, (kr =) SCE ab oc SOO ee nae Ce en ences (22a) The surface-stresses are, on the curved surface r=a, d.” G=R,=nr == Sei SR (Pe Re (23); on the flat end z=1, caper = SENT (le = 1 COaLING ote, ek. nelle (24); on the flat end z=0, GLACIAL) eck Beg Seth Rete On ec cR eas (24 a). By taking k=(2i+1) 5) where 7 is any integer, we get rid of all surface-forces except those on the end z=0 and fp, on the curved surface. Thus from (23) we can determine NV so that (22 a) may represent the solution for the case of a cylinder, one end of which is held, over whose curved surface there is applied any distribution of tangential forces which act perpendicularly to the axis and are symmetrical round it. NV is real or imaginary according as J, (kr J/-1) is real or imaginary, ie. according as J,(kr J—1) is imaginary or real. Since all the terms of J,(kaJ/—1) have the same sign it cannot vanish for a finite real value of k, and so we cannot from (23) and (24) make &, vanish over the curved surface, and 4% arbitrary over the flat ends. This we can accomplish however by using the exponential part of (22), viz. the type j= 18; aialn idl (UP) acdnaeonacstqapcsons56e550000 0000000000000 (22 b). The surface-forces are, on the curved surface Gl iy S = WH rian 240 (80) aa cho aan docason ee aoonceaasboox000 50d (25); on the flat end z=1, GS Ay = mth) (on (ANIL, Wa) copnacscocndsecooocdemoounssonos0a0c¢ (26); on the flat end z=0, GS i Sil ol BT ir ose sde eehedeodcadk Asbaoorcsbcsancadsccan 7567 (26 b). 47 —2 306 Mr C. CHREE, ON THE EQUATIONS OF AN ISOTROPIC ELASTIC By taking for & the roots of J,(ka)=0 we can thus get rid of the tangential force on the curved surface. It is also easily proved that if & and k’ be any two of those roots, then i ; rJ, (kr) J (kr) dr = 0. 0 We can thus expand functions of 7 in a series of terms J, (kr) by the usual method and thus solve the case where, over a flat end of a cylinder, whose other end is held, there are applied arbitrary tangential forces, symmetrical round the axis, perpen- dicular to the radius vector at every point. This solution would thus have an extensive application to the more complicated forms of torsion in circular cylinders. As an example suppose a torsional force 7’r* over the end z=l. Suppose 7r* expanded in a series =nk# cosh kl J, (kr), where & is a root of J, (ka) = 0. Then to determine #, we have Enk cosh kl i “J, (kr)}*dr = 7 | PT or kee ee (27), 0 0 In the right-hand side of (27) using the Bessel’s equation and the relation J, (kr) = — J, (kr), and integrating by parts, we find [rg (kr) dr = pe, (ka) — kad /(ka)} + e {head (ea) ei (ea) tee. eens eee (28). 0 But since J, (ka)=0, we have J, (ka) ' aha be 4J, (ka) = J,’ (ka), 4 and the right-hand side of (28) reduces to ae (ka). Thus using the result* fa : £ B - 1 2]. r iJ, (kr) dr =a & (ka)}? + (1 - aan) {J, (ea) ; and the above values of J, (ka) and J,’ (ka), we finally obtain beer a nk J, (ka) cosh kl’ whence the solution is _ 87 5 J, (kr) sinh kz =e ESS, (ha) cosh Rl CTT estes (29), where the summation includes all values of & satisfying DANY 0 on namigg cosa tony ca incre asap tine ene (80). * Lord Rayleigh’s Sound, Vol. I, Equation (16), p. 270. SOLID IN POLAR AND CYLINDRICAL CO-ORDINATES. 367 The stresses at any point (r, z) in the cylinder are J, (kr) sinh kz =n ay A aN See (3 Re=—8T FC go Fa 1), J, (kr) cosh kz = pee 26 4 = 87 > Paes) cose Gay Wea alia (32). § 97. We now proceed to the second form of solution. Returning to (11), the general solution is the same as before, while the particular solution consists of two parts, the one depending on J,(kr), the other on J, (krJ/—1). For the first part assume wx cosi@J,(kr); then to determine w as a function of 2, we have of which a solution is w=— = (A cosh kz + B sinh kz) cos 16S, (kr). For the second part we get w= i (D sin kz + C cos kz) cos 16S, (kr /—1); therefore the type of the complete solution is w = cos 10 vn (kr) |B cosh kz + A’ sinh kz — ~ (A cosh kz + B sinh ke)} + J, (kr /—1) \D cos kz + C’ sin kz — = (C cos kz + Dsin ka) | es tee (Os =} In obtaining « and v the method to be adopted closely resembles that sketched out in treating the first form of solution. It leads to the typical solution = , d _. (A (m+ 2n) — QnkA’ i. B(m+2n)—2nkB' . “4 wu = cost6 [- ap J, (kr) Dn cosh kz + En sinh kz + 2 (Asinh kz +B cosh ke) we - J, (kr) (A, cosh ke + B, sinh kz) ne d pray [Om + 20) — 2nkC’ , Dim+2n)+2nkD’ . | aF dr J; (kr fe 1) { 2hen cos kz sr en sin kz 2 = (C sin kz — D cos ka)} = : J,(kr J— 1) (C, cos kz + D, sin iz) | ee (34), ; A Qn) — 2nk.A’ B(m+2n)—2nkB’ . v = sin 10 c J, (er) (ar a : cosh kz + ( a= sinh kz + 2” (Asinhkz+B coshe)} 14 | (kr) (A, cosh ke + B, sinb kz) 2nk i dr v —.(C (m + 2n) — 2nkC’ _ , Dim+2n)+2nkD' . 7 mz ie aes, Se — 5 Ii(kr = a cos kz + In sin kz Ink (C sin kz —D coskz) = a (kr =i) (C cose + Dr sin. Kz) |) ojo. .ccrege-a2e-212222 2-70 (35). 368 Mr C. CHREE, ON THE EQUATIONS OF AN ISOTROPIC ELASTIC For the case of symmetry about the plane z=0 we have B=D=B=D =B,=D,=0, and the equations are much shortened. It will be noticed how simple is the relation between wu and v in this solution. $ 98. For the special case i=0, we have the type d = (A(m+2n)—2nkA’ : B(m+2n)— - 2nkB’ = ape (eens Sh LAAT, cosh kz + an inh nke+ = (As noha d CO (m+ 2n) — 2nkC" D(m+2n)+ 2nkD' + 5 J (kr nf =) | ans coskz + ohn inkz—>— (C'sin kz — Deoskz)} Peace eee aaa oe (36). w= J, (kr) {B cosh kz + A’ sinh kz — om (A cosh kz + B sinh ke) +J,(kry j= \p cos kz + C’ sin ke- 9 -(C cos kz + D sin kz)h Mees oe (37), 8=J, (kr) (A cosh kz + Bsinh kz) + J, (kr J=1) (C cos ke + D sin kz)...:..-.2- (38). The independent solution in v is the same as by the former method. In the case of symmetry about the plane z=0, we have B, D, B, D, B,, Dy all zero as before, Of course there is absolutely no connection between the constants in the two different forms of the solution, the same letters being used merely to shew that the two are not distinct solutions. The difficulties in dealing with the surface-conditions are exactly analogous to those occurring in the first form of solution. § 99. A special case of (36) and (37) which seems of interest occurs when the cylinder extends to infinity and is subjected to normal forces on the plane end z=0. In the equations referred to suppose B=—A and B’=—4’, then the solution in ex- ponentials is typefied by Peleg _\ a: (A (m+ 2n)— 2nkA' Amz ee Tne (ker) € | on — oa erate ERROR (39), Se Amz w=—ZJ,(kr)é&* (4’ 7 ) abst sep nauaincle ieee aaa ania ces cere (40), PO acah AL RR IDEN Re Rr 8 Soe (41). The stresses are (m—n) 6+2n S =e J, (kr) {2nkA' + A (mkz—n)} ......ccececeseeven en (42), du dw 4 $5 m+n . n (See) mem J, (ler) 4a (me — mz) - 2nd’ Ache ee (43), SOLID IN POLAR AND CYLINDRICAL CO-ORDINATES. 369 (m—n) 6+ 2n = aug E (kr) {A (2m +n — mkz) — 2nkA'\ iL @/ _, (A (m+ Qn) —QnkA’ Amz ste apie (kh )| ie iar | pieces chateest (44) By taking k such that J,/(ka) =0, we get rid of the tangential traction over the curved m+n the end z=0. The normal traction on the flat end z=0 is then simply AmJ, (kr) . and, as J,’ (ka)=0, the normal pressure on the curved surface reduces to surface r=a; and by putting A’=A we get rid of the tangential traction on SIAN pl UR) GE (USWA). cetnosnetibsnasscsosnoganseconsan20005 (45). If k be large this last expression diminishes rapidly as z increases. Since J,'(ka)=0 we can expand any function of r in a series of Bessels of which J, (kr) is a type, and thus this solution would apply to the most general symmetrical distribution of normal tractions or pressures over the flat end of the cylinder. Writing in the value of A’, the solution is pee. Re mkz i) i (1 -") J, (er) Pratt sve tdete|s(cletareintavetoretertelelots (cfatase| sYatatetatafatsre (46) Aly ss m mkz Es Se Op. mee) J, (kr) lePatetataretsleetelsielpitlelepersie/sisisteletsisye! (47) It is obvious that the strains and stresses become very small, and may be neglected at considerable distances from the bounding plane z=0, especially for the larger values of k. Since the equation determining & gives certain constant values for ka, the solution would be particularly suitable to the case of a cylinder of small radius. ERRATA. p- 260, equation (12), insert 2 in the two last terms. p. 297, equation (30), read 4 for 2. p. 298, equation (33), read 4 for 2 in the last term. XII. On Solution and Crystallization. By G. D. Liveine, M.A., Professor of Chemistry in the University of Cambridge. [Read May 21, 1888.] THE nature of solution has often occupied the attention of chemists, because the homogeneous character of a solution and the definite proportions of saturated solutions at given temperatures point plainly to an analogy between solution and chemical combina- tion. Moreover many substances certainly form chemical compounds with water, alcohol, ‘e., and there is no reason why there should not be, at ordinary temperatures, liquid as well as solid compounds of this kind. The heat evolved in the dilution of many solutions has also been supposed to indicate a concurrent chemical action; and the fact that such an eyolution is most marked on adding water to solutions of those salts which form solid erystallized hydrates lends support to that supposition. The gradual variation in the amount of the substance which will saturate a given menstruum when the tempera- ture is gradually varied is the great stumblingblock in the way of the theory of chemical combination in solution, though this may be pretty well matched by the gradual dis- sociation of many chemical compounds by a rise of temperature. Nevertheless though chemical action doubtless produces its own effects in some solutions, and it is probable that some substances will not dissolve without combining with the solvent, the regular course of solution seems capable of explanation as the result of the molecular energies which we see displayed in surface tension. Between two liquids which mix with each other in all proportions there is no surface tension, and it is well known that the surface tension of liquids varies when substances are dissolved in the liquids. If we consider the case of water and ether; as the water dissolves some of the ether and the ether dis- solves some of the water the surface tension at the junction increases, and when it has acquired a certain value solution ceases. That value must be such that the entropy is a taximum, and this is probably the case, supposing the temperature constant, when the increase of entropy by interdiffusion of the liquids is balanced by the diminution of entropy in the increased surface energy. That an increase of entropy attends solution is certain because work has to be done on a solution in order to separate the substance dissolved from the solvent. Either the latter has to be evaporated away, or cooled below the temperature of the surroundings (supposing it in thermal equilibrium with its sur- roundings to begin with), or something has to be done which implies work. The more completely two liquids are miscible the smaller will be the dissipation of energy which could be effected if we supposed that their interdiffusion were carried beyond the point Pror. LIVEING, ON SOLUTION AND CRYSTALLIZATION. 371 of saturation, and the smaller therefore the surface tension which will balance the tendency to such further interdiffusion. Thus when hydrochloric acid, or hydriodie acid, is dissolved in water, the stronger the solution the less is the amount of heat evolved when more hydrochloric or hydriodie acid is dissolved in it; that is to say the dissipa- tion due to solution diminishes as the strength of the solution increases. This accords with Quincke’s observation that the more freely two liquids are miscible the smaller is the surface tension between them. If the substance dissolved be solid instead of liquid the change of entropy due to its liquefaction will have to be taken into account, but the point of saturation at a given temperature will still be determined by the consideration that the increase of entropy due to further solution is balanced by the diminution arising from increased surface tension. A change of temperature will affect both the surface tension and the amount of change of entropy due to liquefaction, and will consequently affect the point of saturation. The greater the surface tension at the junction of the substance with the solvent the greater must be the availability of the energy necessary to increase that surface tension and the more difficult will it be to increase the amount of the substance dissolved. If the substance is more soluble at higher temperatures, and if, after the solution has been saturated at a given temperature, the temperature be subsequently lowered, there will be a tendency to a reverse action, to a deposition of part of the substance in the solid state. But this can only occur when this deposition will be attended with an immediate increase of entropy. If there is in the solution a piece of the undissolved solid deposition can occur upon this without the development of any new surface tension, since the new surface will be of the same kind as the old and only a very little larger, and there will be no surface tension between the new deposit and the mass on which it is deposited; and this will therefore afford the most favourable circumstance for the depletion of the solution. The entropy will continuously increase until the point of saturation corresponding to the temperature is reached. If there be no undissolved portion of the substance in the liquid deposition may occur at any place where the solution comes in contact with anything else, provided the replacement of the surface tension at that place by the two new surface tensions produced by the deposition is attended with an increase of entropy. When once a deposit has been formed there must be a tendency for this deposit to grow rather than for new deposits to be formed, for the reason above given. The formation of a new surface in the interior of a uniform liquid means a storage of energy which may exceed the availability of the energies in the neighbourhood, that is to say the immediate result of the formation of such a new surface may be a diminution of entropy. Nevertheless the diminution of entropy by the formation of the new surface may be so small in some cases that a trifling mechanical disturbance in the liquid may suffice to render the necessary amount of energy available to produce the deposition, All this is in agreement with what is observed in super-saturated solutions. These remarks about the formation of a new surface in the interior of a solution will apply also to the deposition at any part of the solution where it is in contact with another substance, if it happened that there would be at first a loss of entropy by a deposition at that place. The immediate loss may be very small and the mechanical action of rubbing the Vou. XIV. Part III. 48 372 Pror. LIVEING, ON SOLUTION AND CRYSTALLIZATION. surface, as by rubbing a glass rod against the side of the vessel containing the -solution, may suffice to supply energy enough in a form available to carry the change past the point of minimum entropy. Whenever a substance separates out of a solution the new arrangement of the mass must, if there be equilibrium, be such that the entropy is a maximum, and so far as the external surface is concerned it must tend to take such a shape as shall make the surface energy a minimum. If it separate in the fluid state it will generally appear in spherical drops, which have the minimum surface for their mass. If it separate in the solid state and be a crystallizable substance the arrangement of the particles throughout the mass must be crystalline, because the tendency of erystalloids to take the crystalline structure is a proof that such an arrangement of the particles corresponds to a maximum entropy. A crystalline arrangement of the particles is however consistent with any external form, yet it is a rare thing that we find a crystalloid separating from a solution in globular masses. We must therefore conclude that when there is a crystalline arrangement of the particles the surface tension is a minimum in planes bearing certain relations to the arrangement of the particles. In liquids we suppose the molecules to have a certain freedom of motion amongst themselves whereby diffusion occurs, but in solids the freedom is restricted so that we suppose each molecule, though capable of considerable motion, to maintain on the average the same relative position to the other molecules. This makes a definite arrangement of the molecules possible, and the optical and other pro- perties of crystals render such an arrangement almost a certainty. If then a crystalloid separates from solution in the solid form its molecules arrange themselves in a definite way depending on their mutual actions, and the external surface of the solid assumes such a shape that its surface energy is a minimum. Also, if the surface tension tend to contract the surface, the surface energy per unit of area will, ceteris paribus, be a minimum when the approximation of the molecules of the sur- face to one another is a maximum. The application of these principles gives us, as I hope to shew, a solution to a first, and near, approximation of the problems of the external shape and of the cleavages of crystals. Cubie System. If the substance be one which crystallizes in the regular system we may suppose the law of arrangement of the molecules to be as follows. Let space be supposed divided into cubes of a uniform size by three sets of parallel planes, each set at right angles to the other two, and suppose each corner of a cube, or each point of intersection of three planes, to be the mean place of a molecule of the crystal. This arrangement satisfies the condition imposed by the isotropic character of the crystal. Now of all surfaces which can be drawn through the system of molecules arranged in the way described the planes which form the cubes above mentioned contain the greatest number of molecules per unit of area. For let oa in fig. 1 be one of the cubes and consider the set of planes parallel to a face of it, bac. The distances between contiguous molecules in that plane is the least possible with the arrangement of molecules supposed. Moreover if we take in OX, points #,, 7,, £,, &c, successively at distances all equal to Oz,, and in OY points y,, y,, y,, Pror. LIVEING, ON SOLUTION AND CRYSTALLIZATION. 373 &e., and in OZ points z,, z,, z,, &e., such that their distances from each other are all equal to Oy, or Ox,, every plane which intersects the axis OX in one of the points «,, a,, &c., 7a | ‘ bs 4% Fig, 1. b, 9. ine b, v2 v3 us 5 i 1X é : C C Cc 3 4 5 the axis OY in one of the points y,, y,, &c., and the axis OZ in one of the points z,, z,, &c., will be a surface of maximum concentration of the molecules. For returning to the plane bacy, it meets a succession of molecules c, c,, c,, &e. in the plane YOX, and a similar succession in every one of the planes parallel to YOX through z,, 2, &e. But if we turn the plane bacy, through a very small angle about by, it will pass through none of these molecules except those which lie in the plane ZOY, namely y,, 0, b,, b,, &e., and it will pass through no other molecules nearer than the line in which it intersects the plane ZOX. If we suppose it to meet OX at a point w,, the place of one of the molecules, it will then be a plane of maximum concentration of molecules, for if it be turned either way about y,b through a small angle the molecule nearest to y,b which it will meet will be farther away than «,. Also the nearer a, is to O the less the distance y,#, and the greater the concentration in the plane by,2,. The plane by,#, will intersect every one of the planes parallel to ZOX in a row of molecules, and the successive distances of these rows measured in the plane YOX will be all equal to Y,Xp. Moreover the whole set of planes drawn through 0, Li) DB, .. U5.., all) parallel to by, 2, will together contain all the molecules; and the arrangement of the molecules in each plane of the set will be the same. And if we compare two sets of such planes, parallel to by,x, and by,#, respectively, which pass through unit volume of the crystal, each set 48—2 374 Pror. LIVEING, ON SOLUTION AND CRYSTALLIZATION, will pass through all the molecules, but the number of planes out of each set which intersect the unit volume will be inversely as the distance between the successive planes of the set. Hence the concentration of molecules in each plane of a set will, when p varies, be inversely as the perpendicular distance between successive planes of that set. Similar reasoning will shew that if we make the plane by,«, revolve about y,a, it will come to successive positions of maximum concentration as it intersects OZ in the successive points which are the positions of molecules. Each plane z,y,2, will be a plane of maximum concentration, the whole set of planes parallel to it will contain all the molecules of the mass, and the concentration m each plane, when n is varied, will vary inversely as the perpendicular distance between the successive planes of the set. The same reasoning may now be extended by supposing the plane z,y,x, to be turned about z,7,, every plane z,y,,7, will be a plane of maximum concentration, and the con- centration when n, m and p are varied will vary inversely as the perpendicular distance between the successive planes of the set, which is the same as the length of the perpen- dicular from O upon the nearest plane of the set, because every set has one plane through 0. Every plane 2,7,,2, is by the crystallographic law a possible face of the crystal and if M be the least common multiple of p, m and n, omitting any one of them which is infinite, > = = will be the indices of the face in Millers notation. This shews that for such an arrangement of molecules as is here supposed, plane surfaces following the law of indices will be surfaces of minimum energy. The substance when separating from solution must therefore tend to take a form bounded by such plane surfaces. From the symmetry of the arrangement of molecules it is evident that any interchange of the three symbols will not alter the concentration in the set of planes represented. That is, every face of the same “form” of crystallographers will have the same concentration and therefore the same surface energy when in the same surroundings. If the distance Ox, be taken as the unit of distance the perpendicular from O upon : Ae ae the plane z,y,2, will be the reciprocal of Jataet ae Also since the set of planes parallel to 7, Y,2, passes through every molecule, one plane of the set must pass through each of the molecules z,, 7,...7, and one through each of the molecules y,, y,...y,,, and one through each of the molecules z,, z,...z,, and if none of the numbers p, m, n have a common measure the whole number of the set of planes between O and the plane a,y,z, will be pmn, and in general it will be M, where M is the least common multiple of p, m and n, so that the perpendicular distance between them will be the reciprocal of 1 lata: V2 2 2 EOS bt 3a = or the reciprocal of J (4) oe (2) + (=) , or if h, k&, l be the P91 en Pp m n symbols of the corresponding face, the reciprocal of //*+k*+2. We may designate this distance as P. Pror. LIVEING, ON SOLUTION AND CRYSTALLIZATION. 375 For the cube, 100, 12M for the dodecahedron, 110, = og 2 for the octahedron, 111, IP= ue /3 for the tetrakishexahedron, 210, P= = 5 for the eikositessarahedron, 211, P=~., \ for the triakisoctahedron, 221, P=}, and so on, P diminishing as the indices grow larger, and the concentration also diminishing and the surface energy consequently increasing. It should be observed that although the surface energy must increase as the concen- tration diminishes it does not follow that these quantities are reciprocally proportional : that will depend on the law of force between the molecules. Hence the values of P above given indicate the order only and not the exact amounts of surface energy of the different planes. It will be observed too that the molecules are assumed to be sym- metrical, or to act on each other merely as centres of force, so that their mutual influence will be the same in all directions, otherwise we should have to take into account the orientation of the planes of each set as well as their relative distances. It is probable that to a first approximation we may treat the molecules as so symmetrical, though for the complete solution of the problem we must regard them as unsymmetrical. Hence supposing that the substance is one like common salt which crystallizes in the cubic system, there must be a tendency, when it separates from solution in the solid state, to assume that form which has a minimum surface energy, and as the cube has the minimum surface energy per unit of surface there is a tendency on the part of salt, as well as of all other substances that crystallize in the same system, to take the form of a cube. Salt does form cubic crystals, but other substances such as alum which belong to the same system crystallize more frequently in octahedra. How is this? The law of energy requires the surface energy of the whole solid mass to be a mini- mum, and it does not follow necessarily that for a given mass of the solid the integral of the surface energy over a whole cube will be less than the integral of the surface energy over a whole octahedron when the two figures have equal volumes. In fact for a given volume the surface of the cube is greater than that of the octahedron in the ratio of 2 :./3 or 1:049 to 1. Bearing in mind what has been indicated above, that although the surface energy must, ceteris paribus, decrease as the concentration of the surface increases we do not know as yet the law of force between the molecules and therefore cannot tell the exact ratio in which the decrease occurs for any given increase of con- centration, we can see that it may very well happen that with salt the total surface 376 Pror. LIVEING, ON SOLUTION AND CRYSTALLIZATION. energy of a cube may be less than that of an octahedron of equal mass, while with alum the reverse may be true. Again, by a combination of forms we may easily obtain for a given mass a_ total surface much less than that of the cube. Thus by replacing the edges of the cube by facets of a dodecahedron, and the corners by facets of an octahedron, while retaining the same total volume, the surface will be sensibly diminished. It will be noted that any form of which the surface has minimum energy will be a surface of equilibrium so that any of the forms developed by the crystallographic law are possible. When a crystal has once begun to form there will be a tendency in general to continue to develop the same form, because the development of a face of any other form will imply the development of a surface with a different surface energy. Still for the reasons just given we may have edges and angles replaced by surfaces of greater energy per unit of area than the surfaces of the primary form. When one edge or angle is replaced however other similar edges and angles will be similarly modified, for the following reason. The tensions of a plane surface will have no resultant in any direction out of that plane, but at an edge, or angle, the surface tensions will have a resultant directed to the interior of the mass and, since the arrangement of the molecules is one of equilibrium, that resultant must be equilibrated, either by a corresponding resultant on the opposite side of the crystal, or by an equivalent resultant produced by internal displacement of the molecules. In general such an internal displacement implies that the solid is not formed with the maxi- mum entropy, and hence when one edge or angle of a form is developed all the others will as a rule be developed, and all similar edges and angles will be similarly modified. Occasionally the equilibrium may be attained when only half the faces of a form are developed. Thus in the tetrahedron the resultants due to opposite edges balance one another. Still when hemihedral forms occur in the cubic system we often have evidence from the optical or other properties of the crystal that there is an internal strain or deformation of the arrangement of molecules. Thus cubes of sodium chlorate which have half their angles replaced by faces of a tetrahedron rotate the plane of polarized light, cubes of boracite with similar replacements exhibit pyroelectricity, and so on. Whenever a solid is broken two new surfaces are developed each with its own surface energy. Those surfaces which are surfaces of minimum energy will be produced with the greater ease. Hence cleavage surfaces of crystals must follow the law of the external forms and be planes of maximum concentration of molecules. The only cleavages of crystals of the regular system are parallel either to the faces of the cube, dodecahedron, or octa- hedron, which correspond to the three greatest degrees of concentration. It is plain how- ever, that mere division is not quite the same problem as the development of the complete bounding surface of a mass. In the former case the all important consideration is the surface energy per unit of area of the two new surfaces produced, while in the latter it is the integral of surface energy over the whole solid. Now for many crystals, such as rock-salt, galena, cobaltine, pyrite, we find the cubical cleavage readily obtained, while sometimes a dodecahedral and more often an octahedral cleavage is also obtainable, though less easily. For these crystals the arrangement of the Pror. LIVEING, ON SOLUTION AND CRYSTALLIZATION. 377 molecules above supposed seems adequate to account for the facts observed. But there are some crystals of the cubic system such as fluor, fahlerz, cuprite and alum which do not cleave parallel to the faces of a cube but give, more or less easily, an octahedral cleavage. For them the planes of the octahedron must for some reason be planes of less surface energy than those of the cube. It may be either that the mutual action of the molecules is not the same in all directions but less in the plane of the octahedron than in that of the cube, or that the arrangement of the molecules is different from that above supposed. _ There is another arrangement of the molecules in the erystal which will equally well satisfy the optical requirements of the problem: this is to suppose the molecules to occupy the centres of the several faces of the cubes instead of the angular points of the cubes. The same sets of planes as before will be planes of maximum condensation, but the relative degrees of condensation in the several planes will in some cases be different. In fact the arrangement of molecules will be the same as if three sets of molecules, each set arranged at the corners of equal cubes, were combined. Thus if 0, 0’, O” (Fig. 2), be the centres of three adjacent faces of one of the cubes ag, namely O the centre of the face abfe, O' that of abcd, and 0” that of adhe, we shall have one system of mole- cules which lie in the centres of all those faces of the cubes which are in the plane of abfe or are parallel to it; a second system of molecules, similarly arranged and at the same distances apart, which lie in the centres of all the faces in the plane abed or 378 Pror. LIVEING, ON SOLUTION AND CRYSTALLIZATION. parallel to it; and a third system, similar in all respects to the former two, lying in the centres of all the faces in the plane adhe or parallel to it. Hence for any set of planes belonging to the first system and parallel to a plane with the symbol Akl there will in general be a similar set of planes parallel to them belonging to each of the other two systems. If there were three such sets for all values of hkl the relative condensations in planes having different directions would be the same as in the simpler case first taken. But in some cases two of the three sets of planes coincide, and in others all three sets coincide, and thus in these cases the condensation in the planes which coincide is much increased, Two sets of the planes will coincide when 1. Two of the indices hkl are equal to one another. For in that case that plane of the first set which passes through 2,, z, will also pass through O', that which passes through 2,, z, will pass through a, z,', and so on. In the particular case in which two of the indices are zero the plane parallel to YOZ which passes through O' also passes through 0”, and we have a similar coincidence of corresponding planes throughout the systems. 2. If two of the indices are odd numbers. Thus a plane passing through z,, x, also passes through ,, and a plane through z,, 2, passes through 2,'; and generally a plane through 2,, #,,,,, where n is any integer, passes through z,’. The same will be true if two indices be equimultiples of some odd numbers. When two sets of planes coincide the condensation in every alternate plane will be doubled, while in the remainder the condensation will be that which belongs to only one of the three systems; so that the average condensation in the planes considered will be one and a half times that due to a single one of the three systems. The three sets of planes will coincide whenever they are all three odd numbers. Thus the plane through 2,, y,, z, passes through both 2,’ and y,”’, and so on. Comparing now the relative condensations in the several planes symbolized below, under the arrangement of molecules above described, we find them to stand thus: octahedron ila condensation °5774 or 11548 cube a 100 Be "7 5000 1:0000 dodekahedron qd 110 es 7 3536 ‘7071 m 113 Fi es *B015 ‘6030 q 133 » op 2294 4588 m 112 " A 2041 4083 p 122 5 . 1666 3333 f 0138 “ A 1581 3162 e 012 ore SPOT 2982 se 1238 7 x 1336 2673 w 137 Fs 5 1302 2604 114 " ‘1179 2357 233 5 7 1066 "2132 014 * “ 0808 ‘1616 124 3 F, 0727 1455 Pror. LIVEING, ON SOLUTION AND CRYSTALLIZATION. 379 If then this were the molecular arrangement we should expect the octahedron to be the predominant form, though the cube has a condensation but little less. Perhaps such substances as cuprite, fahlerz, and alum may have their molecules ar- ranged in this way, while others such as most metals, galena, garnet, analcime, and rock salt have the other and simpler arrangement. Pyramidal System. Turning now to the pyramidal system we may as before suppose space divided into similar and equal rectangular parallelopipeds by three sets of parallel planes; the first set parallel to the plane of the two equal crystallographic axes OX, OY, and equidistant from one another at some distance which we may call ¢; the second set parallel to a plane passing through the third axis OZ and through one of the equal axes OX, and equi- distant from one another at some distance a, where the ratio a@:c is that of one of the equal axes to third axis of the crystal; and the third set parallel to ZOY and also at the distance a from one another. If now we suppose the molecules arranged so that one of them is in each point of intersection of three of the planes, the arrangement will satisfy the optical requirements. Also as before every plane which intersects the three axes OX, OY, OZ in points «,, y,, 2, respectively, where z,, y, and z, are the places of molecules, will be a plane of maximum condensation, and therefore a possible face of the crystal. And as before the relative condensations in the several sets of planes of maximum condensation will be proportional to the perpendicular distances between the individual planes of the sets. If c be greater than a, the planes parallel to XOY, that is to the face 001, will have the greatest maximum of condensation, and planes parallel to the faces 100 will have the next less maximum of condensation. If a be greater than c, 100 will be the plane of greatest maximum, and 001 or 110 , ; ‘ ay’. will come next after it according as c or —= is the greater. Ve If P be the length of the perpendicular from O upon the plane through ,. y,, 2, we shall have P equal to the reciprocal of eat Ger ee and if M be the least common multiple of p, g, 7, omitting any of them which may be infinite, P/M will be the perpendicular distance between the successive planes of the set, and will be the reciprocal of a MN SS MN? (a) * Ga) +(e) pa qa re which is the reciprocal of af (“y+ (“) + (<)., a a c where h, k, 1 are the symbols of the face in Miller’s notation. Vou. XIV. Part III. 49 a) is 9) Oo Pror. LIVEING, ON SOLUTION AND CRYSTALLIZATION. Example 1. Idocrase. ¢/a = "5351. Distance between planes parallel to 100 = 1:0000 : . % 110 = ‘7071 r ‘ . 001 = ‘5351 g : r 101 = 4718 P - : 210 = “4472 F r r 111 = -4267 Z i * 201 = °3653 Rs ° 211 = ‘3431 ¥ E ‘ 310 = 3162 " f r 221 = +2950 t . 2 112 = 2502 Now the forms most commonly met with in idocrase are 100, 110, 001, 101 and 310, of which the first four are the four planes of greatest condensation if the molecules have the arrangement supposed. Moreover the cleavages are parallel to 100 and 110 and less distinct parallel to 001. Example 2. Rutile. c/a = ‘6441. (a) P for 100 = 1-0000 P for 310 = ‘3162 (m) , MLO sero » 221 =°3100 (c) , OOl= 6441 » BOl = 2964 (e) , 101= 5416 » 320 = 2774 (s) , lll= -4762 » 321 = 2548 (h) , 210= 4472 , 410 = 2425 , 201= 3950 31321790 , 2l= 3674 The most common forms of rutile are the first five of the above, and the cleavages are parallel to 100 and 110. Example 3. Anatase. If we take the axes as given by Miller, we have c/a=1°7771, and (a) for 100, P = 10000 (i © 5, LLG, TiS ear U0! Cys! 0 PS Thi () , 101, P= ‘8715 vv _ SAO, P= ‘6642 (pp) 53) Ud; P= ‘6570 7) ee UL P= +4813 In crystals belonging to this system there are two variations which are always possible in the crystallographic elements: these are 1°. the pair of equal axes may be turned through half a right angle at the same time that they are diminished (or increased) in Pror. LIVEING, ON SOLUTION AND CRYSTALLIZATION. 381 the ratio /2:1; and 2°. the ratio of each of the equal axes to the third axis may be Increased or diminished in any simple proportion. In either case there will be corre- sponding changes in the symbols of the faces. In the case of anatase if we take the octahedron indicated as (p) in the foregoing list to be the form 101, that is, take as equal axes lines in the plane of the square section bisecting the opposite edges of the octahedron instead of lines joining opposite angles, the ratio c/a will become 275132, and we shall have for the distances between the sets of planes parallel to the several faces above indicated and some others, for (c) 001, P= P5132 (m) 100, P =1-:0000 (a) 110, P= ‘7071 (p) 101, IP = Bey @) FUE P= ‘6807 102, IP = “(Ses (e) 112, [P= Siler 201, P= 4904 210, P= ‘4472 211, P= °4403 Now the most common forms of anatase are those indicated as (c) and (jp), and these are the directions of cleavage. The forms m, q, e are also frequent. It will be observed that whether we adopt the arrangement of molecules which corresponds to the first chosen axis or the other, the form (c) is that for which the condensation is greatest, while in the former case (a) in the latter (m) comes next in condensation. If we adopt the former position for the axes and suppose the molecules to occupy the centres of the faces of the parallelopipeds into which we suppose space to be divided, we shall get the condensation in the planes parallel to the faces as follows: (c) 001, P =°8885 (@) eae P6510 (a) 100, P = 5000 (2) eal} = “4541 (e) 101, P = ‘4358 m 110, 12 = Sie r 115, = ‘3176 1 P = 2766 102, [P= Opie v In = ‘2389 ZANE = ‘2168 u 105, Plows q 201, P ='1604 49—2, 382 Pror. LIVEING, ON SOLUTION AND CRYSTALLIZATION. This arrangement makes the two cleavages coincide with the planes of greatest con- densation; and in other respects tallies with the habit of anatase in bringing comparatively high in the scale of condensation forms such as 115 and 117 which are not uncommon in this mineral but infrequent in others. Example 4. Cassiterite. We find for the distances between the planes parallel to the several faces for (a) 100, P = 1:0000 (m) 110, P= E001 (ec) 001, Olan (e) 101, P= 5329 (5) eeletele P= 4873 (A) 210, P= ‘4472 and that for other faces the order in descending magnitude is 201, 211, 310, 221, 301, 311, 320, 212, 321, 410, 302, &c. The cleavages are parallel to 100 and 110 the planes of greatest condensation, and the forms (a), (m), (e), (s) are of common occurrence. Rhombohedral System. For the rhombohedral system we may suppose space divided by three sets of parallel planes into equal rhombohedra and a molecule to be placed at every point of intersec- tion of three planes. We shall have to take for origin of the axes one of the solid angles contained by three equal plane angles, and for axes the three straight lines containing those three plane angles. Then, as before, every plane which meets the three axes in points which are the positions of molecules will be a surface of maximum condensation. Also all the planes parallel to the faces of the form kl will be planes of equal conden- sation, where the order of the symbols h, k, J is immaterial, provided that when they change sign they all change together. The distance between successive planes of the set parallel to the face hkl will in general, on account of the obliquity of the axes, be different from the distance between successive planes of the set parallel to Akd, and these two sets belong to different crystallographic forms. The calculation of the distances between successive planes of a set is not quite so simple with oblique axes as with rectangular axes, but if = A, B, C (Pig. 3) be the places on the sphere of projection i of the poles of the form 100, X, Y, Z the traces on’ the ‘Cc same sphere of the three axes, V0 the pole of the form 111, and P that of the face hkl, which is parallel to a plane which intersects the axes in z,, y,, and z, respectively, mae Fig. 3. the angles PO and POA can be found by formule given : in Miller's crystallography, OA is the angular element of e ‘ . the crystal and OX can be found from the formula ‘ \ \ tan OX =2cotOA. We have then two sides PO, OX and ¥ 5 cee the included angle POX of the spherical triangle POX, Pror, LIVEING, ON SOLUTION AND ORYSTALLIZATION. 383 whence PX may be found, and the perpendicular distance from the origin upon the plane which passes through a,, y,, 2, 18 pcos PX, Also between the origin and this plane there are (including this plane) in all MW planes belonging to the set where MW is the least common multiple of p, q and r. The distance between successive planes of the set is therefore P cos PX or eee I EEXe M h Example 1. Calcite. Angular element 44°. 37’. Distances between the planes of sets parallel to the faces (r) 001 ‘9448 (0) 111 4425 (a) O11 T767 211 ‘4036 Gaye id 5999 (os) 210 ‘3844 (e) O11 ‘5951 | (m) 311 ‘3260 (b) 207 ‘44.84 (e-) 122 3150 Here the cleavage form (r) 001 is that for which the condensation is far the greatest, and the forms which are high in this list are of very frequent occurrence. Example 2. Chabasie. Angular element 50°. 45’. Distances between the planes of sets parallel to the faces (r) 001 ‘9908 | 210 ‘4620 (a) O11 ‘7387 | (Eyn) g210 4263 (e) O11 ‘6679 211 4.265 (s) 111 5922 | 211 4114, (o) 111 5220 | 211 3743 In this crystal the cleavage is parallel to the faces 001 for which the condensa- tion is greatest, and the most common forms and combinations are 1, er, ers, ersa, ersta: all of which are amongst those for which the condensations are high maxima. The two foregoing examples are crystals which generally assume rhombohedral forms and have easily obtained rhombohedral cleavages, and the rhombohedral arrangement of the molecules satisfies the observed facts. But there are many crystals of this system which generally assume a hexagonal character, and for them we may suppose a different arrangement of the molecules. These crystals may be referred, as is done by many erystallographers, to four axes, of which three are equal to one another and he in one plane and are equally inclined to ove another, while the fourth is at right angles to them, and may be unequal to them in magnitude. Now if we suppose space to be 584 Pror. LIVEING, ON SOLUTION AND CRYSTALLIZATION. divided into equal right triangular prisms by four sets of parallel planes, each set parallel to two of the axes (one of the sets being of course parallel to the three equal axes), and suppose a molecule placed in every point where three planes intersect, we shall have an arrangement which will satisfy the optical conditions observed in such crystals, for we shall have an arrangement which is symmetrical about the axis which is at right angles to the other three; and the surfaces of maximum condensation will, as before, be the planes which satisfy the law of indices. There are always two positions possible for the three equal axes, one set of axes being inclined at 30° to the other set but in the same plane with them. Also it should be observed that a crystal with the molecular arrangement here sup- posed may assume a rhombohedral character so far as its external shape is concerned; because the symmetrical omission of every alternate face of a hexagonal crystal will leave a form which will satisfy the conditions of equilibrium, for it will have no unbalanced edges or angles. If OS, OT, OY (Fig. 4) be the directions of the three equal axes, OW that of the axis at right angles to them, and s, y, t, w be the positions of molecules in those lines at distances from O which are multiples, s, ¢, y, w, of the lengths of the respective axes, and OW be per- pendicular to st, On perpendicular to Nw; then since sO and Ot are known, and the angle sOt is 120°, ON can be found by ordinary trigono- metrical formule, and we have On = w0 sin. tan™ oy wO Then the perpendicular distance between the planes parallel to that through s, y, ¢, w will be On divided by M the least common multiple of s, t, y and w. s/M, t/M, y/M, and w/M, in their lowest terms will be the indices of the face parallel to the plane sytw in Weiss’ notation reduced to fractions with unity as numerator. Example 3. Apatite. Angular element 55°. 40’ in the rhombohedral system. If we choose as equal axes on the hexagonal system the lines which bisect the opposite angles of the hexagonal prism of most common occurrence, to which Miller assigns the symbol 011 (a), and take that form to have the symbol 1010 where the last index belongs to the vertical axis, and make the length of the vertical axis unity, the length of the other axes will be 13660, and the perpendicular distances between the planes of the sets parallel to the faces will be we) co o1 Pror. LIVEING, ON SOLUTION AND CRYSTALLIZATION. Miller’s notation Weiss’ indices reduced Distance. (rhombohedral). (hexagonal), a Oll 0110 11829 an ell 0001 10000 x 120 0111 ‘7636 be oe 1210 ‘6830 r 100, 122 1211 5640 2. 13% 0221 “5090 6 2Bil 0112 4605 eee 0 Nas | 2421 3231 0331 ‘3667 0223 2904. This table includes the planes for which on the arrangement supposed the conden- sation is greatest, and the forms which are most common in apatite. The cleavages are parallel to the first two, that parallel to a being more easily obtained than that parallel to o. Example 4. Greenockite. Angular element on rhombohedral system 43°.37’. Taking as in the last example the equal axes to be the lines bisecting the opposite angles of the hexagonal prisms to which Miller assigns the symbol 011 (a), and assigning to it the symbol 0110, and making the vertical axis unity, the length of the equal axes will be 1:2118, and the perpendicular distances between the planes of the sets parallel to the several faces will be Miller’s notation. Weiss’ indices reduced. Distance. a Oll 0110 10494 a itt 0001 1-0000 x 120 0111 ‘7240 Bb QU 1210 6059 r 100 1211 ‘5183 zZ. 13t 0221 4.648 7 280 0112 4.514 These are the forms which commonly occur in this mineral, and they are those for which the condensation is greatest on the supposition here made as to the arrangement of molecules. The cleavages are parallel to the first two forms. Prismatic System. In the prismatic system there are three axes at right angles to each other and all unequal. We have to suppose, as before, space divided into rectangular parallelopipeds by three sets of parallel planes, the planes of each set parallel to two of the axes and at a distance apart equal to the length of the third axis. A molecule may be supposed placed in every intersection of three planes. If a, b, c be the lengths of the three axes 386 Pror. LIVEING, ON SOLUTION AND CRYSTALLIZATION. we shall have by reasoning precisely the same as is given above for the pyramidal system, P, the perpendicular distance between the successive planes parallel to a face of the crystal h, k, J, on Miller’s notation, equal to the reciprocal of va) +0) +G)- Applying this to particular cases: Example 1. Epsomite. a:b: c=1 : 0:9901 : 05709. The distances between the successive planes parallel to the faces are Miller’s notation. Distance. Miller’s notation. Distance, a 100 1:0000 ip PAW 4.463 b 010 ‘9901 120 4437 m 110 ‘7036 az 021 3740 ec O01 5709 7 2011 3761 v 101 4958 ee ey sital 3516 m O01 4.946 112 "2645 ae Ni la | 4.433 8 122 2403 There is a very perfect cleavage in these crystals parallel to a, and a less perfect cleavage parallel to v. The forms most common are a, m, v, n, z. From the degree of con- densation parallel to b on the supposition here made we should have expected a cleavage parallel to b if not a frequent occurrence of that form. Example 2. Topaz. In this case the easily obtained cleavage indicates the longest axis. We must there- fore take the form which in Miller is designated (y) 201 to be 101, which is equivalent to doubling the length of the third axis, and is always crystallographically admissible. We then get a/b=tan 62°.9'.30", 2b/c=tan29°.5’.30”, and the distances between the successive planes of the sets parallel to the faces are Symbol with Miller’s axes. Symbol with new axes. Distance. ce O01 001 35945 a 100 100 18933 y 201 101 16752 n 101 102 13037 e 203 103 10125 b 010 010 10000 021 O11 ‘9636 w 401 201 9155 2 OL 012 ‘8738 m 110 110 "8842 102 104 8119 ke til 112 ‘7934 dL 210 210 6875 eT Be? 114 63038 n 310 310 5337 Pror, LIVEING, ON SOLUTION AND CRYSTALLIZATION. 387 The forms ¢, a, y, n, 7, m, 0 are very common: the cleavage parallel to ¢ is very perfect, and there are imperfect cleavages parallel to 7 and m. Example 3. Olivine. Here again the cleavage in one direction is easily obtained. If we take the cleavage plane for the plane perpendicular to the longest axis, we must take for the form 101 that to which Miller assigns the symbol (h) 102, which is equi- valent to halving the longest of his axes. We then get a/b =tan 47°.1’. 24”, b/c=2tan38°.27', and for the distances between the planes parallel to the several faces: Symbol with Miller’s axes, Symbol with new axes. Distance. a 100 100 10732 b 010 010 10000 s 110 110 ‘7316 e 001 001 6297 h 102 101 5432 012 O11 5329 Zea) 210 4728 nm 120 120 4532 eo! 201 ‘4084 d O11 021 3915 212 211 3781 e 122 12] 3679 310 310 3368 130 130. 3183 etl 221 3163 The faces a, b, s, c, k, d, e are frequent, and there is an easily obtained cleavage parallel to a and traces of cleavage parallel to 0. We have taken no account of the fact that the concentration of molecules in many of the planes is unequal in different directions; and it must be remembered that the distances given indicate only the average concentration. Further there are other arrangements of molecules which will satisfy the optical conditions as well as that above supposed. For example we may suppose space to be divided by three sets of planes of which one set are parallel to two of the axes a, b, and the other two sets are each perpendicular to the first set, and cut them in rhombuses whose sides are equal to the diagonals of the rectangles on a, b, and are inclined at the same angles as those diagonals. We may suppose the molecules placed in the points where three planes intersect as before. The condensation will be the same as before in most of the planes, but will be doubled in the planes which are parallel to a face Ak2I, when h and & are equal to one another or are both odd numbers, and this arrangement appears to accord with the observed forms and cleavages of some crystals better than the other. Thus Vou, XTV. Parr IU. 50 388 Pror. LIVEING, ON SOLUTION AND CRYSTALLIZATION. Example 4. Nitre. Taking the axes as Miller takes them, we get for the distances between the planes of the several sets parallel to the faces, if the molecules in the planes parallel to that of the axes a, ¢ lie in those diagonals of the rectangles on the axes which pass through the third axis, k& 101 P=1:9425 103 [P= TiO 100 16920 O11 ‘7646 e 001 11861 pe A 6967 301 10187 4 | 201 ‘6888 b 010 10000 210 6459 m 110 8609 a 1102 5597 The forms /, a, m are most common and there is a perfect cleavage parallel to k, a less perfect cleavage parallel to a, and imperfect cleavage parallel to m. Again, Example 5. Baryte. Taking the axes as Miller takes them, and supposing the molecules in the planes parallel to that in which the axes a, b lie to be at the extremities of those diagonals of the rectangles on a, b which pass through c, we get for the distances between the planes parallel to the several faces e O01 (P= is Fp Agha P= 6986 m 110 15506 d 012 6274 a 100 1:2276 201 5736 b =6010 10000 n 210 ‘DAB o 101 ‘9765 mw? VATS u Oll 8497 q 114 7149 @ 102 6734 | ka e118 “3899 There are perfect cleavages parallel to c and m, and the former more easily obtained than the latter, cleavage parallel to a less perfect, and parallel to 0 still less perfect ; but the faces r and gq, for which there is a considerable condensation, are not common and there is no cleavage parallel to them. It may be noted that in the first example given above of a crystal of this system, namely epsomite, if we suppose the same sort of arrangement as that here suggested, with the molecules in the planes parallel to the axes a, ¢ in the diagonals, the face v will come next to @ in the order of concentration, and the face s will come high in the list. This is in agreement with observation since there is a cleavage parallel to v only less perfect than that parallel to a, and s is a form of not infrequent occurrence. Crystals like karstenite with three cleavages at right angles to one another correspond to the first molecular arrangement. In fact in karstenite we have two axes very nearly equal and a very perfect cleavage at right angles to each of them; while the third axis is shorter than the other two and the cleavage perpendicular to it is less perfect though easily obtained. Pror. LIVEING, ON SOLUTION AND CRYSTALLIZATION. 389 Oblique System. Coming now to the oblique system we may suppose space divided as before by three sets of planes, parallel to the axes two and two, into equal right prisms on rhomboidal bases, the height of the prisms being the length of the vertical axis and the sides of the base equal to the lengths of the other axes. We may suppose the molecules placed in the points of intersection of three planes. To find the perpendicular distance between the planes of any set, if OY, (Fig. 5) be the axis which is at right angles to the other two, OX,, OZ, the other two axes, Zt, X,t parallel to OX,, OZ, respectively, OX tZ, will be the rhomboidal base of one of the prisms. Then if Oa, Oc be drawn per- pendicular to X,t, Z,t respectively, and Od ~ perpendicular to Z,X,, Ob perpendicular to 3 Yd, the angles dOa, bOY, and dOc are the three angular elements of the crystal. And if we make the length of the axis OY, unity, Od = cotang Y,dO = cot Y,Ob. Also since d0a=dZ,0O we have OZ, = Od cosecdOa, and so also OX,=Od cosecdOc, which gives the lengths of the axes. Again dOX,=90° —dOc, and therefore a0X,= 90° —dOc—dOa, and Oa the distance between the planes parallel to the form 100 is equal to OX, cosa0X,. Also if a plane meet OX, OY and OZ in X,, Y,, Z,, where OX, is equal to p times OX, and so on, the length of the perpendicular OP from O on Z,, X, can be found, since OX, and OZ, are known; and the two sides of the right-angled triangle Y,OP being known the length of the perpendicular fram O upon Y,P may be found, and this length divided by the least common multiple of p, g, and r will be the distance between successive planes of the set parallel to the plane in question. Applying this to particular cases. Fig 5. Example 1. Gypsum. Angular elements 52°. 16’, 28°.16’, and 71°.51. If OY,=1, OX, = 69222, OZ, ="41450, and the distances between the planes of the sets parallel to the several faces are :— b 010 P =1-0000 | 210 P= ‘3231 a 100 6828 | 021 ‘3165 m 110 5639 | l wD S15 c 001 “4089 2 121 3022 120 4033 121 ‘2741 vy Oll ‘3785 | esr ait 9504 101 3793 | 012 2003 is oaslidill ‘B547 e 103 ‘1381 d 101 ‘3278 50—2 a) 390 Pror. LIVEING, ON SOLUTION AND CRYSTALLIZATION. The cleavage parallel to 6 the plane of greatest maximum of concentration is very perfect and easily obtained, there is a less perfect cleavage parallel to a, and also an imperfect fibrous cleavage parallel to n. The most common forms are }, J, m, h, n, and e. Example 2. Felspar. Angular elements 65° . 47’, 50°. 20’, and 63°.7’. Making OY,=1, we get OX,="6586 and OZ, = 5560. We may however take the length of OZ, double this amount and make corresponding alterations in the angular elements. If we do this we find the following for the distances between the planes of the sets parallel to the several faces. Symbol with Miller’s axes. Symbol with new axes. Distance, a 100 100 5913 b 010 010 1:0000 e 001 001 9983 n O21 oll ‘7065 111 112 *6553 y 201 101 6493 uw ~221 111 5446 m 110 110 5090 cae AUDI 102 5070 O11 012 “4466 120 120 “3815 q 208 103 3672 121 122 3511 h 023 013 “3156 211 212 3088 z 130 130 2903 210 210 "2835 s 131 132 2785 102 104 2779 ge AA 114 ‘2678 212 214 "2456 012 014 2422 There are perfect cleavages parallel to faces 6 and c which have the greatest con- deusation. Besides these faces, 2, m, 0, y, mn, are of frequent occurrence and are all high in the list of condensation. Anorthic System. With regard to the anorthic system it does not seem necessary to say much. We may suppose space to be divided as before into parallelopipeds by three sets of parallel planes, each set being parallel to a plane containing two of the axes, and the edges of the parallelopipeds equal to the three axes respectively, and a molecule to be placed in each point of intersection of three planes. Faces which follow the law of indices will then be surfaces of maximum condensation; and we can generally so choose the axes that the cleavages shall be the planes of greatest maxima of condensation. Pror, LIVEING, ON SOLUTION AND ORYSTALLIZATION. 391 Inequalities of Growth. The unequal development of crystals in different dimensions depends more upon the circumstances than upon the surface energy. When crystallization takes place more rapidly than diffusion the parts of the liquid where the crystals are forming become depleted, and the crystals can only grow into the more saturated parts of the solution. Hence when magnesium sulphate, or oxalic acid, crystallizes out of a hot solution which is rather quickly cooled the crystals run into long needles. At the same time the elongated faces are generally planes of a high degree of condensation. The like may be said of tabular crystals. Thus the tabular crystals of barium chloride have their broad faces perpendicular to the longest axis, the axes being in the ratio 0°9574 : 1 : 15778. The planes perpen- dicular to the longest axis are those of greatest condensation. In silver nitrate the axes are as 09429 : 1 : 13697 and the broad faces of the crystals are perpendicular to the longest axis. Ammonium citrate has axes in the ratio 05746 : 1: 1:3749 and forms tabular crystals with the broad faces perpendicular to the longest axis. How the develop- ment of the crystal depends on circumstances is well seen in the case of the Iodo-sulphate of quinine, sometimes used instead of tourmalines as analysers of polarised light. When the crystals form rapidly from a hot solution which cools and becomes saturated through evaporation at the surface the crystals form broad plates which float on the liquid, but if the dish containing the solution is covered and cooled slowly the crystals form within the liquid in tolerably equally developed prisms with no tendency to a tabular form, Hemihedral forms. In regard to surface tension it is necessary to distinguish the two kinds of hemi- hedral forms, (a) those with parallel faces, and (6) those with inclined, or asymmetric, faces. In the hemihedral forms with parallel faces the stresses due to the surface tension at edges and angles are opposed to similar stresses on the opposite side of the erystal, and therefore in such forms the surface tensions are in equilibrium amongst themselves. In the hemihedral forms with inclined faces the surface tensions at edges and angles will in general give rise to stresses in the nature of couples which can be equilibrated only by opposite couples. It is this kind of asymmetric hemihedral formation which is connected as mentioned above with the properties of rotating the plane of polarisation of polarised light and of pyroelectricity. In order to give this connexion a physical basis it is necessary to suppose that in the formation of crystals having these properties the solidification is attended with a certain internal strain. This strain we may imagine to arise in different ways. It may arise from some want of symmetry in the molecules which prevents their assuming the crystalline arrangement without some strain. This may well be the case where the molecules them- selves seem to produce the rotation of the plane of polarisation as in tartaric acid and other compounds which possess rotatory power in solution and in the amorphous state, Or an internal strain may be left when crystallization occurs in a mixed mass of two or more substances which are not isomorphous, where the crystallization cannot take place with perfect freedom and where portions of one substance become entangled in the crystals of the other. Or a strain might be caused by some stress due to the electric field in 392 Pror. LIVEING, ON SOLUTION AND CRYSTALLIZATION. which the mass was situate at the time of crystallization. Whatever the origin of the strain, if it be one which is capable of being relieved in any degree by a counter stress due to surface energy it will become a factor in determining the development of the erystal. The condition of equilibrium will no longer be that the surface energy of the erystal shall be a minimum, but that this energy together with that of the strain shall be a minimum. In fact if an unsymmetrical development of the external form of the crystal would produce a strain opposite to that which would result from any of the other causes acting independently, such an unsymmetrical development will in general take place. In so far as the surface tension counteracts the internal strain, the latter will be lessened and so will any physical property which arises directly from it. Hence we find that those substances which have rotatory power in the liquid state and form hemihedral crystals lose their rotatory power when crystallized. Sulphate of strychnia which has the rotatory power more strongly in the crystalline form than in solution forms holohedral crystals. There is evidence of another kind that such a strain does exist in some cases. For if a substance, such as dextro-rotatory tartaric acid, can only crystallize with an internal strain, this will tend to impede its crystallization and it will crystallize more readily under circumstances which relieve the strain. Because the strain implies a supply, or a retention, of a certain amount of energy, and crystallization will be facilitated if the necessary energy is supplied from a source independent of the surface energy of the crystal. Now the asymmetry, whatever its nature be, in the molecules of dextro-tartaric acid has its exact counterpart in the molecules of levo-tartaric acid, and we may reasonably suppose that the strain arising from this asymmetry in crystals of dextro-tartaric acid is opposite to that arising from the counter asymmetry in crystals of levo-tartaric acid, and that consequently in crystals formed of a mixture of the two acids in equal proportions the stresses being equal and opposite would produce no torsional strain, and crystallization would be facilitated. This supposition is in fact confirmed; for concentrated solutions of dextro- and levo-tartaric acid, which separately will not crystallize, when mixed form immediately crystals of the non-rotatory acid with an elevation of temperature. When a substance exhibits rotatory power in the crystallized state but not in solution or in the amorphous form we have to look for a common cause for both the rotatory power and the hemihedral crystallization. In sodium chlorate the cause may perhaps be traced as follows. The isomorphism of corresponding sodium and potassium salts is so well marked that we may be almost sure that sodium chlorate is capable of crystallizing in the same form as potassium chlorate, and vice versa, and as the former usually crystallizes in the cubic system and the latter in the oblique, that they are dimorphous, but that under ordinary circumstances the one crystallizes with greater facility in one form and the other in the other form. Now if the particles of sodium chlorate were aggregated partly according to the law of one form and partly according to the law of the other form there would probably result internal strains which would not be in equilibrium amongst themselves, and hence such an aggregation would not be possible unless there were some external stress to counteract them. That stress is supplied by the surface tension of the hemihedral form. This theory is borne out by the structure of boracite. For the examination of slices of boracite with Pror. LIVEING, ON SOLUTION AND CRYSTALLIZATION. 393 a polarismg microscope shews that it is built up of particles which have some of the optical characters of a biaxial erystal. These particles, which have been called “ parasite,” may be boracite in a dimorphous state as supposed by Volger, or they may be parts of the regular crystal which derive their optical characters from a strain as supposed by Klein. Their general arrangement depends on the external form of the crystal, but they are not uniformly disposed throughout the whole mass, and corresponding to this want of uniformity we find that the hemihedral character of the crystals has not been uniformly maintained during their growth. For it seldom happens that the alternate faces of the octahedron are altogether absent, usually they are less developed than their fellows and the alternations of holohedral and hemihedral growth are indicated by roughness of the surface. At those stages of growth, perhaps dependent on temperature, when the parasite was formed the hemihedral crystallization was necessary to the equilibrium, but it would not be necessary at other stages, and then the growth would be holohedral. If a pyroelectric crystal were placed in an electric field which would induce an electric state exactly the opposite of that which the crystal assumes when undergoing a change of temperature the strains due to the changing temperature would be relieved. And conversely if the material forming the crystal were placed during the growth of the crystal in such a field the regular crystalline arrangement of the molecules could not be assumed without strain, and crystallization would be impeded. But if the electric stress could be counteracted by the stress arismg from a hemihedral formation, crystallization could occur in the hemihedral form. When a crystal so formed was removed from the electric field in which it had grown it would be left with a strain, namely the strain due to its hemihedral growth. Action of solvents. The explanation of the forms which crystals take on solidifying from solutions applies equally well to the development of faces on the surface of a crystalline mass which is undergoing slow solution. When the solvent is not nearly saturated differences of surface energy due to the form of the surface will have little effect, because there will always be a sensible increase of entropy resulting from the process of solution whatever the surface energy of the undissolved mass. But when the process of solution involves but little increase of entropy, as is the case when the solution is nearly saturated, those surfaces which have a minimum surface energy will sooner cease to be dissolved than those of greater energy, because there will be an increase of entropy by the transformation of the latter into surfaces of less energy through the process of solution. There are many more details, at some of which I have already hinted, which remain to be worked out. These I hope to make the subject of a subsequent communication. I have presented the problem just as it unfolded itself in my own mind; but it might have been more logically attacked in the inverse way; I might have laid down the principle that cleavage planes and the faces of dominant forms must be surfaces of the greatest maxima of molecular condensation, and thence determined the arrangement of the molecules. This may now be done, and I hope it will be a step towards the solution of the question, What is the law of mutual action of the molecules which causes them to assume such a disposition ? XIV. On Solution and Crystallization. No. Il. By G. D. Liverne, M.A,, Professor of Chemistry in the University of Cambridge. (Plate V1.) [Read Nov. 26, 1888.] IN my previous communication to the Society on this subject I said nothing about the physical causes which determine the arrangement of the molecules when passing from a fluid to a solid state, but only tried to shew that if the arrangements were such as I described the external form and the cleavage would, on well known mechanical principles, be such as we find them to be in fact. I purpose now to try and explain what these causes are. In accounting for the arrangement of the molecules I find it quite unnecessary to suppose that any special force is concerned; it is enough if the molecules attract each other, and so far as force goes the force of gravitation seems to be sufficient to produce the results. I haye to suppose, in accordance with other physical properties of matter, that the parts of every molecule are in continual motion, but that in the solid state the excursions of the parts from the centre of mass of the molecule do not on the average exceed certain limited distances; that these distances are on the average the same for the same kind of matter at the same temperature and pressure, and determine what may be called the molecular volume. Further I assume that the average distribution of the mass of each molecule within the molecular volume is the same for the same substance under the same conditions, but may differ for molecules of different kinds of matter. I assume in fact that the molecules, even of the chemical elements, are, in the solid state at least, complex aggregates, and that they have motions of their own which are of the same sort and on the average of the same intensity for all the molecules of the same chemical substance. To take first the simplest case: suppose that the excursions of the parts of the mole- cule extend on the average equally in all directions about the centre of mass of the molecule. The average space occupied by the molecule will be comprised in a sphere of which the centre will be the average position of the centre of mass of the molecule, and the radius the average distance to which the excursions of the parts from that centre extend. It is not intended to assume that parts of the molecule may never acquire such velocity as to go beyond the limits of this sphere, but that on the average, partly by collision with neighbouring molecules and partly by reason of their mutual attractions, they come up to but do not exceed this limit. If that be conceded, and if moreover the mutual action of the molecules is to attract one another, the problem of finding how Pror. LIVEING, ON SOLUTION AND CRYSTALLIZATION, No. II. 395 the molecules will arrange themselves in equilibrium becomes, by a well known mechanical principle, the problem of finding how the spheres representing the molecular volumes can be most closely packed in a given space, Now if we arrange equal spheres with their centres in one plane as closely as_ possible they will be arranged as the circles of like outline are arranged in fig. 1, so that each sphere is touched by six others. Above them other equal spheres can be arranged in like order and so that each sphere of the upper set, indicated in dotted outline in the figure, touches three of the lower set. Again another set can be arranged above the second set in like order so that each of the third set touches three of the second, and so on. If we call the radius of a sphere 6, the area of the equilateral hexagon circumscribing one of the circles in fig. 1 will be 2/307, and there is one circle for every such hexagon 1 2 V3" plane in which the centres of the first set of spheres lie and that in which the centres of the second set lie, is the length of the perpendicular let fall from one angle of a regular tetrahedron with edges equal to 2b on the opposite side, and this is equal to so that the number of circles per unit area is Also the distance between the 22b. Hence the number of spheres per unit volume is And this appears to 1 4V20 be the maximum number which can be packed in unit volume. We might have started with arranging the first set of spheres so that each one touches four others at points in one plane as in figure 2, and then arrange a second set in similar order above them, as indicated in dotted outline, in such wise that each sphere of the second set touches four of the first set, and so on with successive sets. On com- puting the number of spheres per unit volume we find it to be as before. In i 4 V2b% fact the two arrangements are really identical, and in the figures we are merely looking at the same thing from different sides. The plane through the centres of the row a, a’, a of the first set in fig. 1 which also passes through the centres of the row c, ¢, ¢ of the second set, has in fact the spheres arranged as in fig. 2; while the plane through the centres of the row b, b’, b of the first set in fig. 2, which also passes through the centres of the row e, e, of the second set, has the same arrangement as is shewn in fig. 1. Either arrange- ment is the same as if space were divided into equal cubes*, and a sphere placed with its centre at each angle, and also a sphere with its centre at the centre of each face, of every such cube. It is a combination of the two arrangements suggested for the molecules of a crystal of the cubic system. In this arrangement the relative condensations in planes parallel to the faces of the octahedron, cube and dodecahedron respectively are as ioe! 1 AB} : 3 : 2/2" substances which crystallize in the cubic system. The octahedral cleavage is in general This agrees sufficiently with the observed facts in the greater part of the * The square which forms the face of one cube in the | face. e, e fall in the centres of two faces at right angles plane of the figure is b’cd’c, and c’ falls in the centre of the | to the plane of the figure. Wor XIVa PART Ut. 51 396 Pror. LIVEING, ON SOLUTION AND CRYSTALLIZATION, No. IL. the easiest to obtain, and the cubic next. There are a small number of types of chemical compounds forming erystals in which a cubic cleavage is most easily obtained. These are the haloid salts of the alkalies, such as sodium and potassium chloride, a few sulphides and selenides of the type of galena, and a few sulpharsenides such as cobaltine, to which pyrite may be added. There are also a still smaller number of substances such as blende in which the dodecahedral cleavage predominates. These are, however, the exceptions rather than the rule, and as such their consideration may be deferred. The supposition above made for the sake of simplicity that the excursions of the parts of the molecule from its centre of mass are equal in all directions cannot be the general case. We should expect the excursions to be different in different directions on account of the different masses of the chemical atoms and their different relations to one another in the molecule. The average energy of the motion will however be the same in every direction, but the average molecular volume will be an ellipsoid with three un- equal axes. Further we cannot in general assume that the matter is equally distributed on the average in all parts of the molecular volume. It may for example be more massed about one or other axis of the ellipsoid, or about one or other plane of principal section. Whatever the massing of the matter in the molecule may be, if it be, as the very definite characters of the chemical compounds lead us to suppose, the same in every molecule of the same compound, it will tend to place every molecule in the same position with reference to surrounding molecules. This is as much as to say that it will tend to give all the ellipsoids the same orientation of their axes so that they will be all similarly situated. This appears to be also the arrangement which admits of the packing of the greatest number of equal and similar ellipsoids into a given space. Though I am not at present able to give a direct and complete mathematical de- monstration of the solution of this problem of packing the greatest number of ellipsoids into unit space, I have no doubt what the solution is and can prove it indirectly. The arrangement required is similar to that already indicated for spheres. By no arrangement can every ellipsoid be made to touch more than twelve others. Suppose now the ellipsoids to be all similarly situated and to be arranged as in fig. 3 so that each one is touched by six others at points lying in one plane which is the plane of the figure. Then any one of these, as that with its centre at A, may be touched by three others which have their centres in a plane above the plane of the figure, as the three indicated in broken outline, or the three indicated in dotted outline, as well as by three more with their centres in corresponding positions below the plane of the figure. In this way every ellipsoid will be touched by twelve others. Considering the ellipses which are the intersections of the plane of the paper with the ellipsoids which have their centres in that plane, since they are all equal, similar, and similarly situated, the lines joining the centres of any two which touch each other will pass through the point of contact. Also any such line as AC passing through the point of contact R will bisect the chord PZ between the points where the two adjacent ellipses touch the ellipse having its centre at A. Hence the tangent at R will be parallel to PT and pass through the centres of the adjacent ellipses B, D. Also if Q be the point of contact of the ellipses of which B, C are the centres AQ will be in the line of the Pror. LIVEING, ON SOLUTION AND CRYSTALLIZATION, No. II. 397 diameter conjugate to AT, for it bisects the chord PR which is parallel to BC and AT. Now there is one ellipse for every hexagon such as onsvwy formed by the tangents at the six points of contact of the ellipse having its centre at A with the surrounding ellipses, and the hexagon circumscribing an ellipse will be a minimum when each side is bisected in the point of contact. For if the sides oy, ns, vw be produced they will meet in the centres of the ellipses C, K, Z, and the ellipse having its centre at A and the triangle CXL will be the projections on the plane of the figure of a circle and its circumscribing triangle lying in a plane inclined at some angle to the plane of the paper. In the figure CA, KA and LA produced bisect the opposite sides of the triangle and hence CKL is the projection of the equilateral triangle circumscribing the circle of which the ellipse with centre A is the projection. Also the triangles Lyw, Con, Kvs are all similar to the triangle CXL, and therefore onsywy is the projection of the equilateral hexagon cir- cumscribing the circle. Such a hexagon is the smallest in area which can be drawn circumscribing the circle, and hence onsuwy is the smallest which can be drawn circum- scribing the ellipse with centre A. Hence when the ellipses are all similarly situated the hexagon formed by the tangents at the points of contact is a minimum, and since such hexagons fill the whole area, and there is one ellipse for every such hexagon the number of ellipses per unit area is a maximum. Now if we suppose space filled with the ellipsoids, and suppose the continuous, dotted, and broken-line ellipses in the figure to represent successive sets of ellipsoids with their centres in successive parallel planes, the plane through the centres of the broken-line ellipsoids y, uv which also passes through the centre of the dotted ellipsoid w, and through JZ, will cut the ellipsoids in a series of ellipses of which each is touched by six others just as in fig. 3; so will the plane through the centres of broken-line ellipsoids y, n which also passes through 0, B, C, and the plane through the centres of the broken-line ellipsoids n, v which also passes through s, XK, D. Hence in each of the four planes, namely that of the figure and the three last mentioned planes, the number of ellipses will be a Maximum per unit of area when the ellipses are all similar and similarly situated; and since in four different planes of which no two are parallel the concentration is a maximum under these conditions, I infer that the number of ellipsoids which can be packed into a given space is greatest when they are all similarly situated. If this were in doubt it becomes more certain when it is seen that, if the condition of similarity of situation be satisfied, the orientation of the axes is a, matter of indifference. To prove that statement, return to the hexagon onsvwy. The area of it will bear to the area of the ellipse the same ratio which the area of the circumscribing equilateral hexagon bears to the circle, which is 2/3: a. Hence the area of the hexagon is 2 /3a,b,, where a,, b, are the semi-axes of the ellipse. Again considering one of the ellipsoids, indicated in dotted outline in fig. 3, which touches the three ellipsoids with centres A, B, C, since all the ellipsoids are similar and similarly situated the lines joining the centre of the dotted ellipsoid with A, B, C will pass through the points of contact and be bisected at these points, which will therefore lie in one plane parallel to the plane ABC. Also the centre of the dotted ellipsoid will. lie in the intersection of the three tangent planes at P, Q, R. 5)j_—* 398 Pror. LIVEING, ON SOLUTION AND CRYSTALLIZATION, No. II. For by reasoning similar to that given above to shew that the tangent at Q passes through A it may be shewn that in the plane of B, C and the centre of the dotted ellipsoid the tangent at Q passes through the centre of the dotted ellipsoid, and so also it may be shewn that the tangent planes at P, R also pass through the centre of the dotted ellipsoid. Let fig. # represent a section through this ellipsoid and that with centre dA by the tangent plane through Q, and let H be the centre of the dotted ellipsoid and HO the intersection of the tangent planes through P, R; p the point of contact of the two ellipsoids, and f the point where the ellipsoid with centre A touches a plane parallel to the plane of fig. 3. Then Af will be parallel to HO for they both bisect the chords through p which are parallel to AO, and AM, Af are conjugate semi-diameters. Draw the ordinate pn parallel to Af Then since AH is bisected in p, HO =2pn, and AO =2An. Also in fig. 3 0, M, A are the same points as O, M, A in fig. 4. And AN =4AQ=#4o, and 4do. AN=AM’; therefore 340? = 44M and (in fig. +) 3An* = AM”. pireAne ‘ot. % a Ap? ama) 2 therefore re =1—{=3, and 8Af? = 12pn’ = 3.HO®, and 2 /2Af=/3HO. Now let the inclination of Af or HO to the plane of fig. 3 be a, then HOsina is the perpendicular distance between the plane ABC and the parallel plane through H. And the number of ellipsoids per unit volume is 1 “ V3 Re 1 2/3AM.BQ 2/2Afsina 4./2AM.BQ.Afsina But 84M@.BQ.Afsina is the volume of the parallelopiped whose sides touch the ellipsoid at the extremities of three conjugate diameters, which is equal to 8abe, if a, b, ¢ be the semi-axes of the ellipsoid. Hence the number of ellipsoids per unit volume is on this arrangement, which gives the maximum number, u /2 1 —~ or —=—, Sabe 4, /2abe which is independent of the orientation of the axes. We might have started with the arrangement indicated in fig. 5, in which each ellipsoid with its centre in the plane of the paper is touched at points in that plane by four others, and is also touched by four others, indicated in dotted outline, which have their centres in Pror. LIVEING, ON SOLUTION AND CRYSTALLIZATION, No. II. 399 a parallel plane above the plane of the paper, and also by other four in a similar position below the plane of the paper, twelve in all. - The four ellipsoids with centres in the plane of fig. 5 which touch any one with centre C must touch it at the extremities of conjugate diameters, if each (dotted) ellipsoid, with its centre in the plane above the figure, is to touch simultaneously four below it. The tangents at P, R will then be parallel to BC, AB respectively, and will meet in O the point of intersection of AC, BD, and we shall have BO? + CO? = 2 (OR? + BR’) = 2 (BM? + Cm’). PN? BN? i oe NV. — 9 Lt! — —_—_. = Also, since CO=2PN, and BO BN, and Ome aa Bir il. COX. BOI rig Cm ; BIS” and because the circumscribing parallelogram is double the inscribed parallelogram, Om. BM =2PN.NB; a PN _BM _BM q 4BM* BO’ _ Cm 2NB BO’ *° “BO * BM? 2BM? BM 1 therefore Ror = 1, and BO a5 , and by similar reasoning Cape COR es If now fig. 8 represent a section through the centre H of the ellipsoid which touches each of the four with centres A, B, C, D, the straight lines joining HB and HD will pass through the points of contact p, g, and be bisected in those points; and HO, which bisects BD, also bisects gp, and is the diameter to which gp is ordinate, and is parallel to Bt the diameter conjugate to BM. And by reasoning similar to that above given it may be shewn that HO= V2Bt. The same value for HO would have been found if we had started with supposing the dotted ellipsoid to touch these with centres A and C. Hence it will touch all four ellipsoids with centres A, B, C, D simultaneously. This arrangement is really the same as before and will lead to the same result. In fact the plane through the centres of any two adjacent dotted ellipsoids in fig. 3 as 9, s, which also passes through C, D, cuts the ellipsoids in ellipses arranged as in fig. 5; so does the plane through the centres of dotted ellipses 0, w, and through B, LZ; and so on. While in fig. 5 the plane through the centres of the dotted ellipsoids O, S, which aiso passes through B cuts the ellipsoids in the way indicated in fig. 3; so does the plane through the centres of the dotted ellipses S, 7, which passes through C’; and so on. Further, the hexagon in fig. 3 is the intersection of the plane of the figure with the tangent planes at the points of contact between the ellipsoids. These tangent planes form dodecahedrons circumscribing the ellipsoids; and since each side of the hexagon in fig. 3 is bisected in the point where it touches an ellipse, and the same is 400 Pror, LIVEING, ON SOLUTION AND CRYSTALLIZATION, No. IL. true of the hexagons in the other three planes of section above mentioned, it appears that the points at which the faces of the dodecahedrons touch the ellipsoids are the centres of those faces. Now when this condition of touching the ellipsoid at the centres of the faces is satisfied the volume of the dodecahedron will be a minimum, inasmuch as any small shift of one of the planes forming the dodecahedron by rolling it on the surface of the ellipsoid must increase the volume of the figure. Thus, if ABCD be one of the faces of the dode- eahedron, touching an ellipsoid at its centre O, and it be turned so as to touch the ellipsoid at 0’, and the new position of the face A’B’C’D' intersect the old in Pp, Pp will lie between O and O', and the area ABPp will be greater than the area CPpD, and the wedge AB'p will be greater than the wedge C’PD if the distance OO’ be finite. Hence the circumscribing dodecahedron will have minimum volume when the ellipsoids are all similarly situated. And, since the dodecahedra fill space, it follows that there will be the greatest number of them in unit volume, and _there- fore the greatest number of ellipsoids in unit volume, when they are all similarly situated. Such an arrangement will therefore be assumed by the erystallizmg molecules in consequence of their mutual attractions. Now if we transfer our thoughts to the centres of the ellipsoids, we shall find that their arrangement, when the ellipsoids are situated as above described, will present different degrees of symmetry depending on the relative magnitudes of the axes of the ellipsoids and upon their orientation. For consider the ellipsoids indicated in continuous outline in fig. 5 as having their centres in the plane of the paper, those indicated in dotted outline with their centres in a plane above that of the paper, and again a third set with their centres, which we may call b, ¢, d, e, f &c. corresponding to B, C, D, E, F &c., but in a plane above the dotted set, then the plane of the paper and the parallel plane bdef together with the planes BbdD, EefF, BbfF and DdeF, will form a parallelopiped. And the whole space may be divided by parallel planes into similar and equal parallelo- pipeds which have the centre of an ellipsoid in every angular point and one in the centre of every face. Also every ellipsoid in the space will occupy one or other of these positions. Also if we suppose wos in fig. 3 to be the centres of the dotted ellipsoids lying in a plane below that of the figure, ynv to be the centres of the broken line ellipsoids lying in a plane below wos, and A’ to be the centre of the ellipsoid in a plane still lower which touches the three broken line ellipsoids: then the six planes Aoyw, A’vsn, Awvs, A'yon, Asno, A’'ywv will form a parallelopiped. And the whole space can be divided by planes parallel to these into similar and equal parallelopipeds which have the centre of an ellipsoid in every angular point. The characters of these parallelopipeds will differ according to the orientation of the axes of the ellipsoid, and according to the relative lengths of the axes. In the most general case represented in figs. 8 and 5 each face of the parallelopiped Pror. LIVEING, ON SOLUTION AND CRYSTALLIZATION, No. II. 401 will be equal and similar to its opposite face but to no other, and the erystal with its mole- cules arranged in the same way as the ellipsoids will be anorthic. If the plane of fig. 5 be a plane of principal section of the ellipsoids the edges Bb, Dd, Ee, Ff of the parallelopiped will be parallel to an axis and perpendicular to the plane BDEF. The crystal will in this case belong to the oblique system, if the points of contact in the plane BDEF be not the extremities of axes. But if the points of contact P, R, are the extremities of axes, the crystal having such a molecular arrangement will belong to the right prismatic system. Also in the parallelopiped of which A in fig. 3 is one angular point, the edge As will for certain orientations of the axes of the ellipsoid be perpendicular to the plane Aoyw, and the crystal will be oblique. Further for certain orientations of the axes the edges Ao, Aw will also be equal to one another, or all three edges may be at right angles to one another, and the crystal will be right prismatic. If two of the axes of the ellipsoids be equal to one another, so that the ellipsoids become spheroids, and if the axis of revolution be perpendicular to the plane of fig. 5, the arrangement of molecules will correspond to the pyramidal system, the axis of the spheroids being the axis of symmetry. If the axis of the spheroids be perpendicular to a plane such as Dbe through the diagonals of three adjacent faces of the parallelopiped, the parallelopiped is a rhombohe- dron, and the crystal will belong to the rhombohedral system. The axis of revolution is in this case perpendicular to the plane of fig. 3. In each of the last two cases there are certain relations between the length of the axis of the spheroid and the diameter of the principal circular section which give a cubic arrangement. If in fig. 5 the ellipsoids become spheroids with their axes of revolution perpendicular to the plane of the figure, figure 2 will represent them, and consider the four with eentres c’, c, d, d together with that which touches them, indicated in dotted outline, with centre e, and let fig. 6 represent a section of them through ced’, which will be in a plane perpendicular to the plane of fig. 2. Then if the radius of the principal circular section be a, and b be the semi-axis of revolution, and we take a=./2b, we have in fig. 6, pee aaa pl Tens fai gp tags therefore a =1-}=}, and Vp =b= 7, and pW =5, and eQ = 2pN =a, and e is the centre of the cube described upon the square c’cdd’. 402 Pror, LIVEING, ON SOLUTION AND CRYSTALLIZATION, No. II. And if the whole space be divided into such cubes there will be one spheroid at every corner of each cube and one at the centre of each cube and none in any other position, Hence the crystal will have its external form in the cubic system, and in this arrangement the planes of the rhombic dodecahedron will be those of greatest condensation of molecules and the crystal will cleave most easily in those directions. It may be doubted whether such a crystal would be truly isotropic, but the want of that character in the molecules themselves need not, and for reasons given below I think it does not, influence light passing through the crystals so as to produce double refraction. Again, if in fig. 1 the different circles represent spheroids with their centres in different planes; those in continuous outline with their centres in the plane of the paper; those in dotted outline next above them; those in broken outline next below those in continuous outline; then the planes through the centres c’da’g, c’a‘be, and c’bdf will each be at right angles to the other two if the radius of the principal circular section be double the semi-axis of revolution. For let fig. 7 represent a section through cde, and let a be the radius dM of the principal circular section, 6 the semi-axis of revolution, ¢'@ and pn perpendiculars on dM. Then if the three planes through c’ intersect one another at right angles we shall have de*=2a’*, also Q will be the centre of the equilateral 9 triangle da’b, with sides 2a, and dQ= 5 a. Also c’d is bisected in the point of contact p, and dp, dn are halves of de’, dQ respectively, and + es eee i}e* a a. a 33 a pn? = dp* — dn* = eta Also pa 1, and therefore tae 1—4=8, and ont ze or a = 4b", and a= 20. ae b° Crs Also if space be everywhere divided by planes parallel to the three planes above mentioned passing through the centres of the spheroids, it will be divided into cubes with the centre of a spheroid at every angle. The external form of a crystal with such molecules so aggregated will be of the cubic system, and the most easily obtaimed cleavage will be cubic. It may be doubted whether it would be truly isotropic on account of the want of this quality in the molecules individually. It is however by no means certain that the optical characters of crystals depend directly on those of the molecules. For double refraction wholly disappears when the crystals are fused or dissolved in menstrua. More- over the motions which give the mulecular volume are those which produce expansion when the substance is heated, and are probably distinct from those vibrations which produce radiation in solids as well as in gases. It is therefore quite possible that these two cases in which the arrangement is such as to make the crystal belong externally to the enbic system may be represented by blende with its predominant dodecahedral cleavage, and by rock salt and galena with predominant cubic cleavages. The ordinary laws of mechanics require that the molecules should, when in the most stable equilibrium, be packed as closely as possible, and this appears to give the arrange- ment above indicated and to determine that all the molecules shall be similarly situated. 3ut it does not determine the orientation of the axes of the ellipsoids which represent the Pror. LIVEING, ON SOLUTION AND CRYSTALLIZATION, No. II. 403 molecular volumes, on which the system of crystallization depends. That must be due to the mutual action of the molecules, and, as I have said above, I regard it as dependent on the massing of the matter in the molecular volume. If the mass is chiefly aggregated about any particular diametral plane one would expect that plane to be the plane of fig. 3 in which each molecule meets six others while it meets only three in a plane above and three in a plane below that of the figure. While if the mass is aggregated chiefly about any one diametral line we should expect the diametral plane conjugate to it to assume the position of the plane of fig. 5, since the molecules are symmetrically arranged about a diameter conjugate to the diametral planes represented in the figure, and the position is that which will bring the masses as close as possible. I assume that the influence of the immediately contiguous molecules is much greater than that of those more distant, as we know experimentally that this is the case in capillary actions. To find the relation between the length of the axis of revolution and the diameter of the principal section in any given crystal of the rhombohedral system, if c't in fig. 7 be the trace of the plane through the centres c’a’b in fig. 1 the angle c’tQ will be the angle between the axis of the rhombohedron and the normal to one of the adjacent faces, that is the angular element of the crystal; and if we call this angle D, CQ 2pn_ 22d Woh = Qf dn Ya Also the perpendicular distance between successive planes passing through the centres of spheroids and parallel to a face of the rhombohedron will be E 2 V2 /3ab i) oe V8" + a The distance between successive planes perpendicular to the axis of the rhombohedron, Bt en 2b , is —— eis And the distance between successive planes parallel to the axis is Qt, or are The first of these three distances will be greatest if @ is greater than b but not greater than 8b, and in this case the principal cleavage will be rhombohedral. The second of the distances will be greatest when 6b is greater than a, in which case the principal cleavage will be perpendicular to the axis. The third distance will be greatest when a is greater than 8b, in which case the principal cleavage will be parallel to the axis. It is obvious that the value of D, and consequently the ratio between a and 6, will depend on the choice of the rhombohedron which is taken as the fundamental one. For calcite we have for the cleavage rhombohedron D= 44° ,36'.36" and a:b =1: 34877. Vou. XIV. Parr III. oO bo 404 Pror. LIVEING, ON SOLUTION AND CRYSTALLIZATION, No. II. For quartz, taking the usual value for D, namely 51°. 47’, we find a:b =1 : 4490, which makes the fundamental rhombohedron the form of least surface tension, and the six-sided prism (011) comes next to it. I have now, I hope, sufficiently explained the principles which determine the regular aggregation of the molecules in assuming the solid from the liquid state, and how this leads up to the system of crystallization and the predominant forms and cleavages. There is, however, one arrangement of molecules, suggested in the first part of this communication to explain the forms of hexagonal crystals, which is not accounted for by the principle here laid down. This is the arrangement at the angular points of right triangular prisms. If there were any good reasons for thinking that the molecules would in consequence of their mutual action so place themselves, the facts of hexagonal crystallization would be adequately accounted for. But such an arrangement does not appear to me necessary for the explanation of those facts. Whenever there is no great difference between the surface tension of faces of different forms, there must always be a tendency for the crystal to assume such a combination as will approach to a globular form, as we see in hexagonal crystals. And there is no instance, so far as I am aware, in which a crystal cleaves into a six-sided pyramid. The cleavages in this system are either rhombohedral, or are perpendicular to, or parallel to, the axis, and are adequately accounted for by the rhombohedral, arrangement of molecules. It appears from this that dimorphous substances in their different forms differ in the relative dimensions, or in the orientation, of their molecular volumes. For instance the spheroidal molecules of calcite either have the direction of their axes altered, or become ellipsoidal when the material takes the form of aragonite. Now of all the circumstances which help to determine which of two crystalline forms a dimorphous substance shall assume temperature seems the most important. A change of temperature may alter the relative as well as the absolute distance of the excursions of the parts of the molecule from the centre of mass, and thus alter the relative dimensions of the molecular volume. It may also affect the average distribution of the mass in the molecular volume and so alter the orientation, of the axes of the ellipsoid which represents it. But changes of this sort must, except for the constraint of the solid state, be reversed when the temperature comes back to its former -degree, and in such a case the altered crystalline form must be an unstable one. This is no doubt the case with some substances like sulphur which take one form when crystallized from fusion and a different form when crystallized from solution at a lower temperature. The former crystals are always unstable and become split and cracked by the change which the molecules undergo when cooled. Something more seems wanted to account for such permanent difference as we find between aragonite and calcite. For this I look to the aggregation of matter in the individual molecule. Increase of temperature tends to multiply the number of molecules into which a given mass is divided. We know that it is so in gases, and it may be so in solutions. The number of particles of calcium carbonate each containing one chemical atom of carbon may be greater in a molecule of aragonite than in a molecule of calcite. Each molecule is almost certainly a complicated aggregate containing many chemical atoms of calcium and carbon, and it is Pror. LIVEING, ON SOLUTION AND CRYSTALLIZATION, No. Il. APPENDIX. 405 not unreasonable to suppose that the complication depends on the temperature. There is however no reason to think that this aggregation can be at all readily changed when the substance is solid, so that the crystals formed at one temperature will be permanent at other temperatures within certain limits. The change of the specific gravity of aragonite to that of calcite by a red heat seems to favour this view, though calcium carbonate erystallizes as calcite from cold solutions as well as from fusion, while it takes the form of aragonite when crystallizing at some intermediate temperatures. It seems now possible to acquire a knowledge of the form of the average molecular volume in the case of those substances of which the crystallographic elements are known. To work out the details will be a matter of time, but I think it important that this should be done as it seems to take us a step further in our knowledge of the nature of molecules and chemical combination. ACRE SND xX, HISTORICAL NOTE. M. P. Curiz, in a short paper published in the Bull. de la Societé Minéralogique de France, Vol. vit. p. 145, points out the influence of capillarity on the growth of crystals, and lays down, as I have done, the principle that the development, in order to be stable, must be such that the surface tension is on the whole a minimum. He applies the principle only to the determination of the relative magnitudes of the faces of a crystal which have different constants of capillarity. There is enough in the paper to have suggested the line of thought pursued in the first part of this communication. I had not, however, seen or heard of M. Curie’s paper when my own was written, or I should have mentioned it in connexion with my investigation. M. Bravais long ago investigated the geometrical properties of a system of points regularly distributed in space, and the degrees of symmetry belonging to several arrange- ments, and supposed that in a crystal of given symmetry the molecules must be placed in one or other of the arrangements which have the same degree of symmetry as the crystal. I ought, no doubt, to have been acquainted with M. Bravais’ investigation. He has anticipated me in some points. The arrangements of molecules which I have sug- gested are all, I think, to be found amongst those he has described. He has also pointed 52'= 2 406 Pror. LIVEING, ON SOLUTION AND CRYSTALLIZATION, No. II. APPENDIX. out that the density of the molecules in any plane will be inversely proportional to the perpendicular distance between that plane and the next parallel planes of the set; and concluded that cleavage will be most easily obtained in planes in which the molecular density is greatest. His reason for this conclusion differs from mine. He thinks that cohesion in a direction normal to a set of parallel planes will be less when the perpen- dicular distance between successive planes of the set is greater. Whether that is so or not depends on the law of force according to which the molecules attract one another. It is probable that it is so, and experiments have shewn that cohesion in rock-salt and some other crystals is less in a direction normal to a plane of cleavage than in other directions. The experiments however do not seem to me conclusive, because in rock-salt a direction normal to one plane of cleavage is parallel to two other planes of cleavage in which, according to M. Bravais, the cohesion should be a maximum. The cohesion, per unit of area, between two planes will increase when the number of molecules per unit of area is increased. And my view of the importance of surface tension in relation to cleavage is borne out by observation that when an amorphous solid is broken by a blow, or crushed, the direction of fracture is approximately that of compression. I have lately met with instances of this in the fracture of the conical quartz stoppers used by Prof. Dewar and myself to close the ends of tubes into which gases were compressed. These stoppers were fitted into conical openings in the gun-metal ends of the tubes, and for want of exact fitting sometimes obtained a bearing on only a narrow ring of the hollow metallic cone. When the tube came to be filled with gas at a pressure of 200 or 300 atmospheres, the pressure on the quartz in the plane of the ring became enormous, and when the quartz has cracked it has done so in the plane of the ring approximately. M. Bravais’ idea of hemihedry depending on the polyhedral form of the molecules is essentially different from the dynamical notions which I have advanced on that subject. M. Bravais also supposed that the internal movements which are going on when a crystal is growing play somewhat the same sort of part in regard to the growth of faces which external forces play in regard to cleavage, and tend to develope faces parallel to planes in which the molecular density is greatest. Professor Sohncke has extended M. Bravais’ investigations of the geometrical properties of systems of points regularly disposed in space, so as to include the more complicated cases arising from the superposition of several simple systems; but neither of these investigators have, I believe, attempted, as I have done, to explain on mechanical grounds why the molecules should assume any such arrangement. I am unable to find any mechanical reasons why the molecules should assume the more complicated arrangements. Sohncke has pointed out (Zeitschrift fiir Krystallographie, 1888, p. 218) that in some of the systems of regularly disposed points (not in the most simple) the greatest distance between the successive planes of a set of parallel planes does not always correspond to the greatest molecular density in those planes, and that this will introduce some modification of Bravais’ laws as to the growth of faces and as to cleavages, He also refers to the influence of capillarity as affecting those laws; yet he does not appear to notice the influence of capillarity in regard to cleavage, but only in the development of the natural faces of crystals. He does not notice that the constant of capillarity of any plane surface will depend, not only on the closeness with which the molecules are set in Pror. LIVEING, ON SOLUTION AND CRYSTALLIZATION, No. Il. APPENDIX. 407 that plane, but also on their distances from the molecules in adjoining parallel planes. I regard it as only a first approximation to say that the constant of capillarity will decrease with increasing molecular density in the surface; and it should be observed that the constant will be different in a solution in which the crystal may be growing from what it will be in air. The notion that the molecules might be represented by ellipsoids is not new. Brewster has suggested something of the same kind, and Dana in his Manual of Mine- ralogy, 3rd edition, has pointed out that Haiiy’s parallelopipeds might be replaced by the inscribed ellipsoids. My idea of the molecules is essentially a kinetic one, in which the ellipsoid represents the average molecular volume, not a solid mass. Moreover Dana’s arrangement of ellipsoids, each touched by only six others, will not satisfy the mechanical requirements which I have formulated. At the same time it may be said that there is a certain resemblance between his notion of polar forces acting chiefly at the extremities of conjugate diameters and determining the similarity of position of all the ellipsoids, and my notion of the distribution of the matter in the molecule determining the orientation of the axes of the ellipsoids though not the similarity of their situation. Although I have been anticipated in many points I hope I have added something to the mechanical theory of crystals which may lead to further development on that side of the subject. CAMBRIDGE: PRINTED BY ©C, J. CLAY, M.A. g SONS, AT THE UNIVERSITY PRESS. Tae ITR” UT oh ROTTARLTIAT “YODA ove ae very Yulee ih tT tt i ood aevnpedetl wil) ae Ne “Giratina. WM) Somwals «if? 201) yee OF TOA rouge dni if ‘y eee a ithieitke UI | bil i Mt Veen arlirsoloyn pititevioat, aide Grier. oF yom afl flit ; wone nt ‘mati «J die oe 7. ~ Bry 1h i! Ne t it mf” Jeigtos®. esipenlooy wilt i av Va sid vi cowl fr tered ote: ode, to Serttineaie intone. nh ¥ oe ot atti uy [ol decay | winell sais at betas ined aeiiee, Ne ti edu shel! a Yilanqose st eilrpalonmt aoe ; et qi Be atch i ~ 4 4 f ‘wits role” waned Dike dba oe wait wrt cas yt ote t ? im iz ae ar sou tre telah be ia yA sat ‘e trl i ee hy ea fads lie id “Vaal "Soni oinda tnlt OA Aradatinenyed wend j bl is (hoi ribs me wag % anton wil v>rngod 4 t sn Ye cotitenty To. qittaltoue att yiitiarteiyl hea begp saat 7 uta’ h gitiiimAalsh sidesiom olt.or 3 oMaen oily bu easvidonfivel ly rie ' iteaie. tial! Yo gipieatia mld Jor dey ile piano glia abe 1 id sum Jobhe ered Tn oqnd LL. etatog xpenr ci, Sotorianagerd tated L& gadt we geoogalrrol tdentt op Peo! soy dodiecelatreiasle wel eee o A soe erty Camb. Phil. Soc. Trans.Vol. XIV. Plate VI. Lith.& Imp. Camb. Sci. Inst Co, XV. Systems of Quaternariants that are algebraically complete. By A. R. Forsytu, M.A., F.R.S., Fellow of Trinity College, Cambridge. [Read February 11, 1889.] THE aim of this memoir is to obtain for certain quaternary quantics the respective systems of concomitants which are algebraically complete, that is, are such that every con- comitant of a quantic can be expressed as an algebraical (but not necessarily nor generally an integral) function of the members of the system appertaining to that quantic. The method and the course of development are similar to those in the corresponding investiga- tions relating to ternariants*, in the present case complicated by the presence of the six (non-independent) line-variables. It is shewn that the characteristic equations satisfied by quaternariantst can be reduced to twelve linear partial differential equations of the first order, which are inde- pendent of one another in form; and that these twelve can be reduced to six of them, properly chosen and absolutely independent of one another. The leading coefficient of a quaternariant satisfies three linear partial differential equations, also of the first order; and, when obtained, it uniquely determines the quaternariant by beimg symbolised into the umbral elements of the coefficients of the quantic—this result bemg a consequence of a theorem proved that every quaternariant is expressible as an aggregate of symbolic factors of some of five forms. The number of quaternariants in an algebraically complete system is NM —5, where NV is the number of coefficients in the most general form of the quantic (or of the set of simultaneous quantics) with which the system is associated. A method is indicated by which the leading coefficients of these N—5 quaternariants can be obtained as com- binations of binariants, which belong to binary quantics derivable from the original quantic. And for those quaternariants, which do not involve line-variables and which belong to unipartite quantics in point-variables, it is shewn that their leading coefficients can be expressed as contravariants (and invariants) of ternary quantics derivable from the original quantic. The general theory thus indicated is applied to obtain the special results for the following cases: (i) a quadratic, and (ii) two quadratics, in point-variables; (111) a quantic lineo-linear in point- and plane-variables; (iv) a linear complex; (v) a congruence of * «Systems of Ternariants that are algebraically com- generic concomitant of the quaternary quantic instead of its plete”, Amer. Journ. of Math., vol. xii. (1889). leading coefficient, as suggested by Sylvester, Amer. Jowrn. + I have used the word quaternariant to denote the of Math., vol. v., p. 81. Vou. XIV. Parr IV. D3 - 410 Mr FORSYTH, SYSTEMS OF QUATERNARIANTS two linear complexes; (vi) a regulus of three linear complexes; and (vii) a quadratic complex. In regard to quaternariants which are pure invariants or pure covariants Salmon’s Geometry of Three Dimensions may be consulted, in which there are frequent references to Sylvester and Clebsch; but except the quaternariant which is the complex of lines touching a surface there are hitherto, I think, few cases of quaternariants which involve line-variables. The bibliography of the lineo-linear quantic is given elsewhere*. For complexes the most important references, in addition to the investigations of Cayley and Pliicker, are to be found at the end of Klein’s memoir (all cited in the note to § 1 of this memoir); most of the results, so far as they concern the theory of forms, relate to the canonisation of the complex of the second grade. GENERAL THEORY. The Differential Equations of the Concomitants. 1. In the discussion of the concomitants of quaternary quantics the ordinary (point) variables will be denoted by 2,, z,, «,, 7,3 the linear transformations to which they are subject will be taken to be l 8? TNO ONG ale Ko The variables (plane) contragredient to these will be denoted by w,, w,, u,, %,; and they are subject to the transformations GURU) CS Cle eg, Te One ie ie en) 1 2 3 4 1 1 1 1 2 3 4 ice eu Le Plas rivals ke 11, m, nm, 1 The coefficients in the unit power of the symbolical quantity a, = (a,7,+a,7,+ 4,7, + @,@,) are transformed by the same relations as the quantities wu. The six line-variables will be denoted by p,, p,, Py, Par Dsr Por If Yy> Yor Yor Yai ,, Z, be two sets of point-variables, that is, variables cogredient with the variables a, the line-variables are defined by the equations 7 Z Z 21) “er Pi = Ye? 3 — Yo" 2) Pa = 9:44 — Ye? > Po = Ys? — Vis) Ps = Yrs — Yo?» Ps = 9172 — YP Pa Y3%s — Yes * “Systems of Ternariants” (already cited), § 60. THAT ARE ALGEBRAICALLY COMPLETE. 411 connected by a permanent relation P,Po+ PoP; + PsP, = 0*. The transformations, to which the line-variables are subject in consequence of the variations of the point-variables, are not linear, being Py Par Ps> Ps» Ps> Po ( 1-m,n,, —m,+m,n,, —N,t+mn,, —M,+mM,n,, Ny—M,N,, Mi —hM Nis, bs, fa, bs b., F,): | —1,+1,n,, 1—In,, —n,+ nl, ; L—nl,, nl,—nl,, —n,+n!, | —l,+l,m,, —m,+lm,, 1—ml,, lm,—1m,, —l,+ lm, m,—m,l, | —k,tnk,, k,n, nk,—n,k, , 1—nk,, n,—nk,, n,—n,h, _ ki-m,k,, kym,—m,k,, —k,+m,k,, m,—km,, 1—m ey, m,—k,m, | Lke,— lke, ) aa k, a5 Lk, ? k,— ky ? l, 7a Lk, ? l, a Le, > i Lk, This is not the only definition of the line-variables; for they can be constructed WwW. from two sets of plane-variables v,, v bs Vz, ¥, and w,, w Le w, in the forms 2? Py = UU, -— UU Pz = VW, — VeW,, Py = VW, — UW,; Ps = VQ, — VW,, Ps = Vs, — VWs, Ps = UW, — VW,; these being connected by the same permanent relation as before. * The following table gives a comparison between some of the notations most frequently used for line variables :— g gl 1 p | qd r u | t 8 Cayley, Quart. Journ. Math., vol. iii., pp. 225—236. l | m n r | qd p Cayley, Proc. Lond. Math. Soc., vol. iv., p. 16. a | b c h | g if Cayley, Camb. Phil. Trans., vol. xi., pp. 290—323. | ep | eeery |) ey a, | —a, || a Liiroth, Crelle, t. Ixvii., pp. 130—152. Pes | —Pis| Pio | Poy | —Pin | Pia Weiler, Math. Ann., t. vii., pp. 146—207. Dg | ~Py | Dy my Soy Voss, Math. Ann., t. x., p. 169. Ds | Sano pe | Ds Klein, Math. Ann., t. xxiii., pp. 539—578. | ha | : epee Pi iN | Ge Ds Ps | Pe Notation used in this paper. | | Since only the ratios of the six coordinates are necessary to determine any line, the given definitions imply the existence of an unspecified factor common to all the co- ordinates. Instead of six, Pliicker (in his ‘‘New Geometry of Space”, Phil. Trans. 1865, pp. 725—791, and his ‘‘Neue Geometrie des Raumes” 1868) takes the quantities (= —Pi+s P=P2>s 8=P5+, T=Pg+> 1=P3~) Ps the common denominator being p,; and the permanent relation is n=To — Sp. The introduction of Pliicker’s quantities into concomi- tants would leave only a partial symmetry and would destroy the homogeneity : so that for the present purpose it is preferable to retain some set of six quantities. I should have adopted the set (a, 6, c, f, g, h) as these occur very frequently: but, as there is the disadvantage that all these symbols occur in the usual notation for quadratics, I adopted the system (1, P2, P3; Pg: Ps, P4, the permanent relation being homogeneous in subscript indices) in the text, so as to have the leading term of a concomitant that which involves x, u,, and p,. 53—2 412 Mr FORSYTH, SYSTEMS OF QUATERNARIANTS Strictly, only the ratios of the last six quantities are equal to the same ratios of the first six quantities. But, as will be seen in § 5, we only retain in the variational equations terms which are of the first order in the coefficients of transformation; and as, up to this order, the six quantities in the first definition are equal to the six quantities in the second save as to a factor which is unvarying, we are justified for the present purpose in making the corresponding quantities equal, instead of merely proportional, to one another. 2, The fundamental quantic, when taken to be unipartite and of order mw in the point-variables, may be represented in the symbolical form a,“ or in the explicit form Suet qi rics tat t PImTy sn Agret®, Uz x,, with the condition g+7+s+t=,; and the real coefficients are Aorst = 0,10, 0,0, when expressed in terms of the umbral coefficients. From the last equation the co- efficients of the new quantic are deduced when the linear transformations in the variables are applied. 3. The most general concomitant possible is a function ¢ of the coefficients of the quantic and of the different kinds of variables such that, when the same function ® is formed in the transformed coefficients and in the transformed variables, the equation D(A, X, 2) VU) =A (a; a; p; wu) is satisfied, A being the determinant of transformation and @ an integer which is the index of the concomitant. When in this equation we substitute for A and U in terms of a and w and for x and p in terms of X and P, it comes to be an identity; and, when the two sides are expanded in the coefficients of the transformation, the terms involving the same com- binations of those coefficients on the two sides are equal to one another. Using the expansion by Taylors theorem, we thus obtain from each such combination a differential equation satisfied by the function @ of a, X, P, w identically; after it has been obtained, we may replace X and P by a and p respectively. Also since A=| Leth lero clstrels 'h, m,, Wm, mM, | Tis teas key ae WI Pega a etal | there are no terms in A of the first order in the coefficients of the linear transforma- tion, and the only terms of the second order are those which involve the combinations m,n,, ln, ml, nk, mk,, Uf, 4, Taking then the coefficients of the various combinations of the cvefficients in the covariantive equation, we have ¢=¢ as derived from the terms which are independent THAT ARE ALGEBRAICALLY COMPLETE. 413 of those coefficients; and from the terms which are of the first order we have the following twelve characteristic equations: From the coefficient of < l, the equation L, + u, a = mo = 49) i ar I “ | 2 1 2 6 | 0 a) 0 0 Pee wacsstarstsScreiniys asin DI, +4, aii, = oe a — Pp, aD, + ps ap, | 0 0 ad) 0 L, panosooe s0G00Ce LI, +u, i az, da, Ps Op, + ps ap, | 7 M Clg g + e | Us nonposeanpasaoe MM, + U, Ou x, a, Ps ap, Ps a P, | M+ ‘ é ap cay (fly s6rnoc080000007 SN ei yaaa 18) Op, Ps Op, | é a G a | lifts Sepageoosodsoo M,+u, om = &, or Ds ap, + Po ap, | fi obs P 9 r 5 i, 5 | Meissner s )y n, soe sccccneesens 1 3 Ail 1 On Ps Op, Po ep, | ee 2A caane Ny sree erecceeeees 2 + U, ani =a, On, Ps OD, + Ds Ops } d 0 0 0 | (Dy, Aadasenognodags N,+u, Fai X, a Ps Op, + Ps Op, | “ a GP uae tea Le Aecheeeaneenses K, + 4, ani Lay (ae a | a 0 a a | | I CORES OR oct Oe Ba, Pp, Pp, a a a) 7 We Sescsvavetoneass K, +4, aa isa — Ps a Dr ap, | the literal operators LZ, M, NV, K being oe Gee L,= Co+1, "1, 8, t “ot | M, = 2qa,_ “Lr+l, 6% Gime | L, = Sta Se ae M, = ty, 41, 8, t-1 eu N,= =qa,_ -1,7, 8+, tq | ie = pag be = N, = 21a, 1, 541, fe K, = 21a, »-1,4 i Al N, = ta, ,. «41, 4- 1 Baa | Ky = 28d, ,, 61,041 on | in which the summations extend to all integral combinations such that g+7r+s+t=p4 the order of the fundamental quantic. 414 Mr FORSYTH, SYSTEMS OF QUATERNARIANTS 5. Since the transformations which are applied to the w-variables are linear, it is to be expected that the foregoing twelve equations arising from the linear terms in the covariantive relation will be sufficient for the determination of the concomitant as a function of a (and also of wu). But because the transformations which are applied to the p-variables are not linear, it at first appears as if the foregoing equations would not be sufficient for the determination of the concomitant as a function of p; for the full variation of the variables p is not given by the linear coefficient-combinations. This inference as to insufficiency is not however justified; for, though those linear coefficient- combinations do not give the full variation of the quantities p, they determine it uniquely. And, when the equations derived from a comparison of the terms in coefficient-combi- nations of the second order are formed, they are found to be either quasi-homogeneous equations—satisfied in virtue of the isobaric homogeneity of the concomitants—or equations which are derivable from the twelve characteristic equations earlier obtained. The verification of these statements is not difficult: thus for instance the equation which arises from a comparison of the terms involving m,n, can easily be derived from the two equations which arise from the terms involving m, and n,. The general result therefore is that the twelve equations arising from the terms of the first order in the coefficients of transformation in the covariantive relation form an im- plicitly complete set of characteristic equations *. 6. Though all the characteristic equations can be derivel from the set of twelve retained, it does not follow that these twelve are independent of one another. As a matter of fact, they can easily be reduced to six equations in the following manner. Let any one of the equations be written in the form P=P’, so that for instance L, = @, sees u —p +p : 2 1 Gn,“ Gu, PGp, * Ps dp,’ and let any other be written in the form Q=Q’, it being evident that any one of the twelve literal operators is commutative with any one of the twelve variable-operators. Thus for any function ¢ which satisfies both equations, we have Po= Po, QWM= US. Hence (QP — PQ) $=(QP’— PQ) $ =(PQ-QP)¢ =(PQY- OP) ¢. In every case therefore in which P and @ are not commutative operators, this last equation is different in form from either of the preceding equations and is not evanescent; and it is satisfied in virtue of those two equations. On forming these equations fur the different combinations, it appears that the operators L,, L,, N,, N,, K,, K, are all commutative with Z, and similarly for the variable-operators ; the corresponding derived equations are therefore evanescent. The equation constructed from * The equations agree with those given in a paper by the author in the Proc. Lond. Math. Soc., vol. xix. (1887), p. 42. THAT ARE ALGEBRAICALLY COMPLETE. 415 L,=L,, M,=M/ is a quasi-homogeneous equation satisfied in virtue of the isobaric homo- geneity of the concomitant. But for the others LM,—M,L, = L,, M/L — LM, = L;; L,M,—ML,= Ly, ML, — L/M/ = L;; NL, — L,W, = N,, TRAN 2 NOB feet NC SN RN OBC) TG — ROT so that four of the equations can be derived by combining in the above manner the equa- tion L,= L,' with others of the equations. The aggregate of these relations* is: L,= L,N,—N,L,= L.K,- KL, > TRINA GE TAN RGA, ee TA oie? 5 AM ME = LR KD, ETE Mi SD TR L,= LM,- ML,= L,N,-N,L, LS LON? NL LN M, = M,N, - V,M,=K,- KM, M/ = N/M/— MjN/= K/M/— MK, M,=M.L, —L,M,=MK,— KM, M, = L/M/ — M’L! = K/M/—M/K; M,=MN,—-NM,=ML, - LM, | M/ = N{M/— M/N/= L/M/ —M/L/ | N, =N,M,— MN,= NK, — KN, { Nj = M/N{-Ni/M/=K/N/- N/K; ( N,=N,L, — L,N, = VK, — KN, NWiebN = NTO RON, = NK N,=N,L,— LN, =N,M, - uN, Ni = LN! — NL! = MIN NM K, =KM,— U,K,= KN, -— NK, Kl = MK, KIM’ = N/K) — REN K,=K,L, —- L,K,=K,N, —N.K, | KO SEH! RET. NOR GN K,=K,L, —1,K,=KM, — Mk, |} (Kelp KC Th = Re KM From these it follows that the twelve equations can be reduced to six, for example the set of six constituted by Tea M,=M/, Ml 08, N,=Ny, Sai K.=K.. It is not difficult to see the analytical origin of such a result. The six equations which remain correspond to the changes in the variables caused by the transformations a= X,+1,X,+1,X,+1,X,, x= m,X,+ X,, Ba MA XG, = kX,+ X,. A combination of any two of these transformations, as for example 2, =m,X,+ (1+lm,) X,+ m1,X, + m,),X,, introduces into x, terms corresponding to m,X, and m,X, which in the covariantive relation * The other similar combinations of the twelve operators Those of the operators, which are not lineo-commutative do not lead to any new forms, the operators in such cases are quadrato-commutative according to laws of the form being lineo-commutative ; thus for instance L,? My — 2L, My Ly + UM, L.2=0, ealeg = Talia Ly M,? - 2M, L, M+ M2 L,=0. 416 Mr FORSYTH, SYSTEMS OF QUATERNARIANTS lead to two characteristic equations, M,= MM, and M,=M, shewn to be dependent respec- tively upon M,= M/ and Z,=Z,, and upon M,=M,' and L,= Ly. = 7. This dependence of six equations upon the other six is however of a different kind from the dependence of the characteristic equations derived through terms of the second order in the coefficients of transformation. There, the characteristic equations arise merely from successive application of the operators which occur in the set of twelve; and they furnish no equation new in form. Here, the six dependent equations arise from a functional combination of the operators in the other six; they are similar equations, but are new in form. Since then by means of the twelve all other characteristic equations can be imme- diately deduced, merely by the successive application of the operators they contain and without the introduction of any new operator through combinations, it is desirable to retain the full system of twelve. It will appear immediately that they suffice for the complete explicit development of a concomitant in point-variables and in plane-variables ; but for the full development by this method in line-variables certain combinations will need to be taken. Such combinations are not however here given as the present purpose does not hold in special view the tabulation of the concomitants. 8. The Jacobian conditions that the coexistent equations shall possess common solutions are all satisfied either (i) identically or (ii) in virtue of equations in the set of twelve or (iii) in virtue of quasi-homogeneous equations. Symbolical Representation of Concomitants. 9. Before proceeding further it is necessary to shew that every quaternariant can be exhibited in a symbolical form. The following is an outline of the proof, being similar to the one ordinarily applying in the case of binariants and to that given by Clebsch* as applied to ternariants. (i) Denoting the quaternariant by ¢(a, u, v, p) and introducing five new linear forms v,, Wz, &, Nz, & such that (§ 1) we may take x? = ; = } P, =U, — VM, P_= Us, — UW, Po = Vy — VW, Pp, = Uw, — U,W, Ps = Vs — VW, Py = YW, — VQ, | from the coefficients of two of them v,, w,; and from the coefficients of the other three £., n,, ¢, the relations a. > =e eee ee, Sti Stine aetna Sees, the concomitant changes into $(a, u, & n, € v, w), that is, into an invariant of the simul- taneous system a,, u,, E,, n,, €,; Ue, We: * Crelle, t. lviii., p. 118; see also Crelle, t. lix., pp. 1—62. THAT ARE ALGEBRAICALLY COMPLETE. Al7 (ii) Every invariant of this system is an aggregate of products of the determinants of every four of them, the member a, being repeated any number of times. (ii) The different forms of determinant that may arise are :— (1) (abed); (2) (i) (abeu), (11) (abev), which is merely a ‘polar’ of (abew) and so is not different in nature, and similarly for all the others involving only one of the six sets of quantities u, & 7, 6, v, w; (3) (2) (abvw), (ii) (abuv), which is merely a ‘polar’ of (abvw) and so is not different in nature: similarly for all the others involving two of the six sets of quantities ; (4) @) (a&n$), (ii) (aun), which is merely a ‘polar’ of (a&f) and so is not different in nature: similarly for all the others involving three of the six sets of quantities ; (5) (i) (u&n6), (i1) (v&nf), which is merely a ‘polar’ of (u&é&) and so is not different in nature: similarly for all the others involving four of the six sets of quantities, Hence every invariant is an aggregate of products of determinants of the five different natures given by (abcd), (abcu), (abvw), (a&nf) and (uénf), and of polars of these determi- nants. (iv) When we replace the coefficients in these invariants by their values in terms of the variables as already substituted, the invariants are changed into concomitants: and hence concomitants are sums of products of the transformed values of the five different kinds of factors. These factors now are: (1) (abed) ; (2) (abcu) ; (3) (a,b, i a,b.) Pit (a,b, e a,b,) Pot (a,b, = a5.) Py ar (a,b, a a,b,) Pe + (a,b,— a,b) p, + (a,b, — a4b,) Ps which we shall denote by (abp): it is also the equivalent of a,b, — a,b, where the variables y and z are cogredient with 2; (4) a,; (5) wu,, the universal concomitant. Vou. XIV. Part IY. 54 418 Mr FORSYTH, SYSTEMS OF QUATERNARIANTS *(Hence the jive different kinds of factors which enter into the symbolical expression of a concomitant are of the forms u,, d,, (abp), (abew) and (abcd). (v) It is now necessary to consider the polars of these. (a) Some of the polar variables which occur, as in (iii) (4), can be expressed in terms of the variables x, p, wu, and therefore must be retained. Thus one of the determinants is (aww) =a,’ where “i= (uvw,) = Us Py — Us Py + U,Pe> ry == (u,v,w,) =—U,P, + U,p, + Ups, vy = (u,v.w, = UP,— UP, + UD, @, = — (U,0,W,) = — UP, — Uy, — Ug Py a new set of point-variables, being those of the intersection of the line p and the plane u. Hence, when we have a concomitant which involves the point-variables x, other concomitants can be derived by means of the polar operator 0 rine Payee) Sees ia) 8iOae eee noe aoe repeated any number of times. Thus for instance if U be a quadratic, U=0 is the condition that the point # lies on a quadric surface; 0,U=0 is the condition that the points # and 2’ are conjugate with regard to the quadric surface; and 40,°U=0=U’ is the condition that the point wz’ lies on the quadric surface. The preceding proof, given in outline, is limited to a quantic of the form a,"; but it is similar in the case of a quantic of the form w,”, as well as in the case of a quantic in line-variables which is separately discussed in § 51. In the former of these there arises similarly a set of plane-variables which must be retained: they are U, = (@,4,2,) = X,),—X,),+ 2,),, u, =—(x,y,2,) = —2,p, + 2,),+ %,Do, us, = (x,y,2,)= @,p~,—2,P,+ +2, u, = — (2,Y,%,) =— £,P, — £,P,— Ds, being the coordinates of the plane which passes through the point # and the line p. Hence when we have a concomitant which involves the plane-variables wu, other con- comitants can be derived by means of the polar operator ae ee +U Se Ou, ae Oe) ene. repeated any number of times. Thus, for instance, in the case of the same quadratic as before, there is (§ 20) a concomitant = such that }=0 is the condition that the * (11 March, 1889.) The remainder of this article and § 10 have, in consequence of some remarks kindly made to me by Professor Cayley, been considerably amplified from the form in which they existed at the time the paper was read, with the purpose of rendering more explicit the discussion of the polar variables. THAT ARE ALGEBRAICALLY COMPLETE. 419 plane w touches the former quadric surface; 0,2=0 is the condition that the planes and uw’ are conjugate with respect to the quadric surface; and 40,°.=0=' is the © condition that the plane wu’ touches the quadric surface. Similarly, because we have two points # and a we have a line p’ other than p such that its coordinates are / / / / / / P, =2,%, — LU, , Pg =X,% — ©, Xs, / = / / / = =, / s / = Po = 0, — @,%, , D5 aha | Ue U4; / / / / / fs iy Uy SE 1 = Ui, ey being the line joing @ and «. And because we have two planes uw and w’ we have a line p” other than p and p’ such that its coordinates are ” ’ , ” / ' Pp, =U, Us,— UUs, Dg = Ug Uy — Ugll ; ” / / A Uy / Dag =U, Uy — UUs, Pp; =U, U,—U,u,, ” / / a” Y} U Pz = Ug Uy — UUs; Ps = UU, — UUy ; being the intersection of the planes uw and w’. But it is easy to prove that Di fae = = Dee Sie pane = =, P; Ps so that, even from the point of view of asyzygetic concomitants, it is sufficient to retain only the coordinates of the line p’. Hence, when we have a concomitant which involves the line-variables p, other concomitants can be derived by means of the polar operator ’ ? Op, po 2G a @ j + Ps op + Ps + Ds ap, | Ps Op,’ 3 5 6 2 9 Big, 12 op, repeated any number of times. Thus for instance in the case of the same quadratic as before there is a concomi- tant 7 such that Z=0 is the condition that the line p touches the former quadric surface; 0,77=0 is the condition that the lines p and p’ are conjugate with respect to the quadric surface; and }0°7=0=7" is the condition that the line p’ touches the quadric surface. (8) Some of the polar variables which occur, as in (2), (3), (4), cannot be expressed in terms of the sets of given variables, and therefore are not to be retained. ‘Thus, for instance, if y be the point which is the intersection of the planes wu, 7, € and if q be the line which joins # and y, the value of the concomitant a, (abp) (bung) of the former quadratic is (abp) a,b, =1(abp) (a,b, — a,b,) = 2(abp) (abq), a polar of 7 by means of variables extraneous to the given system. 420 Mr FORSYTH, SYSTEMS OF QUATERNARIANTS Hence all the polar variables that require to be retained are those of the point x’ the line p', and the plane w’. (vi) All functions derived from concomitants, which involve only the variables of the point 2, the line p and the plane wu as given in (iv.), by means of any combinations of the polar operators 0,, 0,, 0, we call polar derivatives; and evidently O.. G50, 0, (abp) = (abp’), @,, (abew) = (abew’). Hence we have as the general theorem : Every concomitant.of the quantic a," can be represented as an aggregate of products of factors of the forms a,, a,/; (abp), (abp’); (abcu), (abeu’); (abcd), together with u,. (vii) But conversely when we have obtained a concomitant which involves not only the variables of the original system a, u, p, but also the derived polar variables a’, uw’, p’, we can pass, uniquely and by a reversible process, to a concomitant which involves only the variables of the original system. For this purpose we merely need to employ the operators which are reverse to 0,, 0,, 0,, v1z pig: © 0 ] 0 Ue ar eat ak Son we + ae 6 ap Se a ee ait ae » Pron? Pop! Ps dp, Ps5 7 Poser Poop’? 0 0/=u : as : -+u gy +u,——< r GL eas) ne ee u 19 each operating as many times as the variables a’, wu’, p' occur respectively; and we obtain a concomitant, the same in literal coefficients and involving only the variables x, u, p alone. If, then, we have obtained all the concomitants which can be represented as aggre- gates of products of factors of the form a,, (abp), (abcu), (abcd), w,, all the concomitants of the system associated with the quantic under consideration can be derived from them as polar derivatives by means of the operators 0,, 0,, 0,; and all concomitants which have the same literal coefficients can be derived from a concomitant with those leading coefficients and involving in its symbolical expression only factors of the form U,, @,, (abp), (abeu), (abcd). We shall therefore speak of these concomitants as the system of fundamental con- comitants appertaining to the quantic; and it is to be understood that with the system must ulways be associated all those polar derivatives which are derivable by means of any comlinations of the operators 0,, 0,, 0,,. THAT ARE ALGEBRAICALLY COMPLETE. 421 10. It follows, then, from the preceding investigation, that the fundamental con- comitants of the system appertaining to a quantic are unique in their expression in terms of the symbolic factors; and therefore that, when these factors are expanded in powers of the variables, the leading coefficient is unique, the leading coefficient being the co- efficient of the term which involves the highest power of 2z,, the highest power of p, and the highest power of u,. And it follows also, from the preceding investigation, that no variables other than «,, w,, p, occur in that leading term of a fundamental con- comitant. Conversely, also, when the coefficient of the leading term of a fundamental con- comitant is known as a function of the coefficients of the quantic, we have one simple method of obtaining the concomitant by expressing that leading coefficient in terms of the umbral elements of the quantic and completing the umbral expression by proper association of the variables). When the fundamental concomitant, thus determined by a leading coefficient, is known, all the polar derivatives can be obtained. ] 11. When the symbolisation of the leading coefficient has been effected, the degree of the concomitant in 2—say its order, its degree in p—say its grade, and its degree in u—say its class, are immediately evident; a partial verification of these can be derived from the isobaric property of the concomitant. We assign the following weights :— to a, the weight o+p+1, Gi. Seeoonsonacendones 8c p+, iPS “Geaseoo0pGconDuUSODO CDS 1, ff) | pBGOIQUGOSCENOIGSS 65958 0; and therefore, since a, and u, are isobaric, to a, and wu, the weight 0, Cnr aNG enc sccce nse o, CORSE T6 [asa Raa ae oc o+p, CATO U5 a ctoaissn sets ot+pt+l1; and the weights to be assigned to the quantities p are: to p, the weight 2+, 7D coopoaoadoobabe 2+pto, /}y actdonaS066880e 2+4+2p+4, (Dy, GbadedoocoSeBen 1, DD Adi biaster ss l+p 422 Mr FORSYTH, SYSTEMS OF QUATERNARIANTS Here p and o may be any magnitudes we please. And it may be noted that, if desirable, an arbitrary magnitude (the same for all) may be added to the weights of the variables uw; and similarly for the variables p. For the present purpose there is no advan- tage in this, as we shall be concerned only with differences of weights. Let the leading term of a concomitant be 2,"p,"u,'4; then another term must be + 2,"p,"u, A’, where A’ is the value of A when the substitutions 2,=X,, 2,=X, are effected on the quantic; and another term must be +2,"p,"u,A”, where A” is the value of A when the substitutions 2,= X,, #,=X, are effected on the quantic. Let W, W', W” be the respective weights of A, A’, A”, which can be determined by the inspection of any term in each of them. Because the concomitant is isobaric, we have the common weight =n(co+p+l)+m(2+p)+W =n(p+1l)+m(2+pt+oe)4+lco+ W' =m(2+p)+l(eo+p4+1)+W"; W'-W and therefore n—- eee : W"—W n — t=————_, o+pt+l two equations determining m and n—l, the values of which should agree with the values derived from the symbolical expression. Equations which determine Leading Coefficients. 12. In the equations (I) let (Z,) denote L, + Piz 2 let (Z,) + u, o be denoted by {Z,}, and so for the others; thus the equations can be 2 Gi = ji aS and so for the others ; written in the form a =2,——. 0a» iJ} Let ¢ be any fundamental concomitant of order n, grade m, class 1; and consider first its expression in powers of #, so that we may write pe tatt a,” of xe $ = 2," bo 09 +--+ 4 21% 21 es ee When this value is substituted in the equations (I) we have {M,} $,, ia $1, st? {Ny} },, ad a p,. e+1,t? {Ky} Pes FP THAT ARE ALGEBRAICALLY COMPLETE. 423 from three of them. Hence if po, ae and n be known, the full development of the concomitant can be obtained ; for br, «2 = UY () {BF do 0,0> and the value of n can be obtained from any of the equations {My bo, 0,0 = 9 = LN} 5,0, 9 = (El $5,0,0° The remaining nine equations give Or, — (UT et + Woe, {L,} ae p= (n—r—s—t+ 1) s$, . 4, aa {L} >, 5 ¢= (w-r—s—t+]l) th, ia {M,} >, st a 8,41, s-1,¢ {WV} p,, s, t e rh,.1, s+l1,¢ | {K,} ?,, eto rh,._4, 8, t+1 {M,} ¢,,, 3, t = thn+1, 8, t—1 {WV} p,, s,t = th, s+1, ¢-1 | {K,} p,, s,t = Sh, s—1, ¢+1 Hence W =, 0,9 satisfies the nine equations {L,} =i0) = {L} z {L,} 3 and the only variables which can enter into W are wu and p. Now let be arranged in powers of u, so that we may write l—r—s—t 4, % » So, t U U, Z,° U, = Fpl 1 (aes he) a SN 490205 ar Te ag t Stncbe : When this value is substituted in the foregoing nine equations, we have (L,) v,, oF Pale V4, s,t = 0, (L,) v,. 8,¢ F V. Bub? 0, (L,) ¥,, 8, ¢ ag ee s,t+1 — 0, from three of them. Hence if Wo, 0,9 and J be known the full development of Ho, 0, 9 can be obtained for v,. sti = G a (L,)” (L,)° (Ly vy, om and the value of J can be obtained from any of the equations Ly hy 0, = O= (Lt ho, 9 0 = Le Yoo, 0: 424 Mr FORSYTH, SYSTEMS OF QUATERNARIANTS The remaining six equations give (MH) Vy, 2 ta, ot, ¢= i } QW) sae t VR, eo Me (My) ¥,. ot MV r-1, 6, 41 = 9 (NO) Vy:0,.2 + Vy, aa, 2 = 0 (K,) V,. neu their, Sota ‘ i (K,) Y,, Bots ty, etl gat 0 Hence y= fp, 0, 9 Satisfies the six equations (M,) =0= (MM); (W,)=0=(N,); (K,)=0= (K,); and the only variables which can occur in y are the variables p. Now let y be arranged in powers of p, so that we may write pip pear pe pe il jikir! st! X= 2X iene G@+jtk+r+s+t=m). When this series is substituted in the six equations, they must be satisfied and relations among coefficients will be obtained; but in this regard we must take account of the permanent relation p,p,+p,p,+p,p,=0, and therefore instead of putting the coefficient of every term equal to zero we make it equal to the coefficient of the same variable- combination in (PPot P2Pst PaPs) Un where U,, is an arbitrary homogeneous function of order m—2 in the variables p. This equality we shall indicate by the congruence symbol, and shall leave the coefficient un- specified; and it is to be remembered that in some cases, e.g. for the coefficient of p,”, the congruence is an equality. We thus have Mx, j,k, r,s,t— "Xi, j,k, r-1, sti, t SON k+1, 1, 8, b M,x;, fbr t= Xajeieet-1 ONG 5 eral t NX, AAC ae ATs LOS ky, j+1, k-1, 1, 8, NV Xe 52, 0,8 = OGRA eae sgh Pe are? K 2Xi, j,k, r, 8, t — ky, j, k=l, r, 8, t+1 *Xi-1, 7, &, r4+1, 8, 0 K. 3Xi, j,k, 1, 8,t — “Xi-1,9,%, 7, 8+1,t — IXi, 7-1, kr, 8, +1? with the limitation?+j7+k+r+s+t=m. These congruences and equations are not, in their present form, sufficient to give the complete development of x»: others would be derived from them for that purpose, but no essentially new equation would (§ 5) be introduced. But our present aim is the derivation of the independent characteristic equations satisfied by the leading coefficient X10,0,0,0,0° Say @; and since all the equations which in form are independent of one THAT ARE ALGEBRAICALLY COMPLETE. 425 another have been retained, the foregoing system is sufficient. Making then j, k, 7, s, t all zero (and 7 therefore equal to 7) and bearing in mind that for these values the congruences are equations, we find that @ satisfies the four equations M,=0, M,=0, N,=0, N,=0; while the two remaining equations are easily seen to give the complete expression of that part of x which involves only p,, p, and p,. But of the four equations satisfied by 6 only three are independent; for by § 6 we have N,=N,M,—M.¥,, and therefore any solution common to WV,=0 and M,=0 necessarily satisfies N,=0. Hence we have the result :— The leading coefficient of every fundamental concomitant, that is the coefficient of the highest power of x,, the highest power of p, and the highest power of u,, satisfies the three (independent) linear partial differential equations Ve OM — ON 0: And as in the corresponding cases of binariants and ternariants, it follows that every solution, which is common to the foregoing three equations and is homogeneous and isobaric im the coefficients of the quantic, is a leading coefficient of some fundamental concomitant of the quantic. 13. The following results relative to the leading coefficients of the eight different kinds of fundamental concomitants can easily be verified :— (3) The independent characteristic equations satisfied by the coefficient of nm u £,"p,"u, in a point-line-plane covariant are VM = — Ne 0 (ii) The independent characteristic equations satisfied by the coefficient of “,"u, in a point-plane covariant are M,= M,=N,= 5, =0. (ii) The independent characteristic equations satisfied by the coefficient of p,"u, in a line-plane covariant are M,= M,= N,=K,=0. (iv) The independent characteristic equations satisfied by the coefficient of 2,"p," im a point-line covariant are M, = ,=N,= L,=0. Vou. XIV. Parr IV. 55 426 Mr FORSYTH, SYSTEMS OF QUATERNARIANTS (v) The independent characteristic equations satisfied by the coefficient of u, in a plane covariant are M, = M,=N,=K,=K,=0. (vi) The independent characteristic equations satisfied by the coefficient of p,” in a line covariant are M, =M,=N,=K,=1,=0. (vii) The independent characteristic equations satisfied by the coefficient of x," in a point covariant are M,=M,=N,=1,=K,=0. (viii) The independent characteristic equations satisfied by a pure invariant are M,=M,=N,=L,= KS K,=0. 14. The foregoing is an outline of the general theory; we now proceed to apply it to a few cases of quantics taken (with one exception) to be unipartite. The sole difference, when the concomitants of bipartite or tripartite quantics are desired, hes in the forms of the literal operators. The first step necessary is the derivation of common solutions of the three charac- teristic equations. In this regard a limitation will be assigned so that, for the most part, only those which are algebraically independent of one another will be obtained; no attempt will be made to obtain the aggregate of ‘irreducible’ concomitants. Since the partial differential equations are all linear, it is a consequence (from the theory of such equations) that all their solutions can be expressed in terms of a finite number of common and algebraically independent solutions. And, as will be seen from the process of solution and from § 18, this finite number is five for a quadratic, is fifteen for a cubic and fifteen for a system of two quadratics, is thirty for a quartic; and for the n* the finite number is 4 (n+1)(n+ 2)(n +8) —5. 15. It is conyenient to arrange the notation for quantics of different orders so that the operators of the characteristic equations are the same so far as common letters occur in them. Thus the quadratic will be taken in the form ag, + 2a,7,0,+ 0,0, + 2a,0,0, + 2a,'0,0,+4,"2%,'; + 2b,2,4, + 2b,0,0, + 2b/x,m, + Cy," ; the cubic in the form 3 2 2 3 F Po | } Uy _ Leet | 2| ” uu | ms ag, +30,0,0, +30,0,0, +4,0, +307) a a, +2a,'0,0,+0,", |+3%,| a, %,+0, 2,)+a, 2, +3b,070,+6b,0,0,0,+ 3b0,0, + 2b/a,2,+ 2b,'x,2, +b, ‘ = 2 2 | Sy 2 +3¢,0,0, +3¢,0,'2, + Cy, BE, We | 2 THAT ARE ALGEBRAICALLY COMPLETE. A427 and so on; the general law being that in the literal coefficient, associated with any variable-combination and the proper multinomial coefficient, the subscript indicates the power of 2,, the number of dashes the power of a,, and the position in the sequence a, b, c,... the power of z,. Thus in the quaternary octavic, the literal coefficient of he eg ty? is ¢/’: the multinomial coefficient associated with it being 8!~+(2!)* With this notation, the operators in the characteristic equations are :— é 0 , 0 , a u é M,= Oa Slap +d, + . +58, aay + % me + b, Bayt +... | uae uh +b, 3 (6, — (5, aa, Cupp as +...) + Gr = ae ton) tes mts 0 a 0 é 0 0 Mi =o. Gath Cape t Mo gene’ titan Shape a a age +... / a) , é pao. “ Cl) +2 (yarn te a ec +b, Tae +...) +38(2, a ton) ten a G ae aC n 0 N= 6,55 hia na +a, a + % ee + a, 367 * 5 SPiap +2(b, im +05 +. +B) + gH ose ten) + 3 (cag t ton) 16. When the fundamental quantic is not unipartite the only difference is, as already remarked, in the literal operators. Taking for example a bipartite quantic, of order w in @ and class v in wu, sym- bolically representable by a,“u,’, we may denote the literal coefficient of Poa Sy tay Kay, Pay To, T ote ta fa fu uPucu, DY Goratepor (with the limitations g+r+s+t=p, «+p+o+7=v) and it is assumed that the cor- responding multinomial coefficient is associated with it; the literal operators are then of the form Cl) a L, = 2rag41, r-l, 8, t, x, p, a5 = DKdy, 7, 8, t,x-1, pt, o, T im se-spae oS Gleaner OG grstepor The same laws hold of these operators as in § 6; and hence all the foregoing theory applies, and the independent characteristic equations determining leading coefficients are 0 M, => (sag, r+1, s—1, t, x, p, 0, 7 — PQq, r, 8, t,x, p—1, o+1, =) Se = 0, qrstxpot M, == (t UE ms g 3 = & (tg, 7-41, 9, t-1, 6, 00,7 — Pa, 75 8, tm e—1 2, 10) eee ma grstkpor N,=> é = = > (1Ay, r-1, 841, t,x, 0, 0,7 — 74g, r, 8, = t,x, p+1, ¢-1, 1) dae qrstkpor 55—2 428 Mr FORSYTH, SYSTEMS OF QUATERNARIANTS Number of Independent Concomitants. 17. The following is a general summary of the process of solution. We first find the independent solutions of M,=0, being one less in number than the number of coefiicients 4 (n+ 1) (n+2)(n+3) of the quantic: let these solutions be A, uw, v, p,.... We now require the functional combinations of A, yw, v, p,... which are such as to satisfy V,=0: let one of them be f(A, pw, v, p,...). Then N,f=0, ae Suv, .fn,,.9¢% v,.o ‘= that is, an oat he Want a ee a Nip+...= 0, the subsidiary equations for which are a _ dy _ dv _ dp _ ND Np Aes Ni being one less in number than the number of the quantities A, w, v, p,--.. But it does uot follow that the number of independent integrals of the equations is the same as the number of the equations; for, as will be seen in the special cases, the quantities NVA, Ny, Ny, Nyp,... are not functions of A, w, v, p,... alone. To obtain integrals it is necessary to take linear combinations of the form AN,o—koN,v (k an integer which varies from one combination to another); and solutions can then be obtained. The effect of this is to diminish the number of equations by one unit; and therefore the number of their independent solutions is less by 2 than the number of functions A, p, », p,. that is, it is ey 4 (n+1) (n+ 2) (n+ 3) -3. Let these solutions, independent of one another and common to the two equations, be p Wien ae y Rp Kaeo In finding the functional combinations of these which satisfy M,=0 we have a similar process to carry out, it being sometimes convenient for purposes of integration to introduce algebraical combinations of 0’, p’, v’, p’,...; and the result is that the number of solutions is again reduced by 2, so that the number of independent solutions is & (n+1) (n+ 2) (n+ 3) —5. In terms of these every solution can be expressed, though not necessarily nor generally as an integral function. For a system of quantics, containing implicitly in their most general form NV co- efficients in the aggregate, the algebraical system of solutions is similarly indicated to contain WV —5 members. 15. That the number of functions, thus indicated as NW —5 by the process of solution, is the complete number of algebraically independent solutions, may be seen by the following considerations. THAT ARE ALGEBRAICALLY COMPLETE. 429 (i) When the general linear transformation is effected, the number of equations connecting the new coefficients with the old is N, the number of coefficients occurring in the most general forms of the quantics that are in question; thus for one quadratic the value of NV is 10, for two quadratics the value of V is 20, and so on. All these NV equations contain the coefficients of transformation. In addition to these we have the following equations: four connecting the variables x with X, six connecting p with P, and four connecting U with u; viz. fourteen in all. And there is one more equation, viz., A= determinant of transformation. Thus the total number of equations is N+14+1=N+15, all of which involve the coefficients of transformation. (ii) These equations are not all independent; in order to obtain the number of independent equations to which they can be reduced, it is necessary to diminish their uumber by an integer equal to the number of relations which subsist among the quantities. These are two, viz., Pit. Ph, Ghee) 2 lee, ae, Sly Hence the number of independent equations in the system is N + 13. (ii) These equations involve a certain number of quantities which do not occur in covariantive relations, viz., the 16* coefficients of the transformation. When, therefore, the 16 quantities are eliminated between the NV +13 equations, the number of independent equations left is N—3. These relations we otherwise know are in general of the type ® (A, X, P, VU) =A‘? (a, «, p, u); every such relation determining a concomitant. But there are two of these NV —3 equations which are independent of the coefficients of the quantic; they are Ue and (§ 1) P: Pot PrPs + PpPi= A (P,P, + P,P, + P,P), by the equations of transformation. Hence the number of independent concomitants which involve the coefficients of the quantics is NV — 5. (The complementary form of the last results can be at once derived from the differential equations. It is easy to prove that every solution of these equations which does not involve the coefficients of the quantic can be expressed as a function of (a) wa, + ue, + U2, + W2,, (8) PyPot+ P2Pst PsPai and these would, from this point of view, be added to the other M—5 solutions of the equations.) * In the introductory paragraphs 1,, m,, 3, ky were all taken to be unity, for the purpose of simplifying the forme- tion of the characteristic equations. 430 Mr FORSYTH, SYSTEMS OF QUATERNARIANTS The general result thus is :— The number of fundamental concomitants, algebraically independent of one another and involving the coefficients of the quantics, is N—5 and every fundamental concomitant can be expressed in terms of those N—5 together with the adjoint form u, and in virtue of the relation p,P~Py+ PxP;+P.P,=9; where N is the number of coefficients in the most general forms of the quantics with which the concomitants are associated. With the fundamental concomitants their polar derivatives must be associated, to form the algebraically complete system of concomatants. APPLICATIONS TO PARTICULAR QUANTICS. I. A single quadratic. 19. With the notation suggested, the characteristic equations are 0 od 0 CG) M, = b, = +b, agit t % ab + 2b, a = 9, 1 1 1 2 G pag 0 a. 98 Nea, ab, +a, ab; +a, 3, + 2b, ah =0, 1 esas deb ae hl oj da, oa, ab, 0a, and we have to find the simultaneous solutions of these three equations. The simultaneous solutions of M,=0 and N,=0, which are independent of one another, can at once be obtained; for those equations are the differential equations satisfied by the concomitants, in the two sets of variables a,, —b, and a,’, —b,, of a binary quadratic with c,, b,, a, as literal coefficients: and every simultaneous solution is one of the system of concomitants. The system of algebraically independent solutions is thus 0=(6,, 6,, a,%a,,—5);, S=(c, 6, a,¥a,, —bha,, —b,'), t =a,c, — b,’, j =a, — a,b,’ ; and the irreducible system of binariants is composed of 0, S, ¢, 7 and 6 where 7=(c,, 6, afe,, —0,)' with the algebraical relation 00 — 3° = #7’. THAT ARE ALGEBRAICALLY COMPLETE. 431 To the set of four solutions, independent of one another, must be added those the variables of which do not enter into either M,=0 or N,=0; these are Wh (hp / , a Sane a ee Hence an algebraically complete system of independent solutions common to M,=0 and N,=0 is constituted by u, uw’, u’, 0,3, t, 7; and an irreducible system—that is a system such that every solution of the equations is an integral rational function of its members— is constituted by u, wu, wu’, 6,3, &, t, 3. We now require the functional combinations of the independent solutions which are such as to satisfy M,=0, say @=0. Carrying out the process indicated in § 17, we have Ou=0, O@=0, Ot=0, so that wu, 0, t are simultaneous solutions of all the three equations; and we have Gu =b,, Ou’ =2b,, OF =5,, with either of the equations Qj = b,b,-—¢,@,, O60 =2b,t, for j and @ can be expressed algebraically, each in terms of the other by means of the other functions. We require two solutions (§ 17) of this set of equations; they are most easily obtained in the forms 2 0 (vt —3) =0 0 (wt — 0’) = 0. Hence we have five solutions, viz., u, t, 0, wt-S, wt-@; and in terms of these every simultaneous solution of the three equations can be expressed. 20. Each of these five is a leading coefficient: we proceed to obtain the concomi- tants. Taking the quadratic in the symbolical form 2,*, we have U=d,=4,". Since this is the coefficient of the highest power of z,, the highest power of w,, and the highest power of p,, it follows that the concomitant is Of SC TH sec being in fact the original quadratic. Next, we have t=a,c, — b,>=4 (2,8, — 4,8,)"; and the concomitant determined by this coefficient is T =} (aBp)* = tp,’ +.... The value of @ being c,a,’—2b,a,b, +.a,),", we have a2 272 S. 2 6 =a,%a," — 2a,4,a,b, + @,"b,” = (4,0, — 2,5,)”, 432 Mr FORSYTH, SYSTEMS OF QUATERNARIANTS or since a,a, — a,b, = 4,8,8, — 4,8,8, = B, (4,8,), we have @ = Byy, (485) (Ass) 3 and the corresponding concomitant is © = 8,7, (a8p) (ayp) = Oa,"p, +... . Next, denoting wt—S by p, we have = 49574 (4,8.%5) = (2,By7,) (2,8,75)+ This is the leading coefficient: the completed form of (4,8,y,) is evidently (aBryu) ; and that of (a,8,y,), viz. of a, (By) +B, (Yo%) +, (485) is evidently a, (Byp) + Bz (y4p) + x (Bp), it being remembered that powers of a,, of u,, and of p, only may be associated with the leading coefficient. Hence the concomitant is P=} (aByu) {a, (Byp) + B, (yap) + yz (@Bp)} = 3, (aBryu) (Byp) = P@,U,p, +... Lastly, denoting u’t— 6’ by oc, we have , GSN 5 5 [os / Big Ways Ge , , ” by; a, a, = a,Byy, (48574) > $ (4,8,7,)° ; and the corresponding concomitant is evidently > =4 (aByu) = cu, +.... Hence all the fundamental concomitants appertaining to the wnipartite quaternary quad- ratic U can be expressed in terms of U, T, @, P, %; to which must be added the universal concomitant u,, and (§ 18) the polar derivatives of the fundamental concomitants, to form the complete algebraical system. 21. Since 66’—%° is divisible by ¢, it follows that if we take p’+o0 we have a reducible combination; and by actual substitution it is easy to shew that us —* (p" + 08) = (unl” —u) t— U6 + Qu’ — 0" =| MU » by a; a, =A, lib Gay te UY My, b, » Ay) a, vee bs, a, a, THAT ARE ALGEBRAICALLY COMPLETE. 433 the discriminant of the quadratic. This function will be otherwise obtained immediately ; but we meanwhile have an illustration of the result of § 19, for 1 = Sp PS A = jap (UST — Pt 30). If it seemed desirable, this equation could be used to replace P by A; for P is deter- minate, save as to sign, as the square root of UST —S0 —uZTA. And then every fundamental concomitant appertaining to the quadratic could be expressed in terms of U, T, O, &, A. 22. As a single illustration of the equations of § 13 and to shew how the dis- criminant arises, independently of the possibility of making a reducible combination of the concomitants in § 18, we shall determine those concomitants into which no w-variable may enter. Their leading coefficients must satisfy K,=0, where wu 0 , 0 F 0 , é K, =a, a, + 2a, aa, +a, oA +b, 2B, We must take such functional combinations of wu, t, 0, p, « as will satisfy A,=0. Using the process of § 17 we have Ku =2u; Kei Or KO =23; Ee 0" KS =9, so that Kp =c; and yu’ =0, Kd =0, so that Kyo =0. Hence t,and o are solutions ; and we must obtain solutions (if any exist) of Ku=2u, K0=28, Kyp=c. From the first two we easily have K, (ut — 0) =2p, which, when combined with the third, gives K, {o (ut — 4) — p*} =0, so that a new solution is o(ut—6)—p’, being in fact tA. That A is an invariant (were it not otherwise known) may at once be seen by symbolizing the determinant A into the form 34 (aBy8)*. Wor, XTV. Parr TV. 56 4354 Mr FORSYTH, SYSTEMS OF QUATERNARIANTS II. Od, = 07 > 06, = 0. O¢, = 0. @v,=0, Ov, =0, @c, = 0; Ov,’ = d, Gy —. B, 00, i tb, 3 08, am | ; Od, = 2b, O¢, = t,B, On," = oe Ov,’ = 2B, | : @6, = 2t,b/)’ O¢,,= 2t,By)’ 06, = (a,C,— b,B,) B, + (¢,B, — 5,C,) A,, + 06,, = (a,C, ri b,B,) By a5 (¢,B, = b,C,) a 06, = (a,C, ae b,B,) b, ak (¢,B, = b,C;) a; | 086, = (a,C, a b,B,) b, ate (cB, a b,C,) a, ate t, H ) Od, = (A,c, — B,b,) b, + (C,b, — B,c,) a, 4O¢, = (A,c, — B,b,) b +(6,b, — B,c,) a’, Of, = (Ac, — B,b,) B,+ (C.D, oan Be) A, ? O¢, = (A,c, — B,b,) B+ (C,b, — Bic.) A,’ + t,b,. From combinations of the last eight equations, we have 0 ( 6, a ,) = tb, \) ( 6, a5 ,) = t,,B, ® (bd, ot 0,) = t,.b, a5 2 \s) (44,, a $,) = t,B als tb, ) THAT ARE ALGEBRAICALLY COMPLETE. 437 In addition to the quantities which are at once seen to be solutions of © =0, viz., ny Ue ag Che Ob CBee, CLS Gals omen other solutions are given by pP,=— 9+ A 7, =—-O,+ oh = o,=-O,+ Be P2=— y+ try) ” T,=— +t)’ o,=—b, +t") A = tyr — Ch 4 ry = £Vn 4, i] d, ; SS une db tv,” — 2¢6,— 6, & = ty ar tv,” = 26, — $, 26. One algebraically complete system for two simultaneous quaternary quadratics is constituted by the fifteen concomitants which have as their leading coefficients the fifteen quantities U5 U5 t, 3 63 Oiont ds; 6, 943 Pi> Po: o,> C5 25 Ee This set is symmetrical so far as the two quadratics are concerned; and every lead- ing coefficient of a concomitant of the irreducible system for the two quaternary quadratics can be expressed algebraically in terms of the foregoing fifteen leading coefficients. In addition to these fifteen, we have obtained other seven, viZ. ties 95, Das Tyr Tes Ay, Ags and we have not yet considered the series of Jacobians j,, j,,..., j,.. Of these we find at once that j,, j,, j, all satisfy ©=0; but the remainder, as well as ¢,, €,, €, €,, €, do not lead to equations admitting, as do those of § 23, of immediate integration. We thus have ten simple leading coefficients tei Os) by5 Ti» Tai May Med Jur Jor Je» subsidiary to the algebraically complete system of fifteen. 27. One remark may be made before passing on to give the full symbolical ex- pression. of the concomitants determined by the leading coefficients which have been obtained. Denoting by U,(=2,0,7+...) and U,(=v,07,7+...) the two quadratics, we have U,+2U, also a quadratic for every value of X. Hence when we form the five con- comitants, which constitute the complete system for U,+2XU,, the coefficients of the various 438 Mr FORSYTH, SYSTEMS OF QUATERNARIANTS powers of X in each of the five must be simultaneous concomitants of the system made up of U, and U,. Adopting the notation of § 18, it is easy to shew that w(U,+AU,) = v, + ,, t(U,+0U,) = t,+ 4, +'%,, 6 (U,+2U,) = 0, + X(20,+¢,) + * (0, + 2g,) + °¢,, p (U,+2U,) = p, + ® (7, +2.) A? (7, 4+A,) + r*p,, o(U,+2U,) =o, + AE, + VE, + Vo, thus verifying the statement. And by means of these we can find expressions equivalent to the five simultaneous invariants similar to that which is equivalent to the discriminant of a single quadratic; for we have td = uct — p®— 0, and all that is necessary is to replace each of these quantities by the corresponding quantity for U,+2U, and then equate coefficients of 2. 28. The full symbolical expressions of which the foregoing are the leading coefticients are as follow :— T, =4 (@8p)* =t,py + Ts (a¢py =typ, + T, =4 (@B'p)y =t,p,! + =B.y.(48p) (ayp) = 8,2,'p, + ©, =4,4,'(48p)(¢Bp) = O,0,'p,?+ ©, =4,8, (24'p) (@B'p) = O,"p, + s = 4,8, (22'p) (Bap) = ag + = 44, (48'p) (a Bp) = b,", , =B,'y. (“B'p) (@’y'p)= a as E, = 4,4,' (¢ap) =e0,"p, + P, = 34. (48yu) (Byp) = p,2,u,p,+ P, = 44. (BY U)(B'Y'p)= pt y,P, + THAT ARE ALGEBRAICALLY COMPLETE. 439 Tl, = 34,' (WaBu) (@Bp) =7,2,u,p,+ = 2%, (aa'B'u) (B'p) =7,0,u,p,+ =, =4 (aByu)? =o, +... >, =4 (a'B’y'u)® Sou +... A, =4, (#B'p) (aa/Bu) =A,z,u,p, + A, =, (@f'p) (WaBu) =r,2,u,p,+ E =4 (wafu)? = Eurt... E, =4(aa’p’u)? = €u7+... J, = Bare (op) (8p) (2'yp) =j,2,"p, + «.. J, = 38.8," (@ap) {(a8'p) (a8p) + (aBp) (2'B’p)} = j,c2p3 + J, = 8.72 (aap) (4 8'p) (ay'p) =janep,o + Further J, +37,,E, =2,2, (ap) (#P'p) (BB'p), J,—31,,E, =4,2, (28'p) (a’Bp) (8B'p); and the symbolical values of the five simultaneous invariants referred to in § 25 are dz (@By8), 4 (@aBy)*, 4(a’R’aR)*, 3 (a'B’y/'a)*, 3 (a’B’n/8'* Hence we have the theorem :— All the fundamental concomitants appertaining to the system composed of two unipartite quaternary quadratics can be expressed algebraically in terms of the members of the set of fifteen algebraically independent concomitants constituted by ee ta tam 7 O1,205,-0;, Gime Pa 3 ey Ss B, ak and to these must be added the universal concomitant u,, and (§ 18) the polar derivatives of the fundamental concomitants, to form the complete algebraical system. 29. The operators of § 13, annihilation of leading coefficients by which implies the absence of variables of some class, are :— 20 Be sled a a ia Be On oe ages Hts age tite og + Sea ast Be apt A, BAY tte KG my — , 0 / 0 uM 0 / , 0 , C) uM 0 / a Bass app ap, 2 apy te = + Ao op t Ar gg +4e’ag7 + 2B. acy » | a ager Ab ae” ule ; palo @ ete cay aay ah ag, +o gar sg sa +B. ag, t2As aA. annihilation by Z, implying the absence of u, by K, of p, by K, of u. 440 Mr FORSYTH, SYSTEMS OF QUATERNARIANTS Thus, to find concomitants which involve at the utmost only # and p, we require to find combinations of p,, py, Oj, Ty» Ts Te Ay Ay» &, & which will satisfy Z,=0; all the other concomitants obtained being evidently such as already satisfy L,=0. We have Le, = 2p, \ LE, = 2 (a, +A,) iL Lic, =2p, | Tbe. = 2 (7, +A,) | Lp, =t,v,—8 il Ld, =t,.v, — 9,— $, ii Lr, = tv, — 9, } Lp, =t,— bs. / Ld, = b,30. — 93 — $5 J Ljr,= tv, —$, all the gieumeeet on the right-hand sides of the last three sets of equations being solu- tions of Z,=0. Thus we have as solutions o, (62%, — @,) a Py» o (i, “ 0.) +o, (t,, = s) = 2/,P.» os (t., of ps) re Po» which are all reducible; there is a similar set connecting the &s, 7’s and 2's; and there is the set of integrals of the last six equations of the same form as (4.2, 5% 0) LL aa (t,, = @,) Py which is reducible, being divisible by ¢,. Similarly for the other classes of concomitants. 30. All the preceding investigations have connected the leading coefficients of con- comitants with the theory of simultaneous binariants. Some of these leading coefficients may, however, be similarly connected with the theory of simultaneous ternariants; and an easy illustration is afforded by the present system of two quaternary quadratics. Thus let it be required to find quaternariants of the system of two quadratics which shall not involve line-coordinates; it is not difficult to prove that their leading coefficients must satisfy the six equations M,=0, M,=0, N,=0 (already rai = 0 é a 0 0 0 i 9! , Ba 9s Bq t aga? rape 2 Tee aay ts 3a, + #: om sigue ee f) : 0 nao » 0 , 0 K,=4, 26, +a,’ . + ay’ "a + 2b 5, +A,’ OB, +A, OB. +A, ap? t Bo aC, = (0; and i ae 5a “9 , 0 ,_o ; ae 0 Fe ty aia, Vat Oo agit he ee a + Avira he tear aye of which, as indicated in § 6, NV, and K, are derivable from the other four: they are how- ever retained here in order to make the set of equations complete in form. Now on comparing these six equations with the six equations which determine ternariants* and associating the six operators D,, D,, D,, D,, D,, D, (l.¢.) with K,, M,, N,, M,, K,, N, respectively, it appears that the preceding six equations are the differential equations of pure contravariants (including invariants) of a system of the two simultaneous ternary quadratics * See the memoir on ‘Ternariants” (§ 1), before referred to. THAT ARE ALGEBRAICALLY COMPLETE. 441 CE + 2b,Ey + ayn’ + 2WlEC+ 2a,'nS+ a,”S CE + 2ByEn + Ay? + 2BiEb+ 2A nt + AE, the variables of the contravariants being two sets, viz. b,, a,, a,; and B,, A,, A,. (It must be borne in mind that there are, in addition, the leading coefficients a,, A, of the fundamental quaternary quadratics; they do not enter into any of the six equatious.) / Hence then we take all the pure contravariants in the two sets of variables b,, a,, 4, and B,, A,, A, of the two ternary quadratics. These will consist of (i) pure contravariants (including invariants) of the two ternary quadratics in by, Gs, (ii) all polars of pure contravariants, of the set (1), with B,, A,, A, as polar- ising variables : @,, 4 as contragredient variables : 0°? (iii) pure covariants of the two ternary quadratics in &=aA,—a,4, n=a,B,—A,b,, €=5,4,—a,B, as ordinary variables : (iv) mixed concomitants in both sets of contragredient variables and in the ordinary variables of (iil): the last two arising from the well-known property that, from two sets of variables co- gredient with one another, a third set contragredient with them can be formed. For instance, by means of the equations ee re ee, Nat Ao eG — 7, (the other combinations of operator and variable giving zero), it is easy to verify that the quadratics satisfy all the six equations. 31. Gordan has given* the complete irreducible system of concomitants of two ternary quadratics in a single set of contragredient variables. If then we take these, bearing in mind the special values of the variables in the present case, and add to the system the concomitants arising from polarisation by the second set of contragredient variables, we shall have all the leading coefficients which are irreducible without the introduction of a, and A,, the leading coefficients of the two quaternary quadratics. Taking then Gordan’s system and notation for the concomitants and retaining the sym- bolical notation for the quaternary quadratic, the fundamental ternary quadratics are f= (4,€ + 5 + a,f)° a ag’, pe = (2, + a,'n WF a,¢)° = ag? ; we shall denote the ternary contragredient variables by W,, W,, W, (=5,, a,, a,), and WowWee Wy (= Be Ae AS), so that F,, = (aBw)?’ = we’, and similarly for the other functions. * Clebsch, Vorlesungen iiber Geometrie, p. 291. Vou. XIV. Parr IV. iy 442 Mr FORSYTH, SYSTEMS OF QUATERNARIANTS These functions have to be transformed so as to contain only the umbral elements of the quaternary quadratics; they are then leading coefficients of quaternariants. We have E=a,A,—4 1% =B,8,' (8,8, — B,8,), =a, B, — 6,4, =B,8, (88, — 88,); c= b,A, . a,B, = 8,8; (8,8, — BB,’), so that Cry cone Men oe F and therefore f = BB, 1%) (4858) (@as74)- This being a leading coefficient, the concomitant determined by it is BB2 722 (4BBU) (ayy U). Similarly we have, for F,,, We =| Ww. W,, WwW, |=) di a, a |=, (4,827,); |) ak meagan ee a lags Basa, Wa ae 78, so that FF, = 1,8, (48 s74) (48,83), and the concomitant thence determined is 9282 (4Bryu) (486u), which can easily be proved equal to a,” (Bydu)* — 4u,? (4B). The first polar of F,, is = weWe Vit (48,74) 07 (4,8y7,/), the corresponding concomitant being Yate (4Byu) (aBy'u), which can easily be proved equal to da" (aByu)® — Sufe (2Byu) (2B YY). The second polar of F,, is Ss = a,'B, (4,8,2,') (2,8,8,), and the concomitant thence determined is a, 8, (aB8a'u) (48R'u). The other functions are similarly obtained with the following results, the (Gordan) symbol on the left-hand side being used, for convenience, to denote the concomitant as well as its leading coefficient. 32. The aggregate is:— T= BB Yate (ABB) (aryry'u) T= BB 1a (4 BRU) (a' yu) P= 282 (Bry) (Bu) *first ‘polar’ of F,=y.y. (aByu) (4By'w) - ; second ‘polar’ of F,,=+,6, (4Byu) (api) * These ‘polars’ are polars of the leading coefficients by means of the literal elements in § 30, (ii); there are, in addition, all the concomitants given as polars of the concomitants, here determined, by means of the variables in § 9. ? THAT ARE ALGEBRAICALLY COMPLETE. 443 and the combination F, (second polar) — (first polar)’, being in fact (O&)*, produces no new concomitant ; FP, = 82 (#B' yu) (a'B'Su) | first ‘polar’ of F.,=y,y,' (a’B’yu) (a’B’y'u) \; second ‘polar’ of F,,=y,'6,' (a’B'y'u) (a B’8'u) A, = (aBryu)° A, a (a’Bryu)? A,,, = (a’B’yu)’ : AR = (a’B'y/uy B,= 7.8.82 (484'u) (a8yu) (4'6B'u)| ‘polar’ of B, = 6,4,'6,' (a8y'u) (a8a'w) (7/86 u) ) B, = 64,6, (a B’yu) (a’B'au) (76'S) ‘polar’ of B,=y,'8,’B, (a’B’y'u) (a Bau) (28’Bu) NW = Br.8,728, (aa Bu) (aryy'u)(a' 85) | ‘polar’ of NV =8,'7,8,7, 6, (aa’ Bu) (ayy'w) (x'88'u) C,=B.6,.r,8, (aa’Bu) (a’'ydu) (ydew) (4X8'x) first ‘polar’ of C, | Bi 5 OReD RAY: pene congruent to 8.6.8, Vx (aa 8B u) (a you) (Sew) (aby u) pry second ‘polar’ of C,=£,8,'y,'6,' (aa'B’u) (a’ydu) (y88'u) (aBry'u) C,=B.7,¢,8, (aa Bu) (ary’s'u) (7/d’ex) Say ccvegruont tof Yet! (a) (anf Bu) (Bu) (aes) | second ‘polar’ of C,=¥,8,'¢,r, (aa'B’u) (ary’S'u) (y8'e'u) (a’yr'v) D = B,¢r, (aa Bu) (ay/Su) (a’ydu) (yew) (y'd'ru) Be r,(aa'8'u) (ary’/S'u) (a’ydw) (ySew) (y/Sru) {> first ‘polar’ of C, is first ‘polar’ of D | congruent to second ‘polar’ of D 4, ie Peg mR , ; BNA Gonemiont to B,€,%, (42 Bu) (ard u) (a'ydu) (Seu) (7/SX'v) third ‘polar’ of Dis =£,/c,/n,/(aa’B’u) (ay’d'u) (a’ydu) (yde'u) (7/8'N'w) FP, = Bry, (44’Bu) (aa’yu) first ‘polar’ of F,,= 8,8, (aa’Bu) (aa’B'u)\ ; second ‘polar’ of F.,=£,'y,/ (aa’B’u) (aa’y'u) N = Ougyy27, 8,€, (488u) (aBeu) sararean Te QipyY2%2 9252 (a88u) (485 uw) second ‘polar’ of V = ypy2x Ox Ex (AB8'U) (aBe'w) | first ‘polar’ of V e ©1 ~ bo 444 Mr FORSYTH, SYSTEMS OF QUATERNARIANTS DT, = Qaay¥e¥r ard (a'B'Su) (aBeu) (SxX'u)) ‘polar’ of T= Qapyyeye es MAe’ (a’B'Su) (aBe'u) (SAN'U)) ” Py = Qaay rere’ Ady (488'u) (a/B"eu) (SDXu) ‘polar’ of Dy = Qaayyeve €x Made (488'U) (2’B’e’u) (SAN'U) A = Oasy Vez Ex€x Nee (4BS'U) (a’BSu) (See’u) (SAU) ; Dy. = Qapy QserYee Mare 3 where Oapy= (aByu) (a'B'y'u) — (aBry'u) (2’B’yu), the value of Gordan’s quantity (aa’r) being the product of y,7, ur in Ou». , by the coefficient of This is the system of fundamental concomitants of the two quaternary quadratics ; with them must be associated (§ 18) their polar derivatives. Many of the concomitants just obtained might be reduced; the reduction however is not an essential part of the present purpose which has been (§ 31) to connect the class of quaternariants involving # and wu with the concomitants of ternary quantics. 33. (a) In regard to the general unipartite quantic of order n in point-variables the process of solution of the three characteristic equations of § 15 is simplified by the following theorem, evident from the forms of those equations :— The simultaneous solutions of the two equations M,=0, N,=0 are binariants of the simultaneous Be quantics 2 , , r 2 “" uw u 2 e 2 ye (65,05, 6 OE BRA » Ye ' y id / 3 a £4) US it Viase (DiC DLA. Wig.) te Wer dessneneaermsenase 4 (05 isi Cas Ds, PO) ets oboe we iteteaeme in sets of variables a,, —0,; a,, —b,; a,”, —b)”; a”, —6,”;... To these must be added M,, Ag, Ag, a", a,”,... Which do not enter into either of the two equations mentioned. To have leading coefficients, which satisfy M,=0 in addition to the two preceding equations, it is therefore necessary merely to form such functional combinations of the preceding binariant system (with a,, a,', a”, a)”, a,””,...) as satisfy M,=0. (b) To obtain leading coefficients of quaternariants, which do not involve line-variables, associated with the unipartite quantic of order n, the following theorem is useful being derivable from the characteristic equations satisfied. These equations, as in § 30, are, in addition to M,=0, M,=0, N,=0, the three : a a 7) 0 a) a) 7) 0 é 0 — => ; . = ard 2 i 2 3a” e " 7p T see N, 1454! b j ler 7 +24, Ay at Saag b, a + 2b, apt One pr ty ae + 3a, age + r a) ’ a) , 3 MG) ma) , 0 We j ” 7 K,=4, a5, ab, a, ae eon 0 ig + "5 35, + a, eee 20, +a, 5,7 + 2 FG: oe ad, +... = 0 0 0 ¢ , bela , 0 , 0 mu 0 2 , 0 K.= a, = +b ah 0 da,’t 2 1 ea, +c,” ° Be, +b, ab,’ + 2b, ab. +a ° da, nt 2a,’ 5a,’ 3a, aa, + ose THAT ARE ALGEBRAICALLY COMPLETE. 445 and, as in § 30, it follows that the concomitants, which do not involve line variables, have as their leading coefficients the pure contravariants (including invariants) in 6,, a,, «,/ (=&, », ) as variables of the simultaneous ternary system (cy b,, a,9X, Y)’+ 2 (by, a, UX, Y)Z + a,’Z’, (d,, ¢,, b, a,9X, Y)°+ 3 (c, b,’, a, YX, YYZ+3(b,", a,"0X, Y)Z?4 "2°, (2,,4,,¢,,6,,0,0X,Y )'+4(d,',¢,',b),a, 0X, Y)°Z+4+6(c.",b,",a, CX, VP Z?+4 (bya, UX, Y )Z*+0,'"Z4, A bipartite lineo-linear quantic. 34. Taking the quantic in the form r=4 U=X(ae,+b.2,+¢2,+d2,) u,, r=1 the characteristic equations satisfied by the leading coefficients are, by § 16, a Ce Ms= brag + Page + Pose, + beac, — “aa, — ab, — % a0, “aa, => UTA INT CRC NO Let eee Ae = erap* taap tap nap, Sag, a ap, °2 Ge, “#Ba, ~ ee ee ae ee ee Cam eon 2 oe agen Hee coh, fee ag, ° 35. To obtain the independent simultaneous solutions of these equations, we first form the usual set of equations subsidiary to M,, which are da,_db,_dc,_dd,_ da, db, de, _ dd, Or cOn nett & Oe Sa. Su bee ad, _da,_db,_ dc, dd, _da, db, de, _ dd, SvOpe Oo nos 0? < 108 * Oro, 9 0 sixteen in number; of these, fifteen independent integrals are required. We adopt the process already used, and we have as immediate integrals = Oss f=, the latter of which is taken as a ‘variable of reference’; 6,=d,, 0,=a,, 0,=b,,, 0,=d,, 0,=a,, 6,=), 0 =d,; go =b, + ¢,. Further, we have 6A0,— 0,A0=—e,, CA0,—20,A0= 20, 0A0,— OAP= 4, 6A0,— 0,A9=—e,, 6A0,— 0,A0= ¢ 6? 446 Mr FORSYTH, SYSTEMS OF QUATERNARIANTS where e, = a,b, + 4,¢,, 0, =4 (c, — 5,) b, — b,¢,, e, =b,d, + ¢,d,. u, = b,c,? —(c, — b,) b,c, — ¢,b,°, &. = be, = be, all being solutions of the equations subsidiary to M,; and the fifteen necessary independent solutions are u, 0; 0,, 9, 9%, 9) 951 9a, 9,3 Ps €5> €» 3 9%, %- Of these fifteen solutions, nine, viz. u, @,, 9,, 9,, , €) €) €» U, all satisfy A=0. Of the set of five equations as given above, subsidiary to the determination of the combinations of @, 6,, 0,, 0,, 0,, 0, which satisfy A=0, we require four independent integrals; these we take in the form 44s hs = (c, — b,)° + 4¢,b,, é= Os tts =a,c,+a,b,, € Oe wis t a,d, —a,d,, and u,=— —— = b,c,a, — $(c, — b,)(b,4, — €,4,) + ¢,4,. The thirteen quantities a; 0, O2 05 Ws eter eyeples Che a Y, are a system of simultaneous solutions of M,=0 and V,=0; and every simultaneous solution of those two equations can be algebraically expressed in terms of these thirteen. 36. If we write c, — b, = 2k, 0 0 rf) we have a, 36 2 Ok’ Oe 0 ac, ob +4 ale? and then the equations M,=0 and NV,=0 become M, =," +,2-0,2 do Gon C, C,) Ons Bag ene eae eee e 0 0 0 a Cb W,=tipy + 55 7 ae tag ok tie that is, they are the differential equations of the concomitants of a binary quadratic having literal coefficients b,, k, —c, and in four sets of variables, viz. c,, —b,; M 4,3 d,, Ty; ¢,, —b,: and since u, 0,, 0, 0, and do not enter into the equations they must be regarded in the complete set of solutions. From this result it is easy to see the constitution of the set of thirteen solutions. Five of them—uw, 6,, 6,, 6, and @—are already accounted for; five of the remainder, viz. » &» € €» €» are determinants of variables; one, #, is the discriminant of the quadratic ; THAT ARE ALGEBRAICALLY COMPLETE. 447 u, 1s the quadratic in one set of variables c, and —b,; and the last one, u,, is the first polar of u, by the variables a, and a,. 2? In terms of the last eight every binariant of the system can be algebraically ex- pressed; and the remaining binariants, helping to constitute the complete irreducible system of binariants, are easily seen to be e,=bd, + c,d,; u, =(b,, k, —o, Ne, — b,0d,, d,), yD k,.— 0,0, — be, —b,), u, =(b,, k, — 6), ,)*, u.=(by k, —¢,§a,, a,0d,, d,), , —¢,0a,, a,0¢, — b,), 37. It is convenient, for the purpose of finding functional combinations which shall satisty @©=0, to modify the system of thirteen algebraically independent solutions. We have €€, = €€; — €,€,, so that we may replace e, by e, in the system. We have UE, = UE, — UE; so that we may replace wu, by u, in the system. We have UE, = UygEg — UAE, so that we may replace e, by wu, in the system. And we have UU — Uy =—te,”, so that we may replace ¢ by uw, in the system. Hence a system of independent solutions is OP he Che GLB (ae Gy Gy & 5 U,, U Us Un, U » € 4, Uy, 10° We now proceed to obtain the algebraically independent combinations of these thirteen quantities which are such as to satisfy @=0. The process is that indicated in § 17; and there must be obtained eleven such combinations. There will then finally be eleven solutions which are common to the three characteristic equations; they are independent of one another and are such that every solution common to the three equations can be expressed in terms of them. In order to obtain these eleven solutions it proves to be desirable to consider some of the other solutions of the irreducible binariant system given in § 36 and to introduce a quantity p=t—t¢°=c,), —c,b,. But all the equations used in this connection are subsidiary to the equations in the above thirteen quantities. 445 Mr FORSYTH, SYSTEMS OF QUATERNARIANTS 38. We have Ou=0, ©00,=0, Gc«,=0, Ou, =90; 06, =), 06, = b, Gc, = 0,0, i Od =—), Ou, = ),¢, Gc, = — b,0, | : Ou, = $,¢, Ge, = b, (b, -— d,) + ¢,),, © (u,— deh) =, (cb, — Cyd.) + A, (C.b, — B,C), for the thirteen principal quantities; and for a subsidiary equation Op = c,b, — b,c, others being useful for other solutions as given afterwards. As solutions we have u, @., €, t,.3 €, + 9,0,; u, — 9,3 6,.+¢; e, + 0,0.; A u' + hed, which are immediately obtainable. For the remaining two which are necessary, we have O(p +¢)= b, ( a 8,), so that 40 (p+) =—2(¢-9,) 9 (6-4), and therefore an integral is 4p + 4e,+(¢ — 6,)°; and lastly © (u, — $¢,6) =b,p + 0,Op = 9 (p8,), and therefore an integral is u, — 4¢,h — pé,. It follows then from the theory that every simultaneous solution of the three character- istic equations can be expressed in terms of the eleven solutions, (evidently independent from the method of integration), given by u; €,+0,0,, «+pt+i(d—9), ¢+9,0,, «3 OSG, $3 u,—9¢, Uutted, u,—teh—p),, Up. 39. Other solutions obtainable by the use of other subsidiary equations are X= hept+u,t+e¢0,+6,(e,+0,0,), u,+40,,—0,t+40,0,7, and so for others; thus ag by ca =u’ + Bu (€, + 8,0,) + 3X + 4(8,+ $)° +3 (8, +9) {e.+p+4($—4,)} +3(u,— 66 — pO,). THAT ARE ALGEBRAICALLY COMPLETE. 449 40. In order to obtain the fundamental concomitants determined by the foregoing leading coefficients, we turn the expressions of these leading coefficients into umbral forms. The quantic itself we take in the form A, Ug =b,Ug=... (using a’, b',... instead of a, b,... to avoid confusion with the real coefficients), Then U=a,=4a,4,, and therefore U = ayy =UdU, +... Next, we have 6,=a,=a,'4,. The completed form of a, is evidently a,. To complete a, we notice that a, in transformation is congruent with the point variables, so that = + w,y,2,a, is a covariantive determinant; one term of it is “,p,%,, and it may be conveniently denoted by (apa). Hence 0, =a, (wpa) = a,x,*p, +... Next we have €, = b,¢,— b,c, = } (a,b, — ay b,’) (a,8, —4,8,). The completed expression of the first factor is (a’b’p). The second factor is the coefficient IN VqWe —Upw, Of v,w,—v,w,, that is, of p, in plane variables; hence we may write the completed expression in the form (ap), and therefore E,=}(aU'p) (a8p) =¢,p, + -.. The remaining leading coefficients may be modified with proper combinations of the orders so as to diminish the order, the grade, or the class, as the case may be. We take Uy = Uy + €,0, = (a,b, = a,b,’) Buy La > and therefore Uy = Ca (a’b'p) (ap) (apy) = u)'a 2p, +... We take €, =€,+ 6,0, +u* = @,'8.b., and therefore Ey = b/g ig = €, LU, +... We take €, = €,+ 0,0,+u0, = b,'a,'B,, and therefore HE, =b,'a, (wp) = ¢,,7p, +... We take 1,=6,+o6+4u = M,, viz. we have 1, =d,=a,+b, +0, +4, an invariant. We take u,’ =u, — 6, = (a,b, — a,b) c.B,y,; and therefore U,' = ¢,/uguty (a’b'p) = u,'p,u,? +... We take U, =U, + deh + UE, = (a,'b,' — a,b,') cx'y,8,, and therefore Uf = ¢yUy (a’b'p) (xp) = Ue, pu, +... Vout. XIV. Parr IV. 58 450 Mr FORSYTH, SYSTEMS OF QUATERNARIANTS We take u, =U, — de, — pd, =|, b,, b, | =% (a,b,c) («.8,y,); GS esl ¢; d., d,, d, and therefore U,' =} (wb'c'u) (aByx) = u,'x,u, +... We take 1,=2 {e+ p+3(d—8,)"}+4 (646)? + uw’ + 2e, + 20,0, = ag by ; viz., we have 1, = dg b,', an invariant. Hence A system of fundamental concomitants of the lineo-linear quantic is given by the eleven algebraically independent concomitants U; u,; %,, 1,; Ey, BZ, E{; 0,; Uy’, Uf, U,’, Uy; and, in order to form an algebraically complete system, their polar derivatives (§ 18) must be associated with them. 41. The canonical form of U is LX,U,+1,X,0,+1,X,U,+1,X,U,. To reduce U to this form we first find 1,, J,, 1,, l,. We have 7,=4+1,41,+1,. Next we have an invariant ; e eo ci e ; j= fos Ox, OU, a3 0x, OU, = om AU Ep which for the canonical form gives jo = 41,4 Ul, + U1, + 11, + 11,4 1,1. We have another invariant : o o° 0° e : (= as ss Oa,0u, 3 0x,0U, ‘ ae a) Uy which for the canonical form gives ‘=U, +E ee ei, 234 341 Lastly we have an invariant 1,=| a, 6, ¢,, d, |=1,11,1, a, \D,, Gaya, a,; 5,, ¢,, a, a, }) bye, 0) for the canonical form. Hence /,, 1,, l,, 1, are the roots of the equation tif +) —i1+7,=0, so that the coefficients of the canonical form are known. THAT ARE ALGEBRAICALLY COMPLETE. 451 To find the variables, we have in the canonical form LX,U,+ 1,X,U,+ 1,X,U, + |X,0, =U, [2X ,U, + igX,U, + 17X,U, + 17X,0, =£,, 17 XU, 4+1,°X,0, + 1,°X, 0, 4+ 17X,0, = 0,73", NG get eae a RO ape When these are solved to give X,U,, X,U,, X,U,, X,U, all that is necessary is to re- solve the right-hand sides, which must be the product of a linear «-factor and a linear u-factor; and the variables for the canonical form are then obtained. 42. As in § 33 we may obtain those concomitants of the quantic which involve only point- and plane-variables, x and u. Their leading coefficients satisfy the three equations M,=0, N,=0, M,=0 as given in § 34, and the three a ap se 0 G cas ger id, Ss oa sami ead anes ca Beco: Bea: 0 0 0 a a é 0 a K,=d, ab, aap: Oop +d, ab, Oaag. = ap, = C, ac, qaag, =() @ a @ A a 8 a a gad Bir Tories eres agai, =cut tsabp << Go, “0d, = of which JV, is derivable from the former three, and K, from K, and the earlier equations. On comparing these with the differential equations of the ternariants of a lineo-linear bipartite ternary quantic, it appears that they are the differential equations of the con- comitants of a ternary quantic (0,X +¢,Y¥+d,Z) U+ (bX +c,¥+d,Z) V + (bX +¢,¥+d,Z) W, a, as point-variables (=X, Y, Z) and b,, c,, d as line-variables (=U, V, W). with a,, ds, To the system we must add a,, which is a solution of all the equations. When we take the ‘irreducible’ system in the foregoing variables for the ternary quantic, we have all the concomitants which if ‘reducible’ can be reduced only by means of the quantic. Now these are given by Clebsch and Gordan*. First, the universal ternary con- comitant XU+YV+ZW is for the present case a,b, + a,c, + a,d, =e,+06,, so that the quaternariant may be taken as in § 40 to be U,+a/= EH, =b. a, up. Next for the ternary form itself, we may write it (LX +1, ¥ +1/Z) (A,U +r,V + r,W) =U). (taking the quaternary quantic as 1,’u,). * Math. Ann. t. 1. p. 373. 452 Mr FORSYTH, SYSTEMS OF QUATERNARIANTS Taking then the irreducible system of ternariants as given (l.c.) by Clebsch and Gordan, we have for their functions in order t=1, =b,+¢,+4d,=0,+ ¢, so that the corresponding concomitant is ut, — U with the notation of § 40 for the quaternariants. Next we have ian =a, (1,4, +14, + U/a,) . Byrd,’ + Ab,’ + 2,0,) = a,B, (la — U,'4,)(b) — 6,0) = 0,/Byla'by’ — 2ua,’l,’r, + uv = a, Blab,’ — Que, + v’, so that we have a new concomitant which we may take FP =¢, 0 Og up. Similarly we have a, =1,'my = (1, + Uy tg + Uy’ y) (a'Ay + My Ay + D4) = (ag’ — a, B,) (ba' — 6,'a,) =i,—2e,+u in the previous notation, so that the ternariant 7, practically leads to the quaternariant ?,, Which is an invariant. Similarly we find Ff, = 4, Bb, cs'd. — Buf" + 3, — u* (where f’ is the leading coefficient of F’) and we therefore have a new concomitant F, = bcs da az'Ug. Instead of 7, we may conveniently take 7” defined (/.c. p. 385) by u =1(0 + 24, — 32,) = 43 (1/m,n) (AMY) =, so that the concomitant thus determined is U,’. In the same way, it is not difficult to prove that the new concomitant determined by @¢ is D' =d,'eskp'a, b,c, (yexx), and that the new concomitant determined by wy is W' = ag db, Ugg, (ceku). THAT ARE ALGEBRAICALLY COMPLETE. 453 Complexes in line-variables. 43. The differential equations of the concomitants of a quaternary quantic in line- variables are similar to those for other quantics, the difference being in the values of the literal operators. If any such complex be denoted by Qe = (:P1 + GeP2t UePot UPst UPs + TPs)” — wee See Amen’ PN THT mie b! =i p! o! op} Canveot Pa P2 Ps Ps Ps Pe » (with the condition A4+p~+v+p+oa+7=n), the laws of transformation of the umbral coefficients g are derivable from those in §1 for the variables p by the relation Q,=g,; and so far as quantities of the first order in the coefficients of transformation, being all that are necessary (§ 5), are concerned they may be taken in the form Q, = oem qa! or Isls 5. qk. ats qk, Q,=-G",4+% —Gn,t+ qh, — qh Q3 =—G% + Golo +s — qk, + Uhs \ .=-—qm,+ ql, +9, +49,M,+ Gls ; = 1M = Gl, +UM +9, + Ole Q; = = Wag + PUL + Gs", + q,™, + UF from which the variation of Quypor(= 4," qo" UP 67 Yo") can be derived. The result is that the characteristic equations M,=0, V,=0, M,=0 are given by 0 M, == (= 2p, 1 =1, v4, po, 7+ POA, u,v, p=, o +1, 7) A =0 Ouvpaor s 0 Nv, =2(— vd, wt, v1, p, 0,7 + OM), p, v, pt, o-1, a) =0 m5 OA) uvpar 0 and M,= 2, (= pari, By v, p-1,0,7 + TAA, w, v+1, p, or- ar =0 ‘Av pot the summation being in every case for values of A, pw, v, p, o, T such that A+e+y+p+ot+T is equal to n, the grade of the complex. When the leading coefficients of concomitants, obtained as simultaneous solutions of these three equations, are to be changed into a symbolical form, with a view to the deduction of the expressions for the concomitant, it will sometimes be convenient to regard the coefficients q as similar to compositions of the umbral elements of two linear powers a, and b,, so that we may take Tee a,b, = a,b,, = 4,0, — a,b,, 2 = Agb, — 4,b,, qs = 4b, — ab,, (B= a,b, = a,b,, Ci a,b, = as, 454 Mr FORSYTH, SYSTEMS OF QUATERNARIANTS only however as an indication of the completion of the umbral forms and not as_neces- sarily implying the relation 9,9,+%%s+%% = 9. We proceed now to obtain the con- comitants of some special complexes. I. The linear complex. 44. We may take the complex in the form 2 = GP, + UePot UsPs + GPs + UsPs + UePo = (abvw) = (abp) = (cdp) =... = symbolically. The characteristic equations are a 0 é é Mar tigg thee. =O Wer rmllnigge Bagg a e=mu = 0 a gadis, Veco. It is easy to shew that every simultaneous solution of M,=0 and V,=0 can be expressed in terms of the three independent simultaneous solutions Gi» Wo W594 + 195° To find the combinations of these which satisfy © =0 we have @q,=0, ©4,= 4» © (9694+ 9296) =— WI so that the solutions can all be expressed in terms of q, and 4,9, + 9,9, + 4%: The first of these evidently determines ©; the second is an invariant, being $(abed). The signification of this invariant is given by Pliicker*. Il. Nah +04, = U9 + 49s + ON» N= Yl + wge = 496 + 129% + 5%: In terms of these seven solutions every simultaneous solution of the three equations can be expressed. 46. The fundamental concomitants thence determined are = op, +... = (abp); OY! = op, +... = (wp); t=Qs +%%s +44 an invariant =} (abed); = G/U 46% +d, 2D Invariant =} (a’b’c'd’); © = a,a,! (W'p) + b,b/ (aa'p) — 4,/b, (ab'p) — a,b,’ (bap) = $271... 5 A =a, (ba’b'u) —b, (aa'b'u) = Awu,+...; AN’ = a, (b’abu) — b,' (a’abu) = Nayu,+...; and it may be remarked that A +A'=(aba'v’) u,, so that we have an invariant ad, +4,a,+ aa, + 4,4, + aa, +a,/a, intermediate between 7 and 7. With these fundamental concomitants their polar derivatives (§ 18) must be associated. 456 Mr FORSYTH, SYSTEMS OF QUATERNARIANTS LUE axel regulus of three linear complexes. 47. We take them in the forms 2 = GP, + Pst UsPst UPs + UsPs + UP = I> OQ'= - OF S956 and then the three characteristic equations are 0 a) 0 0 0 Chee =_— ‘ Pose 4 — ” Oe ” = ) eee a ene gle tages earn r é oo 70 ® 28 eee 83g, ag, — © Bag * © agg 8 age + agp =? ree ee er ee a g oa, 04, ae 04 a 0g, we og. Wg.” Gat age" 48. Of the complete set of solutions some will be, as in the preceding case, quantities which involve coefficients of only one of the three: these are o =% 6=%% +995 +99 3 W=G, 8 =hG +49 +950 5 w= Gq", = Oy'G. +9495 +95 N Some involve coefficients of two of the three: these are P= Wes — se ; Me =U +4 Ge +96 % | a =I +992 +99 ; Ps = 29s — 3% 5 Mes =h Ve +s Ge +94 % 5 =h% +99 +%%5 5 %.,—0, = 9, 4,5 A =O. +99, FU 5 nS =9'Gs + %5 G2 + % Qs - And from these we have three invariants, viz. Nye t Ane 3 Nog + Aros 3 Xa a5 Ne m For functions which involve the coefficient of all three of the congruences we have as simultaneous solutions of M,=0, V,=0 Xi2= Ws — UIs 5 Xe = Ie — We Ge 5 Xs — Is Is “7 UIs - Now we have Ox. =-U% +4 Ip and so for the other two, so that 4; OX + J, X25 is 4; OX = 0, or 1S) (4X0 + OTK + rye) = 0, THAT ARE ALGEBRAICALLY COMPLETE. 457 ? so that we have 6 |.Oien"Gins Gee |) Giri Geo Ie. Gis Gas qt which is a solution new in form. And we also have ©¢.=% ©%=9, 99 =4%> so that © (G.bo5 + Yo ber + Yo Piz) = 9; and thus we have = |"G.) Gas o> Is» We o> I> Is. a solution new in form. The last two are of course expressible, after our theory, in terms of the earlier solutions: thus LPs, aE re Par +2... = 4,0 Udbs, ats Nes Pris ar A2Pos = qe Udy ar Nor Pos ae NosPor 7 qo ) (which reduce the fifteen solutions @, w, 0"; 4, U, 0"; Py, Pogr Pars ® Nes Naame Ae 12? Ayo 3 23? 23) 31? to thirteen ines independent solutions, being the proper number); and similarly O = +1 (Ay Ags — UU) $A qe rece i 23° “23 31 31 23 —w") + qn eer 2 — Ww) 6 / "§ ’ “ wee Ls ~ yb ssPex (1%: q,'6 —-hQa Po, = G9; & Psy — 419; & $,2)- The system of seventeen solutions forms an irreducible system of solutions, that is, every other solution can be expressed as an integral algebraical function of these seventeen. But it does not therefore follow that combinations of the concomitants determined by these are not reducible; reducible examples will actually be found, the reduction taking place owing to a possibility of removal of a factor w,, which for the present purpose is determined by its leading coefficient 1. 49. It is now necessary to determine variables to be associated with these leading coefficients, so as to have the fundamental concomitants. We take Q =q, =(abw) =(cdww) =... YQ’ =q, =(ab'ew) = (cd'ww) Q” = g,” = (ab ww) = (c"d"vw) =... Then we have ie =q, =(a,b, —a,b,) , so that O =op, +... wo =q,/ =(a,b, —a,b,) , so that O’=o'p, + |r = 4," = (a,/b," — a,"b,”), so that 0’ =o"p, + Vor Xk, “Pann ly. 59 458 Mr FORSYTH, SYSTEMS OF QUATERNARIANTS , Ce | t =4 (abcd) , an invariant, (abcd), an invariant, ( be =} (a’b'c'd’), an invariant. Py = 4,4, (b,b,') — a,'b, (b,'a,) — a,,'(a,'b,) + 0,6,’ (a,a,') ®, = $52, P, ++. ®.= $,,2,'0, + +. ®,, = $5:2, P, + --- A. = a, (b,a,b,’) — b, (a,a,'b,') Me = 4,’ (b,'a,b,) — b,' (a,'a,b,) NE= 20 es Ay = ry @u, +... A= AF, + Nyy = gg TU, + +. Ni Ae IS: Ag, = Ag, CU, ++ Also 2,,+2,,’=(aba'b’), an invariant; so that A,,+A,,’ is reducible, being divisible by u,. Similarly for A,,+ A,,’ and A,,+A,/’. Next we have 8 = (a,'a,b,) (b,'4,b,”) — (a,"a,"b,’) (b,/a,b,) and therefore A= (a'abu) (b'a""b"'u) — (a'a’’b"'u) (babu) = $u,7+.... Lastly we have & = a,a,a,” (b,b,'b,") — b,a,a," (a,bb,) — a,b,” (b,a,'b,’) — a,a,'b," (0b a,) fhm Woes t +a,b,'b,” (b,a,a,’) + b,a,'b,” (a,b,'a,") + 8,b,'a,” (a,0,',”) 2°34 pat eat | s bbb," (a,a,a,"), so that A= 0 an: = Xa,a/a,/" (bb'b’u). With these fundamental concomitants their polar derivatives (§ 18) must be associated to form an algebraically complete system. 50. When we consider more than three linear complexes, no functions new in form are obtained beyond those already given; and any new functions which occur are reducible to combinations of those which precede. Thus for instance the eliminant of six linear complexes, viz. POG, a a a can be expressed as a sum of products of the form ae LLL me (9290 Go) (UG Is )s that is, of the form 6,,'6,. Similarly for others. Symbolical Expressions of Concomitants of Complemes. 51. We are now in a position to derive an important inference as to the forms of factors that may enter into the symbolical expressions of concomitants of q,". These sym- bolical expressions give, in accordance with the usual forms, concomitants of a set of simultaneous linear complexes q,, q,, q,',.-. each being represented nm times in the ex- THAT ARE ALGEBRAICALLY COMPLETE. 459 pression. Hence, when the leading coefficients are formed, they are leading coefficients of the concomitants of that simultaneous system. We have seen the kinds of factors that can occur in these leading coefficients: they are of one or other of the eight types O= hp = G96 + 95 + W5%> M= 9,90 + Wee + Ws» X= Hot W542 + 14 JHNENM= GG + UIs + Gee +195 + W692 + 109» $ = 1s. a Ye B= (4,969), 8 = (4.95% )- Hence every leading coefficient of a concomitant of a complex of the nt grade is an aggregate of powers and products of factors of the eight types; there being the necessary restriction that each umbral symbol shall occur exactly n times in any product or power, constituting a term in the umbral expression. To complete the expression of the leading coefficient so as to give the fundamental concomitant we have seen, in the preceding investigation, the combinations of variables that must be associated with the several factors, the respective degrees being as in the following table :— oRDER | GRADE | Lass FACTOR. zi . a o 0 1 0 L 0 0 0 ; nv 1 0 1 r 1 0 1 j 0 0 0 Ci) 2 1 0 Cy 0 0 2 oF 3 0 1 In addition to this it must be borne in mind that certain combinations of factors of the types 6 and 6 are reducible, that is, the completed expression contains some power of uw, as a factor. These results will be used in obtaining the concomitants of a complex of the second grade. 59—2 460 Mr FORSYTH, SYSTEMS OF QUATERNARIANTS IV. A complex of the second grade. 52, Taking the same coefficients as Pliicker (l.c. p. 149), the complex may be written Q=Dp'+ Ep! +Fp’ +Cp' +Bp, + Ap, 4 2Mp,P, AF 2Lp,P, ar 2Tp,P, 53 2Sp,Pz oF 2ND,P, + 2Kp,p, + 2Up,p, + 20p,p, + 2PP.Pa + 2Vp,p, + 2Rp,p, + 20PsPa + 2Gp,p, + 2HP,P. + 2UPPe = (QP: + GP. + Ps + UsPat GPs + VePo) = Vo If instead of taking 2 we take 2- 3 (N+ O+ V) (D,Ps + P.Ps + PsP) the complex may be considered as in its ‘normal’ form, according to Clebsch; we shall however retain the above more general form, for the sake of symmetry. The three differential hatie! which determine leading coefficients are :— a a a M,=B +264, ~F 2-2 Gt B(S- a) + a. a Lag ap +73 aon cAee a 2 a a V0 2g oe = 2K + +(V-0) <5 at (a av) 0 0 0 -M OP tH, zttag a 3 ee @=M,= =F +2005 -D = 7-22 ay t (V- WM sztL (sy ar) ee ee ~ May -Saqt kapt Baz 53. The equations M,=0, N,=0 admit of a transformation the effect of which enables us at once to write down the system of simultaneous solutions. Let new quantities I and W be defined by the relations O0-V=2I, 0+V=2W, 30 that ie Pe sO na 30 ov. of’ the first two equations then become a a 0 a fi) 0 0 0 0 0 =p ORNS 75 loys) eel. ee ye ple 4 M, = Ban +2G 2, — Pap 2K op Raz + Way Lay — Qapt I at Sap aC 0 ,oO 0 a 0 a ol. N,=C=5+2G = bt ox 5405-5 - MA PS +H tT og ag THAT ARE ALGEBRAICALLY COMPLETE. 461 These are the differential equations of the concomitants of a system of three binary quadratics, the literal coefficients of which are CoeG, BE: [Se JK (Of, 1 ay te the variables being the four sets Tete HS Shy = Oe Pei eM: To this system of solutions must be added those, which are simultaneous solutions in virtue of the fact that certain quantities, viz. D, A, N and W, do not enter into either of the equations; these quantities themselves are the proper solutions to take. Among these solutions the following occur (they do not constitute the irreducible system, Jacobians for instance being omitted) :— e,=QT +SP, «= PL— QM) «,= QH + JP, e,= LT + MS \ | c, = HS — TJ, ¢, = HL +S | with the usual relation €,€, = €,€, + 663 —,=(C, G, BYJ, —HY, E&, =(C, G, BYS, —TQQ, P) &,=(C, G, BYJ, — HOS, —T), & =(C, G, BYS, —TYL, M) &,=(C, G, BYJ, —HQQ, P), &, =(C, G, BYQ, P)? 5 £,=(C, G, BYJ, —HUL, M), & =(C, G, BYQ, PUL, M) E.=(C, G, BYS, —T)’, w= (C, G, BYL, MY’ | $,=(E, —K, FUJ, - Hy, | $,,---, ,, functions the same as ¢, in bs coefficients and the same as &,,..., &, | in variables 0,=(U, I, —RYJ, — HY, 0,,..-, @,, functions the same as @, in | coefficients and the same as &,..., &, ' in variables. t,= BC -G@, t,=CF +BE+2GK, t, = EF — K*, t,,=OR—BU +216, t,=UR+T, t,, = FU — ER+2KI, B= WC, BGa bal | 2, —Ke EF Uap oo aes 462 Mr FORSYTH, SYSTEMS OF QUATERNARIANTS being the full set of irreducible invariants of the system of three binary quadratics. Among the irreducible members omitted are, as already remarked, all Jacobians; of which there are ten, involving the coefficients of & and ¢,, ten others, involving the co- efficients of ¢, and @,, and ten others, involving the coefficients of 6, and &. As is well known, the square of each Jacobian can be expressed as an integral function of the foregoing quantities; and that square is therefore a reducible function. An algebraically complete set—that is one such that every solution can be ex- pressed as a function of the constituents of the set—is obtainable by the method of § 17 as composed of D, A, N, W; #¢,, t,, t3 &; E3 dis bys 9%. 9,3 Go Gs yr &> &- By means of the relations which subsist among the irreducible binariants, this set can be changed so as to become the equivalent algebraically complete system composed of D, A, N, W3 ty ty t35 rs G03 bur bo» Pros Or» Fs0s Err ar > os 54. In order to find the functional combinations of these quantities which are such as satisfy © =0 we use the process of § 15; and to obtain the simplest form of solutions it is convenient to retain other members of the system of irreducible binariants, always considered as subsidiary to the algebraically complete system. I do not reproduce the analytical details; the following are a set of sixteen inde- pendent quantities, in terms of which all the simultaneous solutions of the three charac- teristic equations can be expressed :— ana te =2W+N a Ey a 0, aS t,,D Ey = Exot 2e,N + N*D ee) Pr d, = p, + Mt, — 9,— We, .= $, — Nt, 9,, 6'= 0, — We,—Dt,+ DW? i = #D-€, t ti = t, +e, +3W?+2WN + N* e( = 2e, +t, + AD En— 6, 1, e=e« +ND 5 5 6'= 0,— We,—Mt,+NW (= Other solutions of a simple form are $, = $, it eD, p,, = Ae t,A, $, = $,+2¢D+ AD’. THAT ARE ALGEBRAICALLY COMPLETE. 463 55. In order to obtain the fundamental concomitants determined by these leading coefficients, we turn their explicit expression into one which is umbral, and then complete the latter by means of the variables as in § 51. The results are as follow :— } On = qs | Qe ay Sb soc © =I +%%s + 99,» 2D invariant El = Gi (GN Gs ) Q9s + %594 + Yer) B= E/x,u,'P, +... Es = HN" (Ge + UIs + 12) (VGe” + Ws” + 992") ie E02, Pp uy a 0 = UG (4s = IsFo) (GIs — 1542") P= untae dp acer =-G) (4.95%) (Vide + UY0" + 9:42) , &,'p yu, +... » = (U9 9%) (492 Is ) 1 oS 1 i 6- ty = (190 + Io + 92) (1 Ge + 959s + 95 9) YS eT 3 Sided { ®,' = deh, DP; Uy Tee | a = s (1% 98 ) (11% Is ) OF 60a pth \ a, = hh (9.95. = Tee ) (1.96. = U4 at UI ) 8, = 91 PU, Toe { t= (404%) Eb Ae tone | f = (4.95 oo G42) et hpi e = (G06 + 19s +8)? + (GIs + Gt eG)” Ee €, fb, 2b oo. Evidently 2¢, +e,’ is an invariant; and thus 27,’ + E,’ is divisible by w,*. { 6! = (dee — Gos) (Gide + 94 Ue + Ye Ge) = (Ie + Us + 492) (G's — 1.98) i = €,2,'p 144 { & = OG (Nt UGt Gt GIs +99 + 9%) IDE A= Ge/e Se d00 { O, = (419195 ') (9:95 Ie ) @, = 6,2u? +... chet ime * In every case a numerical factor, if it occur on the right-hand side, is omitted. 464 Mr FORSYTH, SYSTEMS OF QUATERNARIANTS A system of fundamental concomitants of the complex of the second grade is composed of the preceding sixteen concomitants ; and with them must be associated their polar derivatives (§ 18) to form an algebraically complete system. 56. The expressions for the other three are :— (b. = 9%" (9s — 542) (91s + %'Ys + 95 Ye) \@/ = b/a3pu, +... (Pe = (9:9s%5 ) (®, = $/2,u,2 +... = =U (GI #95 + Ys) (90% + W9s + 9595) OD! = $, 2,'u,"p? +... Two invariants have been obtained, viz. 6= 9,96 T Ys + 39> 2t! +6, =F (G9 +998 + 9s +U%s + Ie + U9)? and others will be obtained explicitly for a canonical form. The discriminant of 0 is (save as to a numerical factor) LLL Ll (9:90 Us Us Vs” Ve =|) De Me Wie Fin teh U, > O DS DF CHIOV =~ ' N, lee Q, H, 57. The alternative canonical forms of the complex of the second grade, dependent upon relations among its coefficients, are given by Weiler*. The most general of all is the first, in the case of which there is no relation among its coefficients. When we apply the transformations as given in § 1, there are twelve constants at our disposal; with these it is possible to make twelve of the terms in the complex dis- appear. Nine terms will thus be left, the aggregate of which may, on account of the symmetry, be in general assumed to be of the form Dp, + Ep,' + Fp,’ + Cp? + Bp; + Ap,’ + 2Np,p, + 20p,p, + 2V P,P» which includes the 1%, 24, 4%, 6, 7 and 8™ of Weiler’s forms. The further transformations possible on the introduction of coefficients >, pw, v, p (instead of the unities of § 1) do not essentially modify this form; they merely give new coefficients of the complex Dy'v*, Ev'r*, Fr y?, Cv’p*, Bu'p?, Ar’p’, 2Nruvp, 2OrAprp, 2VAmyvp. The preceding form will therefore be taken as the canonical form of the most general complex. * Math. Ann. t. vu., pp. 145—207. THAT ARE ALGEBRAICALLY COMPLETE. 465 58. To obtain the invariants, we might proceed as follows. We know, from § 51, that every invariant is an aggregate of products of umbral quantities which have the form m) (n) (m n) (mr, m) = qrge” + Ga Ge F ade” + Id” + 98g” + 969," each product containing umbral quantities the proper number of times. We have invariants in umbral and canonical forms as follow: I,=4(, 0) =N+0+7; T,=4(0, 1) =N*+AD+0°+ BE+V*+ CF; T,=% (0, 1) (, 2) (1, 2) =N*+3ADN + 0°+ 3BEO+ V*°+ 3CFV ; 1,=4(0, 1), 2)(1, 8) (2, 8) =N*+6ADN*+ AD’ + 0+ 6BEO’+ BE’ + V*+6CFV? + CF’; T,=40 )O, XA 39@ 9B, 4 = N°+10ADN*+5A°D*N + O° + 10BEO*? + 5BE*0 + V? +10CFV? +5C°F'’D : and J,=4(0, 1)(0, 2)(1, 4) (3, 4) (2, 5) (3, 5) =N*°+15ADN*+ 154°D°N* + A°D' + 0° + 15BEO' +15B°E°O? + BSE? + V°+15CFV*+ 15C°F?V? + CoF*. The expressions for these can be modified. When new quantities &, &, &, &, &, & are introduced, defined by £=N+ ora £,=0+ (BE)*| E=V+ a B= NAD) £ = 0-(BE)) £.=-V-(CF)!) so that they are the roots of the equation {(€ —N)*— AD} {(€ — 0)’ — BE} {(€ — V)’ - CF} =0; then the invariants are given by r=6 21,= > &,° (@=1, 2, 3, 4, 5, 6). r=1 And further the coefficients of the foregoing sextic are rational integral functions of the six invariants. 59. When we take the further transformation indicated in § 57, so as to give the complex the most simple explicit form possible, we may take Apwvp =1 And?20? = By’? = Crp? =1, Vou. XIV. Part IV. 60 466 Mr FORSYTH, SYSTEMS OF QUATERNARIANTS. and then the coefficients of the implicitly general complex are AD, BE, CE V1; 1, 2N; 20552. Hence every invariant of the complex is a function of AD, BE, CF, N, O, V. By means of the substitutions which interchange p, and p,, p, and p,, and which leave p, and p, unaltered, it follows that BE and O enter into the invariant in a manner similar to that in which AD and N enter. Similarly for CF and V. Since 2N=E, +8; 4AD=(E,+£,)'—42.8,, it therefore follows that when the invariant is expressed in terms of the six quantities & it is in the first place a function symmetric in & and &; in & and &; in & and &; that is, any interchange of the two members in each of these three pairs does not affect the function. And secondly, since it is symmetric in the three pairs, it follows that the invariant is a symmetric function of the six quantities &; and hence every imvuriant is a rational integral function of the six invariants I,, I,, I,, I,, I,, 1,, which thus constitute the irreducible system of invariants of the quadratic complex. For example, the discriminant is, for the canonical form, = (A D—N*) (BE— 0°) (CF — V*) = eee ee. the integral expression for which in terms of J, is immediately obtainable from well- known formule in the theory of equations. 60. Further, if we write Q=P: Pst P2Ps* PsPs» so that the permanent relation among the line-variables is Q=0 we have I, (Q.+ 2AQ) =(Zo, Lo-1, -:., L,, 3G 1, r)®. For 2+2dQ differs from © only in having N+A, O+A, V+X instead of N, O, V, i.e. in having +2 instead of >A. Hence the theorem. The two conditions necessary that the complex should reduce to the canonical form adopted by Battaglinit which is without the terms in p,p,, p,p,, P,P, are evidently I, (+ 24Q)=0, 1,(2+2AQ)=0, J, (M+ 2aQ) =0, equivalent to the two conditions d,, J, J; 381, —42,)°=0, (Ts) Eos Dy dye Le =O. * Weiler’s (l.c.) first and most general canonical form would be obtained by taking Dy?v?=Ad*p", Ev)? = Byp*, FX u2= Cv*p?. His coefficients \ are then the quantities ¢. + For references, see Klein, Math. Ann. t. xx111., p. 578, Segre et Loria, ib. p. 213. XVI. On the stresses in rotating spherical shells. By C. Cures, M.A., Fellow of King’s College. [Read Feb. 11, 1889.] IN a paper recently printed in the Society’s Transactions I considered amongst other problems the case of a rotating spherical shell of isotropic material. A solution was there obtained involving six arbitrary constants which were completely determined. The solution was however given explicitly only for the case of a thin shell, which was examined in detail. The object of the present communication is to deal with the general case in isotropic materials when the ratio of the radii of the bounding surfaces may be any whatever, and specially to examine into the equilibrium form of the bounding surfaces and the tendency to rupture. In obtaining the solution the surface conditions determining the arbitrary constants were supposed to hold over the truly spherical surfaces existent prior to the strain con- sidered. This introduces a deviation from strict accuracy, and in order that this may be negligible it is essential that the forms of the equilibrium surfaces do not differ much from the spherical. Since in the cases considered here the surfaces are practically spheroids of small eccentricity this condition is secured. In the paper mentioned above two theories as to tendency to rupture were mainly considered. In one it is supposed that if a value be obtained for the difference between the greatest and least principal stresses which exceeds a certain value found by ex- periment then the body will sooner or later rupture. On the other theory, if the greatest strain exceed a certain positive value, rupture will ensue. In both theories the stresses and strains are supposed to be calculated on the hypo- thesis that the laws of perfect elasticity are strictly followed. This does not however necessarily presuppose that the body will remain truly elastic up to the point of rupture. Doubtless in approaching the point of rupture most, if not all, materials cease to follow the laws of elasticity, but there seems reason to believe that in the case of many solids when the limit of perfect elasticity is passed eventual rupture is merely a question of time. If then the experimental values introduced in the above theories be those answering to the limit of perfect elasticity, it follows that rupture will eventually follow the appli- cation of external forces which would call forth, on the theory of perfect elasticity, values for the stress-difference or greatest strain exceeding these critical values. 60—2 468 Mr C. CHREE, ON THE STRESSES IN ROTATING SPHERICAL SHELLS. In materials in which the limit of perfect elasticity can be safely passed, the stress- difference and greatest strain merely indicate the limits within which the application of the equations of elasticity should be confined. For brevity the term “tendency to rupture” will be used here, but strictly it must be interpreted to mean “approach to limits of perfect elasticity.” On what Professor Pearson terms the “biconstant” theory of isotropic elasticity, with its two constants m and n, the expressions occurring for the strains and stresses are so complicated that without assigning special values to the ratio m:n it would be well nigh impossible to apply our results practically. To tabulate all the expressions for a variety of values of m:n would be very tedious and seems scarcely worth the labour, as the general features of the case seem pretty much the same for all reasonable values of this ratio. I have thus mainly confined myself to the special case m=2n,—the only case on the “uniconstant” bypothesis,—which at all events represents an average state of things, and is unquestionably nearly true in some of the commoner metals. A complete analysis of the principal strains and stresses and of the tendency to rupture has been found possible only over the bounding surfaces, which are fortunately the p'aces where the comparison of theory and experiment can best be effected. It has also been found that the tendency to rupture is greatest for a solid sphere at the centre and for a thin shell at the inner surface. This lends considerable support to the view that the bounding surfaces are in general the places most exposed to rupture, and further corroborative evidence will presently be found to present itself. It must be distinctly understood however that this does not amount to a proof. Thus it might not be quite safe to assume that the shell would not rupture when the greatest values we find for the tendency to rupture are less than the critical values, though if they exceed the critical values the shell is certain to break up. Denote by @ and a’ the radii of the bounding surfaces of the shell. Let wu and » denote the strains along the radius vector and perpendicular to it in the meridian plane through the axis about which rotation takes place, and let p and » denote the density and the angular velocity of the shell. : 3 cos" @—1 ryan? Then if P,, = 4 “= ate: : | {24m (m — 2n) (aa’)*(a? — a”)? + (3m — n) (9m + 5n) (a? — a") (a? — a" r — #(4m—n) {om + 5n) (a® —a”) (a? — a") r +5 (3m 4 2n) (aa’)’ (a — a) (a? — a") vr? — 9m (aa’)’ (a* — a”) (a’ — a’*) rf] wp ( ,,d5mtna—a’ | sm+n = 4 es Uae +n 13 & ae -2 i 15(m+n) | Pik gee ag ee ee } ce aes ak (1), * See also footnote on p. 301. Mr C. CHREE, ON THE STRESSES IN ROTATING SPHERICAL SHELLS. 469 2 “= 0 ‘ y= aes cos 8 |- {120 (5m +4n)(aa’y (a’—a" YP +4(5m—n) (9m +5n)(a?—a’*) (a’—a"), 1 + (4m —n) {(om + 5n)(a°— a*) (a — a”) 7 +10n (aa’) (a — a”) (a — a") + 6m (aa’y (a — a”) (a — a”) rt] eS ea OME Soli alaise einissis sid aeiae (2), where D= (9m + 5n) (19m — 5n) (a* — a) (a — a”) — 504m? (aa’)® (a? — a”)? eee (3). Also the dilatation is given by s= eueks |- |stm (aa’)’ (a? — a? + $(9m-+ Sn) (a — a®) (a — a) re +5 (4m —n) (aa’y (a*— a”) (a — a”) | wp { 5m+n a—a*) m+n | ~/6 Tata etatsetatecialalatatalafeleitalatatclal estat e(eteve!ntelat= (5). 5(8m—n) &—a®) 9) ae Soibte In the special case m=2n, writing z for a/a’, we have 2 Jl op?, u=3 —In E (2 —1) (2? — 1) 7° — 322 (2-1) (2-1) Za’ — 560 (2—1) (2 —1) 2? ar? + 252 (2? —1) (2 — 1) ar | 2 5 2 ee 5 a2 ,@e fp, Ue Dewees lar) SS Fae =| obobaoDastanoonacnadscos Lez 45n | 52 (@ —1) ei (ha) Ps: oe |- (67229 (2? — 1) + 207 (2° — 1) (2 — I} 7? + 322 (2-1) (= 1) 2a’ +140 (2 — 1) (27 —1) z7a'r* + 168 (2 — 1) (2° — 1) ear] aici sesceieemisess (Zia) Tie) 5 ON ze b)| (2— I) ON Gee (es a ccaeceenemscscessesce (3a), muapl, af Deas) SNe u (ie 3)— ee INN Rid Ti -2 5-3 iT) — 2{10082 (2? — 1)’ + 23 (2° — 1) (2’ —1)} 7° +-:140 (—- 1) (2-1) za ; 2-1 be + = [pir + 4iz? aT | Be at Raetcs co arenes (5 a). The equilibrium surfaces of the shell being of the form PSC eae le. where c and c, are constants and ¢,/c is small, we may treat each surface as very ap- proximately a spheroid, whose ellipticity may be regarded as the ratio of the difference between the equatorial and polar radii after displacement to the radius of the un- disturbed spherical surface. As a clearer idea of the nature and magnitude of the 470 Mr C. CHREE, ON THE STRESSES IN ROTATING SPHERICAL SHELLS. phenomena may be obtained from the consideration of a particular case, we shall apply our results to a shell whose outer radius is 4000 miles and which rotates with the Harth’s angular velocity. Supposing we take @'pa*/H =-006, where # denotes Young’s modulus, then if p were the Earth’s mean density the material would have the same elasticity as an average specimen of iron. If », and 7, denote equatorial and polar radii, the ellipticities of the two bounding surfaces may be taken as putting r=a in the first expression and r=a’ in the second. The following, Table I., is calculated from (1), noticing that Ye — Ty = — 8 [coefficient of P, in (1)]. We have replaced n by 2# when m=2n, and by }# when n/m is negligible; and e is supposed so small that e° is negligible. TABLE I. 3 fw pa® . / wpa? Particular case when a=4000 miles, and Value of ¢ / —— Value of ¢’, E w*pa2/ i =-006 Gs a Difference in miles on the outer surface on the inner surface between the equatorial when ‘ when — = when ‘ when - and DOE ROG a aa Malus) of. 1 negligible m=2n & negligible m=2n ‘outer inner’ “ outer inner aja m m 7 surface surface surface surface 0 ‘BO ‘BAL 16°36 ‘00205 ‘01 341 1:04 16°36 D ‘00205 ‘00624 ‘ Bo B43 1:05 1-034 16-47 4°96 00206 ‘00621 2 ‘3da9 1-038 17-25 SHIA ‘00216 ‘00625, "25 376 1-050 18:06 12°60 00226 ‘00630 3 47 423 112 1-085 20°31 17°36 ‘00254 ‘0065 4 “482 1:13 23°16 21°78 00289 0068 3 ‘67 "62 131 1:25 29°70 30°06 ‘0037 ‘0075 6 82 143 39°33 41:10 ‘0049 ‘0086 Gj 101 157 48°57 50°32 ‘OO61 ‘0094 $ 1:48 1:90 71:12 72°81 ‘0089 ‘0114 9 2712 188 2°34 211 90°06 91-11 ‘O113 ‘0127 l—e 2°5 2°25 108: ‘0135 Inu the calculations on which the tables are based a closer degree of approximation was usually aimed at than appears in the tables. In each case in the tables the last figure is selected so as to make the result stated as near the closer approximation as possible, but the number of figures retained has been varied according as the rate of change with a’/a of the quantity considered is more or less rapid. [It will be observed how very similar are the conclusions derived from the widely different hypotheses as to the value of m/n; thus the results deduced from the hypo- Mr C. CHREE, ON THE STRESSES IN ROTATING SPHERICAL SHELLS. 471 thesis m= 2n apply doubtless with considerable accuracy to the case of any ordinary isotropic material. Another important conclusion is that the existence of a concentric spherical cavity has extremely little influence on the form of the outer surface so long as the radius of the imner surface does not exceed one quarter that of the outer. The practical constancy in the ellipticity of the inner surface until this limit is reached is also worthy of notice. Some interest may also attach to the fact that in the particular case, even when the sphere is solid to the centre, the difference between the equatorial and polar diameters exceeds three-fifths of what is found in the case of the Earth. Besides converting the spherical surfaces into spheroids of small ecceniricity, the rotation increases the mean radii of the bounding surfaces. These increases are given in the following table. TABLE II. Particular case when aiatna® a=4000 miles, and w*pa*/E=:006 Part of —/ “*" which is independent of P, le = = = r/ E = Increase, in Increase, in = 5 a ay miles, in miles, in Outer surface Inner surface mean radius mean radius Value of Uae ee 9) & roik ~9 of outer of inner aja m negligible oe m negligible aoe surface, Aa surface, Aa’ 0 0 06 16 ‘Ol negligible 06 25 ie 16 ‘066 Al negligible ‘067 "2475 273 1608 65 2 ‘002 ‘069 242 269 165 1:29 3 ‘076 261 1°83 2-09 + ‘0825 257 198 2-47 5 ‘027 ‘096 ‘214 252 231 3:05 6 ‘151 247 314 3°95 8 ‘O94 a 7/Al ‘183 ‘246 4:09 4-72 9 207 247 4°97 5°33 l—e 16 "25 6-0 The small effect on the radius of the outer surface produced by the presence of an internal cavity until the diameter of the cavity approaches to one quarter of that of the outer surface calls for remark. A similar phenomenon has been already pointed out in treating of the ellipticity. It may also be noticed that on the inner surface the increase of the mean radius bears to that radius a ratio which is not far from constant. On the outer surface, especially for small values of a'/a, there is a very decided difference between the results answering to the two values of n/m. This may seem to contradict previous statements as to the representative character of the hypothesis m = 2n; but a comparison of Tables II. and I. will shew that the displacement dealt with in Table II. is a comparatively small one. This will be most clearly seen from the columns in each which deal with the particular case. It will be observed that for no value of 472 Mr OC. CHREE, ON THE STRESSES IN ROTATING SPHERICAL SHELLS. w/a does the increase of the mean radius of the outer surface reach a fifth part of the difference between the equatorial and polar radii. A complete investigation of the state of strain and stress at the bounding surfaces It will thus be best to consider briefly some points connected with the determination of these axes. requires the position of the principal axes there to be known. Let us take, as in my former paper, the radius vector, the perpendicular to it in the meridian, and the perpendicular to the meridian at every point as fundamental axes. Then the strains and stresses are given by the following scheme. Planes of | which relative motion, or Direction of across which relative Components of Components of the force is motion, or the strain stress measured force du du we Sas On , ae R=(m—n)d+2n ai Od 7 u ldv 1 dv | Si bayerite = — 2; —— | ‘ as 18 =(m—n) d+ an (= Aap 5) i ra) Uu v 7 tp cot ® =(m—n) d+ 2n (E+ += cot 6) 70 p | d.2 tbe - rp 7) r— tu Roe=n = at 165 6 dr +r do|°— Sp Te dé If S = {(R- 0) + 4Re} u : v \3 Be ts woe du 7 = 2n Te eeia) oe Tete dos | ccc (7), then it is proved in Section III. of my previous paper that the three principal stresses at any point are R R+0-S R+0+8 = ae 0, = “= Similarly the three principal strains are 1 (du 2 \dr S dv U u 1d 1 (du rtp 07 oat? slartrt 2 1d r db 2n) ee! Uu al Pa ” cot 8. It is supposed that the sign is taken so that S is always a positive quantity, or zero, and so : a dr U gous ti do On Mr C. CHREE, ON THE STRESSES IN ROTATING SPHERICAL SHELLS. 473 Consequently R, is the greatest stress and 5 (ae a . Il Gh mt 2 Fo ee de oa the greatest strain in the meridian plane; and the corresponding principal axis makes with the radius vector drawn outwards the angle Since the expressions for the normal strains and stresses referred to the fundamental axes are all of the general type a, Far + ar? + ag + (a, +4,7 + O97 +, 7°) Pyecavepcrsevecreessns (9), where a,, a, etc. are constants which differ for the several strains and stresses, it is obvious that a complete determination of the principal strains and stresses throughout the shell would be a difficult matter. As depending on P, all the principal strains and stresses will have maximum or minimum values in the equator and at the poles; and the principal strain U v = +—cot 0, Ue To and the corresponding principal stress ®, which are linear functions of P,, can have maximum or minimum yalues nowhere else. For values of r for which S vanishes or is a linear function of P, the same is true of the other principal strains and stresses; but for other values of r we should get a quadratic equation in cos’@ determining polar distances at which additional maxima or minima might exist. The probability of a quadratic equation having roots which lie between 0 and 1 is of course very small, but as the quadratic in question contains the quantities m and n it may not improbably supply possible values of @ for particular kinds of material for most values of a’/a. Referring to (7) it will be noticed that S, as given by the square root of the sum of two squares, can be linear in P, only when one of the squares—that depending on R,—vanishes, and can be zero only when both squares vanish. Now R-®, or its equivalent is given by an expression of the type (9) and so could vanish, for all values of 6, only for values of r satisfying simultaneously two independent equations of the seventh degree. Further, to have any bearing on our problem such common roots would require to. lie between a’ and a. Except for the special case of a very thin shell when terms , in ma ae negligible, in which case R and © both vanish to this degree of approxi- Wor, SW. leer IY: 61 474 Mr C. CHREE, ON THE STRESSES IN ROTATING SPHERICAL SHELLS. mation throughout the entire shell, this contingency is so unlikely to arise except for some chance value of m/n that it need not be further considered. It is different in the case of v Re=n ee oa om dr 7 du) It is not very difficult to prove from (1) and (2) that a 2 i; i 1 du |_ 20*p (4m—n) sin @ cos@(a — a’) (r — a’) (a —7) dr rdé/ nD x | (om + 5n)° 3 - ” (a° + aa’ +a”) +7? (a+a) (a+ a0'+a”) + rad (a+a’)?+ aa" (a+ a’ + 24m (aa’) (a—a’) (ata) r? i +r(ata)t+r(?+ad +a”) +7? (a+a) (+a) ; aa at+a + (" + ) (a*+a*a’+a°a" + aa" + a} | sasha shag topple tena eeanien (10). A glance will shew that the expression inside the square bracket is necessarily positive for all values of r whatever may be the value of m/n. Thus Rg vanishes when r=a and when r=a’ for all possible values of @,—this is in fact one of the surface- conditions ;—but for no other value of 7 can Ry vanish unless in the polar axis or in the equator, where it vanishes for all values of 7 Thus S is a linear function of P, over the surfaces of the shell, and can vanish at no other central distance except in the polar axis or the equator. Consequently the greatest strain and the maximum stress- difference over the bounding surfaces must have their absolutely greatest values either at the poles or in the equator; but for intermediate values of 7 this is not necessarily the case. It should also be noticed that when R, vanishes the fundamental axes are all three principal axes. This is accordingly the case over both surfaces of the shell for all values of @, and also along the polar axis and in the equatorial plane for all values of 7; but it can occur nowhere else. : Referring to (10) it will be seen that R, must have the same sign as = for m cannot well be less than n, and so 4m —n is positive*. Now if D could vanish for any such value of m/n as occurs in real materials, we should have a shell of such material suffering infinite displacements when rotated however slowly. We may thus regard D as necessarily of one sign for all real materials, and it may in fact be found by trial that D is essentially positive. Also, referring to (7), it is apparent that, excepting the bounding surfaces, the polar diameter and the equatorial plane, S—R+© must be positive; hence it follows from (8) that tana has the same sign as sin26. We are thus in a position to assert that the only regions where the fundamental axes are all three principal axes are the * Poisson’s ratio would be negative if m were less than n, and Young’s modulus if 3m less than n. Me C. CHREE, ON THE STRESSES IN ROTATING SPHERICAL SHELLS. 475 bounding surfaces, the polar diameter and the equatorial plane; elsewhere the principal axis in the meridian plane answering to the algebraically greater principal strain and stress in that plane lies between the radius vector drawn outwards and the equatorial plane. The two following tables contain an analysis of the principal strains over the bounding surfaces of a shell for various values of a’/a, the material in all cases being supposed such that m= 2n. and _ stress. TABLE III. In equator Value of E du E ame 1 dv a'/a w*pa* dr wpa*\r ~ r dé E = ( + 2 cot @ wpa \r * +180 “180 182 ‘188 ‘217 243 302 468 664 833 1:00 0 — 0570 — 009 ‘Ol ‘0570 — 009 1 058 — 008 2 062 — 002 3 078 +018 “4 093 + 034 “139 121 + 062 6 185 +088 8 oe + 032 ‘9 "250 — ‘082 l-e 250 — 250 TABLE IV. In equator Bee a a i = ( + ; = ‘0 + 298 — 231 Ol — 068 “418 al ‘0676 “415 Bi; ‘066 ‘417 3 ‘062 437 “4 ‘060 455 & 061 ‘485 ic. 083 ‘521 8 134 475 9 190 380 1) 5c 250 250 ans ¢ oF : cot 6 + 298 621 618 615 623 635 670 A: ‘878 950 1:00 ) ) _ dw w"pa* dr — 019 —-019 — 0185 — 0135 + 0048 + 020 +051 +109 +193 +086 0 E ( u aaa ae wpa* \r The directions of the strains will be found in our scheme of. strain Principal strains on the outer surface =a of a shell for which m=2n. At poles 1 du _ E Le a r d0) wpa?\r 7+ a (c + 029 + 029 + 028 +020 — 007 — 030 — 076 — 163 — 185 — 129 0 Principal strains on the inner surface r=a’ of a shell for which m= 2n. At poles == = ae G ; 3) 2 Fai le a) ; eet 0) SNe 2 oo a 414 621 ‘411 616 “406 609 398 598 B95 592 “384 O70 327 “490 223 334 ‘114 sleral 0 0 When no sign is prefixed to a value it is supposed that it has the sign last attached to an element of the column. The values in Table IV. for a/a=0 refer to the centre of a solid sphere, and the head- ings “in equator” and “at poles ” must here be interpreted to mean “in an equatorial 61—2 476 Mr C. CHREE, ON THE STRESSES IN ROTATING SPHERICAL SHELLS. direction” and “in a polar direction” respectively. The strain here in reality consists of a compression “23le"pa*/E along the axis of rotation, and an extension “298@*pa*/H in all directions perpendicular to this axis. From these tables the principal strains at any point of either surface may be deduced by means of the following formula. Let /(@), (0), f G) represent the values of any one of the principal strains, or stresses, at the angular distances @, 0, 5 respectively from either pole on one of the surfaces: then On the outer surface the state of strain varies at first extremely slowly and with complete continuity as a’/a increases from zero. Along the radial direction there is com- pression in the equator for all values of a’/a. There is compression in the same direction at the poles hkewise until a’/a is about °3. As a’'/a is further increased this is converted into an extension, at first increasing with a’/a, and then decreasing rapidly and finally vanishing in a very thin shell. The following table gives the polar distance at which there is neither extension nor compression along the radius vector. For smaller polar distances there is extension and for greater distances compression, TABLE V. Value of a’/a...... Beaeneecesneae 3 “4. 5 6 8 9 Polar distance of locus of no strain in a radial direction> 13° 45’ Q5r ale Bye py Bye PAR 36° 4 30° 257 on the outer surface The strain along the tangent to the meridian on the outer surface is for values of aja from 0 to ‘2 a compression in the equator an extension at the poles, being very small throughout especially in the equator. For values of a'/a from ‘3 to ‘8 it is an extension in the equator a compression at the poles. As a’/a increases further the com- pression at the poles diminishes and eventually vanishes in a very thin shell, while the extension in the equator becomes converted into a compression which increases very rapidly as the thickness of the shell is reduced. Along the perpendicular to the meridian on the outer surface there is in every case extension in the equator, and this increases continually with a’/a. At the poles the strain in this direction is always the same as that along the tangent to the meridian. The following table gives the polar distance at which there is neither extension nor compression at right angles to the meridian, For smaller polar distances there is compression and for greater extension. Mr C0. CHREE, ON THE STRESSES IN ROTATING SPHERICAL SHELLS. 477 TABLE VI. Wallen Of Gf os ce weinescu esis i I 9) 6 8 ag Polar distance of locus of no strain in a direction per-| 19°45, 19°93’ 6° 89’ ~—«80°: 38’ «27° 48’ «21° 80’ pendicular to the meridian, on the outer surface When a’/a is less than ‘3 the strain at right angles to the meridian on the outer surface is everywhere an extension. In the table for the strains on the inner surface the values for the case a/a=0 are of a wholly distinctive character. There is a complete breach of continuity between this case and that when a’/a has any greater value however small. This is not to be wondered at, because the existence of a cavity completely alters the conditions of the problem. In the solid sphere there is the condition that the strain must contain no negative powers of 7, in the shell the condition that =a’ must be a free surface. Leaving out the case a’/a=0 as exceptional, there is in every case compression along the radius vector all over the inner surface. In the equator it at first diminishes slowly as a’/a increases, attaining a minimum when w/a is slightly less than 1/2, and then in- creases at a much more rapid rate. The compression in this direction at the poles is very much larger than in the equator for small values of a’/a, and remains nearly con- stant until a/a exceeds 1/2. It then decreases comparatively rapidly and becomes less than the compression in the equator before a’/a reaches the value ‘9. Along the tangent to the meridian the strain is in the equator always a compression, having an almost constant value till a’/a exceeds ‘2, then increasing slowly to a maximum when a’/a is about ‘6, and then again diminishing. At the poles the strain in this direction is the same as that perpendicular to the meridian. In this latter direction there is in every case extension in the equator, This at first decreases to a trifling extent until a'/a reaches the value 2, and then increases con- tinually as a’/a is further increased. At the poles there is also extension at right angles to the meridian. For small values of a'/a this is practically the same as in the equator. As a/a increases this strain continually diminishes, becoming decidedly less than in the equator when a/a passes the value -2, and finally approaches zero rapidly as the shell becomes thin. It has been already pointed out that on one theory the tendency to rupture is measured by the value of the greatest positive strain. On reference to the tables this will be seen to have on either surface the perpendicular to the meridian for its direction ; and it is greatest on the inner surface and in the equator. It will be noticed that the existence of a cavity however small more than doubles the tendency to rupture found in the solid sphere. Further slight enlargement of the cavity causes at first a very slight diminution in the tendency to rupture; which has moreover very nearly a constant value 478 Mr C. CHREE, ON THE STRESSES IN ROTATING SPHERICAL SHELLS. over the whole inner surface until a’/a approaches the value -2. As a’/a is further in- creased the tendency to rupture continually increases, at first slowly but then more rapidly as a’/a approaches unity. In a very thin shell the tendency to rupture is nearly 5/3 times that existing when the cavity is vanishingly small, and almost exactly 10/3 times that existing in the absolutely solid sphere. Since the internal cavity has been supposed spherical, we cannot of course expect our numerical results to apply exactly to any other form of cavity; but the following conclusions as to the way in which rupture will take place in a solid sphere, of a material whose limits of perfect elasticity and safety are not far apart, would appear fairly warranted. A crack will begin at the centre, where the absolutely greatest strain is found, and extend in a direction at right angles to the axis of rotation. On the appearance of the crack the tendency to rupture will suddenly increase, and the crack extend right up to the surface. The exact equatorial radius along which the crack propagates itself will in any practical case be determined by slight variations in the structure of the sphere, or deviations from true sphericity in its surface; and, as it will naturally extend in the direction perpendicular to the greatest strain, the crack may be expected to appear on the outer surface as a split along a meridian most apparent at or close to the equator, The value of the dilatation 6 at any point of either surface can be deduced immed- iately from Tables III. and IV. One of the surface conditions is in fact du be eee = (m nye ren 0, and so when m=2n we have over either surface of the shell du eerie Thus to get the dilatation over the surfaces wé have only to multiply the values given for = in the ‘tables by —2; and by writing extension for compression and conversely the remarks already made about the dependence of a on a/a and @ may be at once translated so as to apply to the dilatation. Further, since 6 equals the sum of the three principal strains, it follows that the : 1 dv : sum of the values in our tables for +35 and +0 cot @ added to thrice the corre- du sponding value of ae should be zero. This supplies a very severe test of the accuracy of the tables. It also enables the following tables for the principal stresses over the surfaces to be very simply deduced from the formule u ldv du On48 (C+; oe) ~ arb a ee ae ® = 4H )(7 +) cot) — Fr 4 Mr C. CHREE, ON THE STRESSES IN ROTATING SPHERICAL SHELLS. 479 noticing that when m=2n we have n=2#. The third principal stress, that along the radius, is zero. TaBLeE VII. r=a of a shell for which m= 2n. Value of a'ja Equatorial value of @/w*pa? + 0384 “0384 0398 ‘0478 ‘O77 “102 147 218 “211 135 0 Equatorial value of b/w*pa* + 1899 1899 ‘1914 “200 236 10 @/ w”pa* + a + + + Polar values of 2 and &/w°pa? ‘0384 ‘0384 0369 027 ‘010 ‘040 Principal stresses and maximum stress-difference on the outer surface Greatest value of maximum stress- difference in tons per square inch on the outer surface of a spherical shell for which a= 4000 miles, p=52, and w=earth’s angular velocity of rotation 1s L 15°21 153 16:1 189 2175 272 41-9 575 69-4 S01 Since the maximum stress-difference must have its greatest value either at the poles or in the equator, and @ is positive in the equator while R vanishes, it is obvious that this greatest value is simply the equatorial value of ®. TABLE VIII. r=a’ of a shell for which m= 2n. Value of a'ja 0 © OO SC Ot Bm OO WD ms _ | mn Equatorial value of @/w*pa* Equatorial value Polar values of Greatest value of maximum stress-difference Principal stresses and, maximum stress-difference over the inner surface Greatest value of maximum stress- difference in the special case of last table, measured of b/w? pa* @/w*pa? and &/w"pa* Or Ot Or Or Gt or OO. & & OF = OD DOD Or CO = 683, wpa? 42 831 826 $26 848 ‘873 “926 1-034 1-082 1-064 10 in tons per square inch 340 66°60 66°20 66:15 67:9 69°9 742 82°'8 867 85:2 801 480 Mr C. CHREE, ON THE STRESSES IN ROTATING SPHERICAL SHELLS. In the shell the greatest value of the maximum stress-difference always occurs in the equator, and is =®—®. In the solid sphere the greatest value is found at the centre, where the stresses consist of a pressure ‘038w%pa* along the axis of rotation and a traction ‘3S85e@"pa® along all directions perpendicular to the axis. The conclusions as to the tendency to rupture at the bounding surfaces deduced from the maximum stress-difference are on the whole closely analogous to those deduced from the greatest strain. The tendency is greatest invariably in the equatorial regions of the inner surface, The formation of a spherical cavity however small almost exactly doubles the tendency to rupture found in a solid sphere. As a’/a increases there is at first a very slight falling off in the value of this expression. After a’/a attains the value 1/3 there is a continual increase until a’/a reaches the value ‘8, which is followed by a slight decrease as the shell becomes very thin. This constitutes one rather striking, though numerically small, difference between the conclusions of the two theories; for on the greatest strain theory after a’/a attains the value 1/3 the tendency to rupture increases continually with a’/a, and attains its maximum value in the indefinitely thin shell. It should be remembered however that we have not proved that the absolutely greatest values of either the greatest strain or the maximum stress-difference must occur on the bounding surfaces of a shell, except in the case when the shell is very thin. So in every ease, except the very thin shell and the solid sphere, there may be values for the greatest strain and maximum stress-difference exceeding those of the previous tables. We are con- sequently only entitled to regard our results as supplying inferior limits to the real values of the tendency to rupture. That there are in reality greater values I think is however on the whole improbable, at least when m=2n; otherwise we should not expect to find the values on the inner surface invariably exceeding those on the outer surface. For practical purposes the following rule may be useful. Let 7 be the greatest tractional load in tons per square inch that may be safely applied to a beam of a certain isotropic material of density p; then a superior limit to the number NV of revolu- tions per second which it is safe to allow in a spherical shell of this material, a feet in external radius, is given on the stress-difference theory by 2200 /17\2 /T\? a = v=") ie (F) Se hace (12),* where Y is the number in the last column of Table VIII. answering to the particular value of a’/a occurring in the shell. Possibly the clearest idea of the state of strain and stress in a rotating spherical shell may be obtained from a study of the accompanying diagrams. In every case the abscisse of the curves indicate values of a’/a. The suffix o indicates * On the greatest strain theory replace Y by 80°12 Z, where Z=corresponding number in the fourth column of Table IV. Mr C. CHREE, ON THE STRESSES IN ROTATING SPHERICAL SHELLS. 481 the outer, and the suffix 7 the inner surface. Undashed letters refer to the equator and dashed letters to the poles. The strains and stresses at the centre of the solid sphere are not shewn. In diagram I. the ordinates represent the values of the strains in the equator: . du é e, of dy OD the outer, and e, on the inner surface ; if, © am de On the outer, and f, on the inner surface ; “uv ; g, of Bet cot on the outer, and g, on the inner surface. The curve g, also represents the value of the absolutely greatest strain found on either surface. In diagram IT. the ordinates represent the values of the strains at the poles: du : e, of aoe the outer, and e/ on the inner surface ; , , U ld U / , c — oe Ou ae = and ~+ cot @ on the outer, and f’=g; on the inner surface. In diagrams I. and II. the ordinates multiplied by w’pa’/E give the absolute values of the strains. In diagram III. the ordinates represent the values of the stresses © and ®. The above statement as to the suffixes and the dashed letters will sufficiently particularize the separate curves. There is also the curve M.S whose ordinates represent the greatest values of the maximum stress-difference existing on either surface. As already stated these values occur on the inner surface in the equator. The curve ®, would in like manner represent the greatest values of the maximum stress-difference existing on the outer surface alone. Multiplying the ordinates in diagram III. by wpa’ we get absolute stresses. Vion, XUVe Parr Ve. 62 482 Mr C. CHREE, ON THE STRESSES IN ROTATING SPHERICAL SHELLS. Diagram I. a Equatorial Strains. ot a axis of aja = Polar Strains. Mr C. CHREE, ON THE STRESSES IN ROTATING SPHERICAL SHELLS. 483 Diagram X11. Stresses. Q ‘ ‘ ee ‘ 3 \ a Ny ‘\ ‘ \ ‘\ SSS ‘ a= ‘ ee ~ . 2 2 x ‘ << ™. < x ‘\ a ) \ es S ‘ ee SX ‘ ae SS \ a N \ re NS \ aa ~ 1 ee 0 we = en ora Ne ane Ne “g / bo iv) } 4 / / or lon) lo) ~ Fe ~ ft > ya ~ a =< Mf <) iid ' 7 Ss = / i) \, 8o=8'0 Wy > Fi SK 4 ‘ 4; sS ty. ~. tay ‘A Fad, ~ fs ~ ff = ff S Lied: 2 Ry: fi = ~ Fad, > 4 f Soe “ye ~ fa pee a — ed ———__ = 7p 7 oe 7 SU aSSE ELE —————E SS 3 | see Le ae * i tt —— ss —— ~ — oe O° eZ 62—2 XVII. On the Binodal Quartic and the Graphical Representation of the Elliptic Functions. By Pror, Caytey. [Read May 6, 1889. ] I approacH the subject from the question of the graphical representation of the elliptic functions: assuming as usual that the modulus is real, positive, and less than unity, and to fix the ideas considering the function sn (but the like considerations are applicable to the functions en and dn) then the equation #+iy=sn(«’ +7y’) establishes sponds a single point 2 +%y’, 2’, y’ each positive and less than K, kK’ respectively : and y each positive. 1 (1, 1) correspondence between the ay infinite quarter plane, and the vy’ rectangle and conversely to any such point a’+iy’, there corresponds a single point #+1y, (sides K and K’): viz. to any given point 2+ %y, w and y each positive, there corre- I draw in the a’y'-figure the rectangle A’B’C’D' (sides K and 4’), and in the xy-figure, I take on the axis of a, : il the points B, C where AB=1, AC=7. and the Pe / — x os E, x \ ‘ r \ ‘ . \ ' ae aa 4 GBPME CH point D at infinity. We have thus in the a’y/’-figure the closed curve or contour A'BC’'D'A’: and corresponding hereto we have in the ay-figure the closed curve or contour ABCDA, viz. here D is the point at infinity considered as a line always at Pror. CAYLEY, ON THE BINODAL QUARTIC ec. 485 infinity, extending from the point at infinity on the positive part of the axis of 2, to the point at infinity on the positive part of the axis of y, the contour being thus AB, BC, CD (D at infinity on the axis of x); and then D (at infinity on the axis of y) A. And thus to a point P’ describing successively the lines A’B’, BC’, C’D’, D'A’ there corresponds a point P describing successively the lines AB, BC, CD, DA: to P' at D’ there corresponds P at D, viz. this is any point at infinity from D on the axis of « to D on the axis of y. There is no real breach of continuity: in further illus- tration suppose that P’ instead of actually coming to D’ just cuts off the corner, viz. that it passes from a point y on C’D' to a point a’ on D’A’ (9, @ each of them very near to D’): then P passes from a poimt y very near D on the axis of « (that is at a great distance from A) to a point @ very near D on the axis of y (that is at a great distance from A): and to the indefinitely small arc ya described by P’ there corresponds the indefinitely large are ya described by P. We thus see that if P’ describe any are E’F’ passing from a point E” of A’D’ to a poimt #’ of BC’; then P will describe an are EF passing from a point # of AD to a point # of BC: and similarly if P’ describe any are G'H passing from a point G’ of A’B’ to a point H’ of CD’; then P will describe an are GH passing from a point G of AB to a pomt H of CD. Supposing £’F” is a straight line parallel to A’a’, that is cutting A’D' and BC each at right angles, then HF will be an are cutting AD and BC each at right angles: and so if G'H’ is a straight line parallel to A’y, that is cutting A’B and CD’ each at right angles, then GH will be an are cutting AB and CD each at right angles: and moreover since #’#’ and G'H’ cut each other at right angles, then also HF and GH cut each other at right angles. Supposing as above that E’F” and G’H’ are straight lines, we have EF and GH each of them the are of a special bicircular quartic: the theory was in fact established in a very elegant manner in a memoir by Siebeck, “Ueber eine Gattung von Curven vierten Grades, welche mit den elliptischen Functionen zusammenhangen,” Crelle, t. 57 (1860), pp. 359—370, and t. 59 (1861), pp. 173—184. In particular if P’ describe the line L’M’ lying halfway between A’B’ and D'’ (that is if A’L’=}K’) then P will describe the circular quadrant ZW, radius rain viz. in , VB’ this case the bicircular quartic degenerates into a circle twice repeated: and so if P’ describe successively the lines #’F” and £/F,’ equidistant from JL’M (AE’=4K’—n, AE! =tK'+n) then P will describe the ares EF and £,F, which are the images of each other in regard to the centre A and circular quadrant LM, and which together constitute the quadrant of one and the same bicircular quartic. A bicircular quartic is of course a binodal quartic with the circular points at infinity for the two nodes: there is no real gain of generality in considering the binodal quartic 486 Pror. CAYLEY, ON THE BINODAL QUARTIC AND THE rather than the bicireular quartic, but I have preferred to do so, and I have accordingly introduced the term Binodal Quartic into the title of the present Memoir. I present in a compendious form the properties of the general curve, and I show how the curve is to be particularised so as to obtain from it the special bicircular quartics which present themselves as above in the graphical representation of the elliptic functions. A binodal quartic has the Pliickerian numbers mt nitidurne eile — AES) an) a ela: The number of tangents to the curve which can be drawn from either of the nodes is n—4, =4; and the pencil of tangents from the one node is homographic with the pencil of tangents from the other node. Call the nodes J and J: and let the tangents from J be called (a, b, c, d) and those from J be called (a, Ub’, c’, d’), then if the tangents which correspond to (a’, b’, c’, d’) respectively are (a, b, c, d), they may also be taken to be (b, a, d, c), (c, d, a, b) or (d, c, b, a): and considering the intersections of corresponding tangents, we have thus four tetrads of points, say the f-points, such that the points of each tetrad lie in a conic through the two nodes: and we have consequently four conics each passing through the two nodes, say these are the f-conics. Starting as above with the correspondence (a, b, c, d), (a, b’, ¢’, d’), if the inter- sections of a and a’, b and J’, c and c’, d and d are called A, B, C, D respectively, then we have A, B, C, D for a tetrad of fpoints, lying on the f-conic (ABCD). Writing AB for the two points, the intersections of Jd, JB and of JB, JA tre- spectively, and so in other cases, then the remaining three tetrads of /points are AB, CD lying on the fconic (AB, CD), ACG, BD. 3 3 (AC, BD), AD; «BC «; 53 (AD, BC). The two points AB may be spoken of as the antipoints of A, B: and so in other Cases. Any two of the f-conics have in common the nodes J, J, and they therefore be- sides intersect in two points: at each of these the tangents to the two conics, and the lines to J, J respectively form a harmonic pencil. Consider the two tangents at J and the two tangents at J: we have a conic touching these four lines and passing through the tetrad of /-points, or what is the same thing, intersecting the f-conic in the four f-points: say this is a c-conic. There are thus four c-conics corresponding to the four f-conics respectively. We may consider a variable conic, passing through the points Z and J; and such that the tangents thereto at these points respectively meet on a point of a c-conic: GRAPHICAL REPRESENTATION OF THE ELLIPTIC FUNCTIONS. 487 the variable conic and the corresponding f-conic each pass through the points J, J and they besides meet in two points: and the variable conic may be such that at each of these the tangents to the variable conic and the fconic form with the lines drawn to the points J and J a harmonic pencil. The variable conic, as thus defined contains a single variable parameter; and it has for its envelope the binodal quartic: the binodal quartic is thus in four different ways the envelope of a variable conic. [This is of course Casey's Theorem for the four-fold generation of the bicircular quartic as the en- velope of a variable circle.] One of the fconics say (ABCD) may break up into the line JJ and a line, say the axis (ABCD): we have then the four f-points A, B, C, D on this line. The other f-conics say (AB, CD), (AC, BD), (AD, BC) are as before proper conics: any one of these meets the line (ABCD) in two points: and at each of these the line and the tangent to the conic form with the lines drawn to the points J and J a_ harmonic pencil. The binodal quartic is in this case said to have an axis. But a second f-conic, say (AD, BC) may break up into the line JJ and a line, say the axis (AD, BC): we have then the four fpoints AD, BC on this line. Writing moreover A’, D’ for the two points AD, or say (AI, DJ)=A’, (AJ, DI)=D’; and similarly B’, C’ for the two points BC, or say (BI, CJ)=B’, (BJ, CI)=C': then the four f-points are A’, B’, C’, D’, and the axis (AD, BC) may be called (A’BC'D’). The relation of the f-points A, B, C, D and A’, B’, C’, D’ and of the two axes is as shown in the figure: taking X for the intersection of JJ with (ABCD) and Y for D that of JJ with (A’B’C'’D'), also Z for the intersection of the two axes, then X, Z are the sibiconjugate points of the involution AD, BC and Y, Z the sibiconjugate points of the involution A’D', B’C’. The two axes intersect in Z, and form with the lines Z/, ZJ a pencil in involution. The two remaining f-conics (AB, CD) and (AC, BD), or as they might also be called (A’B’, C'D') and (A’C’, BD’) are as before proper conics: they touch each other 4 (oe) 8 Pror. CAYLEY, ON THE BINODAL QUARTIC AND THE at the points J, J, and have for their common tangents at these points the lines ZI, ZJ respectively. Each conic meets each axis in two points; and at each of these points the axis and the tangent to the conic form with the lines to J, J a harmonic pencil. The binodal quartic is in this case said to be biaxal. The point Z which is the inter- section of the two axes may be called the centre. In the case where an f-conic breaks up into the line JJ, and a line containing four f-points, say an f-line, the corresponding c-conic coincides with the f-conic, viz. it also breaks up into the line JJ and the fline: the variable conic is a conic through the points Z, J such that the tangents thereto at these points respectively meet on the f-line. Moreover the variable conic must be such that at each of its intersections with the fconic, that is the f-line, the tangent to the variable conic and the fline must be harmonics in regard to the lines drawn from the point to the points J, J respec- tively: but this condition is satisfied zpso facto for each of the intersections of the variable conic and the f-line. [This depends on the theorem, that taking on a conic any three points P, J, J, then the tangent at P and the line drawn from P to the pole of JJ are harmonics in regard to the lines PJ, PJ.] Thus we have only three con- ditions for the variable conic, or, as above defined, it would in the case in question (of four f-points in a line) depend upon two variable parameters. There is really another condition—but what this in general is I have not ascertained: and this being so the variable conic in the case in question (of the four f-points in a line) depends upon a single variable parameter, and we have as before the bicircular quartic as the envelope. The foregoing is the axial case; in the biaxial case, the same thing happens in regard to the variable conics belonging to the two axes respectively. Thus in every case we have the fourfold generation of the curve as the envelope of a variable conic: only in the axial case, the variable conics belonging to the axis, and in the biaxal case the variable conics belonging to the two axes respectively, are not by the foregoing definitions completely defined. It will be seen further on how, in the case of the biaxal bicircular quartic, we complete the definition of the variable circles belonging to the two axes respectively. Taking the points I, J to be the circular points at infinity, we have a bicircular guartic. The f-points are the foci, and the f-conics are circles, viz. we have 16 foci situate in fours upon four focal circles) The harmonic relation of two lines to the lines through J, J means of course that the lines cut at right angles: hence the focal circles eut each other at right angles: this must certainly be a known property, but it is not mentioned in Salmon’s Higher Plane Curves, Ed. 3, Dublin, 1879, and I cannot find it in Darboux or Casey: it, is given No. 81 in Lachlan’s Memoir “On Systems of Circles and Spheres,” Phil. Trans., vol. 177 (1886), and I find it as a question in the Educa- tional Times, March 1, 1889, 10034 (Prof. Morley). “Prove that of the four focal circles of a circular cubic or bicircular quartic, any two are orthogonal, and the radii are con- nected by the relation =(w*)=0.” The theorem is not as well known as it should be. The c-conics are confocal conics having for their real foci the so-called double-foci of the quartic (more accurately the common foci are the four quadruple foci of the quartic) ; GRAPHICAL REPRESENTATION OF THE ELLIPTIC FUNCTIONS. 489 we have thus four conics corresponding to the four focal circles respectively, each conic intersecting the corresponding circle in the four foci upon this circle. And we have then thé quartic as the envelope of a variable circle having its centre upon one of these conics and cutting at right angles the corresponding focal circle: the bicircular quartic is thus generated in four different ways. Instead of one of the focal circles, we may have a line or axis, and the quartic is then said to be axial: the foci on the axis may be any four points; and for a real curve they may be all real, or two real and two imaginary, or all four imaginary. The remaining focal circles are real or imaginary circles, cut by the axis at right angles, that is having their centres on the axis, and cutting each other at right angles. But instead of another of the focal circles we may have a line or axis, and the quartic is then said to be biaxal: the two axes cut at right angles at a point which may be called the centre of the curve. The foci on each axis form pairs of points situate symmetrically in regard to the centre. If on one of the axes the foci are real then on the other axis they form two imaginary conjugate pairs; and conversely: but if on one of the axes the foci are two of them real and the other two conjugate imaginaries, then this is so for the other axis also. There are thus only the two cases: 1’, foci on the one axis real, and on the other conjugate imaginaries; 2°, foci on each axis two of them real and the other two conjugate imaginaries: there is however a limiting case where on each axis two foci are united at the centre, the other two foci being real on the one axis and conjugate imaginaries on the other. The remaining two focal circles are real or imaginary circles, cutting each axis at right angles, that is having their centres at the centre; and cutting each other at right angles, that is having the sum of the squares of their radii =0. The biaxal form of bicircular quartic is in fact that which presents itself in the theory of the representation of the elliptic functions. I consider for a moment the case of a variable circle having its centre upon a given line, and cutting at right angles a given circle. The variable circles pass all of them through two fixed points, the antipoints of the intersections of the given line and circle, and which are thus real or imaginary according as the intersections of the given line and circle are imaginary or real. Hence considering any one variable circle and the consecutive variable circle these intersect in two real points, when the given line does not meet the given circle (meets it in two imaginary points); but when the given line meets the given circle in two real points, then the two variable circles intersect in two imaginary points: if however the given line touches the given circle, then the two variable circles touch each other. Taking now the curve of centres to be any given curve whatever, and considering one of the variable circles, and the consecutive variable circle, it at once appears that if the tangent to the curve of centres at the centre of the variable circle does not meet the given circle, then the two variable circles intersect in two real points (which, if the tangent touch the given circle, unite in a single real point): but if the Vou. XIV.* Parr LY. 63 490 Pror. CAYLEY, ON THE BINODAL QUARTIC AND THE tangent to the curve of centres meets the given circle, then the two variable circles do not intersect. It hence appears that the real portions of the envelope arise exclu- sively from those portions of the curve of centres which are such that at any point thereof the tangent to the curve of centres does not meet the given circle. In particular if the given circle be real, and the curve of centres is a real ellipse inclosing the given circle, then the real portion of the envelope arises from the whole ellipse: but if the curve of centres be a real ellipse cutting the given circle in four real points, then drawing the four common tangents of the ellipse and circle, it is at once seen that there are on the ellipse two detached portions, such that at any point of either portion the tangent to the ellipse does not meet the circle: and the real portions of the envelope arise exclusively from these portions of the ellipse. In the case just referred to there are on the ellipse four portions each lying outside the circle and terminating in the four intersections respectively of the ellipse and circle, such that at a point of any one of these portions the tangent to the ellipse meets the circle in two real points. Starting from the extremity of one of these portions of the ellipse and proceeding to the other extremity on the circle, the corresponding variable circles do not intersect each other, but each of them is a circle lying wholly inside that which immediately precedes it; and the variable circle becomes ultimately a point, viz. this point is a focus of the curve: this agrees with the foregoing statement that the f-conic intersects the circle in the four foci upon this circle. For the two portions of the ellipse which lie inside the circle, the variable circle is of course always imaginary. The like considerations apply to the case where the locus of the centre of the variable circle is a hyperbola or parabola. The foregoing remarks illustrate the actual generation of a bicircular quartic as the envelope of the variable circle. Starting now from the equation xr +iy =sn (a +7y') igre __ snw’en ty’ dn vy’ . _ sniy'ena’ dna pastes o= 1 =F sn? a’ sn? iy’’ 1k’ sn’ a’ sn? ty’’ or putting Sic —/p) su —a9), j he a ey 2 Ta ae ey these equations are Ia pa a ae ee. y= qv ce SS : bi . 1+ k’p*g¢ 1+ k*p’¢ pi+¢ . 2' whence also r+y= reg r, (if a+y* be put =7’). These equations, considering therein g as a given constant, and p as a variable para- meter, determine the curve HF: and considering p as a given constant, and q as a variable parameter, they determine the curve GH. But the eliminations are easily effected; we have pa — k*q*r*) ies q, g i k'p*r*) =7 ~ > ‘ SSP, oe Ae, : . ee Let ee i ee WS nee Re a ° we Or a See es Ni Sa : oe i ‘ ‘ " ge at a te any . YOUN ar le anhing CPLA RU re el aaa RS bd i SSNs I0. ; Sar teak ce eek kOe