On systems of complex numbers and their application to the theory of transformation groups.

Transcription

1 Übr Systm complxr Zahln und ihr Anwndung in dr Thori dr Transformationsgruppn Monatsth. f. Math. u. Physik (89), On systms of complx numbrs and thir application to th thory of transformation groups. By E. Study in Marburg (Hssn) Translatd by D. H. Dlphnich Contnts Pag. Basic concpts. Classification of systms of complx numbrs.. 6. Systms with two basic numbrs 8 4. Systms with thr basic numbrs Systms with four basic numbrs. 6. Spcial systms with n basic numbrs On th thory of simply-transitiv groups 7 8. Lmmas from th thory of linar transformations.. 9. Th simplst transformation groups that ar connctd with a systm of complx numbrs 4. A proprty of rciprocal projctiv groups. 8. Projctiv groups whos transformations commut with givn infinitsimal projctiv transformations.. 4. Exampls. 44. Furthr transformation groups that ar coupld with a systm of complx numbrs Th advantags of systms of complx numbrs On th paramtric rprsntation of crtain transformation groups Th rotations of th sphr and th motions in th plan 58

2 Th author has concrnd himslf with th thory of systms of complx numbrs and thir rlationship to th thory of transformations groups in two tratiss that appard in th Notics of th Göttingn and Lipzig Socitis for Scinc ( * ) Th following includs a summary and partly rworkd prsntation of ths invstigations. Only a fw things hav b omittd, and thy chifly rlat to th work of othr mathmaticians ( ** ).. Basic concpts. For th concpt of a so-calld xtnsiv or complx quantity that is composd of n basic numbrs or principal units, w rfr to th first chaptr in H. Grassmann s Ausdhnungslhr (86 dition). W now concrn ourslvs with thos complx quantitis whos multiplication obys th so-calld distributiv law, which is xprssd in th formulas: a (b + c) = ab + ac, (a + b) c = ac + bc. Th totality of all of th xtnsiv quantitis: a = a + a + + a n n, b = b + b + + b n n,... that ar dfind by th basic numbrs,, n with ral or ordinary complx cofficints a i, b k, will b calld a systm of complx numbrs whn it satisfis th following conditions:. Th product of any two of th xtnsiv quantitis must b again rgardd as an xtnsiv quantity with th sam principal units. From th distributiv law, th ncssary and sufficint condition for this to b tru is that thr xist n rlations of th form: () i k = n γ ikss, s= in which th cofficints γ iks rprsnt ordinary ral or complx numbrs.. Any thr of th xtnsiv quantitis must fulfill th so-calld associativ law for multiplication, which is xprssd in th formula: ( * ) Gött. Nachr. 889, no. 9 (pp. 7, t sq.). Sächs. Br. 889, 6 May (pp. 77, t sq.). To som xtnt, Schffrs also arrivd at th rsults that ar rcordd in ths tratiss. Sächs. Br. Jun 889 (pp. 9, t sq.). ( ** ) Gött. Nachr., pp. 7, t sq., pp. 4, rm. pp. 65, t sq., Sächs. Br., pp. 77, rm. pp., rm. pp. 7, t sq.

3 E. Study On systms of complx numbrs and transformation groups. () (ab) c = a (bc). Th ncssary and sufficint condition for this is th xistnc of all rlations of th form: () ( i k ) j = i ( k j ) (i, k, j =,, n); i.., th xistnc of th following systms of quadratic idntitis for th constants γ iks that wr introducd in (): (4) n γ iksγ sjt = s= n γ kjsγ ist (i, k, j, t =,, n). s=. A quantity must xist among th xtnsiv quantitis that satisfis th two quations: (5) x = x, x = x, indpndntly of x. In this, it is only rquird that a quantity xist that satisfis th n quations: i = i = i (i =,, n). If on sts th unknown quantity = α i i thn that will yild th following n quations for th cofficints α,, α n : n αi γ iks = i= ( k s), ( k = s), n αi γ kis = i= ( k s), ( k = s). Ths quations must b compatibl with ach othr and yild a singl systm of solutions. This would still not follow from th conditions that wr statd in () and (). On dos not hav th thorm that th solubility of on of ths systms of quations would imply that of th othr on, ithr. For xampl, on might hav: =, =, =, =. Th dmands () and () ar fulfilld hr, and on will also hav: (x + x ) = x + x idntically; howvr, thr is no quantity that is linarly drivabl from and that will satisfy th scond of th rquirmnts that wr xprssd in ().

4 E. Study On systms of complx numbrs and transformation groups. At th sam tim, th condition (), strictly spaking, rquirs a bit too much: It would alrady suffic to just rquir that th systm of quations: x = x x η = x possss solutions, η, at all. It would thn follow immdiatly that: = η = η ; both solutions would thn b idntical, and both of th systms of quations would possss only this on solution. Th rquirmnt () is compltly subsumd by th othr on that th quations: (6) ax = b, ya = b should b solubl for x and y in gnral ; i.., xcpt for spcial valus of a, or, as on can also say, that division is admissibl in th systm of complx numbrs. In fact, in ordr for this to b tru, it is ncssary that nithr of th dtrminants: (7) γ iksai i, γ iksak k vanish idntically. Now, if th assumption () is fulfilld thn on can tak a = ; on will thn obviously obtain two non-vanishing dtrminants. Howvr, th on can also prov th convrs immdiatly. Namly, if th dtrminants (7) ar non-zro for a crtain systm of valus a,, a n thn on will, in particular, also b abl to solv th quations: aξ = a, ηa = a for ξ and η. Howvr, it will thn follow that for vry valu of x, on will hav: aξx = ax, xηa = xa. Howvr, by assumption, th quations: ax = ax, x a = xa will hav only on solution x, x. It will thn follow that for all valus of x: ξx = x, xη = x, and, as w rmarkd alrady, this will imply that ξ = η. From now on, w will rfr to th quantity that was dfind in () as th numbr on of th systm, and, whn no misundrstanding can aris, xprss its charactristic proprty by th symbolic quation =.

5 E. Study On systms of complx numbrs and transformation groups. 4 W ar justifid in doing this, sinc th numbr will obviously play th sam rol in an arbitrary systm of complx numbrs as unity dos in th ordinary numbr systm. Th solutions of th quations (6) can b writtn in a simplr form, whn on has prviously solvd th simplr quations: (8) ax =, xa =. Th following thorm rlats to this: Th dtrminants (7) of th two quations (8) hav th numbr a as a simpl factor as functions of th cofficints a,, a n. If thy ar non-zro thn both quations will hav th sam solution, which w shall dnot by a. In th othr cas, thr is no numbr in th systm whatsovr that will satisfy on of th quations. Namly, lt th numbr a of th systm b such that th quation ax = posssss a solution x, and lt a =, a,, a k (k < n + ) b linarly-indpndnt powrs of th numbr a, whil a k is xprssibl as a linar combination with numrical cofficints: (9) a k = a a k + + a k a. It is thn ncssary that a k ; othrwis, on could driv anothr form from th rlation (9) by multiplying by x, for which a smallr numbr would ntr in plac of th numbr k; a,, a k would not b linarly-indpndnt thn. If w now multiply a k by x thn that will giv: () x = a [ak a a k - a k a ]; k with that, w will obtain a uniquly-dfind xprssion for x, and indd an xprssion that also clarly satisfis th othr quation xa =. Morovr, it will alrady b compltly sufficint to show that, along with a solution x of th quation ax =, thr will always likwis xist a solution x of th quation xa =. It will thn follow from th associativ law of multiplication that x = (x a) x = x (a x ) = x ; thr will thn xist only on solution of th two quations, and thir dtrminants will always b simultanously non-zro. W will rfr to th numbr a, whn its dtrminant is non-zro, as a gnral numbr of th systm, in contrast to th spcial numbrs, for which th symbol a will no longr hav any maning. If th numbrs a and b ar gnral thn on can, with no furthr ado, driv th solutions to all quations of th form: () axb = c from th solution x = a to th quations ax = and xa =. On will thn hav: x = a c b.

6 E. Study On systms of complx numbrs and transformation groups. 5 If on of th numbrs a, b is spcial say, a thn not just th quations ax = and xa = will b insolubl, but along with thm, also all quations of th form (), whn c is a gnral numbr. Namly, if on sts say ax b = c thn x bc will b a solution of th quation ax =, which is impossibl. In ordr to dcid whthr th quation or th systm of quations axb = c is solubl for a spcial numbr c, on might invstigat th sub-dtrminants of th matrix that ariss from this systm of quations. If that yilds th xistnc of a solution thn it can naturally not b dtrmind: In our cas, a will b what on calls a divisor of zro. Th totality of all numbrs in th systm that satisfy th quation ax =, and likwis, th totality of all numbrs that satisfy th quation ya =, along with th numbr on in th systm, will alrady dfin a systm of complx numbrs, in its own right. Onc w hav rstrictd th domain in which th opration that is opposit to multiplication can b prformd, w can assum that all calculation oprations with th complx numbrs of a givn systm will b valid that consist of addition (including subtracting), multiplication, and division in th units an arbitrary numbr of tims. In ordr to prform th lattr opration, which will b rprsntd by th symbolic quation x = x, w hav only to xclud th domain of spcial numbrs x. In ordr to giv an xampl of th rul abov, w will dfin th two dtrminants that blong to th systm of complx numbrs: In this tabl, th i th horizontal row and th k th vrtical row will b filld in with th valu of th product i k ; for xampl, =, =. If on now computs th dtrminants abov thn on will find that: γ iksai = i a a, γ iksak = k aa ; on will thn obtain two ntir functions with th sam simpl factors a, a. By far, it is, morovr, frquntly th cas that th two dtrminants (7) bcom virtually idntical, which is what happns for quatrnions, for xampl.

7 E. Study On systms of complx numbrs and transformation groups. 6. Classification of systms of complx numbr. Th fundamntal problm in th thory of systms of complx numbrs must b dfind as: Dtrmin all systms of complx numbrs with n principal units. On cannot, howvr, pos th problm as: Find th most gnral systm of complx numbrs with n principal units. It would not b a priori clar thn and, as a closr xamination will show, also not corrct that th systm of constants γ iks that satisfy th conditions (), (), () ( ) will dfin an irrducibl manifold: It will, morovr, dcompos into diffrnt sparat domains whos most gnral rprsntativs ar to b considrd as quivalnt and qually gnral, insofar as non of thm can b obtaind from th othr ons by passing to som limit. Howvr, sinc w fl compltly ignorant of th natur of this domain, nothing will rmain for us to do but to look at th small valus of th numbr n in ordr to immdiatly dtrmin th systms that xist. As a natural classification principl, w shall hncforth appal to th ida that all systms should blong to a class whn thy go ovr to ach othr undr linar transformations. From any of th systms of complx numbrs that on dfins, on can driv infinitly many othr ons, for which on introducs nw basic numbrs,, n in plac of th basic numbrs,, n by way of a linar transformation with a nonvanishing dtrminant, so that all of th numbrs in th systm can b rprsntd as linar functions of,, n. All of th proprtis of th systms that ar obtaind in this way will obviously b known, just lik thos of th systm,, n. W will thrfor considr it to not b ssntially diffrnt from th systm,, n, and will plac it, along with th lattr systm, into on and th sam typ. Howvr, thr is a scond, spcially rmarkabl, way of driving from a systm of complx numbrs, anothr on that has just as many principal units. If w imagin any systm of complx numbrs that is rprsntd by a quadratic multiplication tabl, such as th tabl that is dfind in pag 5, thn w will obviously again obtain a systm of complx numbrs whn w switch any two lmnts in th squar that li symmtrically to th diagonal. In this way,.g., th systm that was givn on pp. 5 will go to th following on: In fact, th opration that was prformd will hav only th ffct of invrting th ordr (squnc) of all multiplications, such that vry product ab f will thn b rplacd with th corrsponding product f ba. Howvr, th charactristic proprtis of a systm of complx numbrs that wr dscribd in (), (), () will obviously not b disturbd. All that will happn is that all of th constants γ iks will b switchd with th corrsponding constants γ kis, from which, nothing will chang in th rlations (4) of.

8 E. Study On systms of complx numbrs and transformation groups. 7 W would lik to rfr to th systm thus-obtaind as th rciprocal systm to th givn on. In many cass, it can also b obtaind from th givn on by introducing nw basic numbrs, and indd, it can coincid with th givn on compltly (namly, whn th givn systm obys th co-calld commutativ law of multiplication); howvr, it can also b diffrnt from th givn on, as in th xampl that was givn. Namly, if th two multiplication tabls that wr givn on pags 5 and 7 can b convrtd into ach othr by th introduction of nw basic numbrs thn th two associatd dtrminants γ iksai and γ iksai i i = γ iksa i = γ iksa k will hav to go to ach othr at th i k sam tim. Howvr, this is impossibl, sinc th on is a product of two squars, whil th othr on is a product of a first and third powr. Just as on has in th individual cass, in any vnt, th rciprocal systm to a givn on is known to b idntical to it in all of its proprtis. W will thrfor b justifid in counting it as likwis having th sam typ. W can now mak th problm that was posd in th bginning of this paragraph mor prcis: Giv all of th diffrnt typs of systms with n basic numbrs. On will obviously thn com to th problm of rprsnting ach typ by thos rprsntativs that will givn th clarst possibl multiplication tabl, which would b similar to th on that was givn on pp. 5. In plac of th concpt of a systm of complx numbr that was dfind to b fundamntal hr, on can also choos a somwhat diffrnt concpt to b fundamntal, and thn onc mor pos a similar problm. Namly, if on rstricts onslf to thos systms of complx numbrs for which th constants γ iks in th rlations hav ral valus thn on will also find onslf daling with only th numbr systms of a class that can b prmutd within itslf by linar transformations with ral cofficints. It dos not sm to m prfrabl to tak such a standpoint from now on, sinc th algbraic charactr of th problm would b corruptd by it. It is, howvr, compltly worthwhil to distinguish th systms within a givn typ that ar associatd with ral valus of th constants γ iks, and which can b prmutd with ach othr by mans of linar transformations with ral cofficints. W would lik to call thm diffrnt forms of th sam typ. Finally, on can also altr th problm in such a way that on prscribs a crtain domain of rationality in which th quantitis γ iks and th transformation cofficints will com from. Howvr, w will not go furthr into this qustion. If on has a systm with n basic numbrs and n > thn a numbr k that lis btwn th limits of and n must xist with th proprty that th k th powr of any numbr a of th systm will b xprssibl in trms of th prvious powrs a =, a, a,, a k, whil th lattr ar linarly-indpndnt of a for a sufficintly gnral choic of a (cf.,, formula 9). W will arrang th numbr systms for a givn valu of n in incrasing ordr of k. In -6, th typs that ar prsnt in th cass n =,, 4 will b dtrmind, along with thir forms, and clarly summarizd. Systms of th sam typ will bar th sam Roman numral. Som forms will b prsntd twic, sinc undr som circumstancs anothr canonical form (i.., multiplication tabl) will sm prfrabl, dpnding upon whthr on is daling with th dtrmination of typs or th dtrmination of th various forms of th sam typ. In such cass, th scond form of a multiplication tabl will

9 E. Study On systms of complx numbrs and transformation groups. 8 charactrizd by an appndd notation a). Th rmaining forms of th sam typ will thn follow with th notation b), c),. Systms with two basic numbrs. Hr, on will ncssarily hav n =. Lt a b linarly-indpndnt of a =, and lt: a = a a + a a, in which a, a ar ordinary ral or complx numbrs. Now, if th quadratic quation: λ = a λ + a has two sparat roots λ, λ thn on will introduc th nw basic numbrs: a λ a λ =, =. λ λ λ λ If it has a doubl root λ thn on will tak th nw basic numbrs to b ths: = (a λ) =, = a λ. On will thus obtain th only two typs of systms that hav two basic numbrs, which ar illustratd in Tabls I and II. Distinct ral systms obviously blong to just th first typ, and indd thr will b two forms, corrsponding to th two possibilitis that th quadratic quations λ = a λ + a has two ral or two conjugat-imaginary roots, rsp. In th scond cas, on will obtain Tabl Ib) by introducing th nw basic numbrs: = +, = ( ) i, which dfins th systm of ordinary complx numbrs. Tabl I itslf can srv as th typ of th numbr systms of th first class. Howvr, it sms prfrabl to writ this tabl onc mor in th form Ia) by th introduction of th nw basic numbrs: = +, =. In Tabls Ia) and Ib), w will thn hav two analogous canonical forms for th two forms of typ I ( * ). ( * ) Th dtrmination of systms with ral basic numbrs was achivd by Wirstrass and Cayly. S. Stolz, Vorlsungn übr allgmin Arithmtik, Bd. II, 5 and Cayly, Proc. Lond. Math. Soc. XV (88-84), pp. 86, t sq. Schffrs was kind nough to mak m awar of this papr by Cayly.

10 E. Study On systms of complx numbrs and transformation groups. 9 I. Ia). Ib) II. As w said, Tabl Ia) will go to I undr th ral substitution: = +, =, whn on, in turn, drops th ovrbar again. Similarly, Tabl Ib) will go to Ia) undr th substitution: =, = i (i = ). Nxt, lt n =, such that on will hav: 4. Systms with thr basic numbrs. a = a a + a a + a a, whil, for a gnral choic, a will not alrady b xprssibl in trms of a and a. W will distinguish thr cass, according to whthr th cubic quation: λ = a λ + a λ + a, has thr distinct roots λ, λ, λ, on doubl root λ and a simpl root λ, or finally a tripl root λ for a gnral choic of th numbr a. In th first cas, on simply sts: ( a λ )( a λ) =, ( λ λ )( λ λ ) and corrspondingly for and, with a cyclic prmutation of th indics. Sinc,, and ar linarly-indpndnt, on th basis of th assumption that was mad, and satisfy th rlations i = i, i k =, in addition, which ar vrifid immdiatly, on will thn obtain th canonical form I that is listd blow.

11 E. Study On systms of complx numbrs and transformation groups. If λ and λ coincid thn and will bcom infinit, and th transformation to th canonical form I will b impossibl. Howvr, on can rmark that + and (λ λ ) will rmain finit. In th limit, on will hav ths xprssions: ( a λ )( a λ + λ) =, ( λ λ ) ( a λ)( a λ) =, ( λ λ ) togthr with th limiting valu for : = ( a λ), ( λ λ ) which ar again linarly-indpndnt numbrs, and on will thn obtain th canonical form II. Finally, if λ also coincids with λ and λ thn this form will also b impossibl. Howvr, +, as wll as (λ λ ) + and (λ λ ), will again rmain finit, and on will hav ths xprssions in th limit: =, = (a λ ), = (a λ ), which ar th basic numbrs of a nw typ III. Furthrmor, lt k =, such that any numbr a of th systm will fulfill an quation of th form: a = a a + a a. If w think of th numbr a as bing xprssd hr in trms of any thr basic numbrs,, in th form λ + λ + λ thn a will bcom a homognous, linar function of λ, λ, λ that dos not vanish whn on sts a qual to th numbr on for th systm. On can thn associat th unity with two basic numbrs, in such a way that th cofficint a that is constructd for an numbr of th form λ + λ will vanish. Th cofficint a will thn bcom a scond-dgr homognous, linar function of λ and λ. It can possss two distinct zros. On can thn introduc two nw basic numbrs and in plac of and that satisfy th conditions: =, =, and th sam thing will happn whn a vanishs idntically. Howvr, if a is a squar thn on can choos th basic numbrs in such a way that:

12 E. Study On systms of complx numbrs and transformation groups. =, =. W nxt considr th scond cas. If w mak th Ansatz: = a + a + a thn by forming th products ( ) and ( ), w will rcogniz that on must hav: a =, a =, a =. If w first tak a =, so = thn it will follow from th associativ law that = ; w will thn obtain Tabl IV. Th othr assumption a = would yild th rciprocal systm, but it will lad back to th first on immdiatly undr th substitution =. If w furthr assum that: =, = thn th associativity law will immdiatly imply that = =. Th cas that was prviously prsntd as possibl, in which th cofficint a vanishd idntically, can thrfor not actually occur; w now obtain multiplication tabl V. W hav thus found, in total, fiv diffrnt systms of complx numbrs that corrspond to th assumption that n =. Thr is no difficulty in also dtrmining all of thir forms. Onc mor, first st k =, so w can hav mad an imaginary substitution only whn th cubic quation: λ = a λ + a λ + a that is charactristic of th systm has two conjugat-imaginary roots. This can occur only in cas I. Whn λ and λ ar th conjugat-imaginary roots, and will bcom conjugat-imaginary. By th substitution: = +, = i ( ), w will thn onc mor obtain a systm with ral constants γ iks, which is rprsntd in Tabl Ib). Of th systms that wr found in th cas k =, th systm V obviously cannot hav two diffrnt forms. Should th systm IV b capabl of a scond ral form thn this could aris only whn on sts =, =, in plac of =, that would thn yild imaginary valus for som of th constant γ iks. W will thn hav to list th following tabls for th cas of n = : =. Howvr,

13 E. Study On systms of complx numbrs and transformation groups. I. Ia). Ib). Tabl Ia) is not ssntially diffrnt from Tabl I, and will go ovr to it undr th ral substitution: = +, =, =. This is only du to th citd analogy with th following tabl Ib). Tabl Ib) will go to Ia) undr th substitution: =, = i, =. II. III. IV. V. Th systm IV will go to its rciprocal systm undr th substitution: =, =, =. 5. Systms with four basic numbrs. First, lt k = 4, so: a 4 = a a + a a + a a + a 4 a. If th bi-quadratic quation:

14 E. Study On systms of complx numbrs and transformation groups. a 4 = a λ + a λ + a λ + a 4 has four sparat roots λ, λ, λ, λ for a gnral assumption on th numbr a thn on can tak th basic numbrs to b,,,, whr,.g.: ( a λ )( a λ)( a λ ) = ; ( λ λ )( λ λ )( λ λ ) on will thn obtain th canonical form I that is listd blow. In th cas of a doubl root λ and two simpl roots λ, λ, on taks th basic numbrs to b th limiting valus of +, (λ λ ),,, namly: ( a λ = {λ )( a λ) λ (a + λ + λ ) + (aλ + aλ + λ λ )} ( λ λ ) ( λ λ ), ( a λ)( a λ)( a λ) =, ( λ λ )( λ λ ) = = ( a λ ) ( a λ ), ( λ λ ) ( λ λ ) ( a λ ) ( a λ ). ( λ λ ) ( λ λ ) Th associatd multiplication tabl will b II. Furthrmor, if λ also coincids with λ thn on will tak th basic numbrs to b th limiting valus of,, +, (λ λ ) : = (a λ + λ )( a λ ) ( λ ) λ, = ( a λ )( a λ ) ( λ ) λ, = (a λ + λ )( a λ ) ( λ ) λ, = on will thn gt Tabl III. ( a λ )( a λ ) ( λ ) λ ;

15 E. Study On systms of complx numbrs and transformation groups. 4 By contrast, if λ is a tripl root and λ is a singl root thn by introducing th limiting valus of +, (λ λ ) +, (λ λ ), on will obtain: = ( a λ) ( λ λ ), ( a λ)( a λ )( a λ + λ ) =, ( λ λ ) = ( a λ) ( a λ ), ( λ λ ) = ( a λ) ( λ λ ), as th basic numbrs in Tabl IV. Finally, if th root λ also coincids with λ thn: +, (λ λ ) +, (λ λ ) +, (λ λ ) will again convrg to finit limits, namly, to: (a λ ), (a λ ), (a λ ), (a λ ). By introducing thn as limiting numbrs, on will obtain Tabl IV. W now turn to th scond principal cas k =. Hr, any gnral, sufficintly-chosn numbr a, along with its squar and a = will togthr dfin a systm of thr numbrs that corrsponds to th typs I, II, III that wr prsntd for n =. Sinc only four linarly-indpndnt units ar prsnt, any two such thr-dimnsional domains will hav a two-dimnsional domain in common viz., a systm of two numbrs (which will thn blong to any systm of thr numbrs, in th sam way). W now first assum that th systm of thr numbrs that blongs to a gnrallychosn numbr a is a systm of th first typ. Th two-dimnsional systm that is dfind by th intrsction of systm I with anothr systm will thn b (triply) wlldfind. Namly, if on xprsss th ida that th squar of a numbr λ + λ + λ of systm I should alrady b linarly rprsntabl in trms of th numbr itslf and + + = thn it should follow that λ, λ, λ ar roots of a quadratic quation such that two of ths thr quantitis must b qual to ach othr. Thrfor, thr ar only thr mutual-quivalnt sub-domains in our numbr systm that hav th dsird proprty; on of thm is dtrmind by th numbrs = + + =, = +. If on introducs ths, togthr with =, as nw basic numbrs

16 E. Study On systms of complx numbrs and transformation groups. 5 and appnds anothr basic numbr that is coupld with and by th sam rlations as thn part of th multiplication ruls that w sk will alrady b known. It will b rprsntd by th formulas: (A) =, =, =, = =, = =. =, Scondly, lt th systm of thr numbrs that is dtrmind by a, a, a b a systm of th scond typ, and also for a compltly gnral choic of a. On can thn associat th numbr = with a numbr in two ssntially diffrnt ways that will both collctivly dfin a systm of two numbrs: = and =. In th first cas, on taks,, and = to b th nw basic numbrs, and appnds a third basic numbr that has th sam rlationship to and as. On will thn obtain th multiplication ruls: (B) =, =, =, = =, = =. =, In th othr cas, th following choic of basic units =, =, = will yild th formulas: (C) =, =, =, =, = =, = =. Finally, for th most gnral choic of a, th systm that is dtrmind by a, a, a might blong to th third typ. On can thn xtnd th numbr = = by anothr numbr in ssntially on way such that and collctivly mak up a systm of two numbrs; this systm will b dtrmind by = and =. If on thn writs for and, appnds a third basic numbr, and drops th ovrbar thn on will obtain a fourth systm of formulas: (D) =, =, =, = =, = =. =, W shall go through th four assumptions (A), (B), (C), (D) individually. W will add to thm, in th most gnral way, rlations of th form: = α + α + α + α, = β + β + β + β, and dtrmin th constants α and β according to th associativ law:

17 E. Study On systms of complx numbrs and transformation groups. 6 (A) =, =, =, =, = =, = =. In addition, if w lt: = α + α + α + α thn it will follow upon multiplying by that α =. Furthrmor, whn on forms th products ( ) and ( ), on will gt: α = α α = α α, α = α, α + α α = ; i.., on will hav ithr: α = α, (a) α =, α =, α =, or (b) α =, α =, α =, or, what amounts to th sam thing: α =, α =, α =, or finally: (c) α =, α =, α =. (a) = + +. If: = β + β + β thn on will find, by th us of th quation ( ) = ( ), or th othr on ( ) = ( ), that β = β = β ; i.., on will hav ithr: or β =, β =, β = β =, β =, β =. At this point, th first assumption will b uslss for us, bcaus it lads back to systm I undr th substitution =, = +, =, =. In th scond cas, w introduc two nw units =, =, in plac of and, which ar dtrmind such that on will hav =, Furthrmor, on introducs two nw basic numbrs and =, + =., in plac of and, which ar chosn such that,, and, whn takn by thmslvs, will dfin th fourth typ of associatd systm of thr units. To that nd, w dtrmin such that

18 E. Study On systms of complx numbrs and transformation groups. 7 =, namly, = + +, and thn dtrmin such that =, namly, =. W will thn obtain multiplication tabl VI. (b) =. If on introducs a nw basic numbr in plac of by mans of th substitution = + thn on will gt back to th cas (a) that was tratd abov. (c) =. If on dos not wish to rturn to th cass (a), (b) thn on must assum that is qual to zro in any cas. Howvr, that would b suprfluous, sinc, by th substitution: =, =, =, =, on would, in turn, obtain Tabl I. (B) =, =, =, = =, = =, = α + α + α + α. =, It would follow upon multiplication by that α = ; furthrmor, upon multiplication by and : α = α α = α α, α α =, i.., on will hav ithr: α = α, α = α, (a) α =, α =, α =, or α =, α =, α =, or (b) α =, α =, α =. (a) =. Lt: = β + β. It will thn follow upon multiplication by that: β + β = ; i.., on will hav ithr β =, β = or β =, β =. Th first assumption is inadmissibl, sinc on would obtain Tabl II undr th substitution =, =, =, =. All that will rmain thn is th assumption that =, which w can also rplac with =, =, in ordr to go to th rciprocal systm. Hr,,, will dfin a systm of thr numbrs that blong to th fourth typ. It will go to its canonical form undr th substitution =, =, = ; if w appnd = thn w will obtain Tabl VII.

19 E. Study On systms of complx numbrs and transformation groups. 8 Starting from th first of th assumptions that wr mad undr (a), w hav arrivd at a systm that it is rciprocal to th systm VII. Had w startd with th scond assumption, w would likwis hav obtaind systm VII itslf. Nithr systm viz., systm VII or its rciprocal can b transformd into th othr on (whil this is still possibl for systm VI, which can b drivd from systm VII by a passag to th limit that is asy to giv). In fact, lt,, b th basic numbrs of th systm that is rciprocal to VII, so its multiplication tabl will mrg from Tabl VII by prmuting th horizontal and vrtical rows. Should,, thn b linarly drivabl from,, with numrical cofficints thn it would nxt follow that =. Furthrmor, sinc on should hav =, =, =,,, must hav th form: = ± + λ, = µ + µ, = ν + ν. Sinc on should hav =, only th lowr sign in th first of ths xprssions will b admissibl. Howvr, th absurd rsult that v = v = would follow from th quation =. (Cf., th rmarks on pag 8, as wll. Th tabl that was givn on pag 6 will go to VII undr th substitution: = +, =, =, =.) (b) If = thn it will also follow that =. This would onc mor b inadmissibl, sinc on would obtain Tabl II undr th substitution =. (C) =, =, =, = =, = =, = α + α + α + α. =, It will thn follow forthwith upon multiplying by,, and that: If on introduc basic numbrs: = =. =, =, =, = in plac of,, thn on will obtain Tabl VIII. (D) =, =, = =, = =, = =, = α + α + α + α. Upon multiplying by,, and, on will find:

20 E. Study On systms of complx numbrs and transformation groups. 9 such that on can thn st: α = α = α =, = α, = β. If on introducs a nw basic numbr = α, in plac of, thn on will gt: =, = (β α), = ( αβ). W now distinguish thr cass: (a) β α, (b) β α =, αβ, (c) β α =, αβ =. (a) On introducs th nw basic numbrs =, =, =, = β α, in plac of,,,, of which, th third on is dtrmind such that = =. On will thn obtain Tabl IX. Th numrical paramtr c that ntrs hr cannot b liminatd by th introduction of nw basic numbrs. (b) = =, = c = c. Hr, if on maks th substitution =, =,, = thn on will obtain th numbr systm X. (c) = =, =. This assumption will giv Tabl XI. With that, th assumption that k = is also dalt with. All that w hav lft to prsnt ar th numbr systms for which k =. Hr, w distinguish btwn th numbr systms for which, along with =, two basic numbrs and ar givn in such a way that on of th products, of,, is linarly-indpndnt (A) of th othr on (B), for which = and two numbrs a and b alrady dfin a systm of thr units in itslf. (A) Hr, with no loss of gnrality, w can assum that ithr: (a) or (b) =, =, =, =. (a) Lt = =. If w thn xprss th ida that:

21 E. Study On systms of complx numbrs and transformation groups. (λ + λ ) should b linarly-xprssibl in trms of (λ + λ ) and = thn it will follow that: + = c, in which c mans a numrical constant. Hr, w introduc a nw basic numbr: = λ + µ, in plac of and sk to dtrmin th constants λ and µ in such a way that on will still hav =, but on th othr hand, on will hav + =. Howvr, this will always b possibl, xcpt whn c = 4. First, lt c 4. W can onc mor rplac = and with nw basic numbrs by th substitution = i, = i. W will thn hav th rlations = =, =. If w st =, and thn dtrmin th xprssions for th rmaining unit products from th associativ law thn w will obtain Tabl XIII, which is th wll-known systm of quatrnions. Scondly, st c =. W thn tak th basic numbrs to b: =, = i, = + λ, =, in which w now dtrmin λ such that on will hav = ; this will yild λ =. On will thn obtain Tabl XIII. Thirdly, lt c =. On will thn com back to th cas c = + that was just tratd by th substitution =. (b) Lt = =. As abov, it will thn follow from th condition k = that: + = c. If c thn w can tak c =, with no loss of gnrality. W can thn immdiatly go back from this cas to cas (a) by th substitution: = +, = i ( ). Howvr, if c = thn w will obtain th nw Tabl XIV. (B) Any two numbrs a, b, togthr with =, dfin a systm of thr units. Hr, thr ar two possibilitis: Eithr this numbr systm blongs to th fourth typ that was prsntd for n = for an arbitrary choic of a, b, or it is a systm of th fifth typ. In th cas, w can assum that th systm of,, has alrady bn brought into th canonical form III (cf., pp. ). If is any numbr that is linarly-indpndnt of,, thn,, will likwis dfin a systm of th fourth typ. W can thrfor

22 E. Study On systms of complx numbrs and transformation groups. tak a nw basic numbr that satisfis th quation = in plac of in this systm. In that way, th systm,, will go to its canonical form. If w again writ in plac of thn,, will now dfin a systm of th fifth typ that will alrady b in its canonical form sinc = =. With that, th xprssions for all of th products of th basic units,,, will alrady b known; w will thn gt Tabl XV. Finally, in th scond cas, w can choos,, such that w will hav = = =. Th last Tabl XVI will thn aris. Th fact that th typs that wr numratd hr ar actually all distinct i.., that it would not b possibl to driv anothr tabl in th sam squnc from Tabls I-XVI by th introduction of nw basic numbrs hardly sms ncssary to mphasiz, in particular. On can also asily xtnd th argumnt that was just dscribd that on will obtain all of th diffrnt forms of any typ. Manwhil, w will go into th tratmnt of that problm, but only giv th rsult. A form viz., th scond of systm X was ovrlookd by m in th original vrsion of this rport. From th rmarks that w mad in our xamination of th cas n =, w do not spcially nd to writ down th substitutions by which th distinct ral forms Ia),, Ic), IIa), and IIb) could aris from th basic forms of thir typs I and II. I. Ia). Ib) II. Ic)

23 E. Study On systms of complx numbrs and transformation groups. IIa) IIb) III IIIa) IIIb) Th form IIIa) ariss from II undr th ral substitution: = +, =, = +, =. Tabl IIIb) ariss from IIIa) by th substitution: =, = i, =, = i. IV V

24 E. Study On systms of complx numbrs and transformation groups. VI VIII Th systm VI will go to its rciprocal undr th substitution = ; th systm VII cannot go to its rciprocal systm. VIII IX Tabl IX will rprsnt infinitly many typs, corrsponding to th diffrnt valus of th paramtr c. Any of ths systms will go to its rciprocal undr th substitution =. X Xa) Tabl Xb) will aris from X undr th substitution = i. XI

25 E. Study On systms of complx numbrs and transformation groups. 4 XII XIIb) Systm XII, which is wll-known undr th nam of th quatrnions, will go to systm XIIb) undr th substitution: =, = i, = i, =. XIIIb) will also mrg from XIII undr th sam substitution. Systms XII-XV will ach go to thir rciprocals undr th substitution: =, =, =, =. XIII XIIIb) XIV XV XVI

26 E. Study On systms of complx numbrs and transformation groups Spcial systms with n basic numbrs. Th xtnsion of th dtrmination of th numbr systms with n basic numbrs that was just givn in som simpl cass to an undtrmind valu of th numbr n can hav its difficultis. Nonthlss, a smingly xtndd class of such systms can b dtrmind in gnral. Thy ar th systms for which th numbr k that was dfind in has th largst possibl valu n. It is obvious that thy will oby th commutativ law of multiplication. Thir totality dfins an irrducibl manifold whos most gnral, and likwis simplst, rprsntativ can b considrd to b a systm with th following multiplication ruls ( * ): i = i, i k = (i k, i, k =,, n). Th totality of all systms of th statd typ will b givn by th following thorm ( ** ): In ordr to find all distinct systms with n basic numbrs for which th powrs A, A,, A n of any numbr A ar, in gnral, linarly-indpndnt of ach othr, on must rprsnt th numbr n as a sum of whol numbrs in all possibl ways. If: n = α + β + + µ is such a dcomposition of n thn on will arrang th n basic numbrs in groups α, β,, µ, and dnot thm by: a,, a α, b,, b β,, m,, m µ. On will thn st th products of any two basic numbrs from diffrnt groups qual to zro, and assum a rul for th multiplication of two numbrs within th sam group, which might b statd, for xampl, as: a i a j = a i+j (i + j α ), a i a j = (i + j > α ), for th first group. Any two numbr systms that ar dtrmind in this way will b distinct; i.., it will not b possibl to tak on of thm to anothr on by th introduction of nw basic numbrs. Howvr, any systm of n numbrs that satisfis th statd condition can b takn to on of th systms that was prsntd by a suitabl choic of basic numbrs. Morovr, th basic numbrs a,, m µ in th givn systms ar not dtrmind uniquly, xcpt for only th simplst cas α = β = = µ =, in which on will com back to th multiplication ruls = i, i k = that wr givn alrady. i On ( * ) On should confr th author s rmarks in Gött. Nachr. 889, pp concrning this oft-tratd numbr systm. ( ** ) According to this, a thorm that was statd by Poincaré [Compts rndus d l Ac. ds Scincs, t. 99, (884), pp. 74] can b corrctd.

27 E. Study On systms of complx numbrs and transformation groups. 6 immdiatly rcognizs that on will obtain th prcisly th sam multiplication ruls whn,.g., on introducs th following basic numbrs: a = λ a + λ a + + λ α a α, a = (λ a + λ a + + λ α a α ),, a α = (λ a + λ a + + λ α a α ) α, instad of a,, a µ. Hr, λ,, λ α man any sort of numrical valus that ar subjct to only th rstriction that λ must b non-zro. Th nwly-introducd basic numbrs ar th most gnral ons that will produc th statd multiplication ruls; th sub-domains a, (a,, a α ), (a,, a α ),, (a α ) will b dtrmind uniquly, if not also th basic numbrs thmslvs. Svral distinct ral forms can mrg from th systms that mrg from th systms that corrspond to th dcomposition: n = α + β + + µ only whn som of th numbrs α, β,, µ ar qual to ach othr, and on if taks th numbr αm tims th first numbr that is diffrnt from α tims m tims th first numbr that is diffrnt from th last two tims m, tc., th on will indd gt: m + m + m + diffrnt forms that can b writtn down immdiatly, as long as [m] mans th largst whol numbr that is lss than or qual to m. For xampl, lt α = β = γ, but γ δ ε k, so thr will b two distinct ral forms. Th canonical form that was givn in th thorm abov can b usd as th canonical form for th first on. Howvr, it sms prfrabl to m to introduc α basic numbrs instad of th basic numbrs a i, b k by th substitutions: a i = a i + b i, b i = a i b i (i =,,, α ), with which, th following multiplication ruls will aris: a i a j = a i+j, b i b j = a i+j, a i b j = b i+j. On will thn obtain th corrsponding tabl for th scond ral form simply by making th substitutions:

28 E. Study On systms of complx numbrs and transformation groups. 7 in th form: a i = a i, b i = b i (i =,,, α ) a i a j = a i+j, b i b j = a i+j, a i b j = b i+j. Naturally, all of th basic numbrs in both tabls will b st qual to zro whn thir indx i + j xcds th largst admissibl valu of α. I will provid th proofs of th assrtions that wr mad in ths paragraphs, with a somwhat waknd formulation of th thorm, in a scond tratis that will trat rcurring sris and bilinar forms that ar closly connctd with our situation. Two diffrnt xprssions for th basic numbrs a,, a α in trms of th numbrs A,, A α will b givn thr. 7. On th thory of simply-transitiv groups. In th following paragraphs, th connction btwn systms of complx numbrs and th thory of transformation groups will b dvlopd. In particular, it shall b shown how on can mak us of th thorms that wr prsntd in 6 in this thory. Bfor w do that, will includ a sction in which a numbr of for th most part, nw thorms will b drivd that rlat to th so-calld paramtr groups that play an important rol in th study of transformation groups, as is known. In ordr to bttr undrstand this, lt it b first rmarkd that w will only spak of continuous groups throughout. If w spak of.g., conformal transformations thn w will man by that only th ons for which th angl is not altrd, and th trms group of rciprocal radii, groups of motions, group of similarity transformations, group of a sconddgr surfac will b usd in a similarly-rstrictd sns. Rgarding th thory of transformation groups, I will rfr to th ground-braking work Thori dr Transformationsgruppn (Lipzig, 888) that was cratd by th collaboration of Fr. Engl and S. Li; in th squl, it will b simply rfrrd to as (Li). In particular, contnts of chaptrs 6,,, 6, 7 will com undr considration by us. From th thory that was dvlopd by Li, any continuous group, along with its paramtr group, is composd th sam as a simply-transitiv group; on th othr hand, two simply-transitiv groups that ar composd th sam will b similar. Thus, any simply-transitiv group will b similar with its paramtr group. That mans that on can introduc nw variabls (or also nw paramtrs, if on prfrs) into any givn simplytransitiv group that is not its own paramtr group in such a way that it will bcom its own paramtr group. In fact, lt a simply-transitiv group G b givn that is gnratd by its infinitsimal transformations and is writtn in trms of th variabls x,, x n and th paramtrs y,, y n. Thn lt y,, y n b th paramtrs of a transformation of th group that taks an arbitrary, but chosn onc and for all, point in gnral position E( x,, x n ) to th

29 E. Study On systms of complx numbrs and transformation groups. 8 point x,, x n such that th variabls y,, y n will bcom indpndnt functions of x,, x n, du to th assumd proprty of th group G. W can thrfor introduc y,, y n as nw variabls. Howvr, th group will go to its paramtr group undr that. Thn, lt y [y, y,, y n ], y, y b th paramtr systm of th transformations S, T, ST of th group G, so th gnral transformation of th paramtr group of G will thn hav th form: or, mor brifly: y i = ϕ i (y,, y n, y,, y n ) (i =,, n), y = ϕ(y, y ). Hr, on should think of y,, y n as indpndnt variabls, y,, y n, as paramtrs, and y,, y n, as dpndnt variabls. Th transformation that was writtn down will b th transformation T of th paramtr group that corrsponds to th transformation T of G. Now, howvr, from th abov, y, y, y will just as much dfin a coordinat systm for th thr points (E)S, (E)T, (E)ST that mrg from th arbitrarily chosn point E undr th transformations S, T, ST, rspctivly, of G. Of ths thr points, th last on will b associatd with th first on by way of th transformation T of G, and indd, for any choic of th transformation S. Howvr, w hav just now sn that th point y,, y n will also corrspond to th point y,, y n undr th transformation T. Th transformations T and T will thn coincid. From now on, w shall assum that th variabls hav alrady bn chosn in such a way that th simply-transitiv group G will b its own paramtr group. Thus, any symbolic formula that dmands th composition of a transformation S in trms of givn transformations of G.g., S = S S S will b capabl of a scond possibl intrprtation, but not in th sam way for othr groups. Namly, any transformation of a crtain point in spac that appars in th formula will now likwis corrspond to th point whos coordinats ar th paramtrs of th transformation. Th formula thus likwis givs a dpndncy btwn diffrnt points of spac. W would now lik to introduc a symbolic notation in ordr to b abl to xprss such a dpndncy simply with formulas. Lt x,, x n b th paramtr systm of th transformation S, w lt th symbol x rprsnt th point that has th coordinats x,, x n. Furthrmor, if w lt y and z b th points that corrspond to th transformations T and ST in th aformntiond way thn w will writ symbolically: z = xy ; this notation will thn srv as an abbrviation for th systm of quations: z i = ϕ i (x,, x n, y,, y n ) (i =,, n). If w lt S, T, R b thr arbitrary transformations of G and lt x, y, z b th corrsponding point thn w will likwis dnot th point that is associatd with th transformation STR symbolically by (x y) z or x (y z), or mor simply by xyz; this symbol will thus rprsnt a point whos coordinats ar givn by th xprssions:

30 E. Study On systms of complx numbrs and transformation groups. 9 ϕ i (ϕ (x, y),, ϕ n (x, y), z,, z n ) = ϕ i (x,, x n, ϕ (y, z),, ϕ n (y, z)). W will furthr rprsnt any point that is associatd with a transformation that is composd from arbitrarily many transformations of G in a similar mannr, which dos not particularly nd to b xplaind in mor dtail. Furthrmor, lt x b th point that corrsponds to th transformation S (viz., th invrs transformation to S), so w st: x = x. This symbolic quation will srv as an abbrviation for a systm of quations that on will obtain as follows: Solving th n quations: will yild th quations: y i = ϕ i (x,, x n, y,, y n ) y i = ϕ i ( y,, y n, x,, x n), in which x,, x n ar crtain functions of x,, x n : x i = ψ i (x,, x n ) (i =,, n), so th last systm of quations will b quivalnt to th symbolic formula x = x. This formula clarly rprsnts an involutory transformation; naturally, on also has, convrsly: x n = ψ i ( x,, x n ) (i =,, n) Finally, as a logical continuation of th notation that was introducd, w will rprsnt th point that corrsponds to th idntity transformation S = T = = by th symbol x, or also by th symbol y, as dsird, or finally, most simply by th symbol. Thus, from th symbolic notation SS = S =, will mrg th othr on xx = x =, which say nothing by th fact that x,, x n and x,, x n ar th paramtr systm imags of invrs transformations of th group G. Howvr, w can now writ th idntitis: x i = ψ i (ψ (x),, ψ n (x)) (i =,, n), simply as: x = (x ), tc. Obviously, all of th ruls that ar tru for calculating with th symbols S, T will also b tru for calculations using th symbols x, y, For xampl, on will thn hav (xy) = y x. No mattr how convnint th symbolic notations that wr introducd might b, thy also show that with thir hlp on can driv a whol sris of ssntial proprtis of simply-transitiv groups which hav bn noticd up to now, for th most part in th simplst way, and mak th formulas mor intuitiv. That will nxt yild a symbolic rprsntation of th group G itslf:

31 E. Study On systms of complx numbrs and transformation groups.. Th symbolic quation: () x = xa, in which x is thought of as variabl, and a, as a paramtr (viz., a paramtr systm), rprsnts th most gnral transformation of th group G. () x = xa is th invrs transformation to (). x = x a b is th transformation of G that ariss from th composition of th transformations x = xa and x = xb. x = xa ba is th transformation of G that ariss from th transformation x = xb whn on introducs nw variabls by mans of th transformation x = xa.. Th symbolic quation: () x = ax givs th gnral transformation of th simply-transitiv group G that is rciprocal to G ( * ).. Th rciprocal groups G and G ar similar to ach othr by mans of th involutory transformation x = x, such that, in fact, any transformation x = xa of G will b associatd with th transformation x = a x of G. In fact, if w lt th transformations x = xa and x = x b dnotd by S and T thn th transformation S T S = STS will b givn by th symbolic quation: x = ((x ) a) = a x; howvr, this is a transformation of G. Morovr, on likwis ss that th composd transformation x = xab of th transformations x = xa and x = xb in G will go to th transformation x = (a b) x = b a x of G, and thus to a transformation that on can obtain immdiatly by composing transformations of G that ar associatd with th transformations x = xa and x = xb. Abov all, G and G ar similar to ach othr by mans of any transformation x = α x β ; in fact, this will tak th transformation x = xa to x = α a α x. 4. Th symbolic quation: (4) x = a x b givs th gnral transformation of a group G,. It will b obtaind whn on prforms th gnral transformation of G and th gnral transformation of G on aftr ach othr. On immdiatly convincs onslf that G and G ar invariant in this group, and ( * ) From now on, w will us th trm rciprocal groups without th slf-xplanatory qualifir simply-transitiv.

32 E. Study On systms of complx numbrs and transformation groups. furthrmor, that th group G, has only n m ssntial paramtrs whn G and G hav an m-paramtr subgroup in common. 5. Th symbolic quation: (5) x = a x a givs th most gnral transformation of a continuous (n m)-paramtr group G that is charactrizd as a subgroup of G, by th fact that its transformations lav th point in gnral position x = fixd. As a rsult of a rmark that was mad in numbr, quation (5) tlls on how th transformations of G can b xchangd with ach othr by mans of th transformations of G itslf. Thrfor, if on kps a point in gnral position fixd in th group G, that is gnratd by th composition of transformations from two rciprocal groups G and G thn th largst continuous subgroup G of G, that is dtrmind in that way will b similar to th adjoint group of G and G. Th transformation x = x commuts with all of th transformations of th group G. If m = thn th group G will hav yt anothr rmarkabl proprty. Namly, whn it is writtn in th form (5), it will thn again hav th group G as its paramtr group. By contrast, if on writs G in th quivalnt form x = a x a thn on will obtain th group G as a paramtr group. On can also immdiatly writ down th continuous subgroup of G, that lavs anothr point b in gnral position fixd: On nds only to introduc nw variabls into th quations of th group G: x = a x a or x = a x a by mans of th transformation x = xb or th othr on x = bx. On will thn obtain two symbolic rprsntations of th dsird group: (6) x = a xb ab and x = ba b xa. Both notations ar quivalnt to ach othr; thy will go to ach othr undr th substitution a = ba b or a = b ab, rsp. Finally, to complt th statmnts, on might giv th rlationship that th infinitsimal transformations of th group G hav to thos of G and G, although w shall not us it latr on. 6. Lt X f,, X m f, X m+ f,, X n f b indpndnt infinitsimal transformations of th group G that ar chosn such that any distinguishd transformation of G will hav th form: λ X f + + λ m X m f. Furthrmor, should th xprssions X i f go to x thn on would hav: X i f + X f = (i =,, m) i X i f by mans of th transformation x =

33 E. Study On systms of complx numbrs and transformation groups. idntically, although X m+ f + X m+ f,, X n f + X n f ar th infinitsimal transformations of th group G. W can also gt this thorm with no calculation. Lt: (7) x i = x i + ξ i δt b an infinitsimal transformation of G that might go to th infinitsimal transformation: (8) x i = x i + ξi δt of G by mans of th transformation x = x. Whn both of thm ar prformd on aftr th othr that will giv a transformation of G [s formula (5) abov]: (9) x i = x i + (ξ i + ξ i ) δt. Now, if (8) is, in particular, a transformation of G that likwis blongs to th group G thn it will b th invrs transformation to (7), so ξ i will thn b qual to ξ i, idntically. Th thorms that wr drivd ar maningful for th gnral thory of transformation groups, sinc a pair of rciprocal groups G, G is indd linkd with vry continuous group of transformations, namly, th paramtr group of th givn group and its rciprocal, rsp. If on writs th transformations of th givn group, in particular, in canonical form: x i = x i + n k X k xi + (i =,, n) k = (cf., Li, chap. 9, 46, pp. 7) thn th associatd paramtr group (viz., a canonical paramtr group) g, its rciprocal g, and th adjoint group G will b thr of th typ that w dscribd. In this cas, th transformation x = x will tak on an spcially simpl analytical xprssion: It will b nothing but th involutory prspctiv transformation: k = k. Thrfor, from our gnral thorm, th canonical paramtr group and its rciprocal group will b similar to ach othr by mans of this transformation. 8. Lmmas from th thory of linar transformations. In addition to th thorms on rciprocal groups that wr assrtd in 7, w will rquir som lmmas from th thory of bilinar forms for our furthr dvlopmnts.