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About Google Book Search Google's mission is to organize the world's information and to make it universally accessible and useful. Google Book Search helps readers discover the world's books while helping authors and publishers reach new audiences. You can search through the full text of this book on the web at |http: //books .google .com/I i, {lOts^'SL \. HARVARD COLLEGE SCIENCE CENTER LIBRARY ^ri. - o ^ Linear Associative Algebra f By BENJAMIN PEIRCB/ LL. D. Late Pbrkins Professor of Astronomy and Mathematics in Harvard University AND Superintendent of the UNitED States Coast Survey. New Edition, with Addenda and Notes, by C. S. Peirce, Son of the Author. [Extracted fnm The American Jdumal of MathemaHcs^l NEW YORK : D. VAN NOSTRAND, PUBLISHER. ^ 1882. / * ] . f \ / J»A — l^ * ' PrB8 op Isaac Frikdbnwald, Baltimore, Md. ERRATA. Page 10, § 31. The first formula should read {A±:B)G=AG±BG. Page 30. The third formula should read k{i — h)=j. Page 36. Foot-note, second line of second paragraph, read Page 40. Last line of foot-note. For e , read I . Page 52. Multiplication table of (/g) . For^i = i, read ji =j. Page 75. Last line of foot-note, insert Z, at beginning of line. Page 86. Foot-note. Add that on substituting k + vj for k , the algebra {aw^) reduces to (00:5) ; and the same substitution reduces (ay^) to (az^) . Page 91. Last line of foot-note. For i, read /. PREFACE. Lithographed copies of this book were distributed by Professor Peirce among his friends in 1870. The present issue consists of separate copies extracted from Ths Ameri- can Journal of Mathematics^ where the work has at length been published.* The body of the text has been printed directly from the lithograph with only slight verbal changes. Appended to it will be found a reprint of a paper by Professor Peirce, dated 1875, and two brief contributions by the editor. The foot-notes contain transforma- tions of several of the algebras, as well as what appeared necessary in order to complete the analysis in the text at a few points. A relative form is also given for each algebra ; for the rule in Addendv/m II. by which such foiTQS may be immediately written down, was unknown until the printing was approaching completion. The original edition was prefaced by this dedication : To My Fbiends. This work has been the pleasantest mathematical effort of my life. In no other have I seemed to myself to have received so full a reward for my mental labor in the novelty and breadth of the results. I presume that to the uninitiated the formulae will appear cold and cheerless ; but let it be remembered that, like other mathematical formulae, they find their origin, in the divine source of all geometry. Whether 1 shall have the satisfaction of taking part in their exposition, or whether that will remain for some more profound expositor, will be seen in the future. B. P. * To page n of this issue corresponds page n+9^ of Vol. IV. of 2^ Jimmai, r\ LINEAR ASSOCIATIVE ALGEBRA. 1. Mathematics is the science which draws necessary conclusions. This definition of mathematics is wider than that which is ordinarily given, and by which its range is limited to quantitative research. The ordinary definition, like those of other sciences, is objective ; whereas this is subjective. Recent investigations, of which quaternions is the most noteworthy instance, make it manifest that the old definition is too restricted. The sphere of mathe- matics is here extended, in accordance with the derivation of its name, to all demonstrative research, so as to include all knowledge strictly capable of dog- matic teaching. Mathematics is not the discoverer of laws, for it is not induction ; neither is it the framer of theories, for it is not hypothesis ; but it is the judge over both, and it is the arbiter to which each must refer its claims ; and neither law can rule nor theory explain without the sanction of mathematics. It deduces from a law all its consequences, and develops them into the suitable form for comparison with observation, and thereby measures the strength of the argument from observation in favor of a proposed law or of a proposed form of application of a law. Mathematics, under this definition, belongs to every enquiry, moral as well as physical. Even the rules of logic, by which it is rigidly boxmd, could not be deduced without its aid. The laws of argument admit of simple statement, but they must be curiously transposed before they can be applied to the living speech and verified by observation. In its pure and simple form the syllogism cannot be directly compared with all experience, or it would not have required an 2 Pbircb : Lmear Associative Algebra. Aristotle to discover it. It must be transmuted into all the possible shapes in which reasoning loves to clothe itself. The transmutation is the mathematical process in the establishment of the law. Of some sciences, it is so large a portion that they have been quite abandoned to the mathematician, — which may not have been altogether to the advantage of philosophy. Such is the case with geometry and analytic mechanics. But in many other sciences, as in all those of mental philosophy and most of the branches of natural history,, the deductions are so immediate and of such simple construction, that it is of no practical use to separate the mathematical portion and subject it to isolated discussion. 2. The branches of mathematics are as various as the sciences to which they belong, and each subject of physical enquiry has its appropriate mathematics. In every form of material manifestation, there is a corresponding form of human thought, so that the human mind is as wide in its range of thought as the physical universe in which it thinks. The two are wonderfully matched. But where there is a great diversity of physical appearance, there is often a close resemblance in the processes of deduction. It is important, therefore, to separate the intellectual work from the external form. Symbols must be adopted which may serve for the embodiment of forms of argument, without being trammeled by the conditions of external representation or special interpretation. The . words of common language are usually unfit for this purpose, so that other • symbols must be adopted, and mathematics treated by such symbols is called algebra. Algebra, then, is formal mathe^natics. 3. All relations are either qualitative or quantitative. Qualitative relations can be considered by themselves without regard to quantity. The algebra of such enquiries may be called logical algebra, of which a fine example is given by Boole. Quantitative relations may also be considered by themselves without regard to quality. They belong to arithmetic, and the corresponding algebra is the common or arithmetical algebra. In all other algebras both relations must be combined, and the algebra must conform to the character of the relations. 4. The symbols of an algebra, with the laws of combination, constitute its language ; the methods of using the symbols in the drawing of inferences is its art ; and their interpretation is its scientijic application. This three-fold analysis of algebra is adopted from President Hill, of Harvard University, and* is made the basis of a division into books. Peiece: Linear AssocicUive Algebra, 5 Book I.* The Language of Algebra. 5. The language of algebra has its alphabet, vocabulary, and grammar, 6. The symbols of algebra are of two kinds : one class represent its fundamental conceptions and may be called its letters, and the other represent the relations or modes of combination of the letters and are called the signs. 7. The alphabet of an algebra consists of its letters ; the vocabulary defines its signs and the elementary combinations of its letters ; and the grammar gives the rules of composition by which the letters and signs are united into a complete and consistent system. The Alphabet. 8. Algebras may be distinguished from each other by the number of their independent fundamental conceptions, or of the letters of their alphabet. Thus an algebra which has only one letter in its alphabet is a single algebra ; one which has two letters is a dovhle algebra ; one^ of three letters a triple algebra ; one of four letters a quadruple algebra, and so on. This artificial division of the algebras is cold and uninstructive like the artificial Linnean system of botany. But it is useful in a preliminary investiga- tion of algebras, until a sufficient variety is obtained to afford the material for a natural classification. Each fundamental conception may be called a unit; and thus each imit has its corresponding letter, and the two words, unit and letter, may often be used indiscriminately in place of each other, when it cannot cause confusion. 9. The present investigation, not usually extending beyond the sextuple algebra, limits the demand of the algebra for the most part to six letters ; and the six letters, i, /, h, Z, m and ti, will be restricted to this use except in special cases. 10. For any given letter anothe)* may be substituted, provided a new letter represents a combination of the original letters of which the replaced letter is a necessary component. For example, any combination of two letters, which is entirely dependent for its value upon both of its components, such as their sum, diflFerence, or product, may be substituted for either of them. *Oiily this book was ever written. [C. S. P.] 4 Peirce : Lmear Associative Algebra. This principle of the svhstitution of letters is radically important, ftnd is a leading element of originality in the present investigation ; and without it, such an investigation would have been impossible. It enables the geometer to analyse an algebra, reduce it to its simplest and characteristic forms, and compare it with other algebras. It involves in its principle a corresponding substitution of units of which it is in reality the formal representative. There is, however, no danger in working with the symbols, irrespective of the ideas attached to them, and the consideration of the change of the original conceptions may be safely reserved for the hooTc of interpretation, 11. In making the substitution of letters, the original letter will be preserved with the distinction of a subscript number. Thus, for the letter i there may successively be substituted ii, *2, is, etc. In. the final forms, the subscript numbers can be omitted, and they may be omitted at any period of the investigation, when it will not produce confusion. It will be practically found that these subscript numbers need scarcely ever be written. They pass through the mind, as a sure ideal protection from erro- neous substitution, but disappear from the writing with the same facility with which those evanescent chemical compounds, which are essential to the theory of transformation, escape the eye of the observer. 12. A ^^re^lgebra is one in which every letter is connected by some indissoluble relation with every other letter. 13. When the letters of an algebra can be separated into two groups, which are mutually independent, it is a mixed algebra. It is mixed even when there are letters common to the two groups, provided those which are not common to the two groups are mutually independent. Were an algebra employed for the simultaneous discussion of distinct classes of phenomena, such as those of soimd and light, and were the peculiar units of each class to have their appropriate letters, but were there no recognized dependence of the phenomena upon each other, so that the phenomena of each class might have been submitted to independent research, the one algebra would be actually a mixture of two algebras, one appropriate to sound, the other to light. It may be farther observed that when, in such a case as this, the component algebras are identical in form, they are reduced to the case of one algebra with two diverse interpretations. Peircb : Lmear Associative Algebra. 5 The Vocahulary. 14. Letters which are not appropriated to the alphabet of the algebra * may be used in any convenient sense. But it is well to employ the small letters for expressions of common algebra, and the capital letters for those of the algebra under discussion. There must, however, be exceptions to this notation ; thus the letter D will denote the derivative of an expression to which it is applied, and 2 the summa- tion of cognate expressions, and other exceptions will be mentioned as they occur. Greek letters will generally be reserved for angular and functional notation. 15. The three symbols J, 9, and 6 will be adopted with the signification J=^/— 1 tl . 9 = the ratio of circumference to diameter of circle = 3.1415926536 . ; 6 = the base of Naperian logarithms = 2.7182818285, I which gives the mysterious formula -^^l^-rr J-»=V 6^=4.810477381. ^^-xC^ 16. All the signs of common algebra will be adopted; but any signification (- '^ . will be permitted them which is not inconsistent with their use in common / ' ^ I algebra ; so that, if by any process an expression to which they refer is reduced to one of common algebra, they must resume their ordinary signification. f 17. The sign =, which is called that of equality, is used in its ordinary sense to denote that the two expressions which it separates are the same whole, although they represent different combinations of parts. 18. The signs > and <] which are those of inequality, and denote **more than " or '* less than " in quantity, will be used to denote the relations of a whole to its part, so that the symbol which denotes the part shall be at the vertex of the angle, and that which denotes the whole at its opening.- This involves the - proposition that the smaller of the quantities is included in the class expressed by the larger. * Thus B^ denotes that there is a class which includes both A and 5,f which proposition might be written B<>A. 19. A vertical mark drawn through either of the preceding signs reverses its signification. Thus A:^B denotes that B and A are essentially different wholes ; A:Jp>B or Been brought to a form which does not contain any other idempotent expression than the basis. It must be, therefore, that all the other expressions are nilpotent. 68. No proAvjct of expressions in the first group of an algebra Tiaving an idem - potent ba>sisj contains a term which is a mvMiple of the basis. For, assume the equation AB= — ad+G, in which A , B and G are nilpotents of the orders m , n and p , respectively. Then, = jI'^+IjB = — xA'^ + A'^C A'^C = xA'^ = A'^G^'^^=xA'^C^=a?A'^G^-'^=x^'^^A'^=x, that is, the term — xi vanishes from the product AB. 69. It follows, from the preceding section, that if the idempotent basis were taken away from the first group of which it is the basis, the remaining letters of the first group xjoould amstitute by themselves a nilpotent algebra. Conversely, any nilpotent algebra may be converted into an algebra with an idempotent basis, by the simple annexation of a letter idemfa/yiend and idemfadervt with reference to every other.* 70. However incapable of interpretation the nilfactorial and nilpotent expressions may appear, they are obviously an essential element of the calculus of linear algebras. Unwillingness to accept them has retarded the progress of discovery and the investigation of quantitative algebras. But the idempotent basis seems to be equally essential to actual interpretation. The purely nilpotent algebra may therefore be regarded as an ideal abstraction, which requires the introduction of an idempotent basis, to give it any position in the real universe. In the subsequent investigations, therefore, the purely nilpotent algebras must be regarded as the first steps towards the discovery of algebras of a higher degree resting upon an idempotent basis. * That every such algebra must be a pure one is plain, because the algebra (a%) is so. [C. S. P.] Peirge! : Linear Associatwe Algebra. 28 71. Sufficient preparation is now made for the INVESTIGATION OF SPECIAL ALGEBRAS. The following notation will he adopted in these researches. Conformably with § 9, the letters of the alphabet will be denoted by i , y , Jc, I, m and n . To these letters will also be respectively assigned the numbers 1 , 2 , 3 , 4 , 5 and 6. Moreover, their coefficients in an algebraic sum will be denoted by the letters a , J , c , d , c and /. Thus, the product of any two letters will be expressed by an algebraic simi, and below each coefficient will be written in order the numbers which are appropriate to the factors. Thus, jl = a^i + b^j + c^k + d^l + e^m +/un, ^ ^ .> while L ^ i ?; =a4» i + *4»y + C42 fe -f ^4, Z + 64, m +/4s^n. ^ In the case of a square, only one number need be written below the coefficient, « thus J(?=a^i + bsj + c^k + d^l + e^m -f/g n . The investigation simply consists in the determination of the values of the coefficients, corresponding to every variety of linear algebra ; and the resulting products can be arranged in a tabular form which may be called the multipli- cation-table of the algebra. Upon this table rests all the peculiarity of the calculus. In each of the algebras, it admits of many transformations, and much corresponding speculation. The basis will be denoted by i . 72. The distinguishing of the successive cases by the introduction of numbers will explain itself, and is an indispensable protection from omission of important steps in the discussion. Single Algebra. Since in a single algebra there is only one independent unit, it requires no distinguishing letter. It is also obvious that there can be no single algebra which is not associative and commutative. Single algebra has, however, two cases: [1] , when its irnit is idempotent ; [2], when it is nilpotent. [1]. The defining equation of this case is 24 Peircb : Linear AssocicUive Algebra. This algebra may be called (oi) and its multiplication table is * («i) i • [2], The defining equation of this case is This algebra may be called (bi) and its multiplication table is f Double Algebra. There are two cases of double algebra : [1], when it has an idempotent expression ; [2], when it is nilpotent. [1]. The defining equation of this case is By §§41 and 50, there are two cases : [1*], when the other unit belongs to the first group ; [12], when it is of the second group. The hypothesis that the other unit belongs to the third group is a virtual repetition of [12]. [1*]. The defining equations of this case are • ■ • . • It follows from §§ 67 and 69, that there is a double algebra derived from (bi) which may be called (%) , of which the multiplication table is J * This algebra may be represented hj i^ A : A ia the logic of relatives, dee Addenda. [C. S. P.] t This algebra takes the form i^zA'.B^ in the logic of relatives. [C. S. P.] I This algebra may be put in the form i = A : A + B : B ^ j=^ A : B . [C. S. P.] Peirce : lAn&ir Associative Algebra. 25 («?) * J ' I • • J • J ' "^ [12]. The defining equations of this case are, by § 41, whence, by § 46, f = 0. A double algebra is thus formed, which may be called (6g) , of which the multi- plication table is * t m • J [2]. The defining equation of this case is in which n is the least power of i which vanishes. There are two cases : [21], when n= 3; [2«], whenn=2. [21]. The defining equation of this case is and by § 60, e=o, ^=j' This gives a double algebra which may be called (cg), its multiplication table being f * This algebra may be put in the form i=i A : A , jzr. A : B , [C. S. P.] t In relative form, i=iA :B + B :C, j = A :C\ [C. S. P.] 26 Peirce: Linear Associative Algebra. • J [2*]. The defining equations of this case are « and it follows from §§64 and 65 that 7 ^- - so that there is no pure algebra in this ctise.*^ Triple Algebra. There are two cases : [1] , when there is. an idempotent basis ; [2], when the basis is nilpotent. [1]. The defining equation of this case is There are, by §§ 41, 50 and 51, three cases : [1^] , when j and k are both in the first group ; [12], when/ is in the first, and h in the second group ; [13], when/ is in the second, and h in the third group. The case of j being in the first, and h in the third group, is a virtual repetition of [12]. [1*]. The defining equations of this case .are ij -=.ji =^ , lie •=. la^=^ k . *Thi8 case takes the form %:=. A :B, jzzC.D. [C. S. P.] I Peirce : Linear Associative Algebra. 27 It follows from §§ 67 and 69, that the only algebra of this case may be derived from (cjj) ; it may be called (03), and its multiplication table is * («s) * Te • m J Tc • J h le • [12]. The defining equations of this case are ji = ij =:y , ih = h , Id = ; whence, by §§ 46 and 67, 3 ;% — m — ^y = 0, j1c = c^le, and there is no pure algebra in this case.f [13]. The defining equations of this case are whence, by § 46, V —Jy ^'* = '^ » J^ — ^^^ = ; P = T^ = Zy = , jk = «^3 i , and there is no pure algebra in this case. J [2]. The defining equation of this case is in which n is the lowest power of i that vanishes. There are three cases : [21], when n = 4; [2*], whenn=3; [23], when 71 = 2. ♦In relative form, t = ^:-4+J5:J5+C:C, jf = .4:B + B:C, k=iA:C. [C. S. P.] t That is to say, i and j by themselveB form the algebra aj , and t and k by themselves constitute the algebra &2 ) while the products of j and k vanish. Thus, the three letters are not indissolubly bound together into one algebra. In relative form, this case is, iz=.A:A+B :B^ j'=.AiBy k^zAiC, [C. 8. P.] t In relative form, I = ^ :A + D:D,j=LA:B,kz:iC:D, [C. S. P.] 28 Peirce : Linear Associative Algebra. [21^. The defining equation of this case is and by § 60 This gives a triple algebra which may be called (63), the multiplication table being * (ftg) i j Jc Jc 1 • J Je Jc [2*]. The defining equation of this case is and by §§ 59 and 64, observing the exception, i*=y, ik = 0, Id = bsij, T^ = bsj . There is no pure algebra when 631 vanishes,f and there are two cases : [2*1] , when J3 does not vanish ; [2^], when b^ vanishes. [2*1]. The defining equation of this case can, without loss of generality, be reduced to This gives a triple algebra which may be called (cg), the multiplication table being J * In relative form, 1= A :J5 + B:C+C:i),y = ^:C+B:D, kzzAiD. [C. S. P.] t This case takes the relative ioTm,i=A :B + B :C, j = A:C, A; = 6s^ :D + D:C. [C. S. P.] tinrelativeform, t = -4 :B+B:C, i = ^:C, k = a. A:B + A:D + D:C. [C. S. P.] Feirce : Linear Associatioe Algebra. 29 (Cs) i k k • J 1 «/ • J An interesting special example of this case is afforded by a = — 2 , when i{k-\-i) lk + i)i (k + if -J 0, so that k-\- i might be substituted for k, and in this form, the multiplication table of this algebra, which may be called (t/j), is * (Ca) i k k • J • J • -J *In relative form, i = A:B + B:C, j=A: C, & = — A :B+B:C+A :D+ D : C. When a = + 2 , the algebra equally takes the form (c',), on subettituting k — i for k . hand, provided a is neither 2 nor — 2 , the algebra may be put in the form (c',') i J k On the other To effect the transformation, we write a = — b — v and substitute t + bk and « + r A; for i and &, and lb — £ Jy ioTJ . Thus the algebra (Cs ) has two distinct and intransmutable species, (cg) and (c, ). [C. S. P.] ■ ■• ■ » a fa 30 Peirce : Linear Associative Algebra. [2^. The defining equation of this case is ;5? = o, and 6ji may be reduced to unity without loss of generality, giving a triple algebra which may be called {d^, the multiplication table being ( ^y = ^* = ^^' = hi^zij + Ps*^ • = A;yi = ciiJd , Cji = = Ay, t7* = M = h^ , Zi = h^i + 64iy + C41A , Ij = Zt* = {b^ + 631^41)/ » =: i^ =zc^ z=c^^ ilk:=J^=zb^\ ZAj = 63^ + 543^ + C43A; , P = a^i + 64/ + C4A; + <^4Z, =7^ = ajc + cjd + dj^ = a4Zi + bjj + ejk + (^4?. But A^ contains no term in Z , so that ^4 = . kl = iP = a4y , 634= ^4 > ^84 = , = Z^= 634A + C4&34J", J34 = a4 = = A^, P = bJ+cJc, kil=Ji?=: b^Jl = , = M = 631^' = 63*1/ = b^i z:zki= lj\ li = b^j + c^ , ZA; = Zt7 = . There are two cases : [231*], when C41 does not vanish ; [2312], when C41 vanishes. [231*]. The defining formula of this case is C4i4=0, and if p is determined by the equation C4ii>* + (C4 — &4l)i> = &4 , we have i{l + pi)='k+pj, {l + pif = (C4 + JPC41) {Tc +pj) , so that 1+ pi and k'\- pj may be substituted respectively for I and A; , which is the same as to make &4 = 0, 40 Peirce: Linear Associative Algebra. and there are two cases : [231'], when C4* does not vanish; [231*2], when c^ vanishes. [221*]. The defining equation of this case can be reduced to This gives a quadruple algebra which .may be called (Z^), its multiplication table being f (li) i j k I • J h ck h I [231*2], The defining equation of this case is * J. e. the neu> C4 , or what has been written c^ +pc^i . In all cases, when new letters of the alpha- bet of the algebra are substituted, the coefficients change with them. [C. S. P.] t When 6 = 0, c =: 1 , we have l(i-'l)'=.{i—l)l'=.(i\ so that by the substitution of t — I for t* , the algebra is broken up into two of the form (e,). When &=.0, 04=1) on substituting ii=i — Z, j\ =:j — cfc, ik, = (c — 1)**, /i = (c — l)i, we have if =ji , t,Zi=0, Zjtizz/Jzzfci; so that the algebra reduces to (r^) . When 6=1, c = , on puttingt'i = t — Z , ji zzj^-k , we have t? = ij/ = , /ti =ji , 2* = I; ; so that the algebra reduces to (^4) . When 6 = 1, e it: , on putting t'l = ^c"^ (^ — 1 ji=j+(c— l)fc, we have t{ =/*=*, t\/ = 0, Kj =ji ; so that the algebra reduces to (p^). When 6(6 — l){6c + 6 — l):t=0, on putting t, = (1 — 6) 6i— (1 - 6) Z, ji = (1 — 6)»(1- 6— 6c)*, *i =:6«(i-.6)(i — 6-.6c)i-6(l — 6)(l-6— c + c»6)*, J, =6(1 — 6)f— 6d, we get the multipU- cation table of (04). When6(6 — 1) + 0, 6c + 6 = l ; on putting t, =6(f— Z), j, =6M1 — 6)i— 6*cfc, ib|=6(l — 6 — e)*, 2|=6i— 2, we get the following multiplication table, which may replace that in the text : {h)i i h I J h I J . i i 1 1 » i 1 Inrelativeform,«= A:B+B: C+ A:D,j::^A:C, kzzAiE, i=A:B+I>:E, [a a P.] J Peirce : Linear Associative Algebra. There are two cases : [231*21], when 641 does not vanish; [231*2*], when 641 vanishes. [231*21]. The defining formula of this case is There are two cases : [231*21*], when % + 1 does not vanish ; [231*212], when c^+l vanishes. [231*21*]. The defining formula of this case is 41 so that ^41+1 + 80 2 f^ni + cJ _ f>id + ^41041^ C41 + I . C41+I ' bjli + C41I J _ ftllfe / b^ii + CJ \^_ ^4 1 + ^ 41 ^41 J + CgJ^ C4I + 1 C41 + 1 ' \ C41 + 1 / • C41 + i • C41 + 1 that the substitution of ^^^^^^ b^l + M ^^^ _bjc ^^^ ^^^j^^j f^^ 0 i j le I Te I • J Te • [23212*]. The defining equation of this case is Z» = 0. This gives a quadruple algebra which may be called {q^, its multiplication table being f (94) * j Te I Tc I • h *InrelatiTeform,t = ^:5 + B:D+C:J^,i=A:2), h^ A\E , l::zA:C-\-C:D. [C. S. P.] t In relative form, i^A: C+ C : 2) , j = ^ : D , jfc = B : D , i = B : C. [C. S. P.] Pbircb: Linear Associative Algebra. 45 [232*]. The defining equation of this case is and we have Id = h^j + 634^ + d^ Ih — b4sj + cjc + d4sl V = hj + cJc + dj. so that there can be no pure algebra in this case if J41 vanishes,* and it may be assumed without loss of generality that There are two cases : [232*1], when d^ does not vanish ; [232^], when rf, vanishes. [232*1], The defining equation of this case can be reduced to which gives = Ar» = ;fcZ=7A; = H=?, and there is no pure algebra in this case.f [232^]. The defining equation of this case is e^ = 0, which gives = ^•' = c^ = Cg, ^ = h^jy = It I '-^ CgiAr + d'^Jvl ^ d^ = 63^31 > = ?^^ = C^ + ^48^:? =(^43 = 63C48 . There are two cases : [232^1], when 63 does not vanish; [232*], when 63 vanishes. [232^1], The defining equation of this case can be reduced to which gives = Cg4 = C43, Jd=:b^j\ lk = b^j, Jc{i-b^:f^=o, *ln tliis case^y, k and 2, might form any one of the algebras (&,), (c,), (d,) or (e,). [C. S. P.] t The case is impossible because A» = and kH = J . [0. S. P.] 46 Peiecb: Linear Associative Algebra. so that I — b^^csxi be substituted for I without loss of generality, which is the * same as to assume kl=Oi and this gives = Z^ = dj^ zzzd^zzz cjk = 04643 = ?A; = C4 , so that there is no pure algebra in this case.* [232^]. The defining equation of this case is which gives ^ = 0, = lj = Pi = dj = d^ , = A;? = c^cl = C34 , hi 0=iPk=: c^lk = C43 , Ik and there can be no pure algebra if c^ vanishes, so that it may be assumed, without loss of generality, that P = k, which gives = P = lk = kL This gives a quadruple algebra which may be called (r^), its multiplication table being f (n) i j k I k I • J • J k [24]. The defining equations of this case are * Substituting i — Z for i , this case is, i = B :Z), i=^ :D, k = A:C+C:D, Izi^AiB. [0. S. P.) U = A:B+B:D+C:D,jz:iA:D,k=:A:E,l=zA:C+C:E. [C. S. P.] Peirce : Linear AssocicUive Algebra. 47 and it may be assumed, from §§ 63 and 65, that ij=zk=^ — ji , t7 = Zi = , which give = ii = ki =^jk = Jcf =ikl=zJJcj = ijl = bje = b^ =jH — — a^h + d^Jl=d^ = a,| , jl= — Jj = c^k, so that there is no pure algebra in this case.* Quintuple Algebra. There are two cases : [1], when there is an idempotent basis ; [2], when the algebra is nilpotent. [1]. The defining equation of this case is t* = I. There are eleven cases : [1^], wheny, hj I and m are all in the first group ; [12], when/, 7c and I are in the first, and m in the second group ; [1 3] , when j and Jc are in the first, and I and m in the second group ; [14], wheny and k are in the first, I in the second, and m in the third group ; [15], wheny is in the first, and k, I and m in the second group ; [16], wheny is in the first, k and I in the second, and m in the third group ; [17], wheny is in the first, k in the second, I in the third, and m in the fourth group; [18], wheny, k and I are in the second, and m in the third group ; [19], wheny and k are in the second, and I and m in the third group ; [10^], wheny and k are in the second, I in the third, and m in the fourth group ; [1 1^] , when y is in the second, k in the third, and I and m in the fourth group. [1*]. The defining equations of this case are ij^=^ji =y, ik =:ki=ik, il =U = l, im z=: mi^= m. The algebras deduced by §69 from algebras (ij to {r^ may be named (a^) to {j\), and their multiplication tables are respectively *«• — i = — A:C+B:E,j=zA:B+C:E+eD:E, k=iA:E, l = — A:D + cB:E. [C. S. P.] 48 Peircg : lAnear Associative Algebra. (rts) i k I m k I m • • J k I m • J k I m k I m I m m 0. (Cb) i t k I m {€,) i t k I k I m • • J k I m • J k I k I I m I k I m m • • J k I m • k I k I m k I or ih) i I m k I m • J k I m • J k I k I I m I I W i k I m («b) * k I m k I m • • J k I m • J k I k I m ak I him • • k I 9\ m • J I k I m k Peibcb: Linear ABsodaHve Algebra. 49 Ud i k I m J h I m m • J k I m • k I h I m oZ ih) » t Jc I m k I m • • k I t I m • k k • I in I k (ffs) i k I m W » i k I m k I m m % • k I m • k k a I n I l-\-ak k I m m m J k I m • k k I m I U.) i k I m k I m • • J k I 1 • J k k I m k I 50 Peirce : Linear Associative Algebra. [12]. The defining equations of this case are ij-=zjiz=LJ^ iJc=iJci = kj ilz=ili=zl^ im = nij m = 0, which give, by § 46 , = my = mk =iml-=- m^, and if A is any expression belonging to the first group, but not involving i , we have the form Am = am , and by § 67, A is nilpotent, so that there is some power n which gives = J.** = A^m = aA'^~^m = a'^m = a = Am , =y?n = Jem = Im ; and there is no pure algebra in this case.* [13]. The defining equations of this case are ij =ji =y , ik = ki=^k, il = 1, im =^ m, Zi = mi = , which give, by § 46, = Z)" = W; = Z' = Zm = m; = mk =ml=^ m^ j and it may be assumed from (ofg), by § 69, that f = k, f = 0. It may also be assumed that jl=zm, whence f kl =jm = . « We thus obtain a quintuple algebra which may be called {k^)y its multiplication table being this : J ♦In fact i and w, by themselves, form the algebra (62), while i, y, *, Z, by themselves form one of the algebras (04), (64), (C4), (d^), the products of m with 7 , k and I vanishing. [C. S. P.] tThisis proved as foUows : = jH=j^m=:d2jl + e2rjm = d2ie2j+ (d26 + ^2%)^- ThusdaftCza = Oand d26 + e2\ = 0;oTd2i=0, 625 = 0, fni=:kl^O. [C. S. P.] Xi = A:A + B:B+C:C, j=A:B+B:C, k = A:C, l = B:D, m=LA:D. [C. S. P.] Peibce : lAnear Associative Algebra. 51 W i • 3 Te I m • ■ • 3 Te I m • J • 3 h m 1c Tc I m [14]. The defining equations of this case are ij :=.ji =y, ik=:Jci=^kj il= l^ mi=^ m, li = im = , which give, by § 46, It may be assumed from § 69 and (a^) that f = k, y' = o, whence 0=jl = M=z mj = mJc =JJm = a^J + 645^- = a^^ = 645 , Im = c^jjc , and there is no pure algebra in this case.* ^ . 4 [15]. The defining equations of this case are iJ = Ji zzzj , %k = A; , il=.l^ im := m ^ Jci=^li = 7ni=^ j which give, by §§ 46 and 67, =y* =:kjz=z^ =1 M = km =zlj =:lk = P = lm =: mj = mk =zml =^ m*. It may be assumed that jk = Z, Jm = ,f whence, j>7=0, and there is no pure algebra in this case. J * i= A : A-i- B :B + C : C , j= A :B + B:C, k = A:C, 1 = A:D, m — cD :C. [C. S. P.] t We cannot suppose jk=.k ^ because j'k=.0. We may, therefore, put I torjk . Then j7 =: . Then, '='p7n =• C2 5^2 5^ + (^2 5^2 5 + C2 r, ) ^ + 62^5m . It follows that j/u = dj 5^ , and substituting m — d^sk for m , we h&\ejm = . The algebra thus separates into (62) and (e^). [C. S. P.] Xi = A:A + B:B,j=zA:B, k=iB:C, l=A:C, m = A:D, [C. S. P.] 62 Peibce: Linear Associative Algebra. [16]. The defining equations of this case are ij '=•3% =y, iAj = A;, il=:l, wii = m , A:i = ?i = tm = , which give, by §§ 46 and 67, =f =jm = Jg = J^ = kl = lj = lk = P = Tnj=7nk = ml = m^, km = a^i + b^j, Im = a^i + h^j, and it may be assumed that and d^ cannot vanish in the case of a pure algebra,* so that it is no loss of generality to assume jk=l, which gives There are two cases : [161], when a^^ does not vanish ; [162], when a^ vanishes. [161]. The defining equation of this case can be reduced to «85 = 1» which gives hn=j, Jcm = i + b^j\ and i + b^' can be substituted for i , and this gives a quintuple algebra which may be called (?^), of which the multiplication table is ih) i Tc % l6 I m • A; V 4 I 3 m *But = m*=Awn&=(a,5t+655j)fc = a35fc+d2a&86'' Hence 0,5 = Oand either d,, or bj^zzO, and in either case there is no pure algebra. The two algebras (Z5) and (1715) are incorrect, as may be seen by comparing Ic . wk with hn , k. [C. S. P.] Peirce: Linear Associative Algebra, 53 [162]. The defining equation of this case is which gives 085 = 0* km = 635^, Z?n = ; and 635 cannot vanish in the case of a pure algebra, so that it is no loss of generality to assume km =y. This gives a quintuple algebra which may be called (mj), of which the multipli- cation table is (Wj) i j k I m k I m • • J k I m J I ^^ m [17]. The defining equations of this case are ij =ji ==y , ik = k, U^=:lj A:i = t7 = im =: mi = , which give, by §§ 46 and 67, =y* ^=jk =jl=zjfn z=kf = l?:=Ij=:P^=:lm=zmj=z mk, kl:=a^i + b^Jj km=^c^k, lk'=^e^m, ml^=d^l, m^=:e^m, 0=jkl = a^j = a^, IE = 634^' = = 643771? = ^43(^54 , kik = huJk-= = ejcm = e^^ , Ikm = c^Ik = e437n* = Cs^e^^m = = e^^ , kml = d^kl = C35H , {d^ — C35) 634 = , km^ = ejcm = c^km , (^5 — C35) c^=0, mV = e^ml = d^ml , (^5 — d^d^ = . 54 Peibcb : Linear Associative Algebra. There are two cases : [171], when eB = l;* [172], wheiiC5 = 0. [171]. The defining equation of this case is which gives m^ = m , U ^— ^^ —^ LrC • There can be no pure algebra if either of. the quantities h^ , c^ or d^ vanish, and there is no loss of generality in assuming ld'='j^ hm'=^h, ml=zl. This gives a quintuple algebra which may be called (wg), its multiplication table being (715) i j k I m • • • le • • J h • Te I I m I m [172], The defining equation of this case is which gives = C35 = d^ = hm = ml ; * But on examination of the assumptions already made, it will be seen that if e^ is not zero, and consequently 643 =0 , the algebra breaks up into two. Accordingly, the algebra (n^) is impure, for i, J , /:and {, alone, form the algebra (/«), while m , Z, A;, J, alone, form the algebra (^4), and tm = mt = . [C. S. P.] Peircb : Linear Associative Algebra, 55 and there can be no pure algebra if either 634 or c^ vanishes, and it may be assumed that This gives a quintuple algebra which may be called (05), its multiplication table being as follows : * (05) i j k I m • • • t k • J m J k • J I I m m [18]. The defining equations of this case are ij =y , iJc = kj il=^lj mi = mj ji z=:ki = li = im=:Oj which give, by § 46 , =y* =^jk=^Jl = kf =ij{? := Jd =:lf =zIk=iP =: mj = mk =zml=:m? But if A is any expression of the second group. which gives Am == ai ; = AmJ =aj =ia=:^ Am =^Jm = km = Im , and there is no pure algebra in this case. [19]. The defining equations of this case are ij=j\ ik = k, li = l, mi=-m^ il=zim=ji=iki=^0 , which give, by § 46. =y* =y^ =zkj=^Ji? = Jj = Ik=:P = Im=: mj = mk = ml=z ?n*. «•• — % = B:B+D:D+F:F,jzzD:F, k = B:C+D:E, l = A:B + E:F, mzzA \C, [C. S. P.] 56 Peirce : Linear Associative Algebra. But if A is an expression of the second group and B one of the third, AB^= ai, which gives = ABj =:aj =:a=: AB =jl =^Jm =:kl=:lmf and there is no pure algebra in this case. [10']. The defining equations of this case are iy=y, iJe = k, li=:l, ji=iJci=:il=::im=:mi=^Oy which give, by § 46, and it is obvious that we may assume y?=o. We have, then, jm = b^J + c^, kl = a^i, km = b^j+cjc, Ij :=: e^m , Ik = c^^m , ml = d^ , ttj? = e^m , There are two cases : [lO'l], when a^ does not vanish ; [10'2], when a^ vanishes. [lO'l], The defining equation of this case can be reduced to kl=:i^ which gives c«5 = , jm=- b^\ There are two cases : [lO'P], whenc5 = l; [10'12], when e^ vanishes. [l(yi*]. The defining equation of this case is and we assume jm=^j\ ml=^l, km:=ky because otherwise this case would coincide with a subsequent one. We get, then, =jlj = e^^Jm = e^=lj\ =jlk=e^m =64^ = Ik, which virtually brings this case under [10'2].* * This does not seem clear. But t = t' =: JdJd = , which is absurd. [C. S. P.] Peibcb : Lirvear AssodcUive Algebra. 57 [10'12]. The defining equation of this case is m» = 0, which gives =ym* = 6315^^ = &w =y^ » = mH = d^ml =a^=^ ml, = km^ = c^km = C35 , fcm = ftggy , IJcl =zliz=:l=z c^ml = , which is impossible, and this case disappears. [I(y2]. The defining equation of this case is There are two cases : [10'21], whene5 = l; [10'2*], when e^ vanishes. [10'21]. The defining equation of this case is m^ = 7W, and if we would not virtually proceed to a subsequent case, we must assume jm =:y, km =z k, ml=:l, and there is no loss of generality in assuming SO that there is no pure algebra in this case.f [10'2*]. The defining equation of this case is which gives = mH = d^ml ^=1 d'^=^ml ; and we may assume ^86 = 0, which gives =jm^ = \Jm = 62B =y^ , = km^ = c^m = c^, km = h^j , = 643m* = Ikm = h^e^m = h^^st ; % * In this case, the algebra at once separatee into an algebra between j, k , { and m , and three double algebras between t and j , t and ft, and t and I , respectively. [0. S. P.] t In fact, = Vdk = e^^yn = 64, = 2A; . So that the algebra falls into six parts of the form (&,). [C. S. P.] tTheauthor omits to notice that = ^2^ = 64 ,Ami = 643635. Thus, either Ann = or (; = Zft=0. The algebra (pt) inyolves an inconsistency in regard to ftZft. [C. S. P.] 58 Peibce : Linear AssocicUive Algebra. and we have without loss of generality Z/z=0, km=:^jy lk=:m. This gives a quintuple algebra which may be called (p^), of which the multipli cation table is (jPs) i j k I m • m • J h m J k • J I I m m • [11']. The defining equations of this case are ij =y, Jci = hf ji = ilc =:il = im = li = mi = ; which give, by § 46, =y* zz: A:* 1= jfc? = hm =ilj ^mj ^ jk = a^i , jl = h^j\ jm = h^J, 1^ = ^^32? + %w , Ik = c^, mk = c^. There are two cases : [ll'l], when I is the idempotent base of the fourth group ; [11'2], when the fourth group is nilpotent. [ll'l]. The defining equation of this case is f = l. There are two cases : [ll'l*], when m is in the second subsidiary group of the fourth group ; [11'12], when m is in the fourth subsidiary group of the fourth group. [ll'l*]. The defining equations of this case are ?7n = m , w/ = ; Peiece: Linear Associative Algebra. 59 which give = m* =:jrf? = h^^^jm = 635 =ywi , = mVc = 053?^^ = C53 = inA: ; and Ogs cannot vanish in a pure algebra, so that we may assume jh = i , which gives jl=jP = h^jl, b}^ = b^=l, lh = l^h = cjk, c^ = c^^z=z\z=z d^, jl=j\ lk = k, kfl=l =kf\ and there is no pure algebra in this case.* [11'12]. The defining equations of this case are Zm = mZ = , which give = jIm = b^jm = 624*85^ = ^84^26 J = Imk = c^^lk = c^j^ = ^43053 , kjl = dy^l = 624^' = b^d^l + b^e^m , Ikf = rf^Z = c^^J = 043^32? + 043^32771 , kjm = e^w? = b^^Jcf = b^^l -f b^^e^m , rrilij = e^m^ = e^J{j = c^l + c^^m , There are two cases : [11'121], when m is idempotent ; [11'12*], when m is nilpotent. [11'121]. The defining equation of this case is m* = 7W, which gives and it may be assumed that 5,4 = 0. But if the algebra is then regarded as having I for its idempotent basis, it is evident from § 50 that the bonds required for a pure algebra are wanting, so that there is no pure algebra in this caae.J * In fact, t , i, k^ I form the algebra (^4), and 2, m , the algebra (^z)* [C S. P.] t The last equation holds by i 68. [C. S. P.] t Namely, d, 2 = , and either 63 3 = 1 , when I forms the algebra (a^) , and i, j\k^m the algebra (94) 1 o^ ®^ ^8 2 = ) when by [18] of triple algebra a, , = , and j and k each forms the algebra (62 ) with each of the letters i , 2 , m . [C. S. P.] 60 Peibce: Li/near Associative Algebrai [11' 12*]. The defining equation of this case is which gives ?n' = 0, =ym' = h^m = hij = h^ =^jm , = nfh = c^mk = c^c = Cgg = wA; , l=6j^ = C43, jl=jy lk = ky = 63,, and there is no piu^e algebra in this case.* ^ [T-1'2]. The defining equation of this case is ^=0, in which n is 2 or 3. We must then have = lm = ml = m*f which give =:jP = b^'P =: bi^jl = b^ ^=jl =-jm =:lk=: mh , = Ttgk = a^ = a^ =^jk , and there is no pure algebra in this case, f [2]. The defining equation of this case is , There are five cases : [21], whenn = 6 [2*], when n = 5 [23], whenn = 4 [24], when w = 3 [25] , when w = 2 . [21]. The defining equation of this case is i^=y, ^ = k, i^ = l, i^ and by § 60, = tn . This gives a quintuple algebra which may be called (55), its multiplication table being * Here, m forms the algebra (&i) , and the other letters form (^^4) . [C. S. P.] t Namely, if n = 2 , j , 2 , A; , form the algebra (d,) (second form), t , j, and i^k, the algebra (bz), and m the algebra (c^). But if n=: 8 , y. k, I and m form an algebra transformable into (J4) or (Af4), while t , J, and t , k form, each pair, the algebra (62 )• [^' 3* P*] i Peirge : lAnear Assodaiive Algebra. 61 (?5) * k I m k I m • J k I m k I m 1 I m m [2*]. The defining equation of this case is and by § 59, There are then by § 64 two quintuple algebras which may be called {r^) and (^5), their multiplication tables being (n) t k I m («6) * k I m k I m J k i k I I 0» I I k I m • J k I 1 k I I • I [23]. The defining equation of this case is and by § 59, 62 Peirce: Linear Associative Algebra. and it may be assumed, from the principle of § 63, that which gives =y? = Id = ill = iP = ilm li =z c^k + ^41? + e^m , P = C4k + dj + 647^ , Im = C45AJ + ^45? + 645171 . There are two cases : [231], when im=^l) [232], whenm = 0. [231]. The defining equation of this case is %m = Z, whence =-jm = km =^jmi =-jml =:jw? = 641 = 64 = 645 , li^ = djij = Zi* =: d^li^ = dlJf? = d^i z=d4x = lj = lk, P =z d^ , = Z* = dj? = CZ4 , Zi = C41& , P = c^k, lm=- cjc + ^ijZ , imi=::U=^c^ik, ini:= c^j + c^ik + d^Jy mj ^=c^{l + d^i)k, mfc = 0, iinZ :=P zizcJCy 7nl=: cj + C54& + CZ54Z , irn^ =:lm = cjc + CZ45Z, w? = 645^ + c^k + cZgZ + ^45771 , =: m^ =: ^45 , lim =zpz=: c^Jmi = =: mli =: CZ54C41& = CZ54C41 , =: miia = cZ54Zm =: CZ54C45 , =: m*Z = cZ54mZ = (£54.* There are two cases : [231*], when C41 does not vanish ; [2312], when c^ vanishes. [231*]. The defining equation of this case is reducible to Zi=: A;. There are two cases : [231^], when C45 does not vanish ; [231*2], when c^ vanishes. [231^]. The defining equation of this case can be reduced to Zrn = ifc, which gives Tffi = & + d^^ + d^^ = A; + cigAj , (^ = cZgi + cZg*! , m' = A; + d^^ + d^d^^ = cZgJfc, cZ|i = — 1 ; * To these equations are to be added the f oUowing, which is taken for granted below : mX = mim : e^^dsift. [C. a P.] \ Peirge : Idnear Associative Algebra. and if x is one of the imaginary cube roots of — 1 , there are two cases : [231*], when(i5i = r; [23P2], when 0^51 = — 1. [231*]. The defining equation of this case is 63 which gives i{m (m {m cj)i = ly l{m - = 0, (m =y+[c.- ^61 = ?, — CriO l=Xh, -C5i(l + r)]* + (2r — 1)?; so that the substitution of w — CgjZ for m is the same as to make Cr, = . There are two cases : '61 [231*], when c^ does not vanish ; [231*2], when c^ vanishes. [231*^]. The defining equation of this case can be reduced to Cb= 1. There is then a quintuple algebra which may be called (^5), its multiplication table being * * The author has overlooked the circumstance that (t^) and (u,) are forms of the same algebra. If in (fj) weput»i=:»— r*Ji ji=j—2v^ky &!=*,;, = — t*Aj + Z, mi = — tV+w, we get (u^). The struc- ture of this algebra may be shown by putting t'x = t* , A = r V ^ Aj^ =: — A? , ij = r V— ' ti , wii == tt — m , when we have this multiplication table (where the subscripts are dropped): (t*») t J m I m • k Xk xl k k I k X'k In relative form, %=iA :B + A:C+ B:E+ C: D + E:G, J= A:D + A: E+B :G, k = A:G, IzzvAiE+CiOy m = x^A:B+A:F+xC:E+D:Q"F:G. [C. S. P.] 64 Peibce : Linmr Associative Algebra. {U) Te I m Te I • k I k k k j + rl {l+x)k Vk {2v-l)l m [231^2]. The defining equation of this case is There is then a quintuple algebra which may be called (u^), its multiplication table being (M5) * j k I m k I • J k I k k k J+vl {l+V)k r* J + {2t-l)l • m [231*2]. The defining equation of this case is ^1 = — 1 , which gives dg = , i{m — C51Z) = 7 , l{m — c^il) = k , {m — c^il)i =zj — /, {m — C5iZ)Z = — k, {m — CgiZ), =j + c^] so that the substitution ofm — c^il * for m is the same as to make C51 = . * The original text has m — Cnk throughout these equations, but it is plain that m^Cul is meant. [C. S. P.] Pbibcb : Linear Associative Algebra. 65 There are two cases : [23P21], when c^ does not vanish; [231^2*], when Cg vanishes. [231^21]. The defining equation of this case can be reduced to There is a quintuple algebra which may be called (vj), its multiplication table being * (^5) i J h I m I Jc I • J k I k k k J -I k j + k m [231^2*]. The defining equation of this case is C5=0. This gives a quintuple algebra which may be called (w^), its multiplication table being * 11 2 * The algebra (Vg) reduces to (w^) on eubstitating I'l = t + ^j + s ' 1 ii =i+ Af,Afi=A?,Zi=gfc + Z, Wi = gj + 5- Z + m . To exhibit the structure of this algebra, we may put p and p' for imaginary cube roots of 1, and substitute in (105)11 =i + p'm, Ji = (1 — p)j+fc+^---8i» fci=8A, ii = (l — p')i+*~ ^ — 8Z , mizzi+pni. Then, dropping the subscripts, we have this multiplication table. m I m J k I m • f k • J • J k k I k m I k 1 In relative form, i = p'A:B + p'C:F+9pD : E, j=.SpA : C + Bp'D : F, fc=8il:D, l^Sp'AiE +8pB:F. m = pA:D + Bp'B:C+pE:F. [O. S. P.] 66 Peibce : Linear AssodcUive Algebra. (w) i Jc I m k I m • J k I k k k j-l k j+k [231*2], The defining equation of this case is fon = 0, which gives mZ=0, m* = 05^ + ^5/, mH:=d^=^ [l+d^i)k, d^=l + df^, and Cgi may be made to vanish without loss of generality. There are three cases : [231*21], when neither d^i nor c^i + 1 vanishes ; [231*2*], when ^51 + 1 vanishes ; [231*23], when d^i vanishes. [231*21]. The defining formulae of this case are There are two cases : d5itO» ^i+-l [231*21*], when Cg does not vanish ; [231*212], when Cj vanishes. [231*21*]. The defining equation of this case can always be reduced to C5= 1. This gives a quintuple algebra which may be called (xs), its multiplication table being * *lti relative form, 1=1 A:B + A:E+B:D + D:F, j= A: D-hB :F. k=zA:F, l=:A:D. m=(l + a)A:B + A:C+A:E+B:D+C:D + D:F-{-E:F. [C. S. P.] Pbibcb : Linear Associative Algebra, 67 (^5) h I m h I 1 1 / Ic h j+al (1 + a)k (1 + a)l m [231*212]. The defining equation of this case is This gives a quintuple algebra which may be called (y^), its multiplication t3.ble being * h I m k I m J k I k Q k * (1 + a)l j-\-al (1 + a)k' [231*2*]. The defining equation of this case is ^1 = — 1 , which gives mi=:j — /, m/=0, m^ = c^k. There are two cases: [23P2*1], when c^ does not vanish ; [231*2^], when Cg vanishes. * The relatiye form is the same as that of (079) ; omitting from m the terms A : E and E : F. [0. S. P.] 68 Peircb: Linear Associative Algebra. [231*2*1]. The defining equation of this case can be reduced to 771* = h. This gives a quintuple algebra which may be called (zj), its multiplication table being * k I m k I m m J h 1 I Tc h j-l I h 1 [231*2']. The defining equation of this case is 7n* = 0. This gives a quintuple algebra which may be called (oag), its multiplication table being f (adg) i j k I m • • Jc .0 I • J h k I h m j-l *In relative form, i = ^:BH-B:C+C:JD, i=^:C+B:D, k=iA:D, l — A:C, mzuBiC + A:E+E:D. [C. S. K] t In relative form, the same as (z^)^ except that mz=:B:C, [C. S. P.] Peirce : Linear AssodcUive Algebra, 69 [231*23]. The defining equation of this case is mi=zj\ which gives = (/ — j)i := (m — i)i ; so that, by the substitution of 1 — j for / and m — i for m, this case would virtually be reduced to [232]. [2312]. The defining equation of this case is which gives mj = , mim = ml = d^ilm , d^^ = , ^54 = ^1045 , mH = d^ifnl = C45A; , C45 = d^iC^ , m^ = d^lTn = d'^ml , d^{c^ — C45) = . There are two cases : [23121], when d^ does not vanish ; [2312*], when d^ vanishes. [23121]. The defining equation of this case can be reduced to which gives % ^^ ^64 5 and it may be assumed without loss of generality that 05 = 0.* There are two cases : [23121*], when C45 does not vanish ; [231212], when C45 vanishes. [23121*]. The defining equation of this case can be reduced to Zm = wZ = k, which gives ^1 = 1. There are two cases : [23121®], when c^i does not vanish ; [23121*2], when % vanishes. [23121®]. The defining equation of this case can be reduced to C51 = 1 . * Namely, by putting Z, = c^k -|- Z , mj = m — cj, [C. 8. P.] 70 Pbircb : 1/inear Associative Algebra. This gives a quintuple algebra which may be called (aftg), its multiplication table being * (065) i j k I m • • ■ k 1 1 I • k 1 Tt I k m k + l k 9 + 1 [23121*2]. The defining equation of this case is Cbi = 0. This gives a quintuple algebra which may be called (ocj), its multiplication table being f *The structure of this algebra is best seen on making the foUowing substitutions: Let l^i and 1^, represent the two roots of the equation x^=ix+l. That is, ^ j = ^ (1 + ^ 6) and ^^ = o" ^^ ""'^ ^^ ' Then substitute t,== ^T*(f+*im), ii =kM (2+*i)i+l^i*+ U + 8^,)Zf, AjiZiJa:, Z^ + ^a* + ( 1 + Sia) Z f , mi = 5a Y (t + JaWi). Then, we have the multiplication table : i j k I m J k I m • k k 0. k I k Ml* 9 In relative form, t=4:B + B:C+C:i>+iji^:(?+ if :2),i=^ :C+B:/>, A;=:A:Z>, l:=iA\F + E:D, m = A:E+E:F+F:D+^i2A:H+G:D, [C. S. P.] t On making the same substitutions for i and m as in the last note, this algebra falls apart into two algebras of the form (&,). [C. S. P.] \ Peibce : Zmear Associative Algebra, 71 {aot) i k I m k I • J k ■ I k , 1 1 • k I ' 1 k J+l m [231212]. The defining equation of this case is 7W? = Zm = . There are two cases : [2312121], when c^i does not vanish ; [231212*], when C51 vanishes. [2312121]. The defining equation of this case can be reduced to C51 = 1 . This gives a quintuple algebra which may be called (ad^), its multiplication table being* • J k I • k • • k I m k-\-dl I * In relative form, i = ^ : B+ B : C'\-C \ D+ E . F+aF:Q . jzn A:C+ B\D + oE'.O , k — A.D, l=zE:Q, m = A:C+E:F+F:0. [C. S. P.] 72 Pbiecb: Lmear Associative Algebra. [231212*]. The defining equation of this case is Cbi = 0. This gives a quintuple algebra which may be called (065), its multiplication table being * (0^5) i j Je I m m % • k I • Te le I m (d I [2312*]. The defining equation of this case is d^ = (i. There are two cases : [2312*1], when C45 does not vanish; [2312^], when % vanishes. [2312*1], The defining equation of this case can be reduced to ?m = ^, which gives There are two cases : [2312*1*J, whend5i = l; [2312*12], when d5i= — l. [2312*1*]. The defining equation of this case is ^1 = 1, which gives C54 = 1 , ml = h . ^ * In relative form, the same as (adj,) except that m = ^ : F+ F:0, [C. S. P.] I Peibce : Linear Associative Algebra. 73 There are two cases : [2312*1'], when c^i does not vanish ; [2312'1»2], when c,! vanishes. [2312*1*]. The defining equation of this case can be reduced to Cji = 1 . This gives a quintuple algebra which may be called (o/j), its multiplication table being* Tc I m Tc I m • Ic I h A; *+Z A; j-\-ck * To show the construction of this algebra, we may substitute I'l = t + m , j^ = !^*+ (a + 1) & + 22 , fti=4^, li'=.2j+(a—\)h—%l^ mi=:i— m. This gives the following multiplication table : h I 3 m k I J k 4 ^ k k 4 '^ k I This algebra thus strongly resembles (065). In relative form, i=iA :B + B : C+C:D+A : O ^^-^0:D,j = A:C+B:D-^!-^A:D, k = A:D,lzzA : F+E iD--^^ A:D, mznAiE + E:F'¥F:D+A:G'-'^^G:D. [C. S. P.] 74 Peibge : Linear Associaiive Algebra. [2312*1*2] . The defining equation of this case is There are two cases : cu = 0. [2312*1*21], when Cj does not vanish; [2312*1*2*], when c^ vanishes. [2312*1*21]. The defining equation of this case can be reduced to C5= 1. This gives a quintuple algebra which may be called (ag^), its multiplication table being * {ag^) i j k I m * • J h I • J h Jc I Ic m I h j + Tc [2312*1*2*]. The defining equation of this case is This gives a quintuple algebra which may be called (oAg), its multiplication table being f * On substituting tj = i + iy + wi , mj = t + ^ /— m , this algebra falls apart into two of the form (5,). [cap.] t On substituting ii = t+ m , wij =i — wi, J^ =y+Z , Zj zzj — l , this algebra falls apart into two of the form (ft,). [C. S. P.] Peibce : Linear Associative Algebra. 76 (dij) ♦ "k I m l6 I fH • J k I Je k I k • J [2312*12]. The defining equation of this case is ^1 ='— 1 , C54 = — 1 , ml = — k . which gives There are two cases : [2312*121], when c^i does not vanish; [2312*12*], when C51 vanishes. • [2312*121]. The defining equation of this case can be reduced to C51 = 1 . This gives a quintuple algebra which may be called (atg), its multiplication table being * (aig) i j h I m m J k I • J k k • I k m k—l k j+ck - ♦In relative fonn, %=:A:C—B:F+C:E + D:0 + E:G, j=iA:E+C:G, k=:A:G, LzzA.F-^B.Q, m-=zA\B+A\D'^B\E+C:F'\'aD\Q+F:Q, [O. S. P.] 76 Feirce : Linear Associative Algebra. [2312*12*]. The defining equation of this case is tni ^ — Z. There are two cases : [2312*12*1], when Cj does not vanish ; [2312*12^], when c^ vanishes. [2312*12*1]. The defining equation of this case can be reduced to Cb= 1. This gives' a quintuple algebra which may be called (a/g), its multiplication table being* (o/s) i j k I m • t • J k I • J h k I k m — I — * j + k [2312*12']. The defining equation of this case is m^=j. This gives a quintuple algebra which may be called (oAtj), its multiplication table being f * In relative form, i = A: C+C:E+ E:0 — B :F, j=A:E+C:G, k m = A:B + B:E+C:F+F:0 + A:D+D:0, [C. S. P.] t In relative form ,i = A:C+C:D+D:F—B:E,j = A:D+C:F,k m=iA:B + B:D+C:E+E:F. [C. S. P.] = A:(?, l = A:F-'B:Q, =:A:F, l=A:E—B:F, Peikce : Linear Associative Algebra. 77 {ak^ i k I m k I • J k I k k — I — k 9 J m [2312^. The defining equations of this case are mZ = ?m =: , m^ =^ c^k. There are two cases : [2312^1], when d^i is not unity; [2312*], when d^i is unity. [2312^1]. The defining equation of this case is which gives i[{l — d^i)m — c^J'] = {l — d^^)l—c^Jc, i[{l — d^i)l—c^Jc'] = 0, [(1 — rf^i) l—c^iJc]i = , [(1 — rfn) m—c^J]i = d^^ [{l—d^)l—Ciik'] , [{l — d^i)l— c^ik] [(1 — dgi) m — Cji/ ] = , [(1 — dpi) m — C5J] [(1 — dji) I— c^Jc] = , [(1 — ^1) ^ — (Hijy = (1 — ^1) V ; so that the substitution of (1 — d^i)m — c^ij for m, and of (1 — d^i)l — c^ik for I, is the same as to make Cbi = . There are now two cases : [2312'P], when c^ does not vanish ; [2312^12], when c^ vanishes. [2312'P]. The defining equation of this case can be reduced to w* =: A. 78 Peirge: Linear Associative Algebra. This gives a quintuple algebra which may be called (aZj), its multiplication table being* (al^ i j k I m • • J k I • k Ic I m dl k I [2312^12]. The defining equation of this case is This gives a quintuple algebra which may be called (a^Wj), its multiplication table being (amj) i j k I m • • J k I • J k k I m dl *Jn Tel&Uve form, i = A:B']-B:C+C:D + dE:F, j:=-A:C+B:D, k = A:D, l=iA:F, m:=:^A:E+B:F+E:D. [C. S. PJ Peibce : Linear Associative Algebra. [2312*]. The defining equation of this case is 79 There are two cases : ^51= 1. [2312*1], when c^i does not vanish ; [2312^^], when Cgi vanishes. [2312*1]. The defining equation of this case is easily reduced to There are two cases : C5i= 1. [2312*1*], when Cj does not vanish ; [2312*12], when Cj vanishes. [2312*1*]. The defining equation of this case is easily reduced to m* = A;. This gives a quintuple algebra which may be called {an^}, its multiplication table being * (an^) i j Tc I m • • Ic I • h h I m l-\-Tc 0. h [2312*12]. The defining equation of this case is 7n* = 0. This gives a quintuple algebra which may be called (005), its multiplication table being f »In relative form, %z:^A:E+ AiB + B\C-\- C\D+ E\F, j=:A:C+B:D+A:F, k = A:D, l = A:F, m=.A:C+A:E+E:D. [C. S. P.] tin relative form, %=iA:B + B:C+C:D + E:F, j=A:C+B:D, k=zA:D, l=:A:F, m — A:C+A:E + B:F. [C. S. P.] 80 Peibce : Linear Associative Algebra. (aoj) i h I m k I • J Je I Je l + lc m [2312^]. The defining equation of this case is mi ^ I, There are two cases : [2312^2], when Cg does not vanish ; [231 2*] , when c^ vanishes. [2312^1]. The defining equation of this case can be reduced to 7n* = A;. This gives a quintuple algebra which may be called (opg), its multiplication table being* (ops) i j Tc I m • m J Te I • Te Te I m I • Te m = *In relative form, i = ^ :B + B :C+C: D+ J?: JP, ; = u4:C+B:D, h A:E+B:F+E:D. [C. S. P.] =:A:D, l=zA:F, Peiboe: Linear Associative Algebra. 81 [2312']. The defining equation of this case is m* = 0. This gives a quintuple algebra which may be called (og^g), its multiplication table being («S^b) ^ j Jc I m I h I m • J k I k I [232]. The defining equation of this case is im = ,* = jm = hm , Zi=0. which gives and it may be assumed that This gives Ij =zlJcz=0 =^iP =^ Pi = ihn = iml = mli = im There are two cases : [2321], when mi = 1; [232*], when mi = 0. [2321], The defining equation of this case is which gives = lmi=: mH=: mi = Z, mj = mk, lm = cjc + ^45? + e^^^m , P = ej, = 1^ = ej? = e^B = ?*, m* = cjc + d4+ e^m , ml=:ej J = mH = e^mH = ^5 = m J ; = Im^ = d4j>m = d '4S * What is meant is that every quantity not inyolving powers of t is nilf aoiend with reference to i. Hence, fZ = , also. [C. S. P.] 82 Peirce : Linear Associative Algebra. There are two cases : [2321*], when C45 does not vanish; [23212], when c^^ vanishes. [2321*]. The defining equation of this case can be reduced to Zm = A;,* which gives 771* = C5A;, (m — c^iy = , so that the substitution of ?n — c^l for m is the same as to make This gives a quintuple algebra which may be called {ar^)y of which the multipli- cation table is (arj) i j k I m • • J k • J k k I k m I [23212]. The defining equation of this case is fon = 0. There are two cases : [232121], when d^ does not vanish ; [23212*], when d^ vanishes. [232121]. The defining equation of this case can be reduced to ^5 = 1. There are two cases : [232121*], when c^ does not vanish ; [23212I2], when C5 vanishes. [232121*]. The defining equation of this case can be reduced to c« = 1. * But = ^1 = r , = i*i + frfA; + i** = *8i + ^(l + 2c4i)» *(* + FO=^+i?/» *(^ + JK/) = ^» (fr + pi)i = bsj + dsil + e^m +pj = (631 + p—pd^^j + ^i(? -^pj) + %w, (Z + 2») i = c?gim, * This line and the first equation of the next can be derived from = (t + J^) '• [C. S. P.] 86 Peirge: Imear Associative Algebra. so that if p satisfies the equation the substitution oi k+pi for h and of 1 + pj for I is the same as to make = 681 = ^8 = 63. There are four cases : [241*], when neither % nor e^ vanishes ; [2412], when % does not vanish but e^ vanishes; [2413], when % vanishes and not e^ ; [2414], when e^ and e^ both vanish. [241*]. The defining equations of this case can be reduced, without loss of generality, to We thus obtain a quintuple algebra which may be called {aw^\ its multiplication table being* (o^b) i j k I m • • J I m • m k xl+m fm m I tm • m [2412], The defining equations of this case can be reduced to *In relative formi==^:B + B:I)+tO:^+tjG?:-F+Gf:F,j==^: 2) + t'C'.JP, h—A\C+B:E •■\-D:F+A',0+QiF, l=zA:E+B:F, m — A\F. To obtain (oaja), omit the last term of fc. To obtain (ay^)^ omit, instead, the last term of t. To obtain (0^5), omit both these last terms. [C. S. P.] Feirge: Lmear Associative Algebra. 87 We thus obtain a quintuple algebra which may be called (ax^), its multiplication table being {cuc^) i j Je I m • • J I m • J m k xl-\-m f*OT I Xm m [2413]. The defining equations of this case can be reduced to We thus obtain a quintuple algebra which may be called (ay^), its multiplication table being Je I m Jc I m a J I m, m tl m fm m xm [2414]. The defining equations of this case are We thus obtain a quintuple algebra which may be called (azj), its multiplication table being 88 Peibce : Imear Associaiive Algebra. {azt) i Je I m 1e I m m J I m m tl fm tm [242]. The defining equations of this case are whidi give K mj tktn mik iJc=zlj i7 =: im = , ild = d^j + CgiZ, = Zt^ = Cgi = li^=lj\ d^jk =:lik = P=:^ikl=:^a^=^c^, Uc = a^j + Cg7 , =1 ik? = W=: c^k = c^ , imi = aji = Cji , mi^ =: tw; = d^Ji + e^iini , = mji = i^fc + iki + ki^=i 681^51 = 2agi + Ojidgi + 681651 , — (hijy = Ji?j=a^ = Jg'=U = ^sAi, imfc =z Ojg =: C53 , = mk? = 653 , ml = Ogdji/ , A^'Aj =,U=. (ogc^i + e^^j + ejidjs? » ^ A:^Z = e^d^ i=lki^= e^ilm = 631035 := Ji?m = 635. % = 0, There are two cases : [2421], when 631 does not vanish ; [242*], when 631 vanishes. [2421]. The defining equation of this case can be reduced to ki^ vn J Peiecb: Linear Assodaiive Algebra. 89 which, by the aid of the above equations, gives = ?ni = kil :=^ml^=. him = m* , a^j = il^ b^J = kik =zkl=i mk , = iJ{? + kik + A*i = A;^ = ag = &53 = A^ = >fc7w = mk = ml ; and if p is determined by the equation = Zm, Aj + ^i, I + pj\ and m+^y can be respectively substituted for A;, ? and m, which is the same thing as to make 63=0. There are three cases : [2421*], when neither d^ nor e^ vanishes ; [24212], when d^ vanishes and not e^ ; [24213], when d^ and Cg both vanish. [2421*]. The defining equation of this case can be reduced to d^=zl. This gives a quintuple algebra which may be called (ftag), its multiplication table being* (ftaj) i j Je I m m t I • J h m l-\-em I m *In relative form, % = A :B + B :C+ A:E , j=A:C, k=D:B+ E:F + D :G + eQ:C+A:E, l=.A:F^m=.D:C. By omittiiig the last term of k and putting e = 1 we get (b&s), and by omitting the last two terms of k we get (be,). [C. S. P.] 90 Peibge : Lmear ABSociative Algebra. [2421*]. The defining equation of this case can be reduced to This gives a quintuple algebra which may be called (bb^), its multiplication table being Jc I m Je I m • I f 1 1 t I m 1 i 1 ■ ■ ■ ■ ' 1 1 1 1 1 1 [24213]. The defining equation of this case is A? = 0. This gives a quintuple algebra which may be called (Jcj), its multiplication table being (ftcg) i j k I m • J I • J h m I m 1 1 Peibge : Linear Associative Algebra. 91 [242*]. The defining equation of this case is 681 = 0. There are two cases : [242*1], when e^ does not vanish ; [242^], when 6g vanishes. [242*1]. The defining equation of this case can be reduced to A? = aji + w , which gives Idle = kl = a^ij , il(? = ZA =: a^' , J^i = a^j + mi = d^kl = a^^^J, = ^i + ik + kik = a^{€l^ + ^i + 1) , fJ^i = (h{<^^ — !)«/ > ^' = ^^^ = ^ , = A? = OgZ + mA; = Os^ri + km , TwA; = — a^I , A»7i = — v^l = f, lk=j\ mk=> — ?, ki=. — A?m = h^ij + rZ, mi = (t* — l)y , m* = — ?/ . There are two cases : [242*1^], when 631 does not vanish ; [242*1*2], when 631 vanishes. [242*1^]. The defining equation of this case can be reduced to ki'=-j + rZ. This gives a quintuple algebra which may be called (icZg), its multiplication table being * ♦In relative form, t = il : D + 2>:J?'+B:^+C:jF',y= A rJ?*, A: = rA:B + rB:C+i): J7— -D:F + ^:F, ^.^=,A\E-'^- A:F+B:F, m = x*A:C—A:D — B:E—C:F. [O.S.P.] t 92 Peibce : lAnear Associative Algebra. bd. ) i • J k I m • ■ I • J k j+xl i + m • ^i j tl I m if-l)J -/ [242*1*2], The defining equation of this case is This gives a quintuple algebra which may be called (ftcg), its multiplication table being * (be^) i j k I m • • I • k xl i + m tj xl I • J m (r'-i)y — I -tj [242*12]. The defining equation of this case is /»,* = tn , which gives 0:=kl=lk=: km = mk zn m^ = J(?i=^ mi . There are two cases : [242*121], when 631 does not vanish; [242*12*], when b^ vanishes. * On adding to the expression for k in the last note the term — A:C, we have this algebra in relative form. [C. 8. P.] Peirce : Linear Associative Algebra. 93 [242*121]. The defining equation of this case can be reduced to This gives a quintuple algebra which may be called (6/5), its multiplication table being * {tfs) i j Ic I m • 1 • J I 1 • J Je j+di tn I m 1 [242*12*]. The defining equation of this case is This gives a quintuple algebra which may be called (bg^), its multiplication table being f (%) i J * I m • • J 1 I • k dl m I m 1 1 ) *In relative form, i = -4 : B+ B : C+i) : JE?, i = ^:C, k=.A \B+dA \ D+ B :E + B \F, l^A'.E, mzziAiE-^A-.F. [C. S. P.] tin relative form, i — A :B+B:C+D:E, j=A : C, k = dA :D + B:E+B':F, l — A:E, mznAiF, The algebra (car^ ) is what this becomes when d = . [C. S. P.] 94 Peibce : Linear Associative Algebra. [242^]. The defining equation of this case is (38= 0, which gives * * It is not easy to see how the author proves that a, = . But it can be proved thus. = ft* = The algebras of the case [242'] are those quintuple systems in which every product containing j or I as a factor vanishes, while every product which does not vanish is a linear function of j and I, Any multiplication table conforming to these conditions is self -consistent, but it is a matter of some trouble to exclude every case of a mixed algebra. An algebra of the class in question is separable, if all products are similar. But this case requires no special attention ; and the only other is when two dissimilar expressions U and V can be found, such that both being linear functions of t , ft and m , r7F= 717=0. It wHl be convenient to consider separately, first, the conditions under which ITF— "n7=0, and, secondly, those under which UV+VU=-0. To bring the subjects under a familiar form, we may conceive of i , ft , m as three vectors not coplanar, so that, writing U'zzict + yft + jCT», V=x'i + y'k + z'm^ we have a;, y, «, and a;' , yf ^ fi ^ the Cartesian coordinates of two points in space. [We might imagine the space to be of the hyperbolic kind, and take the coefficients of j and I as coordinates of a point on the quadric surface at infinity. But this would not further the purpose with which we now introduce geometric conceptions.] But since we are to consider only such proi>erties of XJ and V as belong equally to all their numerical multiples, we may assume that they always lie in any plane ^^ + 5^+08=1, not passing through the origin ; and then a; , |/ , z^ and x' , {^ , ^ ^ will be the homogeneous coordinates of the two points TJ and V in that i^ane. Let it be remembered that, although t , ft , m are vectors, yet their multiplication does not at all follow the rule of quaternions, but that t» = 6jj + d^ , fft = biJ-t- diaZ, »m = &i&; + d^Ji , *» = 6aii + d8i'i ft> = 6J+dJ, ftm = d8j+ 4,6^1 mt = d6ii+ d^^l , mft = 6g J+ d^^JL , to» =6J+ djZ . The condition that TJV— VU=- is expressed by the equations (bi?-b3i)(«^-a?'y) + (&i.--&M)(a^-«'«) + (&.5--&6.)(!/^--y'«) = 0, The first equation evidently signifies that for every value of CT, Fmust be on a straight line, that this line passes through U*, and that it also passes through the point The second equation expresses that the line between ]7and F contains the point e=(^86 -^68)*+ (^51 — di5)ft+(di8 — d,i)m. The two equations together signify, therefore, that Usjid Fmay be any two points on the line between the fixed points P and Q. Linear transformations of J and I may shift P and Q to any other situations on the line joining them, but cannot turn the line nor bring the two points into coincidence. The condition that UV+ VU:=z is expressed by the equations 26iaa/+ 26,j^+ 265^'+ (6i, + 6.i)(«i^+ «V) + (^5 + &5i)(a»'+ a^^^ 2dia»'+2d8OT('+2d5«af'+ (di, + d.i){a^+a?V)+ (di5 + d50(aa^+a?'2j) + (d,8^ The first of these evidently signifies that for any position of V the locus of U' is a line ; that U being fixed at any point on that line, Fmay be carried to any position on a line passing through its original position ; and tiiat further, if O* is at one of the two points where its line cuts the conic 4 1 Peibce : Linear Associative Algebra. 95 = A*i = a^j ^:^a^=iUc=:ml = kl=im^ = e^=^ d^a^ = T^m = e^ , = hmk = a^^l =z a^=ilm . then V may be at an infinitely neighboring point on the same conic, so that tangents to the conic from Vcut the locus of rZat their points of tangency. The second equation shows that the i)oint8 U and V have the same relation to the conic These conies are the loci of points whose squares contain respectively no term in j and no term in I . Their four intersections represent expressions whose squares vanish. Hence, linear transformations of j and I will change these conies to any others of the sheaf passing through these four fixed points. The two equations together, then, signify that through the four fixed points, two conies can be drawn tangent at 17 and Vto the line joining these last points. Uniting the conditions of UV— VU= and UV+ V77=0 , they signify that U and V are on the line joining P and Q at those points at which this line is tangent to conies through the four fixed points whose squares vanish. But if the algebra is pure, it is impossible to find two such points ; so that the line between P and Q must pass through one of the four fixed points. In other words, the necessary condition of the algebra being pure is that one and only one nilpotent expression in i , ft , m , should be a linear function of P and Q . The two points P and Q together with the two conies completely determine all the constants of the multiplication table. . Let S and T be the points at which the two conies separately intersect the line between P and Q . A linear transformation of /will move P to the point pP+ (1 — p)Q and will move S to the point j>S^+ (1 ~ j>) T , and a linear transformation of I will move Q and T in a similar way. The points P and S may thus be brought into coincidence, and the point Q may be brought to the common point of intersection of the two conies with the line from P to Q . The geometrical figure determining the algebra is thus reduced to a first and a second conic and a straight line having one common intersec- tion. This figure will have special varieties due to the coincidence of different intersections, etc. There are six cases : [1], there is a line of quantities whose squares vanish and one quantity out of the line ; [2], there are four dissimilar quantities whose squares vanish ; [8], two of these four quantities coincide ; [4], two pairs of the four quantities coincide ; [5], three of the four quantities coincide : [6], all the quantities coincide. We may, in every case, suppose the equation of the plane tobej; + 2^ + 2 = l- [1]. In this case, the line common to the two conies may be taken as {^ = , and the separate lines of the conies as ;s = and a; =: , respectively. We may also assume 2Pz=.x + y and 2Qzzx+z. We thus obtain the following multiplication table, where the rows and columns having j and I as their arguments are omitted : t km m 8Z m —3 — z 3i+i ■ J l-j [2]. In this case, we may take k as the common intersection of the two conies and the line, i , m , and i^k-\'mwA the other intersections of the conies. We have Q = ft , and we may write P=:5=in+(1— 1> — g)ft + qw, r=:rP+(l— r)g = rpi+(l — rp -rg)ft + fvm. We thus obtain the following multiplication table : 96 Peirge: Linear Associative Algebra. There are two cases : [242^1], when d^ does not vanish ; [242*], when d^ vanishes. [242*1]. The defining equation of this case can be reduced to which gives and if i (^ + b^i+pm) = I + hj = ?n* ; • i k m • f ■ ,i,+ ^)J+r,ir,-l)l ^.f 1 P,[P, %\%^;^'%i k ■ 3(«-8)j+r3(rg-l)l _p(p_8)y_rp(rp-l)J m [3 -p(p +!) + «(«- 8)]i+ [2-rp(rp-\) + rq(rq-\)-\l 1 1 -i>(p+l)i-rp(rp-l)l . [3]. Let A; be the double point common to the two conies, and let t and m be their other intersections. Then all expressions of the form ku + vk are similar. The line between P and Q cannot pass through k , because in that case all products would be similar. We may therefore assume that it passes through i. Then, we have Q = i, we may assume J+(4+i>r»)/ If this analysis is correct, only three indeterminate coefficients are required for the multiplication tables of this class of algebras. [C. S. P.] * See last note. I do not give relative forms for this class of algebrsuB, owing to the extreme ease with which they may be found. [C. S. P.] / Peirce : Luiear Associatioe Algebra, [242^12]. The defining equation of this case is . 99 There are two cases : = 0. [242^121], when igj does not vanish ; [242^12*], when 631 vanishes. [242''*121]. The defining equation of this case can be reduced to 631 = 1 . This gives a quintuple algebra which may be called (6^), its multiplication table being {h%) i Tc I m k I m • I j-\-al hj+cl a'j -If VI dj -\-d!l I [242^12*]. The defining equation of this case is let — — • Cvoi V , There are two cases : 81* [242^12*1], when 651 does not vanish ; [242^12^], when b^i vanishes. [242^12*1]. The defining equation of this case can be reduced to &5i = l. This gives a quintuple algebra which may be called (A/g), its multiplication table being 100 Peirce : Linear Associative Algebra. m .i Je I m k I m • J 1 I al bj+cl j+a'l b'j+il I [242*12^]. The definiDg equation of this case is mi := d^J, ; which can always, in the case of a pure algebra, be reduced to mi = ?. This gives a quintuple algebra which may be called {hh^j its multiplication table being • • I • J Te al bj+cl I m I a'j + 111 I [242*]. The defining equation of this case is »n* = bJ, K 'Peirce : Linear Associative Algebra, 101 and it can be reduced to [242^1] unless ^1 = ^3 = 0, J(^ = hj\ ^31 = — !, ^33 = — cijs; whence it may be assumed that and since when (k + bif = , / + ^; J31 + 63 = , 7^ = 0. it may also be assumed that There are two cases : [242*1], when b^i does not vanish; [242^], when b^i vanishes. [242*1]. The defining equation of this case can be reduced to bsi = l. This gives a quintuple algebra which may be called (64)» its multiplication table being (JZ5) i j Jc I m • % • J I • Tc J-l aj+hl I m • J aj+ hi 9 [242*^]. The defining equation of this case is There are two cases : Id = — I. [242^1], when 635 does not vanish ; [242*], when 635 vanishes. 102 Peibce : Linear Associative Algebra. [242*1]. The defining equation of this case can be reduced to fts8 = l. This gives a quintuple algebra which may be called (img), its multiplication table being* . {bnif) i Jc I m k I m • J I . I j+ai • J bf—al ^ [242*]. The defining equation of this case is b^ = There are two cases : '85 [242*1], when 653 does not vanish ; [242'], when 653 vanishes. [242*1] . The defining equation of this case can be reduced to 653= 1. This gives a quintuple algebra which may be called (6715), its multiplication table being f *Thi8 algebra it) mixed. Namely, if &4:1, it separates on substituting t^ = (1 — &)i + A;, ki = (1 — 6) t+ [a(l — 6) + 1] A;— (1 — 6)«m ; but if 6 = 1 , it separates on substituting ii = at — (a* + a + c)k + m, kizzcd + qk+m. [C. S. P.] t Substitute t'l = f — k^ /^^ = ofc + f7i , and the algebra separates. [C. S. P.] Peirce: Linear Associative Algebra. 103 (bih) i Je I m Jc I m • I * I al J j al 9 [242^]. The defining equation of this case is &53 = 0. This gives a quintuple algebra which may be called (ioj), its multiplication table being * h I m le I m • I I al • J — al 9 [243]. The defining equations of this case are which give = ik = il = im , 0=jIc=Jl=Jm. *' Substitute for m , at + wi , and the algebra separates. [C. S. P.] 104 Peirce: Linear Associative AJgehra. There are two cases : [2431], when hi = /, K=^m, wi = ; [2432], when Jci = 7, K = mi = 0. [2431]. The defining equations of this case are ki = l, K=mf ?ni=0, which give b': = m, Ij = mj = — Uc = mic = ? 0: = i7i? = = iM = ihn = = «3 — «34 = '«35> m = H = = CgZ + efgWi , Mi '=.hm = CgW , = ^m = Cg = Ajyw , = *3 = = b^m + c^TW = = *3 + ^. =zJm=^ ml = 7?r, There are two cases : [2431*], when e^ does not vanish; [24312], when eg vanishes. [2431*]. The defining equation of this case can be reduced to eg = 1 . This gives a quintuple algebra which may be called {hp^^ its multiplication table being* *The structure of this algebra may be exhibited by putting fci =t + a"V — a"'A?, li=:j—a 7, mi = — a~^fn , when the multipUcation table becomes i i k I m 3 k I m • • 3 I m m m Inrelativeform, t = B:C+C:D,i=B:D, k-^AiB + C'.D, l=zA:C, m=:.4:D. [C. S. P.] 1 Pbircb : Linear AjBsoeiaiive Algebra. 105 (*P5) » k I m le I m • 1. I 1 1 1 I m al + m anh I m 1 i 1 1 J 1 1 [24312]. The defining equation of this case is This gives a quintuple algebra which may be called (ftjj), its multiplication table being* Tc I m J 1 * I m — a*j + ,j = A:J5+(l—c«)C:D. k=:A:D. l = A:C + (1 — c-«)B:D, m = A:E. n=i(b+e)B :E+ {b+c''')C:E. [C. S. P.] no Peircb : Linear Aesodative Algebra. {bd.) i I m n Je I m n • • J 1 : k 1 I m n • k 1 ' m k 1 1 1 I ak 1 k hm [132*]. The defining equation of this case is yn = o, and there is no pure algebra in this case. [13*]. The defining equations of this case are / = Aj, lj=h, jk=jl = Jg = ](?=kl=lk = P = 0, which give km =zkn=: 0. There is a sextuple algebra in this case which may be called (ftej), of which the multiplication table is * ^This algebra may be a little simplified by substituting j — Itorj, In relative form, i'^AiA + B:B+C:C+D:D, j = A:D+ B:C, k=zA:C, l = A:B, m=iA:E, n = hB:E+aD:E, [C. S. P.] i Peirge : Linear Associative Algebra. Ill (6ce) i I m n I Tc I m n i J k I 1 m 1 J k am k I * bm 1 1 1 1 [134]. The defining equations of this case are There is a sextuple algebra in this case which may be called (6/e), of which the multiplication table is * (ft/a) i j * ^ ^ ^ k I m n i J 1 k I m n • J k m k III ■ 1 I — k a/m 1 1 ♦Inrelatdveform,i = il:A + JB:B+C:C+D:Z), j = -A:B — C:D, h = A:D, l:=iA:C+B:D, m = A:E, nzzB:E+aC:E. [0.8. P.] 112 Feirce : Linear Associative Algebra, [14]. The defining equations of this case are ij=:ji=ij, ik=z ki=zkj il=z K= ly im =:z m , ni^^rif mi = w = , which give There are four cases defined as in [13]. [141]. The defining equations of this case are which give / = Aj, jh = hi = h jl=}^=kl = lj=lk = P=0, mn = d^l . There is a sextuple algebra which may be called (bg^), of which the multipli- cation table is* {J>9>) I m n % h I m n 1 1 I m • m J h I k I I 1 I n I [142]. The defining equations of this case are the same as in [132], which give *In relative form, i = ^ lil + B :B+C:C+/) :D, i = ^ : B + B :C+ CiD, k — A\C+B\D, l^AiD, mz^AiE, n:=iE:D. [0. S. P.] i Peirce: Iaiwolt AseodoHve Algebra, 113 I I f There is a sextuple algebra which may be called (6^), of which the multipli- cation table is * (hh^ i j k I m n • • m J k I m • J • k * k k I I ak k m k n n [143]. The defining equations of this case are the same as in [13*]. There is a sextuple algebra which may be called (6i^), of which the multiplication table is f {b%) i j Jc I m n • ^ i 1 1 1 j k I m • k k k k I m n .k n 1 * This algebra has two varieties, analogous to those of (Cg). The first is, in relative form, %z=.A\A + B:B+C:C+D:D,j=iA:B + B:C+A:D, k=A:C, l=i'-'A:B + D:C, m = A:E,n=iE:a The second in relative form is the same except iih&tj=A : B + h'^D : C, l=iA:D — hB:C. [C. S. P.] t This algebra may be slightly simplified by putting j — I for/. Then, in relative form, i=zA:A + B:B+C:C.j=B:C, k=iA:C,l=:A:B,m = A:D,n = D:C\ [C. S. P.] 114 Peircb : Linear Associative Algebra. [14^]. The defining equations of this case are the same as in [134]. There is a sextuple algebra which may be called {bj\), of which the multiplication table is * m i I m n I m 71 m % J k I m i 1 m k , 1 j 1 k I ' 1 — Te .i -_ k n , [15]. The defining equations of this case are ij = ji = y , ik=^hi^=k, il= I, im = ?n , in := w , liz= mi = ni=^0 y which give j^ =1 kj =y^ =1 hf =z J^z=i IJ =z !k=z P =zlm=^ In=^ mj ^= mh =^ ml =^ m^ = mn = nj =z nk=z nl = nrn = n*. There is a sextuple algebra which may be called {hk^, of which the multipli- cation table isf *%-A:A + B\B+C\C\D\D, j=A:B—C:D, k = A:D, l=iA:C+B:D, m = A:E, n = E:D. [C. S. P.] tin relative form, i = A:A + B:B+C:C, j=A:B + B:C, k = A:C,l=iC:D,m=:B:D, n = A:D, [C. S. P.] I i \ Pbirce : Linear Associative Algebra, 115 (bkfi) i j k I m n k I m n • J k I m n • h m n h n [16]. The defining equations of this case are ij =ji^j\ ik=:ki=^k, ilznl^ im=^m, ni^^rij li=zmi=zin=^0, which give j^ =:kj =^jk =jn ==Jy'=zIi?=zhn:=lj=^lk=^P=^lm=^mj==^ mk z=:nl=- rr? -^z nj '=' rJc '=^ rd -=- nm = tj?. There is a sextuple algebra which may be called (hl^, of which the multipli- cation table is * (JZe) i j k I m n • m t j Tc I m • J • J h m h h I Je m • n n *In relative form, % = A:A + B :B + C:C, j=A:B + B:C, kzzAiC, l = B:D + A:E, m — A:D,n=iE:C. [C. S. P.] - i 116 Peibge: Lmear Associative Algebra. [17]. The defining equations of this case are ij z=:ji =y , iJc=:ki=^k^ il = l^ mi=z m , li=^ im = m = m = . There is no pure algebra in this case. [18]. The defining equations of this case are ij :=ji =y , ik = kj il=::lj im=^m, in=^n, Jci=^li= 7ni = ni=^0 . There is no pure algebra in this case. [19]. The defining equations of this case are ij =:zji=zj J ik=zk, ilzzzly im = m, ni^^n, i/n^=^kl=^li=zni=in. There is no pure algebra in this case. [l(y]. The defining equations of this case are ij=jiz=zj\ ik=^k, il=^lj mi^=^m^ ni=^n, im = m = A?i = Zi = 0. There is no pure algebra in this case. [11']. The defining equations of this case are i/=yi=y, ik=^k, iZ = Z, mi = ?n, im:=:K:=^in=^ ni=^0. There is no pure algebra in this case. [12']. The defining equations of this case are ij:=zjiz=ij\ ik=:^k, li = l, il^zim =^ in=^ ki=^ mi=^ni=^0 . There is no pure algebra in this case. [13']. The defining equations of this case are y=y, ik=^kf ilzizl, i7n=^7n, in = 71, ji^izki =:li:=:mi=::ni=^0. There is no pure algebra in this case. [14']. The defining equations of this case are ijz=zj\ ik^='kj il=^l, im = m, ni:=^n, ji=:ki=^li =: mi^:=in=^ . There is no pure algebra in this case. [15']. The defining equations of this caee are ^y = y , ik=^kj il=zl^ mi:=my ni=^n, im=^ in =ji = Z?i = Zi = There is no pure algebra in this case. i Peibce: Linear AseodcUive Algebra. 117 [16']. The defining equations of this case are There is no pure algebra in this case. [17']. The defining equations of this case are ij :=y , iJc=::k, K =z I , mi := m ^ ilzzzim^in =^Ji = A?i = ni = . There is no pure algebra in this case. [18']. The defining equations of this case are ij =y , ik=^k, li^=ly ji=zki = il =: im = in = Twi = m = . There are six cases : [18'1], when m^ =^ m , mn = n , nm == , [18'2], when m* = m , mn = , nm = n , [18'3], when m* =^n, mn ^ nm = , n* = m , [18'4], when m^ ^=-my mn = nm = w* = , [18'5], when m^ ^r-n , m^ = , [18'6], when7w*=w* = 0. [18'1]. The defining equations of this case are m^ = m, mn = n, nm = {). There are two cases : [18'P], whenwZ = 0; [18'12], when ?nZ=/. [18'1*]. The defining equation of this case is 7wZ= 0. There is no pure algebra in this case. [18'12]. The defining equation of this case is ml = I , There are two cases : [18'121], whenyrw=y; . [18'12^], whenym = 0. [18' 121]. The defining equation of this case is jm =y. There is a sextuple algebra which may be called (fewi^), of which the multipli- cation table is * Inrelativefonn,i=il:^, j = ^:jB, fcrzArO, l:=.B\A, m — BiB, n = B:C, [C. S. P.] 118 Peirge : Linear Associative Algebra. {bm^) i k I m n k I m n • • Ic 1 • • 1 Te I m n I m n [18'12*]. The defining equation of this case is ym = . There is no pure algebra in this case. [18'2]. The defining equations of this case are TW = m , vnn ^ , nvn = n There are two cases : [18'21], whenmZ = Z; [18'2»], when7nZ=0. [18'21]. The defining equation of this case is ml=il. There are two cases : [18'2P], whenym=y; [18'212], whenym=0. [18'21*]. The defining equation of this case is There is no pure algebra in this case. [18'212]. The defining equation of this case is ym = . There is no pure algebra in this case. I f Peirce : Lmear Associative Algebra. 119 [18'2*]. The defining equation of this case is There is no pure algebra in this case. [18'3]. The defining equations of this case are m* = m , mn = nm = , n^ =zn. There is no pure algebra in this case. [18'4]. The defining equations of this case are m* = w , mn =: nm = w* = . There are two cases : [18'41], when ym=y; [18'42], whenym = 0. [18'41]. The defining equation of this case is jm =y. « There is no pure algebra in this case. [18'42]. The defining equation of this case is y^w = . There is no pure algebra in this case. [18'5]. The defining equations of this case are m* = 71 , m' = . • There is no pure algebra in this case. [18'6]. The defining equations of this case are m* = 71* = . There is no pure algebra in this case. [19']. The defining equations of this case are ij =zj J l-izizJcy ji =zik=z il = im = in = li = mi = m = . There is no pure algebra in this case. [2]. The algebras belonging to this case are not investigated, because it is evident from § 69 that they are rarely of use unless combined with an idempo- tent basis, so as to give septuple algebras. Natural Classification. There are many cases of these algebras which may obviously be combined into natural classes, but the consideration of this portion of the subject will be reserved to subsequent researches. » 120 Peiroe : Linear Associative Algebra. ADDEISTDA. I, On the Uses and Transformations of Lmear Algebra. By B]E!Njamin Peirce. [PreseiKted to the American Academy of Arts and Sciences, May 11, 1875.] Some definite interpretation of a linear algebra would, at first sight, appear indispensable to its successful application. But on the contrary, it is a singular fact, and one quite consonant with the principles of sound logic, that its first and general use is mostly to be expected from its want of significance. The interpre- tation is a trammel to the use. Symbols are essential to comprehensive argument. The familiar proposition that all A is 5, and all B is (7, and therefore all A is (7, is contracted in its domain by the substitution of significant words for the symbolic letters. The A, B, and G, are subject to no limitation for the purposes and validity of the proposition ; they may represent not merely the actual, but also the ideal, the impossible as well as the possible. In Algebra, likewise, the letters are symbols which, passed through a machinery of argument in accord- ance with given laws, are developed into symbolic results under the name of formulas. When the formulas admit of intelligible interpretation, they are accessions to knowledge ; but independently of their interpretation they are invaluable as symbolical expressions of thought. But the most noted instance is the symbol called the impossible or imaginary, known also as the square root of minus one, and which, from a shadow of meaning attached to it, may be more definitely distinguished as the symbol of semi-inversion. This symbol is restricted to a precise signification as the representative of perpendicularity in quaternions, and this wonderful algebra of space is intimately dependent upon the special use of the symbol for its symmetry, elegance, and power. The immortal author of quaternions has shown that there are other significations which may attach to the symbol in other cases. But the strongest use of the symbol is to be found in its magical power of doubling the actual universe, and Peirge : Linear Associative Algebra, 121 placing by its side an ideal universe, its exact counterpart, with which it can be compared and contrasted, and, by means of curiously connecting fibres, form with it an organic whole, from which modern analysis has developed her surpassing geometry. The letters or units of the linear algebras, or to use the better term proposed by Mr. Charles S. Peirce, the vids of these algebras, are fitted to perform a similar function each in its peculiar way. This is their primitive and perhaps will always be their principal use. It does not exclude the possibility of some special modes of interpretation, but, on the contrary, a higher philosophy, which believes in the capacity of the material universe for all expressions of human thought, will find, in the utility of the vids, an indica- tion of their probable reality of interpretation. Doctor Hermann Hankel's alternate numbers, with Professor Clifford's, applications to determinants, are a curious and interesting example of the possible advantage to be obtained from the new algebras. Doctor Spottiswoode in his fine, generous, and complete analysis of my own treatise before the London Mathematical Society in Novem- ber of 1872, has regarded these numbers as quite different from the algebras discussed in my treatise, because they are neither linear nor limited. But there is no difficulty in reducing them to a linear form, and, indeed, my algebra (63) is the simplest case of Hankel's alternate numbers ; and in any other case, in which n is the number of the Hankel elements employed, the complete number of vids of the corresponding linear algebra is 2~ — 1 . The limited character of tKe algebras which I have investigated may be regarded as an accident of the mode of discussion. There is, however, a large number of unlimited algebras suggested by the investigations, and HankePs numbers themselves would have been a natural generalization from the proposition of § 65 of my algebra.* Another class of unlimited algebras, which would readily occur from the inspection of those which are given, is that in which all the powers of a vid are adopted as independent vids, and the highest power may either be zero, or imity, or the vid itself, and the zero power of the fundamental vid, i. e. unity itself, may also be retained as a vid. But I desire to draw especial attention to that class, which is also unlimited, and for which, when it was laid before the math- ematical society of London in January of 1870, Professor Clifford proposed the appropriate name of qtjuid/rates. * This remark is not intended as a foundation for a Ciaim upon the Hankel numbers, which were published in 1867, three years prior to the publication of my own treatise. — B. P. [They were given much earlier under the name of clefs by Cauchy, and (substantially) at a still earlier date by Grassmann. — C. S. P.] I ■ I ti 122 Peiboe: Linear Associative Algebra. m Qtiadrates. The best definition of quadrates is that proposed by Mr. Charles S. Peirce. If the letters A, J?, G, etc., represent absolute quantities, diflfering in quality, I the vids may represent the relations of these quantities, and may be written in the form {A:A){A:B){A:G) . . . {B : A) {B : B) . . . {G:A), etc. subject to the equations {A :B){B:G) = {A: G) {A:B)Ig:D) = 0. In other words, every product vanishes, in which the second letter of the multi- plier differs from the first letter of the multiplicand ; and when these ty?'o letters are identical, both are omitted, and the product is the vid which is compounded of the remaining letters, which retain their relative position. Mr. Peirce has shown by a simple logical argument that the quadrate is the legitimate form of a complete linear algebra, and that all the forms of the algebras given by me must be imperfect quadrates, and has confirmed this conclusion by actual investigation and reduction. His investigations do not however dispense with the analysis by which the independent forms have been deduced in my treatise, though they seem to throw much light upon their probable use. Unity. The sum of the vids {A : A), {B : B), {G : (7), etc., extended so as to include all the letters which represent absolute quantities in a given algebra, whether it be a complete or an incomplete quadrate, has the peculiar character of being idempotent, and of leaving any factor unchanged with which it is combined as multiplier or multiplicand. This is the distinguishing property of unity, so that this combination of the vids can be regarded as unity, and may be introduced as such and called the vid of miity. There is no other combination which possesses this property. But any one of the vids {A : A), {B : B), etc., or the sum of any of these vids is idempotent. There are many other idempotent combinations, such as {A:A) + x{A:B), y{A:B) + {B:B), h{A: A) + h {A: B) + h{B : A) + h{B : B), which may deserve consideration in making transformations of an algebra preparatory to its application. \ Peircb: Linear Associative Algebra. 128 Inversion. A vid which differs from unity, but of which the square is equal to unity, may be called a vid of inversion. For such a vid when applied to some other combination transforms it ; but, whatever the transformation, a repetition of the application restores the combination to its primitive form. A very general form of a vid of inversion is {A yA) ±{B:B)±{C:G)± etc., in which each doubtful sign corresponds to two cases, except that at least one of the signs must be negative. The negative of unity might also be regarded as a symbol of inversion, but cannot take the place of an independent vid. Besides the above vids of inversion, others may be formed by adding to either of them a vid consisting of two different letters, which correspond to two of the one- lettered vids of different signs ; and this additional vid may have any numerical coeflBcient whatever. Thus {A: A) + {B : B) — {G : G) + x{A: G) + y{B : C) is a vid of inversion. The new vid which Professor Clifford has introduced into his biquaternions is a vid of inversion. Semi' Inversion. A vid of which the square is a vid of inversion, is a vid of semi4nversion . A very general form of a vid of semi-inversion is {A:A)±{B:B)±L J{G : G) ± etc. in which one or more of the terms {A\ A), {B : B)^ etc., have J for a coeffi- cient. The combination {A:A)± J{B: B) + x{A : B) + etc. is also a vid of semi-inversion. With the exception of unity, all the vids of Hamilton's quaternions are vids- of semi-inversion. 7%c Use of Gommutative Algebras. Commutative algebras are especially applicable to the integration of differential equations of the first degree with constant coefficients. If i, y, /fe, 124 Pkircb : Linear Associative Algebra. etc., are the vids of such an algebra, while x, y, 2, etc., are independent variables, it is easy to show that a solution may have the form F{xi + yj + zh 4- etc.), in which i^is an arbitrary function, and i, y, k, etc., are connected by some simple equation. This solution can be developed into the form ^(a^* + yj + zk + etc.) = Mi + iV;' + PAj + etc. in which ilf, N, P, etc., will be functions of cc, y, z, etc., and each of them is a solution of the given equation. Thus in the case of Laplace's equation for the potential of attracting masses, the vids must satisfy the equation *^+y* + A? = o. The algebra (oj) of which the multiplication table is % Tc Je m t • J k • J 1c k may be used for this case. Combinations ii , /i , Aji of these vids can be found which satisfy the equation ^+yf + At = o, and if the functional solution J^{^+yji + zk^) is developed into the form of the original vids Mi+NJ+Pk, M, N, and P will be independent solutions, of such a kind that the surfaces for which N and P are constant will be perpendicular to that for which M is constant, which is of great importance in the problems of electricity. The Use of Mixed Algebras. It is quite important to know the various kinds of pure algebra in making a selection for special use, but mixed algebras can also be used with advantage ) I Peibge: Linear Associative Algebra. 125 in certain cases. Thus, in Professor Clifford's biquatemions, of which he has demonstrated the great value, other vids can be substituted for unity and his new vid, namely their half sum and half difference, and each of the original vids of the quaternions can be multiplied by these, giving us two sets of vids, each of which will constitute an independent quadruple algebra of the same form with quaternions. Thus if i,j\ h, are the primitive quaternion vids and w the new vid, let ai = i (1 + t£?) . ij =1 a^i , Then since • hj\ = h = —Jih- a. = h{l — w). h = Oii. h =