ALGEBRAIC FORMS . The subject-See also:

• MATTER

80-94, 95-112) showed by his calculus of hyper-determinants that an See also:

• INFINITE (from Lat. in, not, finis, end or limit; cf. findere, to deave)

The See also:

• YEAR

This enabled See also:

• DAVID
• DAVID (a Hebrew name meaning probably beloved 1)
• DAVID, FELICIEN (1810—1876)
• DAVID, GERARD [GHEERAERT DAVIT], (?=1523)
• DAVID, PIERRE JEAN (1789–1856)
• DAVID, ST (Dewi, Sant)

The quantities ala, a21g ... are called the elements of the determinant; the See also:

• TERM

Similarly it is shown that the transposition of any two columns or of any two rows merely changes the sign of the determinant. Theorem.—Interchange of any two rows or of any two columns merely changes the sign of the determinant. Corollary.—If any two rows or any two columns of a determinant be identical the value of the determinant is zero. Minors of a Determinant.—From the value of A we may See also:

• SEPARATE

+assAnk This theory enables the evaluation of a determinant by successive reduction of the orders of the determinants involved; -1I—5 20 3I+3I0-5 =11 3 -6 I -51+3.2 -51—3.110I =3+30-30—0=3. Since the determinant a21 a22 a23 .•• a2„ an an a23 ••• a2n a31 a32 a33 ••• a3n , having two identical rows, and an2 an3 ••• ann vanishes identically; we have by development according to the elements of the first row aiiAii+a22Al2+a23A13+... +See also:

• A2(z)

Corresponding Minors.—In obtaining the minor Aik in the form ra of a determinant we erased certain rows and columns, and we would have erased in an exactly similar manner had we been forming the determinant associated with Ai., since the deleting lines intersect rk in two pairs of points. In the latter See also:

• CASE
• CASE, JOHN (d. 1600)

Multiplication.—From the theorem given above for the expansion of a determinant as a sum of products of pairs of corresponding determinants it will be See also:

• PLAIN (O. Fr. plain, from Lat. plenum)

+annbnn and all the elements of D become zero. Now by the expansion theorem the determinant becomes ()1+2+3+•••+2nB•C = ( I)n(2n+1)+' C =C. We thus obtain for the product a determinant of order n. We may say that, in the resulting determinant, the element in the ith row and kth column is obtained by multiplying the elements in the kth row of the first determinant severally by the elements in the ith row of the second, and has the expression aklbil +ak2bi2 +¢k3bi3••• +¢knbtin, and we obtain other expressions by transforming either or both determinants so as to read by columns as they formerly did by rows. Remark.—In particular the square of a determinant is a determinant of the same order (bnb22b33...b,n) such that bik=bki; it is for this See also:

• REASON (Lat. ratio, through French raison)

0x3 =A13 1+A236 +A336 +••• and thence . A"'Ei =BI1z x1+ B120x2+ See also:

• BISI

Now the determinant has the value (Ai A11 +A2A22+A3A33 +2A2A3A22+2 A3A1A31 +2AiX2Al2f = —ZA;A,,—2F,A,A.Ar, in general, and hence by substitution t {A1J A11+A2 1 A22+...+X,, Ann) 2. A skew symmetric determinant has ar,=o and a,,= —as, for all values of r and s. Such a determinant when of uneven degree vanishes, for if we multiply each row by — I we multiply the determinant by (—On= -1, and the effect of this is otherwise merely to transpose the determinant so that it reads by rows as it formerlydid by columns, an operation which we know leaves the determinant unaltered. Hence A=—A or A=o. When a skew symmetric determinant is of even degree it is a perfect square. This theorem is due to Cayley, and reference may be made to Salmon's Higher Algebra, 4th ed. See also:

• ART

Further we find x =aX+a'Y+a"Z, y=bX+b'Y+b"Z, z=eX+c'Y+c"Z, and the problem is to See also:

• EXPRESS (through the French from the past participle of the Lat. exprimere, to press out, by transference used of representing objects in painting or sculpture, or of thoughts, &c. in words)

From the differential coefficients of the y's with regard to the x's we form the functional determinant we derive Al, and the adjoint is ayl a 1 ayl aY2 aY2 (Yl, Y2,...ynl R = Z—x2 ... ex — \x1, x2,...xn/ for brevity. Z.-Z. aYs ... ayn O.Z. Oxq ... If we have new variables z such that z,=¢,(yl, Y2,•••y,,), we have also z, = 4,, (x1, x2,...x,,), and we may consider the three determinants (Yl, Y21...yn) / Zl, Z2,...z„ 1 ' (sl, z2, ...z,.) x2,...xn 1\Yi, Y2r.. •yn/ 1\xi, X2, ...xn/ Forming the product of the first two by the product theorem, we obtain for the element in the it" row and kg" column azi ay1 az; ay2 azi ay,, which is a , the partial differential coefficient of z, with regard to xk. Hence the produc\/t theorem (zl, z2,...zn/ fYi, Y2,...yn' = rzi, z2,...zn) IV.. 1\ 1, Y2,...yn xlr x2r...xn xl, x2r...xn and as a particular case (yl, Y2,...yn1 el, x2,...xn = I. \Xi, x2,...xn/ \Yl, Y2,...yn Theorem.—If the functions y1, y2,•..yn be not independent of one another the functional determinant vanishes, and conversely if the determinant vanishes, y1, y2,•••yn are not independent functions of xi, x2,...xn. Linear Equations.—It is of importance to study the application of the theory of determinants to the See also:

• SOLUTION (from Lat. solvere, to loosen, dissolve)

+anµAnµ) = xp. = 0, since from results already giveh the remaining coefficients of x1, x2,...x I, xp+i,•••x„ vanish identically. Hence if A does not vanish xl=x2=•••=x,,=o is the only solution: but if A vanishes the equations can be satisfied by a system of values other than zeros. For in this case the n equations are not independent since identically AIpf 1+A2pf2+•••+Anpfn = 0, and assuming that the minors do not all vanish the See also:

• SATISFACTION (Lat. salisfacerc, to satisfy)

=0, ~(x) = =See also:

• BOX
• BOX (Gr. irl or, Lat. buxus, box-wood; cf. 2r6Eir, a pyx)

Forming the resultant of these equations we evidently obtain the resultant of f and 0. Thus to obtain the resultant of f =aox3+alx2+a2x+a3, 4, = box2+blx+b2 we assume the identity (Box +Bi) (aox3+a1x2+a2x+a3) = (Aox2 + See also:

• AIX

He first divides by the factor x —x', reducing it to the degree m— I in both x and x' where m>n; he then forms m equations by equating to zero the coefficients of the various powers of x'; these equations involve the m powers x°, x, of x, and regarding these as the unknowns of a system of linear equations the resultant is reached in the form of a determinant of order m. Ex. gr. Put (aox'+a1x2+a2x+aa) (box'2+bix'+b2) — (aox''+alx'2+a2x'+aa) (box'+bix+b2) =0; after See also:

• DIVISION (from Lat. dividere, to break up into parts, separate)

= 0, we find that, eliminating x, the resultant is a homogeneous function of y and z of degree mn; equating this to zero and solving for the ratio of y to z we obtain mn solutions; if values of y and z, given by any solution, be substituted in each of the two equations, they will possess a common factor which gives a value of x which, com- bined with the chosen values of y and z, yields a system of values which satisfies both equations. Hence in all there are mn such systems. If, therefore, we have a third equation, and we substitute each system of values in it successively and form the product of the mn expressions thus formed, we obtain a function which vanishes if any one system of values, common to the first two equations, also satisfies the third. Hence this product is the required resultant of the three equations. Now by the theory of symmetric functions, any symmetric functions of the mn values which satisfy the two equations, can be expressed in terms of the coefficient of those equations. Hence, finally, the resultant is expressed in terms of the coefficients of the three equations, and since it is at once seen to be of degree mn in the coefficient of the third equation, by symmetry it must be of degrees npand pm in the coefficients of the first and second equations respectively. Its weight will be mnp (see Salmon's Higher Algebra, 4th ed. § 77). The general theory of the resultant of k homogeneous equations in k variables presents no further difficulties when viewed in this manner. The expression in form of a determinant presents in general considerable difficulties. If three equations, each of the second degree, in three variables be given, we have merely to eliminate the six products x2, ye, z2, yz, zx, xy from the six equations u=v=w=1=1=z =0; if we apply the same process to these equations each of degree three, we obtain similarly a determinant of order 21, but thereafter the process fails. Cayley, however, has shown that, whatever be the degrees of the three equations, it is possible to represent the resultant as the quotient of two determinants (Salmon, l.c. p.

89). Discriminants.—The discriminant of a homogeneous polynomial in k variables is the resultant of the k polynomials formed by differentiations in regard to each of the variables. It is the resultant of k polynomials each of degree m— l, and thus contains the coefficients of each form to the degree (m—1)k-i; hence the See also:

• TOTAL

and, dividing by y1Y2...ym, the discriminant of f is seen to be equal to the product of the squares of all the differences of any two roots of the equation. The discriminant of the product of two forms is equal to the product of their discriminants multiplied by the square of their resultant. This follows at' once from the fact that the discriminant is II(a aXII(1r—iie)2{II(a,.—fie)}2. II. THE THEORY OF SYMMETRIC FUNCTIONS Consider n quantities al, at, aa,...an. Every rational integral function of these quantities, which does not alter its value however the n suffixes I, 2, 3, ... n be permuted, is a rational integral symmetric function of the quantities. If we write (1 +aix) (1+ atx) ... (1 +a,,x) = +alx+aax2+...+anxn, a1, at, ...an are called the elementary symmetric functions. al = al+a2+••••+-a„ =}.•al a¢ = alas+alai+aia3+••. _ 2ala2 1—aix+a2x2—asx3+... which remains true when the symbols a and h are interchanged, as is at once evident by writing —x for x. This proves, also, that in any formula connecting al, a2, a3,... with hl, h2, ha,... the symbols a and h may be interchanged.

Ex. gr, from h2=ai —a2 we derive at=hi —h2. The function Earla2P2...aPn being as above denoted by a partition of the weight, viz.(plp2...pn), it is necessary to bring under view other functions associated with the same series of numbers: such, for example, as P1 E 4' sEa1 Ps P4 Pn_2 = (pi p3)( p.p.... as ...and pn-2). The expression just written is in fact a partition of a partition, and to avoid confusion of language will be termed a separation of a partition. A partition is separated into separates so as to See also:

• PRO

3 of the number n. The expression is (—)v1+v2+va+...(v1+v2+i3+...—)!(n) vl.v2.v3.... _ (—\I1+'2+13+".(J1+j2+J3+...— 1)!(Jl) II(J2)12U / ji J2;J3•••. (J1)it(J2)i2(J3)i3... being a separation of (n1In"2na3...) and the summation being in regard to all such separations. For the particular case (nV Inv2nv3...) = (1n) 1 a (—)"n(n) = (—)il+is+ia+..(j1+j2+j3j+... -1)1(1)il(12)12(Is)h.... jljsa•••.To establish this write 1 +µX1+µ2X2-+3X3+... = 11 (1-1-µalxl+n2aix2+µ3aix3-I-...), the product on the right involving a factor for each, of the quantities al, at, a3..., and /c being arbitrary. Multiplying out the right-See also:

• HAND
• HAND (a word common to Teutonic languages; cf. Ger. Hand, Goth. handus)
• HAND, FERDINAND GOTTHELF (1786-185r)

1XP— "(3)'r1(21)'''2(13)'r3 v1 a2 'See also:

• R2(rt-s2)

(—) v1!v2!v3!... (n) = ()il+i2+i3+..•(j1+j2+j3+...—1)!G )11 (J2)12W 3)1a..., j1;72j3•••• for the expression of Ean in terms of products of symmetric functions symbolized by separations of (n11n$2n33...). Let (n),, (n),, (n),, denote the sums of the nth powers of quantities whose elementary symmetric functions are a1, a2, a3,... ; x1, x2, x3,..; X1, X2, X3,••• respectively: then the result arrived at above from the logarithmic expansion may be written (n),.(n)x= (n)x, exhibiting (n)= as an invariant of the transformation given by the expressions of Xi, X2, X3... in terms of x1, x3, x4,.... The inverse question is the expression of any monomial symmetric function by means of the power functions (r) =s,. Theorem of See also:

• RECIPROCITY (Lat. reciprocus, returning back the same way, alternating, probably from re back and pro forward)

It can be shown that the number B enumerates distributions of a certain nature defined by the partitions (milms2...), (sils~z ), an = alai a3... aa. The general monomial symmetric function is EaP1P2aP3...ann, the summation being for all permutations of the indices which result in different terms. The function is written (Plp2p3•••pn) for brevity, and repetitions of numbers in the bracket are indicated by exponents, so that (PIPi z) is written (pips). The weight of the function is the sum of the numbers in the bracket, and the degree , the highest of those numbers. Ex. gr. The elementary functions are denoted by (1), 12 (12), 1n , are all of the first degree, nd are of eights 1, 2, 3,...n respectively. Remark.—In this notation (0)=tai=(1;(02)=144 =(2'');...(0•) = (; ), &c. The See also:

• BINOMIAL (from the Lat. bi-, bis, twice, and nomen, a name or term)

=...+o(si's2c 2s83...)+•.., (1'1) °1(1 a2) ca(1,3) °a... =... 0(mi'm2 zm$ 3...) +... ; and another by putting x1= x2 = x3 =... =1, for then Xm becomes hm, and we have hih,',3hme... =... +o' (si's= ssa 3...) +..., o' m"`lynµ2ru" 3... Theorem of Expressibility.—" If a symmetric function be symboilized by (Xµv...) and (X2X2X3...), (µ1112113...), (vlv2v3...)... be any partitions of a, µ, v... respectively, the function (XAs ...) is expressible by means of functions symbolized by separation of ()\1X2X3...µiµzµ3... vlv2v3...)." For, writing as before, =}.FO(S°'S°3S°3...)x~'x12x~3 "'1 m2 m3' 1 2 8 1 S $= EPxi 1x1 sxAa P is a linear function of separations of (1'11'2123...) of specification (ml"'m2 2ms3...), and if = EP'x!'x13xi 3..., P' is a linear 1 3 31 S a function of separations of (l21122le3...) of specification (sl's=2sis...). Suppose the separations of (li 1122183...) to involve k different specifications and form the k identities 1aX,'2eX'n3a"`3a... =EP(a)xatxa2x"3 (s=1> 2, •..k) le n'2e I e2 ,3... f where (mi''m2 Rams.'•..) is one of the k specifications. The law of reciprocity shows that t-s P(e)=161at(mµltmP2tre8t ") It Et $t ' a. e~1 viz.: a linear function of symmetric functions symbolized by the k specifications ; and that oa, = Bo. A table may be formed expressing the k expressions Po), P(2),...P(k) as linear functions of the k expressions (m1µ"mµ2a a 2am"+sa 3e...), s =1, 2, ...k, and the numbers oa, occurring therein possess row and column symmetry.

By solving k linear equations we similarly express the latter functions as linear functions of the former, and this table will also be symmetrical. Theorem.-" The symmetric function (mi 18m2 tams 3e .••) whose partition is a specification of a separation of the function symbolized by (li'132133...) is expressible as a linear function of symmetric functions symbolized by separations of (li'122133...) and a symmetrical table may be thus formed." It is now to be remarked that the partition (lA'1A21A3•..) can be derived from 1 2 3 le 2a ae by substituting for the numbers m1e, m2,, m3e,... certain partitions of those numbers (vide the definition of the specification of a separation). Hence the theorem of expressibility enunciated above. A new statement of the law of reciprocity can be arrived at as follows: Since. P(') =µ11!µ2.!µ8al... (J1)'I(J2)12(J3)is- where (J1)I1(J2)t2(Ja)13... is a separation of (1 132133...) of specification (mµ'"mµ2amµ3a...), placing s under the summation sign to denote the le Ya Sa specification involved, µ1.!µ2,!µg8..• (J1)'1j(J2)/2(J3)7a... l!js!js!•• e—1 µlt!/a2t!/1s,!... (See also:

• PIT (O. E. pytt, cognate with Du. put, Ger. Pfutze, &c., all ultimately adaptations of Lat. puteus, well, formed from root pu-, to cleanse, whence gurus, clean, pure)

(1 +a»x) = 1 +alx +82x2 +... +a,.x" multiply each side by I +tax, thus introducing a new quantity µ; we obtain (1+ See also:

• AGE (Fr. age, through late Lat. aetaticum, from aetas)