S21 _ — ai Oaol — a2Ca01 — a13a30+a21 The See also:

• FORMULA
• FORMULA (Lat. diminutive of forma, shape, pattern, &c., especially used of rules of judicial procedure)

+n poxPy 4 + • 1 (1-See also:

• AIX

.1 p! q! S55 ) 2ri! 7r2!...''PlgihP2g2'.. ; )`S ,r hD4= E (Pi 1) '2 ! 'i S 2+ 2—1)! -1 *1 *g l pr! q1! )( p2! q2!... S " 1ri! sr2!...SP1g15P2g2 ... See also:

• DIFFERENTIAL

Writing DPQ=p1q~d od 1, exp(Eodlo+vdoi) = (1+AD10+vDo1+...+, 'v4DP4+...)f; now, since the introduction of the new quantities o, v results in the addition to the function (p1g1p2g2p2q3.••) of the new terms t2iv41(p2g2p3g3•••) +KP2v42(plgip3g3• ••) +113P43(piglp2g2•••) +••• , we find DP1g1(p1g1p2gsp2q3...) = (p2g2psg3...) and thence DP1g1DP242DPags... (plgip2g2p3g3...) = 1 while D„f=o unless the See also:

• PART

See also:

• R2(rt-s2)

Ueber die Resultante eines Systemes mehrerer algebraischen Gleichungen," See also:

• VIENNA (Ger. Wien; Lat. Vindobona)
• VIENNA, CONGRESS OF (1814-1815)

The coefficients ao, a1, ,...an, n+1 in number are arbitrary. If the form, some-times termed a quantic, be equated to zero the n+1 coefficients are See also:

• EQUIVALENT

which have different coefficients, the same variables, and are of the same or different degrees in the variables; we may transform them all by the same substitution, so that they become f (ao, a1, a2,... ; t1, 6), ((ba, bl, b2 ... ; ti, E2),.... If then we find F(ag, a1, a2,...bo, bt, b2, ; E1, 52), s r 'F(ag, a1, e2,...bo, b1, b2, ; x1, Si), and thence µdio+vdoi _ (µ2d2c+2µvdu+v2do2) +••. = log (1+A1)1o+vDe1+...+µvvgDpg+...). From these formulae we derive two important relations, viz. (-)s+g-1(p+q—1)!d '(-)1,r1(See also:

• EA(nar...), (p.s„,.,.)
• EA(psr...), (p, „...)• (pqr...)

A particular quantic of the system may be of the same or different degrees in the pairs of variables which it involves, and these degrees may vary from quantic to quantic of the system. Such quantics have been termed by Cayley multipartite. Symbolic Form.—Restricting See also:

• CONSIDERATION (from Lat. considerare, to look at closely, examine, generally taken to be from con-, and the base seen in sidus, sideris, a star, the word being supposed to be originally an astrological or astronomical term)

We write alaz=ai-laz•bl 2b2, 3 =a1 a2 n-3 .b1 n-3b28.c1n -3 8 a3e2, and so on whenever we require to represent a product of real coefficients symbolically; we then have a one-to-one correspondence between the products of real.coefficients and their symbolic forms. If we have a function of degree s in the coefficients, we may select any s sets of umbrae for use, and having made a selection we may when only one quantic is under consideration at any See also:

• TIME (0. Eng. Lima, cf. Icel. timi, Swed. timme, hour, Dan. time; from the root also seen in " tide," properly the time of between the flow and ebb of the sea, cf. O. Eng. getidan, to happen, " even-tide," &c.; it is not directly related to Lat. tempus)
• TIME, MEASUREMENT OF
• TIME, STANDARD

For this See also:

• REASON (Lat. ratio, through French raison)

In either case (AB) =AIBz—A,BI = (Xs)(ab); and, from the definition, (ab) possesses the invariant property. We cannot, however, say that it is an invariant unless it is expressible in terms of the real coefficients. Since (ab) =aib3—a2b,, that this may be the case each form must be linear; and if the forms be different (ab) is an invariant (simultaneous) of the two forms, its real expression being agbi—aibo. This will be recognized as the resultant of the two linear forms. If the two linear forms be identical, the umbral sets al, a2; bi, b2 are alternative, are ultimately put equal to one another and (ab) vanishes. A single linear form has, in fact, no invariant. When either of the forms is of an order higher than the first (ab), as not being expressible in terms of the actual coefficients of the forms, is not an invariant and has no significance. Introducing now other sets of symbols C, D, ...; c, d, ... we may write (AB)'(AC)~(BC)k... _ Ns) i+i+k+ ...( ab)ac) t(bc)k..., so that the symbolic product (ab)'(ac)i(bc)k..., possesses the invariant property. If the forms be all linear and different, the function is an invariant, viz. the ith power of that appertaining to as and bx multiplied by the :eh power of that appertaining to as and cx multiplied by &c. If any two of the linear forms, say px, qx, be supposed identical, any symbolic expression involving the factor (pq) is zero. See also:

• NOTICE

In order that (ab)i(ac)i(bc)k... may be a simultaneous invariant of a number of different forms See also:

• ATE

If the forms be identical the sets of symbols are ultimately equated, and the form, provided it does not vanish, is a covariant of the form a:. The expression (ab)4 properly appertains to a quartic; for a quadratic it may also be written (ab)2 (cd)2, and would denote the square of the discriminant to a factor pres. For the quartic (ab)4=(albs —a2bi)4=albs—4aia.sbibz+6aa2bib2 _ _ -4aia2bibs+See also:

• ALB (Lat. alba, from albus, white)

(I.) Introduce now new umbrae di, d2 and recall that +d2—di are cogredient with xi and x2. We may in any relation substitute for any pair of quantities any other cogredient pair so that writing +ds, —di for xi and x2, and noting that gx then becomes (gd), the above-written identity becomes (ad)(bc)+(bd)(ca)+(cd)(ab) =0 . (II.) Similarly in (I.), writing for ci, c2 the cogredient pair —y2,+yi, we obtain axby—aybx=(ab)(xy). . ' . . (III.) Again in (I.) transposing ax(bc) to the other side and squaring, we obtain 2(ac) (bc)axbx = (bc)2ai+(ac)2bx— (ab)2cs . (IV.) and herein writing ds,—di for x2, 2(as)(be)(ad)(bd) =(be)2(ad)2+(ac)2(bd)2—(ab)2(cd)2. (V.) As an See also:

• ILLUSTRATION

Asymbolical expression may be always so transformed that the power of any determinant factor (ab) is even. For we may in any product interchange a and b without altering its signification; therefore (ab)sm+141 = — (ab)2m+14 where 41 becomes 42 by the interchange, and hence (ab)2m+1421 = 2(ab)2" (41 — 42) ; and identity (I.) will always result in transforming 41-42 so as to make it divisible by (ab). Ex. gr. (ab) (ac) bxc2, = - (ab) (bc)a.cz =2(ab)c~{(ac)by-(bc)as} =2(ab)2c;; so that the covariant of the quadratic on the left is See also:

• HALF

An important associated operation is 02 a2 020y2—0x2aylr which, operating upon any polar, causes it to vanish. Moreover, its operation upon any invariant form produces an invariant form. Every symbolic product, involving several sets of cogredient variables, can be exhibited as a sum of terms, each of which is a polar multiplied by a product of powers of the determinant factors (xy), (xz), (Yz),•.. Transvection.—We have seen that (ab) is a simultaneous invariant of the two different linear forms ay, b., and we observe that (ab) is equivalent to where f = a=, 4 = bx. If f = a: , 4 =b: be any two binary forms, we generalize by forming the function (m -k)! (n-k)! (Of See also:

• A4(Z)

We may suppose Of, 42 to be any two covariants appertaining to a system, and the process of transvection supplies a means of proceeding from them to other covariants. The two forms a=, by, or 4z, 0g, may be identical; we then have the kth transvectant of a form over itself which may, or may not, vanish identically; and, in the latter case, is a covariant of the single form. It is obvious that, when k is uneven, the k°h transvectant of a form over itself does vanish. We have seen that transvection is equivalent to the performance of partial differential operations upon the two forms, but, practically, we may regard the process as merely substituting (ab)k, (40)k for ayby, 4xk respectively in the symbolic product subjected to transvection. It is essentially an operation performed upon the product of •two forms. If, then, we require the transvectants of the two forms f +af , 4r+µ4', we take their product f4+af'4+µf4'+xpf'4', and the km transvectant is simply obtained by operating upon each term separately, viz. (y y r 4)k+a(f', ~)k+ sc ( r ~')"+)(,/', 0n /k; and, moreover, if we require to find the kth transvectant of one linear system of forms over another we have merely to multiply the two systems, and take the kth transvectant of the See also:

• SEPARATE

Put m- i for m, n- 1 for n, and multiply through by (ab) ; then {(f =(ab)a'aybz'-t'm+ni2(xy)(f,4)2, = (ab)a,'bzkby -4; I 2 (xy) (f,4)2• Multiply byc T1 and for yl, y2 write 4,-ci; then the right-hand side becomes (ab)(bc)az lbs-2cri +m m+ ;- ac:( f,O)', of which the first term, writing cf =0, is a, b,, cs-2(ab)(bc)azcz = - a,by-kc ' 5 (be) 26z+(ab)2cx - (ac)2bs _ -2 a,, (bc)2by s(((c: +cy (ab)2a b: _bs(ac)2aikc;k _-23 (4,42)2•f+( ,42)2.0 —( ,4)2.4J ; and, if (f,4)1=k'"+" 2 (f,4)1 ' 1.c9 1=k5+„-2kyc'' 1• , A See also:

• ACT (Lat. actus, actum)

Xi, X2, /L2 being as usual the coefficients of substitution, let x,- +x2- =D xi— +x2a =D~K, t 2 AA au, u2 µtax,+1+2a a =Dµ,k, Ai— +P2— =DILA, be linear operators. Then if j, J be the original and transformed forms of an invariant w being the weight of the invariant. Operation upon J results as follows: DaaJ=wJ; DA,,J=0; D,AJ =0 jD,.wJ =wJ. The first and See also:

• FOURTH

The second and third are those upon the See also:

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

The second evectant is obtained by similarly operating upon all the symbols remaining which only occur in determinant factors, and so on for the higher evectants. Ex. gr. From (ac)2(bd)2(ad)(bc) we obtain (bd)2(bc)csda+(at)2(ad)cxdi - (bd) 2(ad) azbx - (ac)2(bc) a ,bz =4(bd)2(bc)cydz the first evectant; and thence 4czdi the'second evectant; in fact the two evectants are to numerical factors pres, the cubic covariant Q, and the square of the original cubic. If 0 be the degree of an invariant j Oj = a-=+aa' +... +a„a aj n-1 aj .aj =alaao+al a2aa1+... +a2aa.. and, herein transforming from a to x, we obtain the first evectant p,. ( — )kk,. kaj z xx2 aak k Combinants.—An important class of invariants, of several binary forms of the same order, was discovered by Sylvester. The invariants in question are invariants qua linear transformation of the forms themselves as well as qua linear transformation of the variables. If the forms be az, bi, ci,... the Aronhold process, given by the operation S as between any two of the forms, causes such an in-variant to vanish. Thus it has annihilators of the forms -d -d -d ae b°+a1ab1+a2db2+... 0ciao+bldal+b2da2+...

and Gordan, in fact, takes the See also:

• SATISFACTION (Lat. salisfacerc, to satisfy)

Similarly regarding xi, x2 as additional parameters, we see that every covariant is expressible as a rational function of n fixed covariants. We can so determine these n covariants that every other covariant is expressed in terms of them by a fraction whose denominator is a power of the binary form. First observe that with f'2 = as = b= =...,f, =alas', f 2 = a2 a:-1, (ab)(af)—(bf)a,,. and that thence every symbolic product is equal to a rational function of covariants in the form of a fraction whose denominator is a power of f. Making the substitution in any symbolic product the only determinant factors that present themselves in the numerator are of the form (af), (bf), (cf),...and every symbol a finally appears in the form. = (af)kan,, -k Y'k !Gk has f as a factor, and may be written f. Ilk; for observing that 4'o=f. =f. uo; Y'1=0=f.u1; where uo=1, u1=o, assume that ¢k = (af)ka:2 =f. uk =az.ukk0") Taking the first polar with regard to y (n—k)(af)ka= k-' +k(af)k-ia: (ab)(n—1)bs2by n k,,-2k-1 n-1 k(,.-2) =k(n-2)azux uy+See also:

• NAY, or NEY

Every covariant is rationally expressible by means of the forms f, u2, since, as we have seen uo =1, u1=o. It is easy to find the relations u2 = 2 (f f') 2 u3=((f ,f')2,f"') u4=2(f,f')4•fi- {(f,f')2}2, and so on. To exhibit any covariant as a function of See also:

• U11

The first transvectant, (f,f')i=(ab) asbs,vanishes identically. Calling the discriminate D, the solution of the quadratic as =o is given by the formula a 1 a,, a0 (acx,+a1x2—x2 - doxi+alx2+x2' jI 3 If the form a' be written as the product of its linear factors p,,qz, the discriminant takes the form —(pq)2. The vanishing of this invariant is the condition for equal roots. The simultaneous system of two quadratic forms as, ay, say f and 0, consists of six forms, viz. the two quadratic forms f, 4.; the two discriminants (f, f')2,(0,0')2, and the first and second transvectants of f upon 0, (f, 0)1 and (f, 0)2, which may be written (aa)a,,a,, and (aa)2. These fundamental or ground forms are connected by the relation -21 (f,o)11 2=f2(d,,o')22fo(.f,o)2+0(.f,f )2- If the covariant (f,¢)' vanishes f and 0. are clearly proportional, and if the second transvectant of (f, 4,)i upon itself vanishes, f and 4. possess a common linear factor; and the condition is both necessary and sufficient. In this case (f, 0)1 is a perfect square, since its discriminant vanishes. If (f,4.)' be not a perfect square, and r,,, s,, be its linear factors, it is possible to express f and 4, in the canonical forms ai(r~)2+X2(s,,)2, t.i(r,,)2+µ2(s,,)2 respectively. In fact, if f and .0 have these forms, it is easy to verify that (f, 0)1= (aµ) (rs)r,,s,,. The fundamental system connected with n quadratic forms consists of (i.) the n forms themselves f , (ii.) the (2) functional determinants (fi,fk)', (iii.) the (2i) in-variants (fi, fk)2, (iv.) the (3) forms (fi, (fk, f,,,))2, each such form remaining unaltered for any permutations of i, k, m. Between these forms various relations exist (cf. Gordan, § 134).

The Binary Cubic.—The complete system consists of f =as , ( f,f')2 =(ab)2a,,bz =L1y, (f, A) =(ab)2(ca)bsc2 =Qzs, and (0,0')2 = (ab)2(cd)2(ad) (bc) = R. To prove that this system is complete we have to consider (f,0)2, (0,0')1, (f,Q)i, (f,Q)2, (f,Q)', (A,Q)', (O,Q)2, and each of these can be shown either to be zero or to be a rational integral function of f, 0 Qand R. These forms are connected by the relation 2Q2+A3+Rf2 =0. The discriminant of f is equal to the discriminant of 0, and is therefore (0, 0')2=R; if it vanishes both f and 0 have two roots equal, . is a rational factor off and Q is a perfect See also:

• CUBE (Gr. K46os, a cube)

Iii.) that the substitution, z ~ f, reduces x ~Xl T-x18xz to the form j 21z=2 f2A—A8—3jf3• The discriminant, whose vanishing is the condition that f may possess two equal roots, has the expression See also:

• J2(z)

For, since -2t2=A3—2if2A—3j(—f)3, he compares the right-hand side with the cubic resolvent k2-2ia2k-3jas, of f=0, and notices that they become identical on substituting A for k, and —f for a; hence, if k,, k2, k3 be the roots of the resolvent -212 = (A+k1f) (A+k2f)(A+k3f) ; and now, if all the roots of f be different, so also are those of the resolvent, since the latter, and f, have practically the same discriminant; consequently each of the three factors, of -2t2, must be perfect squares and taking the square root _ 1 t -24)•x•4); and it can be shown that (A, x, ¢ are the three conjugate quadratic factors of t above mentioned. We have A+k1f=Ak2f=x2, A+k3f=#G2, and Cayley shows that a root of the quartic can be expressed in the determinant form the remaining roots being obtained by varying the signs which occur in the radicals (0v, xv, 4) . The transformation to the normal form reduces the quartic to a quadratic. The new variables y,=0 are the linear factors of 4). If ¢=rx.sx, the y2 =1 normal form of as, can be shown to be given by (rs)'.ay = (ar) 4sz+6 (ar) 2 (as) 2rsss+ (as) ; ¢ is any one of the conjugate quadratic factors of t, so that, in determining rx, sx from sl A+klf =o, k, is any root of the resolvent. The transformation to the normal form, by the solution of a cubic and a quadratic, therefore, supplies a solution of the quartic. If (ap) is the modulus of the transformation by which a: is reduced to is the normal form, i becomes (Xu)4i, and j, (au)3j; hence j3 is absolutely unaltered by transformation, and is termed the See also:

• ABSOLUTE (Lat. absolvere, to loose, set free)

The form j is completely defined by the relation (f, j)3 = o as no other covariant possesses this property. Certain convariants of the quintic involve the same determinant factors as appeared in the system of the quartic; these are f, H, i, T and j, and are of special importance. Further, it is convenient to have before us two other quadratic covariants, viz. r = (j, j)2jxjx ; B = (ir)ixrx • four other linear covariants, viz. a=—(ji)2jx; =(ia)is; 7 =(ra)r:3=(rp)rx. Further, in the case of invariants, we write A= (i, i')2 and take three new forms B = (i, r)2; C = (r, r')2 ; R = (py). Hermite expresses the quintic in a forme-type in which the constants are invariants and the variables linear covariants. If a, p be the linear forn}s, above defined, he raises the identity ax(ap)=ax(ap)—px(aa) to the fifth power (and in general to the power n) obtaining (a$) sf = (a,5) See also:

• SAX, ANTOINE JOSEPH

To determine them notice that R=(a3) and then (f, ab) b = — Rb (k1+k2+k3) , (f, a43) 6 = — 5R5 (m1k1+m2k2+m3k2) , (f, a332) b= -10R5 (mlkl+mzk2+m3k3) three equations for determining k1, k2, k3. This canonical form depends upon j having three unequal linear factors. When C vanishes j has the form j = psgx, and (f, j)3 = (ap)2(ag)as =o. Hence, from the identity as(pq)=px(aq)—gx(ap), we obtain (pq)bf=(aq)bpy -5(ap)(aq)4pzgx—(ap)5q', the required canonical form. Now, when C=o, clearly (see ante) R2j=S2pwhere p=3+2Ba; and Gordan then proves the relation 6R4.f = B35+5B34p—4A2p6, which is Bring's form of quintic at which we can always arrive, by linear transformation, whenever the invariant C vanishes. Remark.—The invariant C is a numerical multiple of the resultant of the covariants i and j, and if C =o, p is the common factor of i and j. The discriminant is the resultant of a and a and. of degree 8 in the coefficients; since it is a rational and integral function of the fundamental invariants it is expressible as a linear function of As and B; it is independent of C, and is therefore unaltered when C vanishes; we may therefore take f in the canonical form 6R4f B55 +5Bb4p— 4A2p6. The two equations ALGE'BRAIC'FORMS a simultaneous invariant of the three forms, and now suppressing the dashes we obtain 6(abc +2fgh—aft -bg2—See also:

• CH2
• CH2(OH)

9o-128, 1869, and vi. 436-512, 1873). The complete covariant and contravariant system includes no fewer than 34 forms; from its complexity it is desirable to consider the cubic in a simple canonical form; that chosen by Cayley was ax3+by3+cz3+6dxyz (Amer. J. Math. iv. 1-16, 1881). Another form, associated with the theory of elliptic functions, has been considered by Dingeldey (Math. Ann. xxxi: 157-176, 1888), viz. xy2—4z3-1-g2x2y+g3x3, and also the special form axz2—4byb of the cuspidal cubic. An investigation, by non-symbolic methods, is due to F. C. J. Mertens (Wien.

Ber. xcv. 942-991, 1887). See also:

• HESSE
• HESSE (Lat. Hessia, Ger. Hessen)

Math. lxix. 323- r.). See also:

• AUGUST (originally Sextilis)

Giorn. xiv. 1-14, 1876; also Math. Ann. xi. 30-41, 1877). The system of four forms, of which two are linear and two quadratic, has been investigated by Perrin (S. M. F. See also:

• BULL
• BULL, GEORGE (1634–1710)
• BULL, JOHN (c. 1562–1628)
• BULL, OLE BORNEMANN (18ro-188o)

Such a symbolic product, if it does not vanish identically, denotes an invariant or a covariant, according as factors ax, b5, cx,... do not or do appear. To obtain the real form we multiply out, and, in the result, substitute for the products of symbols the real coefficients which they denote. For example, take the ternary quadratic (aix,+a2x2+a3x3)2 =a2 x, or in real form ax:+bxs+cx:+2fx2x3+2gx3xi+2hxix2• We can see that (abc)asbxcs is not a covariant, because it vanishes identically, the interchange of a and b changing its sign instead of leaving it unchanged; but (abc)2 is an invariant. If ax, b:, c= be different forms we obtain, after development of the squared determinant and See also:

• CONVERSION (Lat. conversio, from convertere, to turn or ,change)

This is of degree 8 in the coefficients, and degree 6 in the variables, and, for the canonical form, has the expression -9m6(x3+y3+z3)2 — (2m +5m4+20m7) (x3 +y'+z2)xyz — (15m2+78mb -12m2)x2y2z2 + (1 +8m3)2(y3z3 +z3x3 +x3y3)• Passing on to the ternary quartic we find that the number of ground forms is apparently very See also:

• GREAT

F. Bull. xviii. 1-8o, 189o) ; Rosanes (Math. Ann. vi. 264) ; ' and Gerbaldi (Annali (2), xvii. 161-196). Ciamberlini has found a system of 127 forms appertaining to three ternary quadratics (Batt. G. See also:

• XXIV

J. xii. 1-6o, I15-160, and Camb. Phil. Trans. xiv. 409-466, 1889). He-proves, by means of the six linear partial differential equations satisfied by the Concomitants, that, if any concomitant be See also:

• EXPANDED

16) and by Cayley (Amer. J. iv., 1881). Clebsch, in' 1872, in papers in Abh. d. K. Alead. d. U. zu See also:

• GOTTINGEN

It has been shown above that a covariant, in general, satisfies four partial differential equations. Two of these show that the leading coefficient of any covariant is an isobaric and homogeneous function of the coefficients of the form; the remaining two may be regarded as operators which cause the vanishing of the covariant. These may be written, for the binary n' mkak_1- -x2— =0; m(n—k)aa i-k—xia--d =0; or in the form St—x2ax-=0, O where d d =ao i+2aiaa } d d d O = naidao+ (n -1)a2aai +... +a,, . Let a covariant of degree a in the variables, and of degree 0 in the coefficients (the weight of the leading coefficient being w and nO -2w =e), be Cox: +ecixi 1x2+.... Operating with 0—xo -we find SlCo=o; that is to say, Co satisfies one of the two partial differential equations satisfied by an in-variant. It is for this reason called a seminvariant, and every seminvariant is the leading coefficient of a covariant. The .whole theory of invariants of a binary form depends upon the solutions of the equation SZ=o. Before discussing these it is best to trans-form the binary form by substituting I !al, 2 !47,2, 3 ! a2,...n !an, for al, a2, aa...a,, respectively; it then becomes aexi+naixi 1x2 } n(n—1)a~xi~4 F...--n!anx2; and 0 takes the simpler form ag-cii +alaa2+a2aaa+'"+a~~-i do One See also:

• ADVANTAGE

Hence, excluding ao, we may, in partition notation, write down the fundamental solutions of the equation, viz.-. (2), (3), (4),...(n), and say that with ao, we have an algebraically complete system. Every symmetric function denoted by partitions, not involving the figure unity (say a non-unitary symmetric function), which remains unchanged by any increase of n, is also a seminvariant, and we may take if we please another fundamental system, viz. ao,(2), (3), (22), (32),...(21") or (321("-a)). Observe that, if we subject any symmetric function (pip2Ps...) to the diminishing process, it becomes aoi-172 (P2P3...). Next consider the solutions of 0=o which are of degree 0 and weight w. The general term in a solution involves the product av°aila= ...a*"wherein Mir=o, 157r, =w; the number of such products that may appear depends upon the number of partitions of w into o or fewer parts limited not to exceed n in magnitude. Let this number be denoted by (w; o, n). In order to obtain the seminvari-ants we would write down the (w; 0, n) terms each associated with a literal coefficient; if we now operate with S2 we obtain a linear function of (w—I; B, n) products, for the vanishing of which the literal coefficients must satisfy (w—I; o, n) linear equations; hence (w; 0, n)—(w—l; B, n) of these coefficients may be assumed arbi- trarily, and the number of linearly independent solutions of 12 =o, of the given degree and weight, is precisely (w; B, n)—(w—I; 0, n). This theory is due to Cayley; its validit _depends upon showing that the (w—l; 0, n) linear equations satisfied by the literal coefficients are independent; this has only recently been established by E. B. Elliott.

These seminvariants are said to form an asyzygetic system. It is shown in the See also:

• ARTICLE (from Lat. articulus, a joint)