ALGEBRAIC
290
FORMS
by recalling the definition of hpq. From a previous result we may write '(*■,..) 2( ttCtTo!...
(D ,(2)...being a notation for the successive binomial coefficients n, -n(n - 1),.... Other forms are , + n(n-l)cx™ 2x2 + ax, +nbx.
_^(-)2,r ^Stt-I)!^ g172 ... V Poth =2 TT, ! TTo!.. which must be compared with
the binomial coefficients (j)being replaced by slQ), and +fjbx71-1x +,1^cx71-22 x +
2( 2(-)S7r 1(27T -1)! D-1 D-2 .... TTj ! 7T2!... Ml ^252 Now, just as Sw = 2(^1?]7rip//27r2...)1 so Dpg=■). and substituting for Sp^, Sp2g2, ... their partition expressions we obtain, from the quantity relation, the important formula 1 (_)^ (s7r - 1)! ’r1!7r2!... x2j -1, j where (J1)A(J2)-/2... is a separation of (ppl^Pz'lv2---) > so in the operator relation, by substituting for the weight operators their expressions in terms of partition operators, we obtain the new formula, TT^TTz'.... ~^ i D 1(JT1)D , j'z,2(j2 2j ji-jJ DCJj), D(J2),... being partition D operators Reversing the two formulae we obtain r r _2(_)2J-1 ; !j(Stt, 2-l)!.. !...7-l)!(S7r i ’7ri !...7 !7r : 1 2
1
2
21
22
sh^s^Ja)... ;
(-)27r_iD(i515'17rii52?27r2-..) 1 -1)!(S7t2-1)!... ^i( Ji) ^2( J ) • • • , 2 J ii'j'2 '■ ••■7rn ■ ’’‘la ■•••7r2i! where (Jj) — {Pi^lu^P^h^^-• ■) > (^2)= {V'rih^^pyp-bzi^-- ■) > &cWe have thus complete correspondence between the algebras of quantity and differential operation. Observe the particular results ~£)lPrlxP&‘Z!‘l-• • )si>i3isM2' • •=: t. ■1| — cfh(J1)^2(J;2)... (J.. = 1. 3^-— References for Symmetric Functions. — Girard. Invention nouvelle en Valgebre. Amsterdam, 1629.—Waring. Meditationes Algebraicoe. London, 1782.—Lagrange. M6m.de Vacad. de Berlin, 1768.—Meyer-Hirsch. Sammlung von Aufgaben aus der Theorie der algebraischen Gleichungen. Berlin, 1809.— Serret. Gouts d’algebre superieure, t. iii. Paris, 1885.—Unferdinger. Sitzungsber. d. Acad. d. Wisscnsch. i. Wien. Bd. lx. Vienna, 1869.—L. Schlafli. “ Ueber die Resultante eines Systemes mehrerer algebraischen Gleichungen,” Vienna Transactions, t. iv. 1852.—Mac Mahon. “Memoirs on a New Theory of Symmetric Functions,” American Journal of Mathematics. Baltimore, Md. 1888-90 ; ‘'Memoir on Symmetric Functions of Roots of Systems of Equations,” Phil. Trans. R. S. London, 1890. III. The Theory of Binary Forms. A binary form of order n is a homogeneous polynomial of the nth degree in two variables. It may be written in the form , , n-% +, cxn-2x2 +... . ; axln+bx 1 2 1 2 or in the form , Ti-1 x +. rn)cx7171-2 2 . ax1n+{ijbx 1 2 1 which Cayley denotes by {a, b, c, ...){x1, x2)n
1
2
1
2
the special convenience of which will appear later, For present purposes the form will be written — 71-1 , /7t— ti-2 2 . n a c71+, /7l— d 1 )aixx x2+)aF:i x2 + ...+anx2, the notation adopted by German writers ; the literal coefficients have a rule placed over them to distinguish them from umbral coefficients which are introduced almost at once. The coefficients a0, ax, a2,...an, 71 +1 in number are arbitrary. If the form, sometimes termed a quantic, be equated to zero the n + coefficients are equivalent to but n, since one can be made unity by division and the equation is to be regarded as one for the determination of the ratio of the variables. If the variables of the quantic f {xx, £c2) be subjected to the linear transformation X = a 1 lfl + a12^2 > X2 = a2l£l + a22^2 > £i> £2 being new variables replacing Xj, x2 and the coefficients an, cti2, a21, a^, termed the coefficients of substitution (or of transformation), being constants, we arrive at a transformed quantic ~'*7ir , rn—'n-l^ +, en—' n-2y . ,~z' >n f (£1, £2) = a0£x + Cl £‘2 K2 )a2y£x £2 "b + an£2 in the new variables which is of the same order as the original quantic ; the new coefficients a0, a^, a2...an are linear functions of the original coefficients, and also linear functions of products, of the coefficients of substitution, of the nth degree. By solving the equations of transformation we obtain r sl = avFx - a12^2 > r£2 — — a21Xx + anX2, where r — aan a“12 = ana22—i I 21 22 I r is termed the determinant of substitution or modulus of transformation ; we assume aq, aq to be independents, so that r must differ from zero. In the theory of forms we seek functions of the coefficients and variables of the original quantic which, save as to a power of the modulus of transformation, are equal to the like functions of the coefficients and variables of the transformed quantic. We may have such a function which does not involve the variables, viz. :— F(ao, a^, a2,...an)=r E(&o, oq, u2,...an), the function F(a0) «i> is then said to be an invariant of the quantic qud linear transformation. If, however, F involve as well the variables, viz. :— F(a0, alt a2 5 £1, «i> a2, ; Xx, x2), the function F^, a:, a2,...= ; Xx, x2) is said to be a covariant of the quantic. The expression “ invariantive forms-” includes both invariants and covariants, and frequently also other analogous forms which will be met with. Occasionally the word ‘ ‘ invariants ” includes covariants ; when this is so it will be implied by the text. Invariantive forms will be found to be homogeneous functions alike of the coefficients and of the variables. Instead of a single quantic we may have several a /(ao> 2--- 5 xi> x2)> <t>(K b2,... ; x,x, x^), ... 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{a0, a^ a2,... £2), (f>[bQ, b^ *2,...; £x, £2),... If then we find F(a0, Gj, a2,...by, b^, b2,...,... ; ^2), —r F(ct0, a1; u2,...bo, bx, b2,Xx, x2), the function F, on the right which multiplies rK, is said to be a simultaneous invariant or covariant of the system of quantics. This notion is fundamental in the present theory because we will find that one of the most valuable artifices for finding invariants of a single quantic is first to find simultaneous invariants of several different quantics, and subsequently to make all the quantics identical. Moreover, instead of having one pair of variables aq, se2 we may have several pairs yx, y*', %
