ALGEBRAIC
FORMS
291
We pass to the symbolic forms addition, and transform each pair to a new pair by substitutions, having the same coefficients aj2, a21, a22 and arrive at functions <4 - (“i^i + a‘F-2?, A| = (Ajq A^2)2, of the original coefficients and variables (of one or more quantics) which possess the above-defined invariant property. A particular by writing for quantic of the system may be of the same or different degrees in a0, oq, a2 the symbols al, axa2, a the pairs of variables which it involves, and these degrees may Aq, A1; A2 ,, Af, AjA^ A^ vary from quantic to quantic of the system. Such quantics have and then been termed by Cayley multipartite. A o — a ^ X j -f- 2oqa2X1X2 + a 2 X | = (a^X^ + —tcy, Symbolic Form.—Restricting consideration, for the present, to binary forms in a single pair of variables, we must introduce A x = (ajXj^ + a2X2)(a1M1 + a^p^) = the symbolic form of Aronhold, Clehsch, and Gordan ; they write A2 = (ajMi + a^)2 - 5 the form r , a
anxn , rn n-1 n-1 , . n
n so that (a^i + F.2) = 1 1 +li >! + • • • + a2 a:2 =% A = a +a t = aUi + 2« AA£2 + ( ^i n02 > wherein av a2 are umbrae, such that whence A1( A2 become a^ respectively and ^(0=(fA1+«A)2are symbolical representations of the real coefficients a0, oq,... The practical result of the transformation is to change the umbrae an_i, an, and in general a™ is the symbol for ak. If we ai, a2 into the umbrae restrict ourselves to this set of symbols we can uniquely pass ax - a^ + a2X2, a^ = oq/q + from a product of real coefficients to the symbolic representations respectively. of such product, hut we cannot, uniquely, from the symbols By similarly transforming the binary n form ax we find recover the real form. This is clear because we can write A0 = (ajX] + a2)n = e4 = A™, a = while the same product of umbrae arises from Ax = (aiX1 + a2X2) (aiMi A ^2^2) ~ u '^'i Aa ’ n n-3 % 2rt-3 3 a0a3 = a1.a1 «27- . . . .n-Je, , s,h n-k k_An-k/,k Hence it becomes necessary to have more than one set of umbrae, A*—(flqXj-FoA) (aiMi + a‘2tx-2) ak 1 '^2’ so that we may have more than one symbolical representation of so that the umbrae Aj, A are a^, a^ respectively. 2 the same real coefficients. We consider the quantic to have any Theorem.—When the binary form number of equivalent representations a^=bx^c^=... So that a£ = (a1aq + a2ce2)"', /c an121212 '~kak=bn~lcb =cn~kc^... —at; and if we wish to denote, by is transformed to
umbrae, a product of coefficients of degree s we employ s sets of A £ — (Ai£i + A2?2)U> umbrae. by the substitutions -i Ex. gr. We write » a aq = Xjfj + Mi?2 > x2 ~ ^•2sx + M2?2» m-3 n 3,3 n-3 3 3 the umbrae Aj, A2 are expressed in terms of the umbrae aq, a2 by a=a al.b the formulas and so on whenever we require to represent a product of real Aj — Xj®! + X./q,, A2 = /q®i M2®2* coefficients symbolically ; we then have a one-to-one correspond- We gather that A1; A2 are transformed to aq, a2 in such wise that ence between the products of real coefficients and their sym- the determinant of transformation reads by rows as the original bolic forms. If we have a function of degree s in the coefficients, reads by columns, and that the modulus of the we may select any s sets of umbrae for use, and having made a determinant transformation is, as before, (Xq). For this reason the umbrae selection we may when only one quantic is under consideration at A!, A2 are said to be contragredient to aq, a:2. If we solve the any time permute the sets of umbrae in any manner without alter- equations connecting the original and transformed umbrae we find ing the real significance of the symbolism. Ex. gr. To express the (Xq)( — a2) = X^ — A2) + /qAj, function aaa2 - af, which is the discriminant of the binary quadratic (Xq)®! = X2( — A2) + q2Ai, a{)x + 2«1a;1x2 + a2xl = = ^4, in a symbolic form we have and we find that, except for the factor (Xq), --a2 and -f®! are 2(ffi0a2-af ) = a0a2 + a0a2- 2% . = + ab - 2ala2b1b2 transformed to — A2 and + Aj by the same substitutions as aq and x2 are transformed to iq and For this reason the umbrae = (a A - «A)2- ®2, a1 are said to be cogredient to aq and a:2. We frequently Such an expression as aA - «A, which is meet with cogredient and contragredient quantities, and we have 0a* cbx ?iax cbx in general the following definitions:—(1) “It two equally numerous sets of quantities x, y, z, ... x', y', z', ... are such that dx1 0x2 0a;2 dxx ’ whenever one set x, y, z, ... is expressed in terms of new is usually written (ab) for brevity ; in the same notation the quantities Y, Z, ... the second set x', y', z’, ... is expressed determinant, whose rows are aq, a2, «3 ; bx, b2, b3 ; <q, c2, c3 respec- in terms ofX,other new quantities X', Y', Z', ..., by the same tively, is written (abc) and so2 on. It should be noticed that the scheme of linear substitution two sets are said to be coreal function denoted by (a&) is not the square of a real function gredient quantities.” (2) “Twothesets of quantities x, y, 3, ... ; denoted by (ab). For a single quantic of the first order (ab) is £, n, f, ... are said to be contragredient when the linear substitu the symbol of a function of the coefficients which vanishes iden- tions for the first set are tically ; thus a: = XjX + qjY + »qZ + ..., (ab) «q&2 — — U/Qaq aqaQ — 0 y = X2X + qA + jqZ + ..., z = X3X + q3Y + ^Z + ..., and, indeed, from a remark made above we see that (ab) remains unchanged by interchange of a and b ; but (ab) = - (ba), and these two facts necessitate (ab) = Q. To find the effect of linear transformation on the symbolic form and these are associated with the following formulae appertaining of quantic we will disuse the coefficients au, al2, a21, a22, and em- to the second set, S=Xjf+t)+X3f +..., ploy Xj, /q, X2, /q. For the substitution H = q1f + q2^+q3f+ , X = 1 = v^+v2y+v^+ ... , X] — Xjiq + Mis2 > 2 ^2S1 + M2S2 > =: of modulus I| ^ (V4)) a2 ^ Mg I! = (y-2 ~ the quadratic form a0cef + 2a1x1x2 + a<pcl = al=f(x), becomes Aoli + SA^fc + A2?i =• A|= ^,(1), where Aq — c^qX '(-f- 2(Xj XjXo -f- a2X 2, Ai = "iXjMi + a1(X1M2 ‘; X2Mi) + a2X2/42, A2 = a0Mi + 2a1MiM2 + •
wherein it should be noticed that new quantities are expressed in terms of the old, as regards the latter set, and not vice versa." Ex. gr. The symbols citic ay az ■■■ are contragredient with the variables x, y, z, ... for when (%> y> ...):=(X1, qj, vlt ...)(X, A, z,...), >> M2) »'v2> ••• ^3, Ms; it •••
