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ALGEBRA I c
dition o-jTjiij = A1B1d1 = 0, where A1 + B1 + C1 = 0. We have a choice of six products for our base, viz., B^, B^, A^, A^, A®B1, A^, and these yield respectively | a0(l2)j(l)c | , | a0(l)b(l22)c, I (IVo (l)c I , I (l)a&0 (l )c I , 1 (l2)a(l)6 C0 |,| (l)a(lVo I , Taking the product A2B1 and the corresponding fonn I (12)«(1V0 i we find | (l)a(l)i | c0— I (l)a(l)6 I C1 + | *0(1)5 | C2 = (^2^1 — 3ct3&o)Co — (Uj&j — 2a2&o)Ci + (Uq^I — ®l^o)^2 ) the simplest perpetuant of the kind ; the general form, given by the generator (-zf being |(ill+2)„(V'+1)bi co I Similarly for 4 binary forms the simplest perpetuant, of degrees 1, 1, 1, 1, is (corresponding to product A^B2^) | (i4)a(i2)&(i)A |; and has the expression “f" 4* 2 biG'^y 4~ 4" ^2v b | c0^4 4‘1 b.2c0dx + b^d^ — 3 b) 4“ 34- 3b3 b^g4“ ^b^c^d^ boG'fb^ 4” 3b1 4- 66^0^2 + 4&1c1S1 - 8&2«o^i + 3bzc0dQ) 4- a5{ - lOigCjdj 4- lO^CgC^j - 561c1c?0 4- 10J2c0fZ0) 4- a6( 4- ISVi^o “ 15 Vo^o)Y. Restricted Substitutions, We may regard the factors of a binary nic equated to zero as denoting n straight lines through the origin, the co-ordinates being Cartesian and the axes inclined at any angle. Taking the variables to be x, y and affecting the linear transformation cc=X1X4-yRiY, y='Kc!lL + , so that Y Xl y-X2 « Y y + X ; Y’ Xy it is seen that the two lines, on which lie (x, y), (X, Y), have a definite projective correspondence. The linear transformation replaces points on lines through the origin by corresponding points on projectively corresponding lines through the origin ; it therefore replaces a pencil of lines by another pencil, which corresponds projectively, and harmonic and other properties of pencils which are unaltered by linear transformation we may expect to find indicated in the invariant system. Or, instead of looking upon a linear substitution as replacing a pencil of lines by a projectively corresponding pencil retaining the same axes oi co-ordinates, we may look upon the substitution as changing the axes of co-ordinates retaining the same pencil. Then a binary w*0, equated to zero, represents n straight lines through the origin, and the x, y, of any line through the origin are given constant multiples of the sines of the angles which that line makes with two fixed lines, the axes of co-ordinates. As new axes of coordinates we may take any other pair of lines through the origin, and for the X, Y corresponding to x, y any new constant multiples of the sines of the angles which the line makes with the new axes. The substitution for x, y in terms of X, Y is the most general linear substitution in virtue of the four degrees of arbitrariness introduced, viz., two by the choice of axes, two by the choice of multiples. If now the nic denote a given pencil of lines, an invariant is the criterion of the pencil possessing some particular property which is independent alike of the axes and of the multiples, and a covariant expresses that the pencil of lines which it denotes is a fixed pencil whatever be the axes or the multiples. Besides the invariants and covariants, hitherto studied, there are others which appertain to particular cases of the general linear substitution. Thus, what have been called, seminvariants are not all of them invariants for the general substitution, but are invariants for the particular substitution X 1 ~ ^l£l + Ml?2 >
- 2 =
^2^2 • Again, in plane geometry, the most general equations of substitution which change from old axes inclined at w to new axes inclined at u = fi-a, and inclined at angles a, /3 to the old axis of x, without change of origin, are
FORMS
a; = sin Q-tt)x + sin(w-p)Y 5 sin w ^ sin w sin a^. sin STT y=-—x 4- sin ——^Y, sin w u a transformation of modulus sin w' sin w' The theory of invariants originated in the discussion, by Boole, of this system so important in geometry. Of the quadratic ax2 + 2bxy 4- cy2, he discovered the two invariants ac-b2, a- 2bcos w4-c, and it may be verified that, if the transformed of the quadratic be AX2 4- 2BXY 4- CY2, AC-B2=fSA-^y(ac-62)
sm u> / v
A - 2B cos u’ + C=( Ssm ^.n Uw/ (a-2b cos w 4- c).' The fundamental fact that he discovered was the invariance of cc2 4-2 cos w cci/4-1/2, viz.— £C2 4- 2 cos a) xy + y2 = X2+2 cos w'XY 4- Y2, 2 from which it appears that the Boolian invariants of ax + 2bxy 4- y2 are nothing more than the full invariants of the simultaneous quadratics ax2 4- 2bxy 4- y2, xz + 2 cos co xy + y2, the word invariant including here covariant. In general the Boolian system, of £he general nic, is coincident with the simic ultaneous system of the n and the quadratic x2 4- 2 cos uxy --y2. Orthogonal System.—In particular, if we consider the transformation from one pair of rectangular axes to another pair of rectangular axes we obtain an orthogonal system which we will now briefly inquire into. We have cos w'^cos w = 0 and the substitution Xj = cos flXj - sin 0X2, x2 = sin gXj 4- cos 0X2, with modulus unity. This is called the direct orthogonal substitution, because the sense of rotation from the axis of X1 to the axis of X2 is the same as that from that of xx to that of x2. If the senses of rotation be opposite we have the skew orthogonal substitution xl = cos tfXj 4- sin 0X2, =sin tfXj - cos 0X2, of modulus -1. In both cases and -L are cogredient with dd&Y doc *2 xx and x2 ; for, in the case of direct substitution, d d ■ d -7—= cos 0-TTr - sm a , doC]' ' d • nd . d ^=sm<’jxl+c°si,dx1: and for skew substitution d a& , ■ ad dxi dXx dXi d ■ d a d -7—= sm 0-77Fftxg dXi - cos 0-t^dX . 2
Hence, in both cases, contragrediency and cogrediency are identical, and contra variants are included in covariants. Consider the binary nic, (a1x1 + a2x<2)n—a,‘, and the direct substitution Xi = XXi — fiX2, X2—yXi 4- XX2, 2 2 where X 4-/i =1 ; X, jn replacing cos 0, sin0 respectively, In the notation ax = ajXi + a2x.2, observe that aa = a2+a2, ab = aib 4- o,2b2. Suppose that ax—bx -—Cx—1... is transformed into Ax —Bx—Cx=... then of course (AB) = (a6) the fundamental fact which appertains to the theory of the general linear substitution ; now here we have additional and equally fundamental facts ; for since Ai = Xosj -)- ga^, A2 — — y.a 4- Xet^, Aa = A2 4-A| = (X24-M2)(a! +al) = aa ; AB = AjBi 4- AgBg^ (X2 -f y?){aibi 4- a^jb^) = ab ; (XA) = XiA2 - X2Ai = (Xxi 4- [Mi 4- Xa2) - ( - y.Xi 4- Xa32)(Xai 4- ya2) — (X2 4- y2){xia2 - x2ai) = {xa);
