AIB4 (AiB2+A1Bs+B4) (—Bs—A1B2B3 —ATB4) =0. Selecting the product A;B4B3BQ, we find the simplest perpetuant
(14).(4322)b — (13). (43221)6+(12),(432212)6 — (1).(432213)1, +ao(432214)b, and thence the general form
(1A1+4), (4µa+13.3+1201}2) b —... t a0(4'44+l3µ3+l2µ2+21A1+4)1,, due to the generating function
so that
+µ2X y a1x—a2
Y=
=
x Y'X y'
Xi+µ1X µ2—µ1x
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 of coordinates, we may look upon the substitution as changing the axes of coordinates retaining the same pencil. Then a binary n", 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 coordinates. 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 it" 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 tt
xl= XItt Ei+111 ,
x2 = P,26.
Again, in plane geometry, the most general equations of substitution which change from old axes inclined at w to new axes inclined at w'=/3—a, and inclined at angles a, 13 to the old axis of x, without change of origin, are
x—sinwa)X+sin(w—li)y
sinw sin w
__sin aX y sin w a transformation of modulus
sin co' sin w'
The theory of invariants originated in the discussion, by George Boole, of this system so important in geometry. Of the quadratic ax2+2bxy+cy2,
he discovered the two invariants
ac—b2, a—2b cosw+c,
and it may be verified that, if the transformed of the quadratic be
AX22BXY+CY2,
(sin 0 2
AC—B2= \co) (ac—b2),
A2Bcosw'+C= (sin w'1 ?(a—2bcosw+c). \sin w f
The fundamental fact that he discovered was the invariance of x2+2 cos co xy+y2, viz.
x2+2 cos w xy+y2=X2+2 cos w'XY+Y2,
from which it appears that the Boolian invariants of axe+2bxy—y2 are nothing more than the full invariants of the simultaneous quadratics
ax2+ 2bxy+y2, x2+2 cos w xy±y2,
the word invariant including here covariant. In general the Boolian system, of the general n'°, is coincident with the simultaneous system of the n"° and the quadratic x2+2 cos w xy+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=o and the substitution
xi = cos 8X1—sin OX2
x2 = sin 8X1 +cos 8X2,
with modulus unity. This is called the direct orthogonal substitu
tion, because the sense of rotation from the axis of XI to the axis
of X2 is the same as that from that of x1 to that of x2. If the senses
of rotation be opposite we have the skew orthogonal substitution j
xl =coseXi+sinOX2,
x2 = sin 8X1—cos8X2,
of modulus—1. In both cases dal and dal are cogredient with xi and x2; for, in the case of direct substitution, d
axi = cos 8aX 1— sin BdX 2'
d dx2 = sin BaXI +cos BdX 2'
and for skew substitution
adl=cos0Xl+sinead 2X d
dx2=sin 0 —cosed 2
Hence, in both cases, contragrediency and cogrediency are identical, and contravariants are included in covariants.
215
215
1—z.1—z2.1—z3.1—z4'
The series may be continued, but the calculations soon become very laborious.
V. RESTRICTED SUBSTITUTIONS
We may regard the factors of a binary n'° equated to zero as denoting n straight lines through the origin, the coorstinates being Cartesian and the axes inclined at any angle. Taking the variables to be x, y and effecting the linear transformation
x = XiX+µlY,
y = X2X+µ2Y,
sin l3
+sin wY'
Consider the binary n`a. (aixi+a2x2)^=a=, and the direct substitution
x, = )X1 —µX2,
x2=µXi+XX2,
where a2+µ2 =1; X, µ replacing cos 0, sin 0 respectively. In the notation
a: =71x1+72x2, observe that
as = +a2,
ab =aibi+a2b2. Suppose that a:==c,=...
is transformed into
Ax=BX=Cx=...
then of course (AB)=(ab) the fundamental fact which appertains to the theory of the general linear substitution; now here we have additional and equally fundamental facts; for since
A, = +µa2, A2 = +Xaz,
AA=Ai+Al= (X2+) (4+a2) =aa;
AB = A, B,+A2B2 = (X2+µ2) (a1 b, +a2b2) = ab;
(RA) = X1A2 —R2A1= (Xxi+µx2) ( —µa,+Xa2)
 (µxi +axe) (aai +µa2) = (a2+µ2) (xias  x2ai) = (xa) ;
showing that, in the present theory, ¢a, ab, and (xa) possess the invariant property. Since x;+xi=x. we have six types of symbolic factors which may be used to form invariants and covariants, viz.
(ab), a,,, ab, (xa), as, xa.
The general form of covariant is therefore
(ab) (ac) b2 (bc) ba...a~1b`2c`a aa1a7g6~a
a b c c ••
X (xa) k1 (xb) k2 (xc) k' ...a."
= (AB) 61 (AC) k2 (BC) ba...AABBC...A ACB..: X (RA) 'hi (RB) k2 (RC) ha...A7[Bj1Cj[...XI.
If this be of order a and appertain to an 00
Xk+El+2m = e,
hi +h2+...+2i, +j,+j2+... +ki+li = n,
hi+ha+...+2i2+ji+is+...+k2+12=n,
h2+h3+... +2'la+j2+ia+...+ka+la =n ;
viz., the symbols a, b, c,... must each occur it times. It may denote
a simultaneous orthogonal invariant of forms of orders n1, n2, na,...;
the symbols must then present themselves n2, na...times respectively. The number of different symbols a, b, c,...denotes the degree 0 of the covariant in the coefficients. The coefficients of the covariants are homogeneous, but not in generalisobaric fuhctions, of the coefficients of the original form or forms. Of the above general form of covariant there are important transformations due to the symbolic identities:
(ab)
s 2 2 2
as a consequence any even power of a determinant factor may be expressed in terms of the other symbolic factors, and any uneven power may be expressed as the product of its first power and a function of the other symbolic factors. Hence in the above general form of covariant we may suppose the exponents
h1, h2, ha,...k1, k2, ka,...
if the determinant factors to be, each of them, either zero or unity. Or, if we please, we may leave the determinant factors untouched and consider the exponents ji, j2, j3,•••11, 12, 18,... to be, each of them, either zero or unity. Or, lastly, we may leave the exponents h, k, j, 1, untouched and consider the product
.1 t2 :a m
to be reduced either to the form g, where g is a symbol of the series
a, b, c,... or to a power of x,. To assist us in handling the symbolic products we have not only the identity
(ab)c,+(bc)a.+(ca)b: =0, but also
(ab)x,+(bx)a2+(xa)b==0, (ab)a,+(bc)aa+(ca)ab=0,
and many others which may be derived from these in the manner which will be familiar to students of the works of Aronhold, Clebsch and Gordan. Previous to continuing the general discussion it is useful to have before us the orthogonal invariants and covariants of the binary linear and quadratic forms.
For the linear forms aox,+alx2==b. there are four fundamental forms
(i.) ate=70x1+7,x2 of degreeorder (1, 1),
(ii.) x = x1 +x2 „ (0, 2),
(iii.) (xa) =—aox2 „ (1, 1),
(iv.) ab=a02+ai (2, 0),
(iii.) and (iv.) being the linear covariant and the quadrinvariant
respectively. Every other concomitant is a rational integral function of these four forms. The linear covariant, obviously the Jacobian of aZ and xx is the line perpendicular to as, and the vanishing of the quadrinvariant ab is the condition that a. passes through one of the circular points at infinity. In general any pencil of lines, connected with the line as by descriptive or metrical properties, has for its equation a rational integral function of the four forms equated to zero.
For the quadratic aoxi+2axx1x2+72x9, we have
(i.) as =aixi+2a,xix2+a.,xi,
(ii.) xz=xl+x9,
(iii.) (ab)2=2(¢oaa—ai),
(iv.) a,,=ao+a2,
(v.) (xa) as = a1x + (a2 70 xi x2
— a1x Z .
This is the fundamental system; we may, if we choose, replace (ab)2 by 4 =74+2—4 +—a; since the identity aabb 4 b = (ab)2 shows the syzygetic relation
(ao+72)2—1Io+2ai+4) =2(¢oa~—ai)•
There is no linear covariant, since it is impossible to form a symbolic product which will contain x once and at the same time appertain to a quadratic. (v.) is the Jacobian; geometrically it
denotes the bisectors of the angles between the lines ay, or, as we may say, the common harmonic conjugates of the lines ay and the lines xy. The linear invariant as is such that, when equated to zero, it determines the lines a, as harmonically conjugate to the lines xe; or, in other words, it is the condition that al may denote lines at right angles.
End of Article: AIB4 (AiB2+A1Bs+B4) (—Bs—A1B2B3 —ATB4) 

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