Page:EB1911 - Volume 01.djvu/680

apparently, a choice of four products. ${\text{A}}_{2}^{2}{\text{B}}_{1}{\text{B}}_{2}$ gives $(2^{2})_{a}(21)_{b}-(2^{2}1)_{a}(2)_{b}$ and ${\text{A}}_{1}{\text{A}}_{2}^{2}{\text{B}}_{2}$, $(2^{2}1)_{a}(2)_{b}-(2^{2})_{a}(21)_{b}$; these two merely differ in sign; and similarly ${\text{A}}_{2}{\text{B}}_{1}{\text{B}}_{2}^{2}$ yields $(2)_{a}(2^{2}1)_{b}-(21)_{a}(2^{2})_{b}$, and that due to ${\text{A}}_{1}{\text{A}}_{2}{\text{B}}_{2}^{2}$ merely differs from it in sign. We will choose from the forms in such manner that the product of letters ${\text{A}}$ is either a power of ${\text{A}}_{1}$, or does not contain ${\text{A}}_{1}$; this rule leaves us with ${\text{A}}_{2}^{2}{\text{B}}_{1}{\text{B}}_{2}$ and ${\text{A}}_{2}{\text{B}}_{1}{\text{B}}_{2}^{2}$; of these forms we will choose that one which in letters ${\text{B}}$ is earliest in ascending dictionary order; this is ${\text{A}}_{2}^{2}{\text{B}}_{1}{\text{B}}_{2},$ and our earliest perpetuant is

$(2^{2})_{a}(21)_{b}-(2^{2}1)_{a}(2)_{b},$

and thence the general form enumerated by the generating function ${\frac {z^{7}}{(1-z)(1-z^{2})^{2}}}$ is

$(2^{\lambda _{2}+2})_{a}(2^{\mu _{2}+1}1^{\mu _{1}+1})_{b}-(2^{\lambda _{2}+2}1)_{a}(2^{\mu _{2}+1}1^{\mu _{1}})_{b}+...$$=(2^{\lambda _{2}+2}1^{\mu _{1}+1})_{a}(2^{\mu _{2}+1})_{b}.$

For the case $\theta =1$, $\theta '=3$, the condition is

$\sigma _{1}\tau _{1}\tau _{2}\tau _{3}(\sigma _{1}+\tau _{1})(\sigma _{1}+\tau _{2})(\sigma _{1}+\tau _{3})={\text{A}}_{1}{\text{B}}_{3}^{2}+{\text{A}}_{1}^{2}{\text{B}}_{2}{\text{B}}_{3}=0.$

By the rules adopted we take ${\text{A}}_{1}^{2}{\text{B}}_{2}{\text{B}}_{3}$, which gives

$(1^{2})_{a}(32)_{b}-(1)_{a}(321)_{b}+a_{0}(321^{2})_{b},$

the simplest perpetuant of weight 7; and thence the general form enumerated by the generating function

${\frac {z^{7}}{1-z.~1-z^{2}.~1-z^{3}}},$

viz:—   $(1^{\lambda _{1}+2})_{a}(3^{\mu _{3}+1}2^{\mu _{2}+1})_{b}-...\pm a_{0}(3^{\mu _{3}+1}2^{\mu _{2}+1}1^{\lambda _{1}+2})_{b},$

For the case $\theta =2$, $\theta '=3$, the condition is

$\sigma _{1}\sigma _{2}\tau _{1}\tau _{2}\tau _{3}(\sigma _{1}+\sigma _{2})(\sigma _{1}+\tau _{1})(\sigma _{1}+\tau _{2})(\sigma _{1}+\tau _{3})(\sigma _{2}+\tau _{1})(\sigma _{2}+\tau _{2})(\sigma _{2}+\tau _{3})\times (\tau _{1}+\tau _{2})(\tau _{1}+\tau _{3})(\tau _{2}+\tau _{3})=0.$

The calculation results in

$-{\text{A}}_{2}^{4}{\text{B}}_{3}{\text{B}}_{2}{\text{B}}_{1}^{2}+2{\text{A}}_{2}^{3}{\text{B}}_{3}{\text{B}}_{2}^{2}{\text{B}}_{1}^{2}-{\text{A}}_{2}^{2}{\text{B}}_{3}{\text{B}}_{2}^{3}{\text{B}}_{1}^{2}+{\text{A}}_{2}^{4}{\text{B}}_{3}^{2}{\text{B}}_{1}-2{\text{A}}_{2}^{3}{\text{B}}_{3}^{2}{\text{B}}_{2}{\text{B}}_{1}-{\text{A}}_{2}^{2}{\text{B}}_{3}^{2}{\text{B}}_{2}{\text{B}}_{1}^{3}+{\text{A}}_{2}^{2}{\text{B}}_{3}^{2}{\text{B}}_{2}^{2}{\text{B}}_{1}+{\text{A}}_{2}{\text{B}}_{3}^{2}{\text{B}}_{2}^{2}{\text{B}}_{1}^{3}+{\text{A}}_{2}^{2}{\text{B}}_{3}^{3}{\text{B}}_{1}^{2}-2{\text{A}}_{2}{\text{B}}_{3}^{3}{\text{B}}_{2}{\text{B}}_{1}^{2}+{\text{A}}_{2}{\text{B}}_{3}^{4}{\text{B}}_{1}=0.$

By the rules adopted we take ${\text{A}}_{2}^{4}{\text{B}}_{3}{\text{B}}_{2}{\text{B}}_{1}^{2}$, giving the simplest perpetuant of weight 15, viz:—

$(2^{4})_{a}(321^{2})_{b}-(2^{4}1)_{a}(321)_{b}+(2^{4}1^{2})_{a}(32)_{b};$

and thence the general form

$(2^{\lambda _{2}+4})_{a}(3^{\mu _{3}+1}2^{\mu _{2}+1}1^{\mu _{1}+2})_{b}-...\pm (2^{\lambda _{2}+4}1^{\mu _{1}+2})_{a}(3^{\mu _{3}+1}2^{\mu _{2}+1})_{b},$

due to the generating function

${\frac {z^{15}}{(1-z)(1-z^{2})^{2}(1-z^{3})}}$

For the case $\theta =1$, $\theta '=4$, the condition is

$\sigma _{1}\tau _{1}\tau _{2}\tau _{3}\tau _{4}(\sigma _{1}+\tau _{1})(\sigma _{1}+\tau _{2})(\sigma _{1}+\tau _{3})(\sigma _{1}+\tau _{4}){\text{II}}(\sigma _{s}+\tau _{t})=0;$

the calculation gives

${\text{A}}_{1}{\text{B}}_{4}({\text{A}}_{1}^{2}{\text{B}}_{2}+{\text{A}}_{1}{\text{B}}_{3}+{\text{B}}_{4})(-{\text{B}}_{3}^{2}-{\text{A}}_{1}{\text{B}}_{2}{\text{B}}_{3}-{\text{A}}_{1}^{2}{\text{B}}_{4})=0.$

Selecting the product ${\text{A}}_{1}^{4}{\text{B}}_{4}{\text{B}}_{3}{\text{B}}_{2}^{2}$, we find the simplest perpetuant

$(1^{4})_{a}(432^{2})_{b}-(1^{3})_{a}(432^{2}1)_{b}+(1^{2})_{a}(432^{2}1^{2})_{b}-(1)_{a}(432^{2}1^{3})_{b}+a_{0}(432^{2}1^{4})_{b},$

and thence the general form

$(1^{\lambda _{1}+4})_{a}(4^{\mu _{4}+1}3^{\mu _{3}+1}2^{\mu _{2}+2})_{b}-...\pm a_{0}(4^{\mu _{4}+1}3^{\mu _{3}+1}2^{\mu _{2}+2}1^{\lambda _{1}+4})_{b},$

due to the generating function

${\frac {z^{15}}{1-z.{\text{ }}1-z^{2}.{\text{ }}1-z^{3}.{\text{ }}1-z^{4}}}.$

The series may be continued, but the calculations soon become very laborious.

V. Restricted Substitutions

We may regard the factors of a binary $n^{ic}$ 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 effecting the linear transformation

$x=\lambda _{1}{\text{X}}+\mu _{1}{\text{Y}}$
$y=\lambda _{2}{\text{X}}+\mu _{2}{\text{Y}}$

so that

${\frac {y}{x}}={\frac {\lambda _{2}+\mu _{2}{\frac {\text{Y}}{\text{X}}}}{\lambda _{1}+\mu _{1}{\frac {\text{Y}}{\text{X}}}}},{\frac {\text{Y}}{\text{X}}}={\frac {\lambda _{1}{\frac {y}{x}}-\lambda _{2}}{\mu _{2}-\mu _{1}{\frac {y}{x}}}};$

it is seen that the two lines, on which lie $(x,y)$, $({\text{X}},{\text{Y}})$, 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 co-ordinates, we may look upon the substitution as changing the axes of co-ordinates retaining the same pencil. Then a binary $n^{ic}$, 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 co-ordinates we may take any other pair of lines through the origin, and for the ${\text{X}},{\text{Y}}$ 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 ${\text{X}},{\text{Y}}$ 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 $n^{ic}$ 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

$x_{1}=\lambda _{1}\xi _{1}+\mu _{1}\xi _{2},$
$x_{2}=$      $\mu _{2}\xi _{2}.$

Again, in plane geometry, the most general equations of substitution which change from old axes inclined at $\omega$ to new axes inclined at $\omega '=\beta -\alpha$, and inclined at angles $\alpha ,\beta$ to the old axis of $x$, without change of origin, are

$x={\frac {\sin {(\omega -\alpha )}}{\sin {\omega }}}{\text{X}}+{\frac {\sin {(\omega -\beta )}}{\sin {\omega }}}{\text{Y}},$
$y={\frac {\sin {\alpha }}{\sin {\omega }}}{\text{X}}+{\frac {\sin {\beta }}{\sin {\omega }}}{\text{Y}},$

a transformation of modulus

${\frac {\sin {\omega '}}{\sin {\omega }}}.$

The theory of invariants originated in the discussion, by George Boole, of this system so important in geometry. Of the quadratic

$ax^{2}+2bxy+cy^{2},$

he discovered the two invariants

$ac-b^{2},a-2b\cos \omega +c,$

and it may be verified that, if the transformed of the quadratic be

${\text{AX}}^{2}\div 2{\text{BXY}}+{\text{CY}}^{2},$

${\text{AC}}-{\text{B}}^{2}=\left({\frac {\sin {\omega '}}{\sin {\omega }}}\right)^{2}(ac-b^{2}),$

${\text{A}}-{\text{2B}}\cos {\omega '}+{\text{C}}=\left({\frac {\sin {\omega '}}{\sin {\omega }}}\right)^{2}(a-2b\cos {\omega }+c).$

The fundamental fact that he discovered was the invariance of $x^{2}+2\cos {\omega }~xy+y^{2}$, viz.—

$x^{2}+2\cos {\omega }~xy+y^{2}={\text{X}}^{2}+2\cos {\omega '}~{\text{XY}}+{\text{Y}}^{2}.$

from which it appears that the Boolian invariants of $ax^{2}+2bxy+y^{2}$ are nothing more than the full invariants of the simultaneous quadratics

$ax^{2}+2bxy+y^{2},x^{2}+2\cos {\omega }~xy+y^{2},$

the word invariant including here covariant. In general the Boolian system, of the general $n^{ic}$, is coincident with the simultaneous system of the $n^{ic}$ and the quadratic $x^{2}+2\cos {\omega }~xy+y^{2}$.

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 $\cos {\omega '}=\cos {\omega }=0$ and the substitution

$x_{1}=\cos {\theta }{\text{X}}_{1}-\sin {\theta }{\text{X}}_{2}$
$x_{2}=\sin {\theta }{\text{X}}_{1}+\cos {\theta }{\text{X}}_{2},$

with modulus unity. This is called the direct orthogonal substitution, because the sense of rotation from the axis of ${\text{X}}_{1}$ to the axis of ${\text{X}}_{2}$ is the same as that from that of $x_{1}$ to that of $x_{2}$. If the senses of rotation be opposite we have the skew orthogonal substitution

$x_{1}=\cos {\theta }{\text{X}}_{1}+\sin {\theta }{\text{X}}_{2},$
$x_{2}=\sin {\theta }{\text{X}}_{1}-\cos {\theta }{\text{X}}_{2},$

of modulus $-1$. In both cases ${\frac {d}{dx_{1}}}$ and ${\frac {d}{dx_{2}}}$ are cogredient with $x_{1}$ and $x_{2}$; for, in the case of direct substitution,

${\frac {d}{dx_{1}}}=\cos {\theta }{\frac {d}{d{\text{X}}_{1}}}-\sin {\theta }{\frac {d}{d{\text{X}}_{2}}},$
${\frac {d}{dx_{2}}}=\sin {\theta }{\frac {d}{d{\text{X}}_{1}}}+\cos {\theta }{\frac {d}{d{\text{X}}_{2}}};$

and for skew substitution

${\frac {d}{dx_{1}}}=\cos {\theta }{\frac {d}{d{\text{X}}_{1}}}+\sin {\theta }{\frac {d}{d{\text{X}}_{2}}},$
${\frac {d}{dx_{2}}}=\sin {\theta }{\frac {d}{d{\text{X}}_{1}}}-\cos {\theta }{\frac {d}{d{\text{X}}_{2}}}.$

Hence, in both cases, contragrediency and cogrediency are identical, and contravariants are included in covariants.