Page:EB1911 - Volume 01.djvu/681

Consider the binary $n^{ie}$, $(a_{1}x_{1}+a_{2}x_{2})^{n}=a_{x}^{n},$ and the direct substitution

$x_{1}=\lambda {\text{X}}_{1}-\mu {\text{X}}_{2}$
$x_{2}=\mu {\text{X}}_{1}+\lambda {\text{X}}_{2}$

where $\lambda ^{2}+\mu ^{2}=1$; $\lambda ,\mu$ replacing $\sin \theta ,\cos \theta$ respectively. In the notation

$a_{x}=a_{1}x_{1}+a_{2}x_{2}$,

observe that

$a_{a}=a_{1}^{2}+a_{2}^{2}$,
$a_{b}=a_{1}b_{1}+a_{2}b_{2}$.

Suppose that

$a_{x}=b_{x}=c_{x}=...$

is transformed into

${\text{A}}_{\text{X}}={\text{B}}_{\text{X}}={\text{C}}_{\text{X}}=...$

then of course $({\text{AB}})=(ab)$ the fundamental fact which appertains to the theory of the general linear substitution; now here we have additional and equally fundamental facts; for since

${\text{A}}_{1}=\lambda a_{1}+\mu a_{2},{\text{A}}_{2}=-\mu a_{1}+\lambda a_{2},$

${\text{A}}_{\text{A}}={\text{A}}_{1}^{2}+{\text{A}}_{2}^{2}=(\lambda ^{2}+\mu ^{2})(a_{1}^{2}+a_{2}^{2})=a_{a};$
${\text{A}}_{\text{B}}={\text{A}}_{1}{\text{B}}_{1}+{\text{A}}_{2}{\text{B}}_{2}=(\lambda ^{2}+\mu ^{2})(a_{1}b_{1}+a_{2}b_{2})=a_{b};$

$({\text{XA}})={\text{X}}_{1}{\text{A}}_{2}-{\text{X}}_{2}{\text{A}}_{1}=(\lambda x_{1}+\mu x_{2})(-\mu a_{1}+\lambda a_{2})$ $-(-\mu x_{1}+\lambda x_{2})(\lambda a_{1}+\mu a_{2})=(\lambda ^{2}+\mu ^{2})(x_{1}a_{2}-x_{2}a_{1})=(xa);$

showing that, in the present theory, $a_{a},a_{b},$ and $(xa)$ possess the invariant property. Since $+xZ=xx$ we have six types of symbolic factors which may be used to form invariants and covariants, viz.—

$(ab),a_{a},a_{b},(xa),a_{x},x_{x}.$

The general form of covariant is therefore

$(ab)^{h_{1}}(ac)^{h_{2}}(bc)^{h_{3}}...a_{a}^{i_{1}}b_{b}^{i_{2}}c_{c}^{i_{3}}...a_{b}^{i_{1}}a_{c}^{i_{2}}b_{c}^{i_{3}}...$

$\times (xa)^{k_{1}}(xb)^{k_{2}}(xc)^{k_{3}}...a_{x}^{l_{1}}b_{x}^{l_{2}}c_{x}^{l_{3}}...x_{x}^{m}$ $=({\text{A}}{\text{B}})^{h_{1}}({\text{A}}{\text{C}})^{h_{2}}({\text{B}}{\text{C}})^{h_{3}}...{\text{A}}_{\text{A}}^{i_{1}}{\text{B}}_{\text{B}}^{i_{2}}{\text{C}}_{\text{C}}^{i_{3}}...{\text{A}}_{\text{B}}^{i_{1}}{\text{A}}_{\text{C}}^{i_{2}}{\text{B}}_{\text{C}}^{i_{3}}...$

$\times ({\text{X}}{\text{A}})^{k_{1}}({\text{X}}{\text{B}})^{k_{2}}({\text{X}}{\text{C}})^{k_{3}}...{\text{A}}_{\text{X}}^{l_{1}}{\text{B}}_{\text{X}}^{l_{2}}{\text{C}}_{\text{X}}^{l_{3}}...{\text{X}}_{\text{X}}^{m}.$

If this be of order $\epsilon$ and appertain to an $n^{ic}$

$\Sigma k+\Sigma l+2m=\epsilon ,$

$h_{1}+h_{2}+...+2i_{1}+j_{1}+j_{2}+...+k_{1}+l_{1}=n,$
$h_{1}+h_{3}+...+2i_{2}+j_{1}+j_{3}+...+k_{2}+l_{2}=n,$

$h_{2}+h_{3}+...+2i_{3}+j_{2}+j_{3}+...+k_{3}+l_{3}=n;$

viz., the symbols a, b, c,... must each occur n times. It may denote a simultaneous orthogonal invariant of forms of orders $n_{1},n_{2},n_{3},...$; the symbols must then present themselves $n_{1},n_{2},n_{3}...$times respectively. The number of different symbols $a,b,c,...$denotes the degree $\theta$ of the covariant in the coefficients. The coefficients of the covariants are homogeneous, but not in general isobaric functions, 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:—

$(ab)^{2}=a_{a}b_{b}-a_{b}^{2};(xa)^{2}=a_{a}x_{x}-a_{x}^{2};$

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

$h_{1},h_{2},h_{3},...k_{1},k_{2},k_{3},...$

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 $j_{1},j_{2},j_{3},...l_{1},l_{2},l_{3},...$ to be, each of them, either zero or unity. Or, lastly, we may leave the exponents h, k, j, l, untouched and consider the product

$a_{a}^{i_{1}}b_{b}^{i_{2}}c_{c}^{i_{3}}...x_{x}^{m},$

to be reduced either to the form $g_{g}^{i}$ where g is a symbol of the series $a,b,c,...$ or to a power of $x_{x}.$ To assist us in handling the symbolic products we have not only the identity

$(ab)c_{x}+(bc)a_{x}+(ca)b_{x}=0,$

but also

$(ab)x_{x}+(bx)a_{x}+(xa)b_{x}=0,$
$(ab)a_{c}+(bc)a_{a}+(ca)a_{b}=0,$

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 ${\bar {a}}_{0}x_{1}+{\bar {a}}_{1}x_{2}=a_{x}=b_{x}$ there are four fundamental forms

(i.)	$a_{x}={\bar {a}}_{0}x_{1}+{\bar {a}}_{1}x_{2}$ of degree-order		(1, 1),
(ii.)	$x_{x}=x_{1}^{2}+x_{2}^{2}$	”	(0, 2),
(iii.)	$(xa)={\bar {a}}_{1}x_{1}-{\bar {a}}_{0}x_{2}$	”	(1, 1),
(iv.)	$a_{b}=a_{0}^{2}+a_{1}^{2}$	”	(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 $a_{x}$ and $x_{x}$ is the line perpendicular to $a_{x},$ and the vanishing of the quadrinvariant $a_{b}$ is the condition that $a_{x}$ passes through one of the circular points at infinity. In general any pencil of lines, connected with the line $a_{x}$ by descriptive or metrical properties, has for its equation a rational integral function of the four forms equated to zero.

For the quadratic ${\bar {a}}_{0}x_{1}^{2}+2{\bar {a}}_{1}x_{1}x_{2}+{\bar {a}}_{2}x_{2}^{2},$ we have

(i.)	$a_{x}^{2}={\bar {a}}_{1}x_{1}^{2}+2{\bar {a}}_{1}x_{1}x_{2}+{\bar {a}}_{2}x_{2}^{2},$
(ii.)	$x_{x}=x_{1}^{2}+x_{2}^{2},$
(iii.)	$(ab)^{2}=2({\bar {a}}_{0}{\bar {a}}_{2}-{\bar {a}}_{1}^{2}),$
(iv.)	$a_{a}={\bar {a}}_{0}+{\bar {a}}_{2},$
(v.)	$(xa)a_{x}={\bar {a}}_{1}x_{1}^{2}+({\bar {a}}_{2}-{\bar {a}}_{0})x_{1}x_{2}-{\bar {a}}_{1}x_{2}^{2}.$

This is the fundamental system; we may, if we choose, replace $(ab)^{2}$ by $a_{b}^{2}={\bar {a}}_{0}^{2}+2{\bar {a}}_{1}^{2}+{\bar {a}}_{2}^{2}$ since the identity $a_{a}b_{b}-a_{b}^{2}=(ab)^{2}$ shows the syzygetic relation

$({\bar {a}}_{0}+{\bar {a}}_{2})^{2}-({\bar {a}}_{0}^{2}+2{\bar {a}}_{1}^{2}+{\bar {a}}_{2}^{2})=2({\bar {a}}_{0}{\bar {a}}_{2}-{\bar {a}}_{1}^{2}).$

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 $a_{x}^{2},$ or, as we may say, the common harmonic conjugates of the lines $a_{x}^{2},$ and the lines $x_{x}.$ The linear invariant $a_{a}$ is such that, when equated to zero, it determines the lines $a_{x}^{2}$ as harmonically conjugate to the lines $x_{x};$ or, in other words, it is the condition that $a_{x}^{2}$ may denote lines at right angles.

References.—Cayley, “Memoirs on Quantics,” in the Collected Mathematical Papers (Cambridge, 1898); Salmon, Lessons Introductory to the Modern Higher Algebra (Dublin, 1885); E. B. Elliott, Algebra of Quantics (Oxford, 1895); F. Brioschi, Teoria dei Covarianti (Rome, 1861); W. Fiedler, Die Elemente der neueren Geometrie and der Algebra der binären Formen (Leipzig, 1862); A. Clebsch, Theorie der binären Algebraischen Formen (Leipzig, 1872); Vorlesungen über Geometrie (Leipzig, 1875); Faà de Bruno, Théorie des formes binaires (Turin, 1876); P. Gordan, Vorlesungen über Invariantentheorie, Bd. i. “Determinanten” (Leipzig, 1885); Bd. ii. “Binäre Formen” (Leipzig, 1887); G. Rubini, Teoria delle forme in generale, e specialmente delle binarie (Leue, 1886); E. Study, Methoden zur Theorie der Ternären Formen (Leipzig, 1889); Lie, Theorie der Transformationsgruppen (Leipzig, 1888–1890); Franz Meyer, Bericht über den gegenwärtigen Stand der Invariantentheorie; Jahresbericht der Deutschen Mathematiker-Vereinigung, Bd. i. (Berlin, 1892); Encyklopädie der mathematischen Wissenschaften, Bd. i., Heft 3, 4, by Heinrich Burkhardt and Franz Meyer (Leipzig, 1899); J. H. Grace and A. Young, The Algebra of Invariants (Cambridge, 1903).  (P. A. M.) 

ALGECIRAS, or Algeziras, a seaport of southern Spain in the province of Cadiz, 6 m. W. of Gibraltar, on the opposite side of the Bay of Algeciras. Pop. (1900) 13,302. Algeciras stands at the head of a railway from Granada, but its only means of access to Gibraltar is by water. Its name, which signifies in Arabic the island, is derived from a small islet on one side of the harbour. It is supplied with water by means of a beautiful aqueduct. The fine winter climate of Algeciras attracts many invalid visitors, on whom the town largely depends for its prosperity. The harbour is bad, but at the beginning of the 20th century it became important as a fishing-station. Whiting, soles, bream, bass and other fish are caught in great quantities by the Algeciras steam-trawlers, which visit the Moroccan coast, as well as Spanish and neutral waters. There is also some trade in farm produce and building materials which supplies a fleet of small coasters with cargo.

Algeciras was perhaps the Portus Albus of the Romans, but it was probably refounded in 713 by the Moors, who retained possession of it until 1344. It was then taken by Alphonso XI. of Castile after a celebrated siege of twenty months, which attracted Crusaders from all parts of Europe; among them being the English earl of Derby, grandson of Edward III. It is said that during this siege gunpowder was first used by the Moors in the wars of Europe. The Moorish city was destroyed by Alphonso; it was first reoccupied by Spanish colonists from Gibraltar in 1704; and the modern town was erected in 1760 by King Charles III. During the siege of Gibraltar in 1780–1782, Algeciras was the station of the Spanish fleet and floating batteries. On the 6th of July 1801 the English admiral Sir James Saumarez attacked a Franco-Spanish fleet off Algeciras,