Yoz

YOZ, parallel to OX, is the same-say x =a; and similarly for y= b, z = c. Hence these three lengths (fig. 6) or their num bers a, b, c, or the three planes through P, or their equations are co-ordinates of P. Clearly any equation of the first de gree in x, y, z, as lx+my+nz+c = o, is depicted by a plane, with axial intercepts -c/l, -c/m, -c/n, and conversely. Also an equation of the second degree in x, y, z, symbolized by is depicted by a quadric (ellipsoid, paraboloid, hyperboloid), and so on for higher degrees and surfaces.

If there be four such independents (x, y, z, u), any set of values named will specify an element in a manifold of four dimensions, and all such would constitute the total fourfold extent. Any single equation binding together the four would be depicted by a threefold extent, a section of the fourfold, as a curve is a tion of -a surface, and a surface a section of a solid. Similarly for any number of co-ordinates or values determining elements. Although no logical difficulty arises, the geometric representation offers a serious one; for the constructive power of our space sense does not reach beyond three dimensions, and so we cannot envisage a fourfold, although the assemblage of musical notes appears to be such, since we distinguish a note by four marks high, loud, long, rich bre), that is by four co-ordinates. But we do not imagine a fourth axis U perpendicular to X, Y, Z, to the whole XYZ space, as Z is to XOY, to the X Y-plane. In spite of this culty, however, the abstract reasoning is undisturbed. Thus, in transforming to polar co-ordinates, consider this example : in XOY draw OP or pi to P (x, y), inclined 0, to Y, so that x= P1 sin 01,y =Pi cos 0i. So in O - X YZ draw OP or to P(x, y, a), inclined 02 to Z, 90°- 02 to the X Y-plane, on which its gonal projection is sin 02; i.e., a inclined as above, so that, on putting sin 02 for pi we have x=p2 sin 02 sin 01, sin 02 cos obi, and besides z cos 02• Quite similarly, in four di mensions, x= sin 03 sin 02 sin 01, y =P3 sin 03 sin 02 cos 01, Z = p3 sin 03 cos 02, u = cos 03; and so on for other dimensions.

Triangular Co-ordinates.-Thus

far only parallel (especially rectangular) and polar co-ordinates have been considered, During the past century, there have been devised many others both linear and curvilinear. The first are linear combinations of parallel co-ordinates introduced by Mobius (1827) in Der barycentrische and then by Plucker (1828) in his Analyt isch-geometrische Entwicklungen-of signal importance in pro jective geometry. Mobius imagined three masses (pi, p2, p3) at the vertices of a A ABC, as co-ordinates of their own mass centre P. Of course, only the ratios :p2 :p3 are significant. If 2, 3 be (fig. 7) the altitudes of ABC, and d1, 2, 3 the distances of P from BC, CA, AB, we have the static equalities: = (pi+p2+p3)di, and so for the indi ces 2, 3; whence at once :P2 = di/hi : d2/122 = d3//i3, that is, the p's vary as the d's multi plied by the height-reciprocals, a special case of general trilinear co-ordinates in which the assumed three lines must form a triangle. In any L1 ABC (fig. 7) take Id any point P(x, y) . Then d, = a =x sin aa+bf3+cy = (i) If P lie beyond any side, as at P', the corresponding distance is negative. We may now take a, 0, y strictly a :/3 :y as trilinear or trimetric co-ordinates of P, under condition (i) . The quotient (aa+b0+c7)/2 i and may serve as multiplier to make all terms of an equation explicitly homogeneous in a, g, y. If for aa/2 Cy/2 0 we put x, y, z, called areal co-ordinates the quotient becomes x+y+z=1 and serves to make such equa tions homogeneous, x, y, z being any numbers rightly signed and proportioned. The advantages of trilinear co-ordinates lie in homogeneity, adaptability, and the treatment of infinity (Klein) .

Extensions.-Epoch-making were the works of J. Plucker (1828, 1831, 1835, 1839, 1846, 1868-69), broadening the notion not only of point-co-ordinates but also of co-ordinates in general, by regarding any geometric (or algebraic) form as a co-ordinate base and reckoning with the co-ordinates so defined. Thus he regarded any rational integral combination of point-co-ordinates denoted by a single letter, as itself a co-ordinate. Hence geomet rically the most general co-ordinates of a point in three-space would be any three definite surfaces (of three surface-systems) meeting in the point. "The most general system of co-ordinates of a point consists of three sets of surfaces, on one of each of which it lies." (Thompson and Tait, Nat. Philos., ii. 202.) Curvilinear Co-ordinates.-We now reach the type of curvi linear co-ordinates, as especially treated by Lame in the period from 183o to 186o. Any sur face-systems Ui = (i= I, 2, 3) , the C's being constants, whose functional-determinant U; y, z does not equal o, by their inter sections (in twos of two systems) determine points in space just as do three systems of parallel planes in three-space (or two systems of parallel lines in two space, the plane). But this they do only within a region free from singularities of any such co-ordinate surface, which hardly affects their availability in mathematical physics. Of these curvilinear systems the elliptic (announced in 1839 by both Lame in Liouville's Journal and Jacobi in Crelle's Journal) are of note as applicable throughout three-space, unhemmed by singularities. They are in a sense the three roots Xi, X2, X3 of the cubic in, symbolizing three confocal quadrics (for the three roots) through the point (x, y, z) . For — oo < X < c > o, an ellipsoid E; for c < X < b, an hyperboloid of one nappe ; for b < X < a, an hyper boloid of two nappes; for a nullibi (nullteilig) quadric without real points. Such confocals intersect at right angles, forming an orthogonal system. So through each point P there pass three such surfaces (E, all intersecting by twos in P, and by Dupin's theorem each two meet on curvature-curves common to the two-space-curves of the fourth order, each in general of two branches.

Specially useful in the study of spheres are pentaspheric co ordinates, in which five spheres are assumed—one always a (nullibi) sphere without real points, if the five are all to meet at right angles, forming an orthogonal system. If Si(i= 1, 2, 3, 4, 5) be the five powers (squared tangent-lengths from point to spheres) of a point P as to the five spheres, then the five sphere co-ordinates (strictly, their four independent ratios, which alone count) are fixed by crxi=kiSi there v denotes the proportionality-factor and ki the arbitrary constants. These five are homogeneous, and since three-space has only a fourfold of spheres (an assemblage around each point as centre), some homogeneous equation of the second degree must connect the five; it is written Si = o. Just here emerges a notable distinction. Hitherto, in point-and-plane geometry, the co-ordinates used in determining an element of space have been quite independent of each other, wherefore the space as so constituted is called a linear manifold; but in the point-geometry of pentaspheric co ordinates as in various other types, there appear supernumerary co-ordinates which must satisfy a quadratic equation, wherefore the space so constituted is known as a quadratic manifold. Made generally known by Darboux (1873), but already used since 1869, pentaspherics have been studied and developed mainly by Sophus Lie and Felix Klein.

Other Elements.

Thus far the point P has been treated as the only geometric irreducible, but others are often useful. If ix-I-my+ 1= o be a line L, then its intercepts (— l , — m) on OX, OY fix it and are constant for any one line, while x and y vary from point to point. But now suppose P (x, y) (as 3, 4) and let 1, m vary; then as the intercepts (— l m) change, the line will swing round P (x, y). Hence the equation no longer symbolizes a row or range of points all on the same line L, but a sheaf or pencil of all lines in the plane on (through) the same point P. Thus the notions of P and L are inter changed, giving lines on a point instead of points on a line, the equation remaining unchanged; hence 1 and m are line co ordinates; that is, co-ordinates of the lines on (or flat pencil through) the point (x, y). So in lx+my+nz+ o, on holding 1, m, n fixed and letting x, y, z change, we get the twofold of value-triplets depicted by all points P (x, y, z) of the plane; but on holding P fast (say 2, 3, 6) and letting 1, m, n loose, we get the twofold of value-triplets correlated with the (bundle of) planes on (through) the point P (2, 3, 6). So 1, m, n become plane co-ordinates, specifying all planes lying on a point (x, y, z).

Duality.—This discussion leads to the recognition of the principle of duality (Gergonne, 1771-1859) or reciprocity. An equation is understood in two ways, as a symbol both (1) of a line (or plane) total of points, in point co-ordinates, and (2) of a point-total of lines (or planes) in line (or plane) co-ordinates. And so on prolifically; to every such theorem in either type of co-ordinates corresponds the dual or reciprocal in the other; proving either proves both, as in the theorems of Pascal and Brianchon. (See CONIC SECTION.) But the theorem must be such, i.e., concerning positional relations, and not metrical, as lengths or sizes of angles. Order (number of intersections by a line) in the point-co-ordinate locus corresponds to class (number of tangents from a point) in the line (or plane) co-ordinates. In the line (or plane) co-ordinate envelop, each number being given by the degree of the equation in the respective co-ordinates; these latter co-ordinates are also called tangential, since they are tangent to the forms they envelop and thus determine.

Thus there appear as many lines as points in a plane, and as many planes as points in space. This is not strange, since there are infinitely more circles in the plane and spheres in space than points in either; in fact, around each point as centre a countless number of circles and spheres. It is no question of room or of crowding, but only of definition, of distinction one from another. Equations in point-co-ordinates (x, y, z) sym bolize forms as aggregates of points (P) co-ordinated in specific ways; equations in line or plane-co-ordinates symbolize such forms themselves as units co-ordinated into other aggregates, each individual in which is specified by a special set of values (1, m, n). The mind makes space of more dimensions in com plexes than in simples. In the line geometry proper of Plucker (1868-69), the line displaces the point as "space-element." But it is fixed by four co-ordinates (not three), as in two pro jections on co-ordinate planes (traces of two planes through the line and perpendicular to the co-ordinate planes), which are (say) x = az+a, y = bz+ f3; hence a, a, b, 13 are the line's four co-ordinates, though the preferred line-co-ordinates are six homogeneous co-ordinates connected by one quadratic equation. Hence, as an aggregate of lines our three-space is fourfold. According as these four independents are joined by I, 2, 3 equations, there result three configurations: (I) a triple complex, as the total of tangents to a surface, a plane or sheaf of tangents at each point of the twofold surface of points; (2) a double congruence, as the total of common tangents to two surfaces, as quadrics; (3) a simple regulus (or skew), as the system of tangents to H, an hyperboloid of one sheet, whichever half of the total be regarded.

The query will at once arise, may not still other geometric wholes, as circle, conic or quadric, be taken as space-elements, raising still higher the dimensionality of our triple space of points? The answer is that this is possible and that it has been done, in a way, with the circle (by C. Stephanos, 1881), the aggregate of circles in space appearing as a sixfold extended domain of the fifth order in a ninefold extent. If the quadric be taken as space-element, our three-space attains nine dimen sions, and so on. These brief hints may suggest the range and significance of the concept co-ordinates, and how the mind develops the notion of space, constituted at will of forms differing endlessly in complexity and degree, but all obeying the same logical laws and all united into one self-consistent whole. (See also ANALYTIC GEOMETRY.) BIBLIOGRAPHY.-N. M. Ferrers, An Elementary Treatise on Trilinear Bibliography.-N. M. Ferrers, An Elementary Treatise on Trilinear Coordinates (1861) ; W. A. Whitworth, Trilinear Coordinates and other Methods of Mod. An. Geom. of two Dimensions (1866) ; K. Zindler, "Algebraische Liniengeometrie," Encyklopadie der Math. Wis sensch. iii. bd., ii. teil, heft 8 (1922) ; F. Dingeldey, "Kegelschnitte u. Kegelschnitt-systeme," ibid., heft 1 (19o3) ; F. Enriques, "Prinzipien der Geometrie," ibid., iii. bd., i. teil, heft 1 (1907) ; E. Duporcq, Pre miers Principes de Geometrie moderne (1924) ; P. Humbert, Fonctions de Lame et fonctions de Mathieu (1926) ; D. B. Mair, Fourfold Geom etry (1926) ; Sophus Lie, Geometrie (2 vols., 1927). (W. B. SM.) CO-ORDINATION, in chemistry, denotes a mode of linking between atoms first recognized by Alfred Werner in 1893. In accordance with the co-ordination theory, which affords a simple and comprehensive explanation of the chemical constitutions of complex salts, amines (q.v.) and mordant dyes, certain salts of boron, beryllium, chromium, cobalt, copper and many other ele ments should exist in optically active forms. This prediction has since been verified experimentally in many instances. (See STER EOCHEMISTRY.) An electronic interpretation of co-ordination was provided by N. V. Sidgwick in 1923, thus bringing this chemical conception into line with modern views of the constitution of matter. (See VALENCY.)