COOKDINATES. 382 COOBDINATES. sides o, 6, c of the triangle ABC, they are con- nected by the relation aa; + 61/ + c« = fc = 2 X area of ABC, or x sin A + y sin B + s sin C = a constant. (See Trigonometky. ) We may also take X, y, :: olilique to 0, 6, c. each forming the same angle with its corresponding side. Equa- tions between the coordinates of two or more points in this system are homogeneous, as iiix + py -{- qz = 0. the equation of a straight line. Such coordinates are called trilinear or homo- geneous coi'.rdinates. These form a special case of barycentric coordinates, the first homogene- ovis coordir^ates in point of time, introduced by ilfibius (q.v. ) in his Der iarpcentrische Calcul ( 1827 ) . Tetrahedral space coordinates belong to the same class. If in the above figure rectangu- lar axes are also assumed, and if P„ P,. P3 are the perpendiculars from the origin upon the sides a, 6, c, respectively, then X BPC y _ ACP , 2 _ BAP P,- CBA' Pj— CBA' "^""^ P, — CBa5 these expressions may be designated by x'. y', s^, and be employed to determine the position of point P (P denoting the point at which x, y, and s meet). Since they are expressed in terms of areas, they are called areal coiirdinates. In cither the trilinear or areal system a point is determined if the ratios only of the coordinates are 1-cnown. If Ix + my --nz = is the tri- linear equation of a straight line (L in the figure), then by making x, y, z constant and I, m, n variable, the equation is called the tan- gential equation of the point O {x, y. z) . Since I, m, n are varialilc, the equation represents any straight line passing through O. If ?., /i, v are the perpendiculars from A, B. C, upon the line L, and Pi, P2. Pr, the altitudes of the triangle ABC, the equation of the point O is Pi P2 Pi When the perpendiculars ?., fi, v are taken for the coordinates of the line, the coefficients be- come the areal coiirdinates of the points referred to the same fundamental triangle. Any homo- geneous equation in I, m, n as tangential coordinates is expressed in terms of ?., /i, v by substituting ->£->_> for 7, m, n respectively. Pi Pi P3 An equation in /. ;^, v of a degree higher than the first represents a curve such that /, n, v are always the perpendiculars upon the tangent. The curve must therefore be the envelope (q.v.) of the line (?., /i, v) . Tangential coordinates are often called Boothian coordinates, in honor of James Booth, who invented them. Bicircular coiirdinates are magnitudes which determine a point with reference to two series of circles which intersect one another at a constant angle. Generalized, Lagrangian, Eulerian, and Rod- rigue's coordinates are .special systems used in treating certain problems of mechanics. From the idea of coordinates of an element in a plane we easily pass to the notion of coordinates of an element in geometi'j' of three dimensions. The determination of a point in .such space re- quires three coordinates. In the Cartesian sys- tem these are represented by x, y, z. An origin being taken (as O in the figure), and three axes, OX, OY, OZ mutually at right angles to one another, the point is referred to the three planes through these axes. Here s or PX is its distance above the plane YOX ; y or NM is its distance from tlie plane XOZ ; and x or Oil is its distance from the plane ZOY. In three dimen- sions, as in two, the problem may be stated to be: Given the law of the motion of P, to express the law of variation of its coordinate ; the alge- braic expression of the latter law is the equa- tion of the surface traced by the point in mov- ing over all the space it can traverse consistent- ly with the law of its motion. Thus, the equa- tion of a sphere referred to its centre O is x' -- y' -- z' =^ r'. As in plane geometry, if the axes are taken oblique to one other, the coordi- nates are called oblique coordinates. Likewise, the polar coordinates in space corresponding to the polar coordinates in a plane are p, 8, <f>. In the above figure p = OP, the distance from the origin : # = ^ POZ, the angle between OP and OZ, and = ^ XOX, the angle between planes XOZ and POZ. As in plane geometry there are intercept equations for straight lines, so in solid geometry there are intercept equations for planes. And as certain equations of plane geometry are made homogeneous by the introduc- tion of a third coordinate, so in solid geometry certain equations are made homogeneous by the introduction of a fourth coiirdinate. Thus, in tet- rahedral or barycentric four-plane coordinates, the four faces of a tetrahedron (q.v.) of reference are taken as the coordinate planes. The equa- tion of any plane in this system is Ix -{- my-- nz + rw = 0, in which I : in : n : r = — J — : — ; ■ — 1 '^i* i'i> '^i» **^i Pi Ps' P3 Pi being the perpendiculars upon the plane from the vertices A, B, C, D of the tetraliedron of reference, and />,, p.,, p^, pt, the altitudes of the tetrahedron from A, B, C, D respectively. For the coordinate systems for space of »!-dimen- sions, and for the transformations from one sv3-
