LOCUS
, is some line by which a local or indeterminate problem is solved; or a line of which any point may equally solve an indeterminate problem.
Loci are expressed by algebraic equations of different orders according to the nature of the Locus. If the equation is constructed by a right line, it is called Locus ad rectum; if by a circle, Locus ad circulum; if by a parabola, Locus ad parabolam; if by an ellipsis, Locus ad ellipsim; and so on.
The Loci of such equations as are right lines or circles, the ancients called plane loci; and of those that are conic sections, solid loci; but such as are curves of a higher order, sursolid loci. But the moderns distinguish the Loci into orders according to the dimensions of the equations by which they are expressed, or the number of the powers of indeterminate or unknown quantities in any one term: thus, the equation
denotes a Locus of the 1st order, but , or , &c, a Locus of the 2d order, and , or , &c, a Locus of the 3d order, and so on; where x and y are unknown or indeterminate quantities, and the others known or determinate ones; also x denotes the absciss, and y the ordinate of the curve or line which is the Locus of the equation.
Forinstance, suppose two variable or indeterminate right lines AP, AQ, making any given angle PAQ between them, where they are supposed to commence, and to extend indefinitely both ways from the point A: then calling any AP, x, and its corresponding ordinate PQ, y, continually changing its position by moving parallel to itself along the indesinite line AP; also in the line AP assume AB = a, and from B draw BC parallel to PQ and = b: then the indefinite line AQ is called in general a geometrical Locus, and in particular the Locus of the equation ; for whatever point Q is, the triangles ABC, APQ are always similar, and therefore , that is , and therefore is the equation to the right line AQ, or AQ is the Locus of the equation .
Again, if AQ be a parabola, the nature of which is such, that , or , and therefore is the equation which has the parabola for its Locus, or the parabola is the Locus to every equation of this form .
Or if AQ be a circle, having its radius AB = a, the nature of which is this, that , or or ; therefore the Locus of the equation of this form , is always a circle.
In like manner it will appear, that the ellipse is the Locus to the equation , and the hyperbola the Locus to the equation ; where t is the transver<*>e, and c the conjugate axis of the ellipse or hyperbola.
All equations, whose Loci are of the first order, may be reduced to one of the 4 following forms: ; ; ; ; where the letter c denotes the distance that the ordinates commence from the line AP, either on the one side or the other of it, according as the sign of that quantity is + or -.
All Loci of the 2d degree are conic sections, viz, either the parabola, the circle, ellipsis, or hyperbola. Therefore when an equation is given, whose Locus is of the 2d degree, and it is required to draw that Locus, or, which is the same thing, to construct the equation generally; bring over all the terms of the equation to one side, so that the other side be 0; then to know which of the conic sections it denotes, there will be two general cases, viz, either when the rectangle xy is in the equation, or when it is not in it.
Case 1. When the term xy is not in the proposed equation. Then, 1st, if only one of the squares| x2, y2 be found in it, the Locus will be a parabola. 2d, If both the squares be in it, and if they have the same sign, the Locus will be a circle or an ellipse. 3d, But if the signs of the squares x2, y2 be different, the Locus will be an hyperbola, or the opposite hyperbolas.
Case 2. When the rectangle xy is in the proposed equation; then 1st, If neither of the squares x2, y2, or only one of them be in the equation, the Locus will be an hyperbola between the asymptotes. 2d, If both x2 and y2 be in it, having different signs, the Locus will be an hyperbola, having the abscisses on its diameter. 3d, If both the squares be in it, and with the same sign, then if the coefficient of x2 be greater than the square of half the coefficient of xy, the Locus will be an ellipse; if equal, a parabola; and if less, an hyperbola.
This method of determining geometric Loci, by reducing them to the most compound or general equations, was first published by Mr. Craig, in his Treatise on the Quadrature of Curves, in 1693. It is explained at large in the 7th and 8th books of I'Hospital's Conic Sections. See this subject particularly illustrated in Maclaurin's Algebra. The method of Des Cartes, of finding the Loci of equations of the 2d order, is a good one, viz, by extracting the root of the equation. See his Geometry; as also Stirling's Illustratio Linearum Tert<*> Ordinis. The doctrine of these Loci is likewise well treated by De Witt in his Elementa Curvarum. And Bartholomæus Intieri, in his Aditus ad Nova Arcana Geometrica delegenda, has shewn how to find the Loci of equations of the higher orders. Mr. Stirling too, in his treatise above-mentioned, has given an example or two of finding the Loci of equations of 3 dimensions. Euclid, Apollonius, Aristæus, Fermat, Viviani, have also written on the subject of Loci.
