# 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|
*x*^{2}, *y*^{2} 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 *x*^{2}, *y*^{2} 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 *x*^{2}, *y*^{2}, or
only one of them be in the equation, the Locus will
be an hyperbola between the asymptotes. 2d, If both
*x*^{2} and *y*^{2} 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 *x*^{2} 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.