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Locus

line, parallel, draw, equation, unknown and fig

LOCUS geometricus, denotes a line, by which a local or indeterminate problem is solved. See Loc.u. PROBLEM.

A locus is a line, any point of which may equally solve an indeterminate pro blem. Thus, if a right line suffice for the construction of the equation, it is call ed locus ad rectum; if a circle, locus ad circulum ; if a parabola, locus ad parabo lam ; if an ellipsis, locus ad eilipsin ; and so of the rest of the conic sections.

The loci of such equations as are right lines, or circles, the ancients called plain and of those that are parabolas, hy perbolas, &c. solid loci. But Wolfius, and others, among the moderns, divide the loci more commodiously into orders, ac cording to the numbers of dimensions to which the indeterminate quantities rise. Thus, it will be a locus of the first order, if the equation is x = ; a locus of the second or quadratic order, if e=--a x, or ; a locus of the third or cu bic order, if y3=-_co x, or y3=a &c.

The better to conceive the nature of the locus, suppose two unknown and va riable right lines A P, P M (Plate VIII. Miscel. fig. 4 and 5) making any given angle A P M with each other; the one whereof, as A P, we call x, having a fixed origin in the point A, and extending it self indefinitely along a right line given in position ; the other P M, which we call y, continually changing its position, but always parallel to itself. An equation on ly containing these two unknown quanti ties, x and y, mixed with known ones, which expresses the relation of every va riable quantity A P, (x), to its correspon dent variable quantity P Al, (y) : the line passing through the extremities of all the values of y, i. e. through all the points Al, is called is geometrical locus, in general, and the locus of that equation in particu lar.

All equations, whose loci are of the first order, may be reduced to some one of the four following formulas : 1. y bx b xb x a y 3. y — a C. 4.

b x y =-_ c —. Where the unknown quan

a tity, y, is supposed always to be freed from fractions, and the fraction that mul tiplies the other unknown quantity, x, to be reduced to this expression and all a.

the known terms to c.

The locus of the first formula being al ready determined : to find that of the se bx cond, y = ; in the line A P, fig. 6, a take A B =. a and draw HE = 6, A Da= c, and parallel P M. On the same side A P, draw the line A E of an indefinite length towards E, and the indefinite straight line D M parallel to A E. Then the line I) M is the locus of the aforesaid equation, or formula ; for if the line M P be drawn from any point .31 thereof paral lel to A Q, the triangles A 13 Eond A P F, will be similar : and therefore A B (a) : B E (b) :: Al' (x) P F ; and con e sequently PM (y) PI (= x) + (c), To find the locus of the third form, y r-- b x a c, proceed thus : assume A B = a (fig. 7); and draw the right lines B E = b, A I) = c and parallel to P M, the one on one side A P, and the other on the other side : and through the points A E, draw the line A E of an indefinite length towards E, and through the point D, the_ line D M. parallel to A E : then the inde finite right line G 31 shall be the locus sought ; for we shall have always P (y) = P F =-- — F M (c).

Lastly, to find the locus of the fourth formula, y = c — b x; in A P .(fig. 8) : take A B = a, and draw B E b, A D= c, and parallel to P 31, the one on one side A P, and the other on the other side ; and through the points A, and E, .draw the line A E indefinitely towards E, and through the point D draw the line D TSt parallel to A E. Then D G shall be the locus sought ; for if the line M P be drawn from any point M thereof parallel to A Q, then we shall always have P M b