LOCUS. 



fuion, and by its intorfcflion with the line A B, gives the 

 point E, with the pofition of D E, and thence the whole 

 triangle as before. Here it may be obferved, that the 

 anjjle D of the triangle E D F, given in fpecies touching 

 a given point D, and another of its angles touching A C, 

 the line A E here found is the locus of the third angle E. 



Of the h'l^hir Order of Lcc't. 



Loci are very co:nmodi<)u(ly divided into orders, accord- 

 ing to the dimeiifions to which the variable quantity rife<> 

 in the formula which exprelTcs the equation of the curve. 



Thus it will be a locus of the f.rfl order, if the equation 

 he X ^ a y ; a locus of thefuond or quadrate order, it ^' = 

 ax, ot y =^ a — x'. Sec. ; a locus of the third or cubic 

 order, it jr' = a x, or y =^ a x' — x , &c. 



The better to conceive the nature pf the locus, fuppofe 

 two unknown and variable right lines A P, P M {f^s. 1 2 and 

 1 ^ .) makin^j any given angle A P M witli cacii otiier ; the one 

 whereof, as A P, we call x, having a fixed origin in the point 

 A, and extending itfelf indefinitely along a right line given in 

 pofition; the other P M, which we call j, continually 

 changing its pofnion, but always parallel to jtfi-lf; and 

 moreover an equation only containing thele two unknown 

 quantities x and \, mixed with known ones, which exprefTes 

 the relation of every variable quantity A P (x) to its cor- 

 refpondent variable quantity P M (y) : the line pafiing 

 through the extremities of all the \-alues of y, i. e. through 

 all the points M, is called a geometrical locus, in general, 

 and the locus of that equation in particular. 



All equations whofe loci are of the fijl order, may be 

 reduced to fume one of the four following formulas : 



I . y = 



.y :^ —-r e 



3 -y = ( 



a 



4 ■ y 



■ ', where the unknown quantity j is fuppofod always to 



a 



be freed from fniftions, and tlie fracl-ion that multiplies the 

 other unknown quantity x to be reduced to this expreffion 



— , and all the known terms to this c. 



a 



The locus of the firft formula being already determined- ; 

 fince it is evident, that it is a right line whichcuts the axis in 

 A, and wh'ch makes with it an angle, fuch that the two un- 

 known quantities x,y, may be always to one another in the 

 proportion of a to b ; to find that of tlie fecond, y = 



— -\- c. In the line A P (jfj. f4.) take A B = <?, and 

 a 



draw B E = i, A D = #, parallel to P M. On the fame 

 fide A P, draw the hue A E of an indefinite length towards 

 E, and the indefinite flraight line D M para'lel to A E. 1 

 fay the line D M is the locus of the aforefaid equation or 

 formula t for if the line M P be drawn from any point M 

 thereof parallel to Q A, the triangles A B E, A P F, will 

 be fimilar; and therefore A B (<:) : B E (^) : : A P (.v) 



: PF = — ; and coinequently P M Cy) =.P F (— } 



+ F M (<i). 



b: 



fide A P, and the other on the other fide ; and through the 

 points A, E, draw the right line A E of an indefinite lenglls 

 tovvards E, and throngh the point D the llnel') M parallel 

 to A E : I fay, the indefinite right line (r M ftiull be the 

 locus fought ; for we fliall have always PM (_v) = P F 



(7)-^ 



FM 



hx 



Laflly, to find the locus of the fourth formula, ^ = c — 

 ; ill A P ifg. 16.) take A B = a, and draw B E = ^, 



AT> ~ e, parallel to P M, the one on one fide A P, and 

 the o'heron the other fide ; and through the points A, E, 

 draw the line A E indefinitely tovvards E, and through the 

 point 1) draw the line 1) M parallel to A E. I fay D G 

 fiiall be tlie locus fought ; for if the line M P be drawn from 

 any point M thereof, parallel to A Q, then we fliaU have 



akays P M {y) = F M {e) - V Y (-~)- 



Hence it appears, that all the loci of tha firfl degree ara 

 ftraigh: lines ; which may be cafily found, becaufe all their 

 equations may be reduced to foine one of the foregoinjj 

 formulae. 



All loci of the fecond degree are conic feftions, 'Ciz. 

 either the parabola, the circle, ellipfis, or hyperbola ; if art 

 equation therefore be given, whofe locus is of the fecond 

 degree, and it be required to draw the conic fe£\ion, which 

 is its locus, firft draw a parabola, ellipfis, and hyper- 

 bola; fo that the equations exprcffing the n.ittires there- 

 of may be as compound as polTible ; in order to get gc» 

 ncral equations, or formulx, by examining the peculiar pro- 

 perties whereof we may know which of tliefe formula; the 

 given equation ought to have regard t-o ; that is, which of 

 the conic fedHons will' be the locus of the propofed equation. 

 This known, compare all the terms of the pr<>i)ofed equation 

 with the terms of the genoral formula of that conic leClion, 

 which YOU have found wiltlx' the locus of the given equation ; 

 by which means you will know how to draw the fedion-, 

 which is the locus of the aquation. given. 



For example :. let A P (.v), P M (_)■)» be unknown,. and 

 variable (Iraight lines (f.g. 17. ), and let m, >i, p, r, s, lie 

 given right hnes : in the line A P take A B = ni, and draw 

 BE— N, AD. = r, parallel to PM; and through the 

 point A draw A E — f, and through the point 1) the inde- 

 finite right line D G parallel to A E. Ih D G take 

 D C = J, and with C G. as a diameter, having its ordinates 

 parallel to P M, and the line C H = f>, as the parameter, 

 defcribe a parabola C M, and it will be the locus of the fol- 

 lowing general formula :. 



I 



xy + 



Zry + M + r z= o. 



X. + pj. 



For irfrom any point M there he drawn the right line M P, 

 making any angle A P M with A P'; the triangles A B E, 

 A P I, fiiall be fimilar ; therefore A B (m) : A E (f ; : : 



^ , , , , r ,. u- J r '^ ■■'■ , A P (.0 : A F, or D G = — ; and A B (;,<) : B E (n) 



To find the lociK of the third form, ;( = <-, proceed ■" ■ ' ' „, ^ ^' 



thus. AffumQ A B = a (fg. 15.) and draw the right 

 Unes B E = £» A D = <) gariUlel to P M, tke one on orx 



: : A P (.r) : P F = — . And confequently, G M or P M 



-PF 



