LOCUS GEOMETRICUS. 



compound as possible. In order to get 

 general equations, or formulas, by exam- 

 ining- the peculiar properties whereof we 

 may know which of these formulas the 

 given equation ought to have regard to ; 

 that is, which of the conic sections will 

 be the locus of the proposed equation. 

 Tills known, compare all the terms of the 

 proposed equation with the terms of 

 the general formula of that conic sec- 

 tion, which you have found will be the 

 locus of the given equation ; by which 

 means you will find how to draw the sec- 

 tion, which is the locus of the equation 

 given. 



For example ; let A P = x, P M = y, 

 be unknown, and variable straight lines 

 (fig. 9); and let m, n, p, r, s, be given 

 right lines : in the line A P take A 13 = 

 m, and draw B E = n, A 1) = r and paral- 

 lel to P VI ; and through the point A 

 draw A E = e, and ihrough tiie point 1) 

 the indefinite right line D G parallel to 

 A E. In D G take D C = s, and with 

 C G, as a diameter, having its ordinates 

 parallel to P AI, and the line C H =/>, as 

 the parameter, describe a parabola C M : 

 then (he portion thereof, included in the 

 angle P A D, will be the locus of the fol- 

 lowing general formula : 



2 nx y . n n xx n 2 ' 



in 

 ep_x_ 



m ' " 



For if from any point M of that por- 

 tion there be drawn the right line M P, 

 making any angle A P M with M P ; the 

 triangles A B E, A P F, shall be similar ; 

 therefore, 



A B : A E :: A P : A F or D G ; that is, 



m : e : : x : . And A B : B E :: A P : 



in 



P F ; that is, m : n :: x : . And cdnse- 

 m 



quently, G M or P M P F F G = y 



_!L?_ r . And C GorDG DC = 



m 



s. But from the nature of the para- 

 bola G M* = C G X C H ; which equa- 

 tion will become that of the general for- 

 mula, by putting the literal values of 

 those lines. 



Again, if through the fixed point A you 

 draw the indefinite right line A Q (fig. 

 10), parallel to P M, and you take A B 

 = m, and draw B E = n parallel to 

 A P, and through the determinate points 

 A E, the line A E = e; and if in A P you 

 take A D = r : and draw the indefinite 

 straight line D G parallel to A E, and take 



DC s: this being done, if with the 

 diameter C G, whose ordinates are paral- 

 lel to A P, and parameter the line C H 

 = p, you describe a parabola C M ; the 

 portion of this parabola contained in the 

 angle BAP shall be the locus of this se- 



cond equation, or formula : 



xx- nyx --\ nn y 



m r m m 



o r x 4- 



nry 



For, if the line M Q be drawn from 

 any point M, therein, parallel to A P ; 

 then will A B : A E :: A Q or P M : A F 



or D G ; that is, m : e ::#:; and A B 





: B E :: A Q : Q F ; that is, m : n :: y : 







And therefore G M or Q M Q F 

 F G == x ^ r; andC G or D G 



And so by the common property of the 

 parabola, you will have the foregoing se- 

 cond equation, or formula. So likewise 

 may be found general equations for the 

 other conic sections. 



Now if it be required to draw the para- 

 bola, which we find to be the locus of 

 this proposed equation yy 2 a y bx 

 -j- e c = o ; compare every term of the 

 first formula with the terms of the equa- 

 tion, because y y in both is without frac- 

 tions: and then will = o, because the 

 in 



rectangle xyuot being in the proposed 

 equation, the said rectangle may be es- 

 teemed as multiplied by o ; whence n = o, 

 and m = e,- because the line A E falling 

 in A B, that is, in A P in the construc- 

 tion of the formula, the points B E do 

 coincide. Therefore destroying all the 



terms adfected with - in the formula, 



m 

 and substituting m for e, we shall get y y 



2ry p x -j- r r -j- p s = o. Again, 

 by comparing the correspondent terms 



2 r y and 2 a y, as also p x and 



b x, we have r = a, and p = b; and 

 comparing the terms wherein are neither 

 of the unknown quantities x y, we get 

 7- r -j- ps = c c ; and substituting a and 



b for r and p, then will s = 



which is a negative expression when a is 

 greater than c, as is here supposed. 

 There is no need of comparing the first 

 terms y y and yy, because they are the 

 same. Now thp values of n } r, p t s, be 



