Dec. 16, 1881.] 



KN.OVi^LE]>GE 



147 



^■' 



(Buv i^latOnnatiMl Column. 



THE WITCH OF AGNESI. 



[Reply to E. H. B., Mathematical Qaery, 2.]— The Witch of 

 Affoesi is a cnrve of the third order described in the " Analytical 

 liistitntos " of Maria Gaetana Agnesi (1748). It may be thns drawn 

 geometrically ; — 



Take a circle OBA and draw a diameter OA ; then if QN por- 

 pendicalai' to 0.1 be produced to the point P, snoh that 



NP : NQ . : AO : no 



P is a point on the " Witch of Agnesi." 



'■ It is evident that the curve will extend indefinitely on either side 



of OA (we only show one branch) ; and that the tangent to the 



circle ABO, at 0, will be an asymptole to both branches. Again, 



if C be the centre of the circle; OC, CA, bisected in m and n; and 



f SC, qn, and tm, perpendicnlar to OA, axe produced to meet the 



. emre in b, p, and r, it is obvious that 



bC=0A = 2a (putting OC = n) 



V3a 



-3pn 



OA J -4 



Ott 3 a 



,m = ^^. t7n = 4to = 2v'3a = 

 Oft 



Thus, a straight line drawn from Atop and produced will jmss 

 through r. 



Again, if we draw AkM a tangent at A and LPS, IQk parallel to 

 OA, it follows from the fundamental property of the curve that 



rect. LN^rwt. lA, 

 whence Q is on the diagonal OK. Tims, we have a simple 

 geometrical method of drawing the cur\-e, as follows ; — 



On AM take any point K, and join OA' cutting circle OBA in Q ; 

 then parallels KP, QP to OA and OY, intersect in P a point on the 

 curve. (We give the construction for the point a on the cnrve, 

 joining MO, which cuts OBA in 0, and drawing the parallels 

 Ma, Go.) 



For the ec|uation to the curve, take OA as axis of ir, and OL as 

 axis of V. Then, we have — 



ON.PN=qA.QN; that iB 

 xy = 2ay/2cu — x' ; OTxy'=4a^(2a — x). 

 l''rom tliis equation to rectangular co-ordinates we can deduce the 

 properties of the cnrve. Differentiating — 



^=-=^(2cu^-x=)- 



dy __2a'^ 



^y = ^ (3o - 2x) (2a<E - x') " 



(i) 

 (ii) 

 =0, ov ! 



From (ii) we see that — disappears when 3n — 2; 

 so that there is a point of inflection at p. 



Putting ,c successively equal to— , a, and — in (i) we find: — 



d,j 



dy 



at pi. 



- -(•■-?)-'-- 



V3 



'11 at pt. b = 



J-^tpt.p = -^[s- 



From these values, remembering that 



— (2a=-a=)"»= - 2 



4 / 3\/3 



= 2\/3a; bC = 2a; and pn = 



v^3 



we see that the tangent at r cuts O^at a point, [i, such that mz = — ; 



4 

 the tangent at 6 passes through A ; and the tangent at p cuts OA at 



3o 

 a point X such that nx-= — , 



A geometrical construction for finding the point x in which a 

 tangent at a point p cuts ^the axis of x is easily obtained. For we 

 dy 2a^^Q„..,_„,.j — i 



have— = 



d.t 

 whence fi 



(2a^-x») 

 nco ijn = v^2nr — 



it follows that 

 OA.OO ^ pn.On ^ OA.OC 

 On. in' 



OA. (jn O A. OC 



or {qny=OC. 



Hence, if we take ntn = OC, a perpendicular at q to mq will cut OA 

 (produced, if necessary) in the required point x, such that xp is a 

 tangent at p. 



We have not by us any treatife bearing on the history of this 

 curve, and are, therefore, unable at the moment (o state the pro- 

 perties which led to its invention. In Brando's " Dictionary of 

 Science and Art" the definition of the curve is given, and we have 

 based on that the above investigation. Most probably the cnrve 

 was one of those which mathematicians were fond of inventing (in 

 the 17th and 18th centuries) for the purpose of squaring the circle. 

 (Such a curve they would call a quadratrix.) It is manifest that 

 the cnrve can readily be swept out mechanically. Thus, let there 

 be a semicircular groove OBA (Fig. 2) and a straight groove 

 OT square to OA, and let bars Nn', LI slide between the partUlels 

 .4T, Oa (square to 0-4) so as always to bo parallel to ^ITand 0.4 

 respectively. Then if a rod OG swinging around 0, carry 

 points along AT and ABO, which respectively bear with them the 

 parallels LI, n'N, a pencil so carried as to lie always at the inter- 

 section of these parallels, will trace out the curve APb. 



It is readily seen that the cnrve is a quadratrix. For, take any 

 two points, KJ. near together on 0.4, and about as centre take 

 circular arcs Kq', Jq to the semicircle AqO. Draw OqL, Oq'L', as in 

 fig, 2, and parallels L'Pl', LPl, mPq'M, n'PqN, giving two points, F, P, 

 on the curve. Draw qn perp. to inM. Let Oq LI, cut mM in e' and i 

 respectively, arc q'K cutting Oq in o. Draw the quadrant Ara about 



