r U| 



Tod ^tilur f(|iiiititiii of the conchoid draw 



AOY \*>T\*>\ i :.lllrl, 1,. OX. illld Irt 0.4.*; 



//) be any | -si? i,, i, .>f the generating point. 

 .\ thr oniinatr ////'. ilu-n, from the similar triangle* 



.1/7: ///' M7 177V 

 : Vtf 1 y 1 : | 



:h is the equation sought. 



The definition of the eon -hoid, as well as the equation just 

 1, shows that the || symmetric with regard to 



the y-axis ; that it lies wholly between lines y mm a 



and y = <i ; ami that it lias four infinite branches to each 



- an asymptote. 



N<> M polar equation of the conchoid. Let A be toe pole, 



the initial lim*, ami 7' = (p, 6) (or /**) any position of the generating 

 point ; then 



p = ,</>=. I O.l.sretf .sv. 



-<eae*d 



is the desired equati.. i. 



1 may also be readily constrocted by continuous 



motion as follows : By means of a slot in a nig over a |>in at .1 . 



the motion of t ::-! . if now a guide pin at 



5. and a tracing point at 7', l attached t.. tin* rul.-r, then the point /' 

 will trace out the conchoid when the guide point 6* is moved along the 



By means of a conchoid, any given angle may be trisected. 1 

 I BC be any angle, on one side (HA) take any distance, as IIH. and 



It to evident that, if A0< OB, <.., if 0<* the com hat an oval below 

 A as shown in Fig. 185; if c = a, this oval closes op to a point ; and if c>, 

 both parts of the curve lie wholly above A. 



t For the insertion of two mean proportionals between two given lines by 

 means of the conchoid, see Cantor, Uesohlcbte dec Malhemstik. & 



