CHAP. IV.] 



MELOID AND APIOID. 



101 



focus, in a greater proportion than the power of the greater focus to 

 that of the less, the curve is convex toward the greater focus at that 

 point, but if the proportion is less, concave. 



Let AD:T>E::p:q. 



If AD : DB > m : n, ODE is convex towards A ; but if AD : 



DB < m : n, it is concave towards A Take C and E near D, 



and CD = DE. Join CE, CE cuts the axis in 0. Draw the circle 

 CLE from B, and CPE from A. Also draw HTK as in Prop. 3 



Then BD : DT : : m : n : : BC : CH, but BC = BL, and OH = PT 

 .-.BD:DT::BL : PT .'. BD : BL : : DT : PT . . BD :BL-BD :: 

 DT : PT - DT . . BD : DL : : DT : DP . . BD : DT : : DL : DP . . DL : 

 DP ::m:n. 



As E and C are very near D, AD : BD : : AC : BC, but PE = LE 

 and PCE : 2 right angles : : PE : circumference of CPE, and LCE 

 : 2 |J_ : : LE : circumference of CLE, but circ. CPE : circ. CLE : : 

 AC:BC::AD:BD ::p:q and PE = LE.\ PCE: 2 [L: :q PE : q 

 circ. CPE, and LCE : 2 [L_ : : p PE : Q? circ. CLE or) q circ. CPE .*. 

 PCE: LCE :: 2 PE :^PE : : q :p .'. PCO : LCO : :q :p and PO : OL 

 ::q:p; and if p : q > m : n, OL : OP > DL : DP, and D is nearer to 

 A than the line COE, and CDE is convex toward A ; but if p : q < m : n, 

 D is on the opposite side, and it is concave. 



PROPOSITION 6 PROBLEM. 



To draw a tangent to a meloid or apioid from a focus without : 

 Take m for the power of the greater focus, and n for that of the 

 less, and find the angle ADC (Prop. 5 of the Oval) upon AB describe a 



