CHAP. IV.] 



OVAL CURVES. 



Let m power of A, and n = power 

 of B, n BH = constant quantity = n BC 

 + n CH, and n BC + m AC = constant 

 quantity = n BC + n CH, take away n BC, 

 and n HC = m AC, therefore HC : CA : : 

 m : n. QED. 



Cor. 1. When the powers of the 

 foci are equal the curve is an ellipse. 



Cor. 2. When the less focus is at an infinite distance the curve is 

 an ellipse, for the circle becomes a straight line ; and 



Cor. 3. When the greater focus is at an infinite distance, the 

 curve is an hyperbola for the same reason. 



PROPOSITION 4 THEOREM. 



When the less focus is in the curve, an 

 angle will be formed equal to the vertical 

 angle of an isosceles triangle, of which the 

 side is to the perpendicular on the base, as 

 the power of the greater focus to that of the 



For let a circle be described, as in Prop. 3. 

 it is evident that it will pass through B. Take indefinitely small arcs 

 CB = BD, join CA and DA, join EH. EC : EB = power of B : power 

 of A, and EC = BO, therefore EB : BO, power of A : power of B. 



PROPOSITION 5 PROBLEM. 



A point A, and a point B in the line BC being given, to find a 

 point in the line as D, so that m AD + n BD may be a minimum. 

 Take a line HP, raise HX perpendicular, and from X as a centre 



describe a circle with a radius = ' so that m : n : : XP : XH. 



