i8 KA^s'SAS ACADEMY OF SCIENCE. 



lie on equal fixed "central" circles having a common "central diameter" and 

 on opposite sides of the center; then are the other two points of intersection on 

 an oval to which the circle is tangent at the extremities of the major axis. 



1. Three points, A, P, B, are taken on an ellipse whose center is C. Parallels 

 to the tangents at A and B drawn through P meet CB and GA respectively in 

 the points Q and R. Then is QR parallel to the tangent at P. 



2. Three points, A, P, B, are taken on an oval whose center is C. " Central " 

 circles drawn through P and having common "central diameters" with the 

 "central" tangent circles at A and B respectively, meet the lines CB and CA 

 respectively in the points Q and R. Then the " central " circle through Q and R 

 has a common "central diameter" with the "central" tangent circle at P. 



1. The sum of the distances from any point on an ellipse to the two foci is 

 constant. The ellipse may be described mechanically by the use of this property. 



2. The sum of the distances from any point on an oval to the two foci bears a 

 constant ratio to its distance from the center. The oval, also, may be described 

 mechanically by the use of this property. 



1. To draw a tangent at any point of an ellipse, bisect the extei-nal angle be- 

 tween the focal radii: to draw a normal, bisect the interior angle. 



2. To draw a tangent line to an oval, bisect the external angle formed by the 

 two "central" circles through the point and the two foci respectively: to draw 

 a normal, bisect the interior angle. 



1. The subtangent of an ellipse is equal to the corresponding subtangeat of 

 the circle described upon the major axis. 



2. Given an oval and a circle described upon its minor axis. " Central " tan- 

 gent circles are drawn at the points where the oval and circle are cut by a " cen- 

 tral " circle that intersects the major axis at right angles. Then will the two 

 "central " tangent circles cut the transverse axis at the same point. 



Consider now the case where e > 1, and the equation of the curve is 



a^ x^ — b'' i/~ = (a;2 + i/^f (6') 



The development of the properties of this curve is much the same as for the 

 quartic first considered. Analytically, it is only necessary to change b' to — b^ 

 in the equations of the former to obtain the corresponding equations in the latter. 



The curve cuts the axis of coordinates at the points (+ a, 0) {0, 0); and a and 



b may be called the minor and major semi-axis respectively. 



The distance from center to either focus is —. 



e 



The origin is a real double point. The equation of real tangents at the origin is, 



a'-.v- — b-i/'^ = 0. (7') 



The circular points at infinity are imaginary double points. 

 The curve has four foci, two real and two imaginary. 

 The origin is a point of inflection. 

 When e = 1; a = 0; and the equation breaks up into two imaginary circles, 



x' + y^=±byV~i. (17') 



When e > ^, the shape of the curve is that of a figure 8 extending along the 

 axis of abscissae. 



When e = yT\ a = 6, and the curve is identical with the lemniscate of Ber- 

 aouilli. 



When e := CO ; b =: 0, and the equation breaks up into 



x~ +y-^±ax; (17") 



that is, two circles of radius la, tangent to the axis of ordinates at the origin. 



