TWO NOTES ON THE ELLIPSE 



Edward L. Beown 



I. 



Salmon, in his Conic Sections, Art. 179, draws attention to the 

 fact that we can draw a pair of conjugate diameters intersecting at any 

 other angle, making use of the truth that diameters parallel to 

 any pair of supplemental chords are conjugate. For, if we describe 

 on any diameter a segment of a circle, containing the given angle, 

 and join the points where it meets the curve to the extremeties of the 

 assumed diameters, we obtain a pair of supplemental chords inclined 

 at the given angle, the diameters parallel to which will be conjugate 

 to each other. 



From the above, readers are almost always of the opinion that 

 the circle will in every case intersect the ellipse in points other than 

 the extremeties of the assumed diameter, and that the construction 

 is possible for any angle whatever. This, however, as is well known, 

 is not true. The following is a discussion of the construction: 



Let A A^ be the assumed diameter. At A construct an angle 

 A ^, which is to be equal to the angle between the conjugate 



diameters. Join A and B. Draw I* A 

 perpendicular to A X. A P will be the 

 radius of the circle to be constructed, a seg- 

 ment of which will contain the given angle. 

 If the angle P A B \^ greater than 

 the angle P B A, then will the line P B 

 be longer than the line P A. In this case 

 the circle will intersect the ellipse in the points A and A^ only. 

 For, if the circle cut the ellipse in points other than A and ^', they 

 would be symmetrically situated with respect to B. This is impos- 



