46 UNIVERSITY OF COLORADO STUDIES 



sible, since the radius is less than P /?, and no part of the circle can 

 lie below B. Nor can the circle intersect the upper part of the 

 ellipse; for, if it did, the points of intersection would be symmetrical 

 with respect to the point B. This would cause an arc of the circle 

 to pass between P and i>', which is impossible, since P A^ AO^ 

 O B^ ^ P B\ Therefore, when the angle P A B \q greater than 

 the angle P B A, the circle intersects the ellipse in the points A and 

 ^4\ and in no other points. In this case the construction is impos- 

 sible. 



If the angle P A B equals the angle P B A, then will P A = 

 P B^ and the circle will be tangent to the ellipse at B. In this case 

 the circle intersects the ellipse in four points — A^ A\ and two 

 coincident points at B. Since it intersects the ellipse in these four 

 points, it cannot intersect the ellipse above the X-axis, as is also 

 evident since the angle 6^ ^ ^ is greater than the angle B AO. In 

 this case the construction is possible. 



If the angle Z^ ^ ^ be less than the angle A B P, then 

 will P B he shorter than P A. In this case the circle must inter- 

 sect the quadrants A B and A B^ in points symmetrical with respect 

 to B. It is evident that this will be the case ; for the line A JT, 

 perpendicular to A P, will always intersect the ellipse. And since 

 the circle described with A P as a, radius must lie in the angle P A X, 

 it must intersect the ellipse between A' and B, and in a point in the 

 quadrant B A\ In this case, also, the construction is possible. 



We have seen that the greatest value the angle P A B can have 

 is the value of the fixed angle A B P. From this it follows that the 

 angle A P B i& a minimum when A P = B P. This is the mini- 

 mum value of the constructed angle A JT, which is equal to the 

 angle A P B, leaving their sides perpendicular to each other. 



Now, let us find the value of this constructed angle when a 



minimum. 



O AX= 180 — 2<t> 



2 ah 

 . • . Sin O A X = Sin 5 « = 2 Sin cb cos <b == ^ , ,.^ 



This same result is easily obtained from «' ¥ sin d = ah, where 



