WO Al'l'l \ />! \ 



hyperbola through P } Insert ~ ior angle between F l P l nnd / ' /' . 



while the tangent /','/'. to the ellipse through this same point bisects the 

 external angle formed l>y the*- t\\o lines (cf. Part I, Arts. 148, 1 

 these tangents are therefore at right angles, hence (cf. Part I, Art. 100) 

 the conies intersect at right angles. 



Tin- fact could also have been readily proved analytically by compar- 

 ing the equations of the two tangents. 



REMARK 1. It is easily seen that as \ approaches - ft* from the positive 

 the ellipses represented by equation (2) grow more and mow flat 

 (because the length of the semi-minor axis y/b* + A approach* 

 approaching, as a limit, the segment F l l-\ of the indefinite straight lino 

 igh the foci. On the other hand, if A approaches b 2 from t,rl<r, 

 then the hyperbolas grow more and more flat, approaching, as a limit, 

 the other two parts of this line. Again, if A approaches - a 1 fn.m 

 above, the hyperbolas approach the y-axis as a limit. 



Hi M \I:K 'J. Since through every point of a plane there passes one 

 ellipse and one hyperbola of the con focal system represented by equation 

 .ind but one of each, therefore the two values of A which determine 

 these two curves may be regarded as the coordinates of this point; they 

 are known as the elliptic coordinates of the point. If the rectangular 

 coordinates of a point are known, the elliptic coordinates are easily found 

 by means of equation (2). 



E.g., let P l = (r p y t ) be the point in question, then the elliptic < 

 di nates of P l are the two values of A, which are the roots of equat ion (:?). 

 So, too, if the elliptic coordinates are given, the Cartesian coordinates can 

 be found. 



REMARK 3. The above observations concerning confocal conies are 

 easily extended to confocal quadrics, i.e., to quadric surfaces v. 

 principal sections are confocal conies. They are represented by the 

 equation 



* 



