.!;// \///.\ 



ftindamental conic, and represented by . , 

 other conks of the system may be designated 



.. then eaeh of the 

 bj iu sppropriats Tain* 



Through any assigned point, P, s(x,, y,), of the plane, there passes one 

 ellipse and one hyperbola of the system represented by equation 

 For substituting the coordinates r, and y l of />, in equation (.'), it gives 

 the quadratic equation 



for the determination of X. Equation (3) gives two rallies of X 

 two conies of this confocal system pass through P,. That one of these 

 is an ellipse and the other an hyperbola is shown as follows : the quad- 

 ratio function in A 



1 



is negative when X = + ce. and, as X decreases from + oo to - <r, thk 

 function becomes positive just before X = - W ; negative again just after 

 X = - * and positive again just before X = - o; hence, of the two 

 roots of equation (3). one lies between - and , and the other between 

 -o f and -&; and therefore of the two confocal conks which pass 

 through /, . one is an ellipse and the other an hyperbola. Moreover, the 

 two confocal conks whkh pass through any given point, as P, a (*,. jr,), 

 of th. T.:.M- intersect at Hght angles. This k easily seen geometrically 

 thus: connect /, with the foci F, and F,. then the tangent P 4 r, to the 



