( 327 ) 



little larger than 1, this circumstance is to l>e expected; then onlv 

 negative values of e, and f a exist. 



5. The intersection with the Ej axis. 

 From (2n -f sj' = 4n s / 3 (1 -f e x ) follows : 



fl = 2n (n/ 2 — 1) ± 2nl' Vy P - 2n -f l;. 



For P = l, we get *! = (") and f 1 = 4« (n — 1). For /' < 1 these 



-" — ! 

 values approach each other, and they coincide for r= as was 



"In— 1 (n— l) s 



stated above. For r<[- or 1 — r]> positive values off, 



3 

 no longer occur. For r = - e.g., the ellipse will just touch the s, 



axis for n = 2, and that in a point for which e,=2; but for 



smaller value of n the ellipse does not cut the s, axis; for larger 

 value of ft, on the other hand, it does. 



6. The intersection with the line PQ of tig. 36. 

 If in : 



( 2n + Sl + n' e,)"-' = 4n 2 F (1 + ej (1 + e.) 



we substitute the value e t -f- w * 6 i = ( n — l) 1 , we find for the deter- 

 mination of f, the equation : 



(n- + 1)' 

 «V + 2 (n-l)e a + / ' - [1 + (n-m = 0. 



4n t 



i/t ~r^ xi 



When /* <^ t , 2 ri / 1 -^- the two values of f 3 are negative. II 



4n s [l-{-(n— l) s ] 



/ 3 is greater, a point of intersection is found at positive value ofe, ; 



for / 2 just equal to the given value the ellipse passes exactly through 



the point Q, and the same relation exists between A' and //, as is 



also found when we substitute the value (n — l) a for e l in the 



equation of 5. 



While one of the values of f, is positive the ellipse intersects 



not only the line PQ, but also the tirst parabola. For smaller value 



of 1- or larger of n, the line PQ is no longer intersected in the 



positive quadrant; intersection of the ellipse with the first parabola 



will, indeed, be still possible, till the two points of intersection 



coincide with further decrease of /'"'. Then the ellipse touches the 



d'tp -/-'if' 



parabola, and the possibility that - =0 and =0 no longer 



'/./■'-' ilr' : 



intersect, vanishes. 



