( 326 ) 



with the e s axis, the tangent of which is equal to /r, hence 0' M is 

 parallel to the axis of' the first parabola. For the coordinates of the 

 centre we find : 



(» — I) 2 , (n— l) a 



(1 + .,)* = ~- and (1 + , ,),/ = ^Ï3]ij " 



2. Highest and lowest point. 

 For these points fe, = or 



- (n - 1)' + (1 + ej -f n' (1 + e.) = 2P (1 + «.) 

 and so 4/-V (1 + e,) (1 + e,) = 4/< (1 -4- gj 8 . 



Hence 1+^ = for the point 7; and (1-f-eJ = — (1 + *,) 



(r* — 1)* 

 for the point 7i' ; from this follows for IV the value (l-f^i)/." = rp- 



/' J (n—l)' , (n — l) 5 



and (1 + e t )B> — - — - — and for B is 1 -f e = — . 



1 — r n n 



3. The points A and A'. 

 For these points fs t = or 



_ (n - 1)' + (1 + f,) f rC- (1 + *,) = 2JV (1 + 8,) 



and so ^Vd+fJ^^VIl+F^l + g. 



Hence for J 1 + s , == and 1 + f, = (« — 1)' and for A' holds 



(1 | Bl ) = /•-"•-• (1 + e.) or (1 + *,U = " '~ " : ,, and : 



// l — r 



4. The points of intersection with the f, axis. 

 From (2n + n*£,)* = 4nV s (1 + e s ) follow?: 



_ 2 (;,-/ 3 ) dr Z l/(n — IT - - (1— P) 

 '-=." "^ 



So while ! — J 2 <C ( w — 1) ! there are two points of intersection 

 with the f 3 axis both on the negative side of the origin. For I s = 1 

 one of the points of intersection lies in the origin, and the other 



n-1 



point of intersection at s, = — 4 , wliich value = — 1 tor n = 2, 



71- 



and for all other values of n not so large negative. For 1 — I* =(2 — l) s 

 the two points of intersection with the f, axis coincide, and for 

 1 — /- ^>(n — 1)'-' they are imaginary; then the whole ellipse has 

 descended below the horizontal axis. For the case that n is but 



