( 433 ) 



So wc have ,r ^> and J ,/'^> , and (lie equation 



// — 1 n — 1 



i 



ire. 



(n-iy (n-iy _ 1 



x 1 — x 



holding for the values of x of the points of the locus, we find, after 



substitution of uf> and 1 — # ]> : 



« — 1 n — 1 



ra — 1 n — 1 



a relation which exists indeed for points of the space OPQ below 



the parabola. 



But now we have to make the following remark about the equa- 

 te 



Hon which indicates the value of x for the points where — = 



dr 



for the closed curve. For this equation (#>'") we found the following 



form : 



I dA) l dA) 



n—\—n\/\A — as — =f V \A + (1 — as) — = 

 | das ) | das \ 



or 



t /c i 1 — as da ) t /c ( as da ) 



(n--l)-n* \/ - 1+- -- =F(l--'j / - 1 - - =0. 



{/ a [ a das ) y a y a ax ) 



da da 



a + I 1 — «0 y rt — •'• T- 



If we seek the values of and of - - , we 



a a 



a s — c (1 — x)' 2 a 1 — cas s 



find and - for this. These quantities must be 



a a 



positive, because they occur under the radical sign. And this gives 



d\: 

 a restriction for the values of x for which — can be = 0. Ifa,]>c 



das 



the former of the values mentioned is positive for all the values 



«s ft S (l-|-£ 2 ) 



from x = to x = l. The quantitv is equal to , and 



c (n — 1)- 



so certainly greater than 1 for positive e t . The quantity a l — ex* is 

 positive, when x" <[ ■- and negative for #*]>—. So if -< 1, 



c '' c 



values of # lying near 1, cannot exist. This will be the case, as 



a. 1 -)~ € i ^ 



soon as 1 ]> — or 1 ^> , or w* — 2ra > f , . If we put the 



c (n — l) 3 



