f 436 ) 



point of (</)'"). At the final point (<p'") = 0. At \x == 0, {<p'") is positive. 



So for the intermediate minimum value (</>'") is negative. 



The function <;" z= (), the relation through which we may know 



dv 

 the value of x for the points in which = for the closed curve, 



dx 



must also be satisfied by the value of x of the point in which the 



closed curve has contracted to a single point. To show this we have 



to substitute the values x 



■//— 1 



and (1 — ,0 



I *, . 



Ill : 



(n— 1) — nx (I — #) 



XC 

 (1--T) 



a 



-1 = 



c.r' 



•(1 .»■)■-' 



where if appears that this equation is satisfied. That we only retain 



the sign - - for the third term is in accordance with our conclusion 



dv 

 thai r<^/> 2 for the whole curve. V i u 1 that — = must also be satis- 



dx 



tied in the isolated point, - the point to which the whole curve 



has contracted, - follows from the circumstance (hat for such a point 



dv 



has an arbitrary value. The quantity 

 dx cx(l—x) A 



1 

 — — — is equal to 



a x {\ — 2(-f-« s « — ex. I — 2) a, a s 



cx{\ — x) ex t'(l — .'') 



1 = 



l + «i 



»(l + 



(w-l)l/6 l (m-I)^, 



— 1 



or 



Further 



9(1 - •)' 



— 1 — 



Substituting these values we find : 



(n-1) 



(n-1) 



1 n 



+ 



and 



n 1 



+ 



_lA 2 j/ Cl . 



l/g, |/c s 



Let us write the equation of the closed curve in the following 

 form : 



V ^\{l-A)-2 V - + (1+J3) = f 



representing by 5 



*(!-*) 



(n— 1)» *•(!— ar) 



(1— #) 2 -f- 2n.c(l- *■) + nV 



