( 202 ) 



When B is negative, this equation has certainly a positive 



n — 1 



root, when viz. the quantity B -f- Cx is positive, which is required 



fdp\ 

 for the occurrence of the line I — I = 0. The negative root is 



without significance for the problem. The quantity f — — — B J 

 positive may be written as follows : 



n 



ft,* — a i 



Let us call the case - < n the first case, and — — - > n 



the second case. 



v — b 

 Iu the first case the quadratic equation in — — has the factor of 



dx 

 the first power of {v — b) negative, and the known term positive. 

 Two real positive values of v — b can then satisfy this equation. 

 These roots are however real only if: 



C 



B 



C V n— 1 



x ^ l - x) '(BTciJ- U{1 ~ x) 



B+Cx 



or 



p-j.^Cf-i) 



1 \B-\-Ca: 



C 



So the roots are imaginary if: 



--l-(f-i)l + '^-i)>'' 



For values of x which are nearly or nearly 1, they are therefore 

 imaginary. From this follows that the locus of the points of inter- 

 section of ( — | = and ( — ) = is a closed curve in the first 

 \dxjv \dx* J v 



case. By dividing by x (t — x) we have only lost the values x = 

 and x = l, which, however, only correspond to v — b = and 

 T—0. This closed curve can contract to a single point, and even 

 disappear altogether. Contraction to a single point takes place, if: 



