( 432 ) 



between the parabola and the axes. It' we mil (1— xY = 



(n-1)" 



n*8 1 -\-k 

 and >)''.! 2 = — — — , these two equations, when k has been properly 



determined, will hold for the value of x of the root. By application 

 of (1 — x) -\- x = l, we l)ring the condition for the determination of h 

 in the following form : 



+ - —, — = i. 



n — 1 n — 1 



If for the binary mixture s 1 and e, were such that: 



w — I w — 1 



the point (c, , 6,) lies on the parabola, and the whole loens reduces 



d{(f ) 

 to a point. Bui then it appears that for the root <>l = the 



il.r 



quantity k must be = 0, and thai the value of.?' for this root 

 coincides with the point in which the locus has contracted. Iff, and 



£ s have such a value thai 1 — — <[ 1, the point ? , , ? , lies in 



» — 1 // 1 



the space OPQ, and there is a locus between two values a , and ,r. r 



/■ k 



It we add liolli n» s, and to s, . then may be chosen in such 



a way thai the condition - = is satisfied, and so also: 



da 



\/< 



/■ | / k 



n 2 k n 



f- - = 1 



/< — 1 it — 1 



The addition of an equal amount to e, and to p 2 involves, of 



course, a shifting of the point (* 1 , £ a ) in a direction which makes 



an angle of 45' with the axes, and thai in such a way that the 



k 

 projection of the shifting on each ot the axes is equal to — We 



suppose k to be positive. So we find the value of k by taking if 

 times the amount which is to be added to the projections of the 

 said point to reach the parabola. If the point («, , a s ) lies in 0PQ,k 

 is positive. But for points within the parabola, k is negative. But 

 as for tlie case that the closed locus exists, the point (s l , f 2 ) must 

 lie inside OPQ, we have only to deal with positive values of k. 



