( 323 ) 



arrived on p. 320, so near x = 1, when n*e s is small compared 

 with f, , and near x = when /r>-, may be large compared with c r 

 But now we have lo consider the question whether such sets of 

 values of f, and f, can actually occur for mixtures. As we do not 

 know any rule as yet which indicates the value of a l% for given 

 value of a, and a si we cannot give a perfectly decisive answer to 

 this question. But we shall examine what may be derived about this 

 from the rule which is of frequent application: 



or 



or 



or 



>«»' 



M +*,)(!+*,)> [n + ^LJJl 



4n'(l +e,)(l +«,)>[(! + *i) + * 8 (l 4" *,) - (»-l)T 



(1 + O -f »' (1 + *,) 



4 — TTÏ (1 f ^J (1 + *,)> 

 («- l) 4 



1 



(n-1)' 



■W) 



If we think for a moment the sign ]> replaced by the sign = 

 the locus (d) is perfectly equal to (y), but with shifting of the two 

 ordinates in the negative direction over an amount equal to --1. 

 So if we draw two lines, one parallel to the horizontal f 3 axis al a 

 distance equal to --1, and one parallel to the vertical f, axis at a 

 distance — 1, and if we construct the same parabola for these two 

 lines, so that also the points P, Q, S are replaced by P,Q f , S', and 

 we have, accordingly, a line i J/ Q' \ (d) is satistied for all points 

 lying within this parabola. 



For the points of the line P' Q' the second member of (<f) is equal 

 to or a 15 = 0, and for the points lying below P'Q' a lt would be 

 negative. Accordingly these points will furnish no realizable sets of 

 values of e x and f 2 . But leaving (his for the moment out of consi- 

 deration, we may say that the series of points which the two parabolas 

 mentioned have in common, fulfil the two requirements thai they 

 furnish sets of values of s, and f, which admit of no intersection of 



= with - = U, and for which n, a 

 dx* dv* ' 



a la *. This holds equally 



for the points lying above the first parabola, but within the second. 

 The second parabola enters the positive quadrant of the f, and f, 

 coordinates in the origin, louches there a line e t - ns t = 0, and so 

 cuts the first parabola in a point represented by R in fig. 36. The 

 equation of the second parabola may, viz. be reduced to the form : 

 (6, — n-s,Y = in (n — 1) (e, - »«,). 



