( 322 ) 

 the direction of f lie above straight line being given by : 



So these two directions are symmetrical with respect to the axes 

 mentioned. In fig. 36, in which the e, axis has heen drawn horizon- 

 tally, the value of n, which is always larger than 1, is not supposed 

 to be very large. 



The calculation of the place of the top of the parabola may, 

 among others, be made, by making use of the property that in the 

 top the tangent to the parabola is normal to the direction of the 



ds. e l —n i e t -{-(n—iy 1 



diameter. So — = n' — — - — = — , from which follows 



de.j s t — ^'f, — (n—ly n 



n 4 — l 

 that for the top s l —n , s a = — (/? — 1)*— . This is the equation of 



n A -\ 1 



the axis: it cuts the f, axis in a point in which e l = and c, = OS= 



(// — 1)' // 4 — 1 ^ o n*-\ 



= — — -. Hence OS = op . . So tor very small values 



n* rr-j-l " 4 + l 



of n also OS is small, hut for larger value of n OS approaches to OP. 

 All the points inside this parabola give values for f, and f,, for 



which equation ( t ?) is satisfied; thus this equation reduces for all the 



points of the line PQ fig. 36 to : 



*: +*,«•#■ >0 



and for sets of value- of e, and e„ belonging to points lying within 



'/'-'if« -/'-if' 



the parabola, there is. therefore, never intersection of =Oand =0. 



Summarizing we arrive at the following result. All the points in 



the positive quadrant of the f 3 , f, axes of fig. 36, lying above the 



line PQ, represent sets of values of ?-, and *-,, for which (a- A' 



,/'U' d'tp 



must always he positive! no intersection of — = and — =0can 



take place. The points lying below PQ, hut within the parabola, 



also represent such sets. The points below PQ, lying exactly on the 

 parabola represent sets of f, and s a , for which the locus of the 

 points of intersection of the two curves mentioned reduces to a 

 single point. And finally the points below PQ and below the parabola 

 represent sets of values of g, and s, for which the two curves 

 yield a locus of points of intersection. The point to which the 

 locus of the points of intersection has contracted lies at a value of' 



1 _ gl 4 rc 2g » 



(n - l) 2 l/fj 



N = — - = - , at which result we had alreadv 



1 — x « a t a n,y e s 



