( 732 ) 



curves 



= 



— = and (— 



dx 2 \d x , 



intersect, which then necessarily 

 takes place in 2 points, the shape 

 of the </-lines is much more intri- 

 cate. Then numerator and deno- 

 minator are equal to zero in 



dx' 



and I — | 

 das 



to be determined from this equa- 



. do 



Then ( — 



dx 



Fig. 5. 



d z %p \ fdvy 

 dxdv 2 ) \dicjc, 



+ 2 



tion. 



mined from : 

 iv 



d> 



dx*dv) \(l,i' 



+ 



d'ty 



dx 



must be deter- 



0. 



In the discussion of the shape of the /Mines we came across an 



d*ip , d'tp 



analogous case when the curves — — = and — — - = intersect ; 

 ° </c dxdo 



and there we found that the two points of intersection had a different 



character. For one point of intersection the /;-line has two different 



<IS}i <l'Up f (/'if' 

 real directions, depending on the sign of 



dv 3 dvdas* 



ilxilr 



[f 



this expression was negative, t'.ie 

 loop-isobar passed through that 

 point of intersection. In the 

 same way, when from the above 



(dv 

 quadratic equation for I — 



we write the condition on which 

 the roots are real, we find the 

 condition : 



~M««=° 



dx 1 dxdv* 



dx 2 dv 



negative; 



which may be immediately found 

 from the condition for the loop 

 of the loop-isobar, as require- 

 ment for the loop of the loop- 

 a-Une, when we interchange x 

 and v. 



Fig. 6. 



