( 133 ) 



g-line must have such a course in the upper plaitpoint that they 



descend towards the right, which moreover could be put a priori. 



dp 

 For every «-line when it still lies above the line — = 0, and does 



dx 



not pass through — = 0, descends when it proceeds towards the 

 dx 1 



right. But in the lower plaitpoint, i. e. in the plaitpoint with the 



dp 



larger volume that lies below the line — =0, the (/-line which touches 



dx 



in that plaitpoint, must descend as it proceeds to smaller value 



of '■, in accordance with the course of the nodal lines. We should 



also have found this course of the nodal lines confirmed, if we had 



paid attention to the course of the yij-lines. 



Everything we have discussed in this V th communication rests on 



dp d*ib 



fig. J4; so we have thought that — = and — =0 intersect. There 



° dx dx' 



is, however, also a possibility, and it will even be the rule, that the 



two curves exist, but do not intersect. Then two cases are to be 



,. . d*q 



distinguished, viz. that =0 remains restricted to smaller volumes 



dx' 



d,> 

 than those of — = 0, or to larger ones'). When tracing the two 

 dx 



curves with respect to each other we must take care that the points 



./'■.!• 

 in which tangents may lie drawn to - = parallel to the y-axis, 



dx' ' 



, , d 'P ''/> 



lie on the line — = 0, and that also the point in which — = () 

 dx* dx 



has minimum volume, lies on this line too. Now the line — =0 



dx* 



dv 

 has a simple course. The value of — for this line is equal to 



dx 



ill) 3 ,!"'/, 



— - — 7. ri'om this follows that (his line — = consists of a single 



d,- i + 2 h_ & 



V 



branch, which from a point of the 1 st axis moves regularly to the 



right to points of continually larger volume. So if the line — =0 



dx 



d*q> 



cuts the line - — = 0, the two points in which tangents parallel to 



dx' 



J ) See These Proceedings April 20, 1007, p. 833 seq. 



