( 138 ) 



dp 



ctp 

 the detached closed'longitudinal plait is cut by the line — =0, and 



dx 



has the above discussed plaitpoints. 



But though on t he supposition of this way of splitting up we do 



not meet with a definite contradiction, yet there is one circumstance 



which makes me doubt whether it will occur frequently or universally. 



If we draw the point of intersection of the mentioned loci on which 



iPv d?v 



— = and = 0, we find a point which lies on the left side 



dx*p </■''*,/ 



of — = 0, whereas after the detaching we should sooner expect 

 da 7 



the place of the plaitpoint with the largest volume, according to the 



d*ip 



course of the nodal lines, on the right side of = 0. Indeed, 



dx 



another way of detaching is possible. The splitting may take place 



dp d'jp 



in a point on the left of — = 0. Then - — == 0, which curve must 

 1 dm das* 



disappear in a point of — = 0, must already have contracted so 

 dx 



dp 

 far that it lies entirely m the region where — is positive. As we 



d.v 



d*v 

 observed before, there runs a branch on which — — - = also there 



da> p 

 d*v 

 and for the loop-line on which — = (These Broc. March 30 1 907 



p. 736) there must be a closed figure, which has got detached from 



the branch right of — = 0, because the double point, the point in 

 dx 



which — = and — = 0, no longer exists. Then we have again a 



dx dx t 



detached closed longitudinal plait, but one which is not intersected 



h v JL = and which has two plaitpoints in which the p- and 

 J dx 



o-lines which touch have -—=-—= positive, in accordance with 



J dXp dXq 



the course of the nodal lines. In fig. 25 the circumstances after the 

 splitting have been represented for this case. First of all the lines 



— = and — = occur in the figure; further — = 0, which 

 dx dv dx 



passes through the point where j- = has minimum volume. 



