the A parabola has its opening turned downwards (this case is repre¬ 
sented in fig. 2, where besides some osculating curves the enveloping 
parabolae are also given). 
Special cases. 
§ 9. At the close of $ 5 we saw that two special cases may 
occur, viz. when K= 0 and when K= ±— 1/3. 
A. K — 0. We deduce from the. relation 
$Vl^cosy=:K 
three possibilities: 
1. 5 = 0. The movement remains confined to the FZ-plane. 
2. 5 = 1. The movement remains confined to the X^-plane. This 
form of motion - however proves to be impossible when x =}= 0 and 
y = 0 is substituted in (5). 
3. cos y = 0, therefore y = — or y = - - invariably. The os- 
2 2 
culating curves have their nodes at 0. The form of movement 
approaches asymptotically to a motion in the FF-plane. What‘beeomes 
of the enveloping parabolae has been represented in fig. 3, in which 
some osculating curves have been drawn too. 
B.K=±~ |/3. Then 5 X = 5, = > A == j- Now cos y «=‘±1: 
invariably, thus y= 0 or y = rr. The same parabola is continuously 
described, in which also the 5 a and 5, parabolae have coincided. 
(Fig. 4). When K undergoes a slight change, 5 X and 5, fall close 
together. So this form of movement is stable. 
S— 3 , ~ is of order 
$ 10. The expansions in series written down by Prof. Korteweg 
h 
lose for S = 3 their convergency as soon as — passes into order 
~ (page 18 of his paper) or i. o. w. as soon as sinks into order ~ . 
We shall now discuss this case. 
We again take as first approximation : 
((2n-j t -f- 4n l ft,) ; 
42 * 
