821 
system has, therefore, the same appearance as that of a binary 
system; consequently it exists, as 2 (I), of one twocurvical and two 
onecurvical bundles. One of the curves of this figure must represent 
now the three coinciding singular curves. 
When the singular curves are represented by one of the curves 
of the twocurvieal bundle, then fig. 8 arises; when they are repre- 
sented by one of the two other curves, then fig. 6 arises. 
In main-type II curve (M) is bidirectionable; the two other sin- 
gular curves coincide therefore in opposite direction | fig. 2 X, 3°X) 
and + (X)]. 
In main-type ITA curve (M) is a middle-curve of the (J/)-bundle 
(fig. 3(X)]. The type of P,7-diagram consists of: 
(M)-bundle + 22 other bundles 
viz. « bundles on each of the sides of the (J/)-bundle. | In fig. 3 (X) 
is == 2 |y The (M)-bundle itself consists of one curve at the one 
side and three curves at least at the other side of the invariant 
point; it consists, therefore, of four curves at least. | In fig. 8 X of 5}. 
When we take an (J/)-bundle of 4 curves, then, as 5 curves occur 
in the invariant point, 4 + 22—5, consequently «= }. An (M)- 
bundle of four curves cannot exist, therefore. When we take an 
(M)-bundle of 5 curves, then 5 + 227 == or w — 0. Consequently 
the P,7-diagram consists only of an (M)-bundle of 5 curves; we 
obtain, therefore, a diagram as in fig. 4. 
In main-type Il B curve (M) is a side-curve of the (J/)-bundle 
lig. 4 NJ. The type of P,7-diagram consists, therefore, of: 
(M)-bundle + (2.7 +1) other bundles 
viz. w bundles at the one side and (rm +1) bundles at the other side 
of the J/-bundle. [In fig. 4 (X) is #1]. The J/-bundle consists of 
two curves at least at each side of the invariant point; consequently 
it consists of four curves at least. [In fig. 4 (X) of 6}. 
When we take an (J/)-bundle of four curves, then 4-+ 27+ 1—5, 
consequently «0. At the one side of the (J/)-bundle is situated, 
therefore, one curve [viz. « +1—1] on the other side not a single 
curve is situated [viz. «=0]. Now we obtain the type of P,7- 
diagram of fig. 2. 
_ In communication (X) we have deduced the rules: 
1. The* two indifferent phases have the same sign or in other 
words: the singular equilibrium (J/) is transformable into tle in- 
variant one and reversally. Curve (J/) is monodirectionable; the 
three singular curves coincide in the same direction | fig. 1 (X)]. 
