In order to find the different types of the P,7-diagrams, we shall 
use the following theses. 
In a P,T-diagram always a same number of bundles is situated 
at the right and at the left of each bundle of curves. 
In every P,7-diagram the number of bundles of curves is always 
odd and three at least. 
In accordance with this property, which we shall show further, 
in a P,T-diagram occur, therefore, 3 or 5 or 7 ete. bundles of 
curves and diagrams with 2 or 4 or 6 bundles cannot exist. 
In the cases treated up to now, we see the confirmation of these 
rules. In each of the figs. 1 (I), 2 (I), + dD, 6 (If), 2 (ID, 6 (III) 
and 8 (III) we find three bundles in each of the figs. 2 (ID and 
4 (II) and in the symbolical diagram 21 (IV) we find five bundles. 
We may deduce the above-mentioned rules a.o. in the following 
way. First it is apparent that a P,7-diagram with only one 
single bundle cannot exist; in this case viz. one region at least 
should exist with a region-angle, larger than 180°, which is not 
possible in accordance with our previous considerations. 
Now we take in an arbitrary P,7-diagram a bundle of curves ; we 
call the stable part of this bundle a, the metastable part >. Now we 
go from a towards 6 along a curved line which does not go through 
the invariant point. Starting from a this curve intersects first a 
metastable, afterwards a stable, further again a metastable, afterwards 
again a stable bundle, etc. Arrived at 4, we have consequently 
intersected just as many metastable bundles as stable ones. | At least 
1 metastable and 1 stable bundle.) Therefore, when we find 7 
metastable bundles going from « in righthandside direction towards 
b, then we find there also nm stable bundles. As, however, every 
metastable bundle, which is situated at the right of a, is the pro- 
longation of a stable bundle, which is situated at the left of a, 
consequently at the left of « also n stable bundles must be situated. 
Therefore we find: at the right and at the left of every bundle of 
curves always a same number of bundles is situated. Hence it 
follows immediately that the total number of curves is always odd 
and three at least. 
Now we shall deduce the different types of P,7-diagrams with 
the aid of these properties. 
1. Unary systems. (One component; three curves.) 
In an unary system three curves occur, which have to be divided 
in accordance with our previous considerations over an odd number 
of bundles (three at least). This can take place only in one single 
way, viz. in such a way that three onecurvical bundles arise, 
