( 634 ) 
B=w, + 4u,7 = 0 is represented a pencil (8) of curves of order six 
with nine double points. To a pencil of curves of order nine belong 
in general 3(n—1)’ curves possessing a double point; this amount 
must however be diminished by seven for each common double 
point which the curves possess in the base points. So we can expect 
that there will be twelve points, which can appear as tenth double 
point of a curve out of the pencil (8). It seems however desirable 
to prove that in this case too where one of the curves is a ¢, 
counting double the number of these points is twelve. 
The points indicated are found by elimination of 4 out of the equations : 
dp dp dp : 
ae i) — —0 and — = 0, or 
dz, dz, dx, 
dw du dw du dw du 
+4—=0, —+14—=0, — _——0. 
dx, dz, dz, dz, de, dx, 
By this elimination we find : 
dw dw dw 
en 
du du du ’ 
de, dx, dx, 
which equations represent three curves, whose common points of 
intersection — if only differing from the base points of (3) —- are 
the demanded double points. (The factor «=O, which we have 
omitted» means that each point of w, can be regarded as a double point, 
We write them in this form: 
dw du dw du 
ate Ss Oe so op wi AEN 
de,de, dz,dz, 
dw du dw du 
— — 0 n EA 5 . = - (2) 
de, de, de, da, 
dw du dw du 
tE) dte Te 
dz,dz,  de,de, 
The curves represented by (1) and (2) have forty-nine points of 
intersection; among these there are however ten which do not lie on (3), 
dw 
du 
== (and Per The remaining thirty- 
viz. the points which satisfy 
& 
a, 
nine points must still be diminished by the points of intersection 
lying in D,, D,...D,. If again we allow the vertex O, of the 
triangle of coordinates to coincide with D, and if we note down 
the equations of w, and wu, ranged according to the descending 
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