Mathematics. — ‘“Degenerations in Linear Systems of Plane Cubies’’. 
By Prof. K. W. Rureers. (Communicated by Prof. JAN pe VRIES). 
(Communicated at the meeting of November 30, 1918). 
1. The number of curves with two double points in a net of 
plane curves is given by the formula: 
RDE 6 API DTE PS 
where D represents the number of free points of intersection of two 
elements of the net, o the number of base points, p the genus of 
the curves. 
For a net of plane cubics this is therefore the number of degene- 
rations into a conic and a straight line; in a net without base points 
the formula gives a number of 21; each single base point reduces 
the number given by the formula by one. 
If there are single base points the question can be raised in how 
many degenerations the straight line passes through two, through 
one, or through none of the base points. If we take one of the base 
points as an angular point (2, == 0, «‚ =0) of a triangle of coordi- 
nates and if we make the condition that the straight line «, = ma, 
must be a part of a cubic of the net, it is easily seen that 6 values 
are found for m, that therefore there generally pass through a single 
base point 6 straight lines, parts of degenerations. From this follows 
the solution of the problem in question. | 
Another solution is found in the following way. The net is defined 
by a curve c, and a pencil to which c, does not belong. If D is 
the number of free points of intersection of two curves of the net, 
c, must pass through 9—D base points A; of the pencil. The latter 
cuts c, in an involution y of order D. Now the following is clear: 
a. The number of degenerations into a straight line A; Az and a 
corresponding conic is 4 (9—VD) (8—-D). 
6. A straight line through one of the base points can form a 
degeneration with a conic through the remaining 8—D. Thesystem 
1) GAPORALI, „Sopra i sistemi lineari triplamente infiniti di curve algebriche 
piane’’, Collectanea mathematica in memoriam Chelini, p. 182. The letter NV 
stands there instead of D. 
