( 543 ) 



The cases (4, 2) and (6) arc to be regarded as included in the 

 preceding. By allowing two of the vertices or the three vertices of 

 the triangle just considered to coincide we find for case (4, 2) the 

 connecting line of the two bascpoints counted four times and the 

 tangent to the conic in the basepoint of highest multiplicity counted 

 two times, for case (6) the tangent to the conic in the point counting 

 for six basepoints counted six times. That there can be no locus proper in 

 the last case ensues also from the fact, that_ ajgenuine cubic with 

 cusp allows of no sextactic point. 



b) The case (3,3). If the six basepoints coincide three by three in 

 two points of the conic, then c 6 consists of a part improper, the 

 connecting line of the two points counted four times, and a part 

 proper, a conic touching the conic of the ^basepoints in these points. 

 The new conic lies outside the conic of the basepoints. 



c) The case (1, 5). This case agrees in many respects with the 

 preceding. We find a part improper, the tangent in the point counting 

 for five basepoints drawn to the conic of the basepoints, and a part 

 proper, a conic touching the conic of the basepoints in these points. 

 The new conic lies inside the conic of the basepoints. 



6. Of course it is possible to call forth by the curve c 12 succes- 

 sively all the different special cases which can put in an appearance 

 by the parabolic curve s la of the various surfaces >S 3 . As this would 

 lead us here too far, we limit ourselves to a single remark, which 

 can eventually facilitate an analytic investigation of this idea. 



According to the general results with respect to a linear system of 

 curves c n obtained as early as J 879 by E. Caporali the locus 

 c 4(2n— 3) f f] ie cusps f this system has in each r-fold basepoint of 

 the system a 4(2r— 1) fold point and besides 6(n — l) a — 22(3r*— 2r+l) 

 nodes C. Each of those points C is characterized by the property 

 that each curve of the system passing through this point is touched 

 in this point by a definite line c. 



For the case under observation, // = 3 of the cubics, the number 

 of points C is represented by 24 — 6/>, when /> is the number of 

 basepoinis. 



If we wish to investigate analytically what peculiarity the locus 

 of the cusps shows in a basepoint of the system, or how a line 

 through three basepoints separates from it, then the result — and 

 this is the remark indicated - will be independent of the fact, 

 whether the remaining basepoints occur or nol, if in the former case, 

 that some of these basepoints appear in a real or in an imaginary 

 condition, we assume that these points both with respect to each 



