VAN DER VRIES. — MULTIPLE POINTS OF TWISTED CURVES. 521 



(2). If the tangents lie by threes in six planes, as in 22, the point is 

 equivalent to 18 actual and 3 apparent double points. This will be ex- 

 plained with the following case. 



(3). If the tangents lie by threes in five planes, as in 23, the point is 

 equivalent to 16 actual and 5 apparent double points. The branches of 

 a sextuple point of kind (2) before they unite to form the sextuple point 

 have their tangents lying as edges of a tetrahedron, as in 1 1. Adding 7 

 in the plane of 3 and 5 causes the intersection not only of 7 with 3 and 

 5, and of 3 with 5, but also of 1 with 6. The lines 1, 2, 3, 4, 5, and 6 

 now meet in one point. This septuple point is thus equivalent to a IV 

 (2), four actual and two apparent double points, that is, to 16 actual and 

 5 apparent double points. If moreover 7 is added in such a way that it 

 also lies in the plane of 1 and 6, it meets 1 and 6 and two more actual 

 double points must be added. A septuple point of kind (2) is thus 

 equivalent to 18 actual and 3 apparent double points. 



(4). If six tangents lie in one plane, as in 24, the point is equivalent 

 to 16 actual and 5 apparent double points. This can be obtained from a 

 IV (1). 



(5). If four tangents lie in one plane, and the others as in 25, the 

 point is equivalent to 15 actual and 6 apparent double points. This can 

 be obtained from a IV (2). 



(6). If the tangents lie as in 26, the point is equivalent to 15 actual 

 and 6 apparent double points. The tangent lines before the branches of 

 the curve unite to form the septuple point lie as in 27, which have 

 twelve points of intersection. 



(7). If five tangents lie in one plane, and the other two lie in a plane 

 with one of these, as in 28, the point is equivalent to 13 actual and 8 

 apparent double points. This can be obtained from a IV (3). 



(8). If six of the tangents he by threes in four planes, and the seventh is 

 not in a plane with two others, as in 29, the point is equivalent to 13 actual 

 and 8 apparent double points. This can be obtained from a IV (2). 



(9). If five tangents lie in one plane, and the other two do not lie in a 

 plane with one of these, as in 30, the^ point is equivalent to 12 actual and 

 9 apparent double points. This can be obtained from a IV (3). 



(10). If four tangents lie in one plane and the other three lie in a 

 plane with one of these, as in 31, the point is equivalent to 12 actual and 

 9 apparent double points. This can be obtained from a IV (4). 



(11). If four tangents lie in one plane, and the other three as in 32, 

 the point is equivalent to 12 actual and 9 apparent double points. This 

 can be obtained from a IV (5). 



