520 PROCEEDINGS OF THE. AMERICAN ACADEMY. 



these lines to meet in one point. The line 6 is added in such a way that 

 it lies in the plane not only of 2 and 5 but also of 3 and 4. The lines 

 before we have reached the limit then lie as in the tetrahedron in Figure 

 11. The actual intersections are 12, 13, 14, 15, 23, 25, 26, 34, 36, 45, 

 46, and 56, and the apparent intersections are 16, 24, and 35; thus 

 twelve actual and three apparent double points in the composition of the 

 sextuple point. 



(3). If five tangents lie in one plane, as in 12, the sextuple point is 

 equivalent to 11 actual and 4 apparent double points. This can be 

 obtained from a III (1). 



(4). If four tangents lie in one plane, and the other two lie in a 

 plane with one of the four, as in 13, the point is equivalent to 9 actual 

 and 6 apparent double points. This can be obtained from a III (2). 



(5). If the tangents lie by threes in three planes, as in 14, the point 

 is equivalent to 9 actual and 6 apparent double points. This can be 

 obtained in the same way as case (2) above. The lines before we 

 reach the limit lie as in Figure 15. The actual intersections are 12, 13, 

 14, 15, 23, 25, 26, 45, and 56, and the apparent intersections are 16, 24, 

 34, 35, 36, and 46. 



(6). If four tangents lie in one plane and the other two do not lie in 

 one plane with one of these, as in 16, the sextuple point is equivalent to 

 8 actual and 7 apparent double points. This can be obtained from a 

 III (2). 



(7). If three tangents lie in one plane and two lie in a plane with 

 one of these, and the six in no plane with two others, as in 17, the sex- 

 tuple point is equivalent to 7 actual and 8 apparent double points. This 

 can be obtained from a III (3). 



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

 that does not contain one of the first three, as in 18, the point is equiva- 

 lent to 7 actual and 8 apparent double points. This can be obtained 

 from a III (4). 



(9). If three tangents lie in one plane and none of the remaining lie 

 in a plane with two others, as in 19, the point is equivalent to 6 actual 

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



(10). If no three lie in one plane, as in 20, the point is equivalent to 

 5 actual and 10 apparent double points. This can be obtained from a 

 III (5). 



V. Septuple Points. 



(1). If the tangents all lie in one plane, as in 21, the point is equiva- 

 lent to 21 actual double points. This can be obtained from a IV (1). 



