522 PROCEEDINGS OF THE AMERICAN ACADEMY. 



(12). If the tangents lie by threes in four planes, as in 33, the point 

 is equivalent to 12 actual and 9 apparent double points. This can be 

 obtained from a IV (5) by adding 7 so as to lie in a plane with G and 4 ; 

 thus causing 6 and 4 to meet. 



(13). If four tangents lie in one plane, and two lie in a plane with one 

 of these, and the seventh lies in no plane with two others, as in 34, the 

 point is equivalent to 10 actual and 11 apparent double points. This can 

 be obtained from a IV (7). 



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

 plane not containing one of the first four, as in 35, the point is equiva- 

 lent to 10 actual and 11 apparent double points. This can be obtained 

 from a IV (8). 



(15). If the tangents lie as in 36, the point is equivalent to 10 actual 

 and 11 apparent double points. This can be obtained from a IV (5). 



(16). If four tangents lie in one plane, and none of the other three lie 

 in a plane with any two, as in 37, the point is equivalent to 9 actual and 

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



(17). If the tangents lie by threes in three planes, as in 38, the point 

 is equivalent to 9 actual and 12 apparent double points. This can be 

 obtained from a IV (7). 



(18). If the tangents lie as in 39, the point is equivalent to 8 actual 

 and 13 apparent double points. This can also be obtained from a 

 IV (7). 



(19). If three tangents lie in one plane, and three in another plane, 

 and the seventh in no plane with two others, as in 40, the point is 

 equivalent to 8 actual and 13 apparent double points. This can be 

 obtained from a IV (8). 



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

 four lie in a plane with any two, as in Figure 41, the point is equivalent 

 to 7 actual and 14 apparent double points. This can be obtained from a 

 IV (10). 



(21). If no three tangents lie in one plane, as in 42, the point is 

 equivalent to 6 actual and 15 apparent double points. This can also be 

 obtained from a IV (10). 



VI. Multiple Points in General. 



Considering multiple points in general, it is evident that a £-tuple 

 point is equivalent to at least Jc — 1 actual double points. For, every 

 branch of the curve has in the limit at least one actual intersection with the 

 totality of 7c — 1 other branches. It is also evident that if the tangents 

 at the multiple point lie in a number of planes of which no two have a 



