VAN DER VRIE3. — MULTIPLE POINTS OF TWISTED CURVES. 519 



the cross section of a plane that contains the lines whose bases are 1, 4, 

 and 5 ; where the five lines have not yet united to form a quintuple 

 point. The lines then have six actual intersections, viz. 12, 13, 23, 14, 

 15, and 45, and four apparent intersections, viz. 24, 25, 34, and 35. 

 Considering these lines as the tangents to a curve we see that this quin- 

 tuple point can in the limit be considered as composed of six actual and 

 four apparent double points. 



(4). If three tangents lie in one plane and the other two do not lie in 

 a plane with one of these, as in 7, the quintuple point is equivalent to 5 

 actual and 5 apparent double points. This quintuple point can be 

 obtained from a II (2) by adding a new branch with its tangent in the 

 limit not in the plane of two other tangents, or from a II (3), by adding 

 a new branch with its tangent in the limit in the plane of two other 

 tangents. 



(5). If no three tangents lie in one plane, as in 8, the quintuple 

 point is equivalent to 4 actual and 6 apparent double points. This is the 

 kind of quintuple point mentioned on page 516. It can be obtained from 

 a II (3) by adding a new branch in such a way that its tangent does not 

 in the limit lie in the plane of two other tangents at the quadruple point. 

 It does not matter whether this new branch is added in such a way that 

 the five tangents in the limit lie on a quadric cone that contains an 

 arbitrary line as an edge or on a cubic cone that contains this line as a 

 double edge. The multiplicity of the point on the monoid only affects 

 the composition of the multiple point on the curve by placing an upper 

 limit on the number of tangent lines that can lie in one plane. The 

 only relations between the tangent lines that directly affect the number 

 of actual and apparent double points to which the multiple point is 

 equivalent are the number of lines that lie in one plane and the number 

 that lie in two or more planes containing two or more other tangent 

 lines. 



IV. Sextuple Points. 



(1). If six tangents lie in one plane, as in 9, the sextuple point is 

 equivalent to 15 actual double points. This can be obtained from a 

 III (1). 



(2). If the tangents lie by threes in four planes, as in 10, the sex- 

 tuple point is equivalent to 12 actual and 3 apparent double points. 

 For let 1, 2, 3, 4, 5, and 6 represent not only the bases of the tangent 

 lines, as in 10, but also the tangent lines themselves, where the branches 

 have not yet united to form the sextuple point. We suppose 1, 2, and 3 

 to lie in one plane, and 1, 4, and 5 in another plane, but no three of 



