500 PROCEEDINGS OF THE AMERICAN ACADEMY. 



Thus, as a multiple point at which the tangents do not lie in one plane 

 is a point on S, a number of apparent double points disappear when it is 

 formed on the curve. A multiple point, however, at which the tangents 

 all lie in one plane, can be obtained, as we have seen, by means of a 

 monoid that has it as an ordinary point. It is therefore a 0-tuple point 

 on S and has no apparent double points in its composition. This will 

 be shown more in detail later. 



Every K-tuple point, however, whether its tangents do or do not lie 

 in one plane, has the same effect on the possible number of apparent 

 double points ; each reduces it by £ k (k — 1). Thus a curve of order m 

 having a number of K-tuple points can never have more than 



l( OT _l)( m _ 2 )-^2K(K-l) 



apparent double points ; where the summation includes all points for 

 which 2 < k < m — 2. 



5. There is an upper limit to the necessary order of the cone on which 

 the tangents to a curve at a multiple point lie. The equation of a cone 

 of order k + 1 that has a certain line as a A:-tuple edge has 2 k + 2 

 arbitrary constants, that is, such a cone can in general be made to pass 

 through 2 k + 2 arbitrary lines through the multiple point. We can 

 therefore always pass a cone of order k + 1, having an arbitrary line 

 through the K-tuple point of the curve as a fr-tuple edge, through the k 

 tangent lines at the K-tuple point of the curve, if k < 2 k + 2, i. e. if 



K 2 K 



— - — < k, or - < k + 1. We need, therefore, never take k greater 



than — - — if k is even, nor greater than — - — if k is odd. The tangents 



may, however, in some cases lie on cones of lower orders than those 



given by the expressions above. We can in fact classify multiple points 



according to the orders of cones of the lowest order that can be passed 



through the tangents at them, this order taking any value from 1 to 



k k + 1 



- or — - — , according as k is even or odd. We must also bear in mind 



that the order of the cone places a limit on the number of the tangent 

 lines that can lie in one plane. If A tangents at a K-tuple point of a 

 twisted curve of order m lie in one plane, the curve is met by that plane 

 in at least 2 A + k — A or k + A points. We must therefore have' 

 k -j- A < m, i.e. k < m — A, or A ^ >» — k. A curve of order six can 

 therefore not have a quadruple point, three of the tangents at which lie 



