48-1 PROCEEDINGS OF THE AMERICAN ACADEMY. 



Every point of multiplicity Ick' of this kind reduces the possible number 



k k' (k -\- k' 2) 



of actual double points of the curve by ^^ — -. Therefore: — 



A kk' -tuple point on a curve that is the complete intersection of two 

 surfaces that contain the point as points of multiplicities k and k' , respect- 

 ively, is equivalent to ^^ — actual and ^^ — 



^ Ji 



apparent double points. 



The quadruple point on the sextic curve mentioned on page 483 is 

 equivalent to four actual and two apparent double points. This will be 

 discussed more in detail later. 



6. It is also possible to find an upper limit to the number of actual 

 intersections of the two components of orders m and ^?^' into which the 

 curve may break up ; where m -^ m' ^ ^v. Two curves of orders m 

 and m' have m m' intersections, actual and apparent. A point that is a 

 p-tuple point on one curve and a p'-tuple point on the other curve counts 

 as pp' intersections of the two curves. We have then 



m 



m' = t + H» + pp<, (IV) 



where t is the number of actual intersections of the two curves. Sub- 

 stituting the value of H ivova (III) into (IV), we get 



", avU-\){v-\) kk> ik-\)(k'-\) , ,, 

 t = mm'-pp'- '^-^ ^ ^ + ^ ^ ^ + /^ + h'. 



The maximum or minimum number of actual intersections of the two 

 curves thus corresponds to the maximum or minimum number of 

 apparent double points of the two components. A twisted curve of 

 order m with a p-tuple point can be projected into a plane curve of order 

 m with a p-tuple point. This plane curve of order m cannot have more 

 than I (m — 1) (m — 2) — i p (p — 1) double points. The component of 

 order m cannot therefore have more than J (m — 1) (m — 2) — ^ p (p — 1) 

 actual and apparent double points. If it have 8 actual double points, we 

 must have 



k = i^(m-l){m-2)- ip (p - 1) - 8. 



Similarly, for the component of order m' with a p'-tuple point and 8' 

 actual double points, we have 



h> 5 i (m' - 1) (m' _ 2) - ^p' (p' - 1) - 8'. 



