VAN BER VRIES. — MULTIPLE POINTS OF TWISTED CURVES. 515 

 If we use a and b, where a + b = m — 1, in (I), we obtain 



P a + Pb + (m — 2 b) apparent double points = P a +i + Pb— 1« 



If m is odd, we have 



pi (m - 1) (m- 1) (m + 1) -] 

 =? + ^ + L 2 4 J 



apparent double points 



_ [- w (m - 3) (m - 3) (m - 1) -) 



- n^ + <r»_* + |_ 2~ 4 j 



apparent double points, i. e. 



2 P m _[ + 1 apparent double point = P m+1 + P m _^. 



~2~ ~T~ ~2~ 



Thus, if m = 2 a -\- 1 



2 P a + 1 apparent double point = P a +\ + Pa— l • 



If m is even, we have 



m (m — 2) m (m — 2)' 



2 2 I 



p m Q - 4) (m- 4) (m - 2) ] 



— ■» m+2 "T -* m— 4 T o 4. • 



apparent double points, i. e. 



P,„ + P )iL _2 + 2 apparent double points = P,„ +2 + P„-^. 



2 2 2 L 2 



Thus if m = 2 a, 



-Pa + -fa— l + 2 apparent double points = -Pa+i + Pa— 2' 



Therefore, in general, 

 If m = 2 a + 1, 



2 P a + A; 2 apparent double points = Pa+k + Pa—k', 

 also, 

 Po+i- -1- P a -k + g (g+2Jc) apparent double points = Pa+k+g + P a -k-g '• 



If m = 2 a, 



Pa + Pa-\ + k (Jc + 1) apparent double points = P a +k + Pa-k-i ; 



_. „ [m (m — 2) m (m — 2)"| , , , 



P m + P m _ 2 + — ^—^ r apparent double points 



2 ~2~ L " 4 J 



