VAN DER VRIES. — MULTIPLE POINTS OP TWISTED CURVES. 515 

 If we use a and b, where a + 5 = TO — 1, in (I), we obtain 



Pa + Pb ■\- (^« — 2 b) apparent double points = Pa-\-\ + Pb—i- 



If m is odd, we have 



p ^ p , y m (m - 1) (m- 1) (m + 1) -| 

 ^-1 + ^"-ti + L 2 ~ 4 J 



apparent double points 



_ p m (m - 3) _ (m - 3) (m - 1) -| 



apparent double points, i. e. 



2 P,„_i + 1 apparent double point = P^+i + PmS' 



~2~ 2 2 



Thus, if m = 2 a + 1 



2 Prt + 1 apparent double point = Pa+i + -^a— i • 



If m is even, we have 



^ „ [m (m — 2) m (m — 2)"| . , , 



Pm + Pm-2 + — ^^ —J apparent double points 



_ P A. P _L r ^ (^- ^) _ (w-4) (m- 2) 1 



— -» m+2 + -^ m— 4 ' O 4 1 



apparent double points, i. e. - 



P,,, + Pm-1 + 2 apparent double points = P^+j + Pm-A- 



2 ~2~ 2 1,2 



Thus if m = 2 or, 



Pa + /'a— 1 + 2 apparent double points == Pa-\-i + Pa— 2- 



Therefore, in general. 



If m = 2a + 1, 



2 Pa -\- /t" apparent double points = Po-ffc + Pa-k'-, ' 

 also, 

 Pa-\.k + Pa-k + 9 (g + 2 k) apparent double points = Pa+k+g + Pa-k-g ■ 



li m = 2 a, 



Pa + Pa-i + 1<^ (^" + 1) apparent double points = Pa-\-k + Pa-k-i ; 



