VAN DER VRIES. — MULTIPLE POINTS OF TWISTED CURVES. 503 



that is, ^ must always be taken as great as the quantity on the left side 

 of this inequality. The smallest value of this quantity occurs when 

 o- = 0, and otherwise depends on the values of k (2 k — k + 2), that is 

 on the orders of the cones on which the tangents at the multiple 

 points lie. 



K 2 



I). If k is even, k can take any value from zero up to — - — inclusive. 



The greatest value of k (2k — k — 2) will then be — ( — - — ) ( — - — ] 

 if k — 2 is an evenly even number, or 



-(^)GM«-G)vM}- 



if k — 2 is an oddly even number. We must therefore always have 



^^ + \ j/(m-2) 2 -2(«-2) 2 ^ h, ^ every K = 2 n, 

 m + 2 



or — f- i a/ (>n — 2) 2 — ^ K ( K ~ 4) < /<, if every k = 4 /?'. 



(Here n is an odd number, whereas n' is any integer.) We can thus 

 never take fj. less than the quantities on the left-hand sides of these 

 inequalities, if the k's are all even numbers. It is also evident from 

 (VI) that, if the points are all such that the tangents at them lie in 



planes or on cones of order — , we must have 



m + 2 



+ J V(m - 2f ^ /,, 



m _ 



The lower limit for p when the k's are all even numbers thus varies 



"ill tH I A / ~ " ~~ ' ' . _ , " " ~ 



between — and — \- \ a/ (m — 2) 2 — ^ (k — 2) 2 , according to 



the way in which the tangeuts to the curve at the different multiple 

 points lie. 



