ing as 



504 PROCEEDINGS OF THE AMERICAN ACADEMY. 



k — 1 

 II). If k is odd, k can take any value from up to — - — inclusive. 



The largest possible value of k (2 k — k + 2) for any multiple point will 



then be -(^-j— \ \~^r\ ° X ~ \V) ( Sr~ ) ' aCC ° rdi 



k is one more than an oddly even or an evenly even number. The 



two products are, however, equal. The greatest possible value of 



? k (2 k — k + 2) will thus occur when all the points are such that 



k = — - — or — - — , according as k is one more than an oddly even or 

 an evenly even number. If k's are all odd numbers, we must thus have 



^4^ + 1 j/(™ - 2) 2 - ^ (k - 1) (k - 3) ^ p. 

 It is evident from (VI) above that 



^4^ + * /o - 2 ) 2 + 4 2(«- *) < ^ if * =1 ir for each k; 



m + 2 



or — h i VC?" — 2) 2 < ,u, if £ = for each k. 



The lower limit for p when k is odd thus varies between 



^ + I |/(™-2) 2 - 2 ("-1)(" - 2 ) 



and 



7W + 2 



+ i |/(m - 2) 2 + 4 J (k - 1), 



according to the way in which the tangents to the curve at the different 

 multiple points lie. 



It is thus evident that ju can in no case be taken less than 



m + 2 



+ i |/0» - 2) 2 - 2 (k - 2) 2 . 

 Using the formulae found above we can tabulate as follows : — 



