502 PROCEEDINGS OF THE AMERICAN ACADEMY. 



ing values of k are determined by — - — or — — , according as k is odd 



or even, we must have in general 

 1°. If the k's are all even, 



m — 2 



(H) 



-~- + i J(P - 2) 2 - ^ x 2 + 4 < is ', 

 2°. If the k's are all odd, 



2^=i + J |/(m-2/-2K 2 +5 ^ ^ (Ill) 



It is to be noticed that ^ (k + \) (k — k — 1) in (I) above is always 



greatest when the K-tuple points are all of the most general kind. The 

 sufficient value of (* is therefore always least in this case, that is when ft 

 is given by (II) or (III) above. The quantity (k -f 1) (k — k — 1) in 

 (I) is never less than zero; the quantity under the radical sign is thus 

 never greater than m 2 — 4 m + 8 ; therefore ,u = m — 1 will always 

 satisfy the conditions. We need therefore never take fi greater than 

 m — 1. 



7. The inequalities above give values of ft greater than which we 

 need never take the order of the monoid. We can, moreover, find values 

 of fi smaller than which we can never take the order. As M^ has /i 

 (^ — 1) lines on it, we must always have 



h + ^k(k+l) + a^u (/i-1); (IV) 



where h corresponds to the ordinary lines of 71/ M that are double lines on 

 K m , ^9k(k+ 1) to the &-tuple lines of M^ that are K-tuple lines on 

 K m , and o- to the ordinary lines of M„, that are ordinary lines on K m . 

 As K m intersects M^ in m Qi — 1) lines, we must have 



2 a - 2 * * + °- = m <> - - 1 )- ( v ) 



By eliminating h from (IV) and (V), we obtain 



2* (2k- k- 2) + a = (2(i-m) (jt- 1), 



i. e . r HL+l + i^/(r n -2f+S^k(2k-K-2) + Sa^ ti i (VI) 



