( 294 ) 



number of triple systems for arbitrary n (if only e 1 or 3, mod. 6), 



i.e. to classify all diagonalless 



Cff 



2 



namely : 



1. by starting from all possible 



Cff !(/*-3)„_7 , 



V 2.3 J, ' 



(n-3)(n- 7) 



6 



2. bv enumerating the Cff. of its diagonals, those are : 



Cff (m-3), 



3(n— 3) 



3. by investigating in how many ways each of these Cff. can 



n — 3 

 break up according to its lines into three principal - sides, 



4. by every time assuming the lines of such a poly-side to be 

 convergent to one point and these three points to be collinear, 



5. by arranging the obtained systems in types. 



Already for n = 15, however, this method is checked by the 

 absence of the classitication ot Cff. (12 4 , 16 3 ) necessary for 1, of 

 which only some six forms have been enumerated '). Let us restrict 

 ourselves to the best known and most regular form, of Hesse : 



A 12 3 4 



."> 

 G 



7 



8 



i 



in which each point of the quadruple! abed is collinear to \\\^ points 

 of the two other quadruplets in the same row of column, then three 

 lypes of triple systems of 15 elements appear : 



I. Complement xab, .red, ./■ 1 2, x 34, #56, #78, 



y a c, ybd, y 13, y 24, yol, // 68, 

 zad, zbc, ': 14, z 23, z 58, z 67, 

 x ii z. 

 All rests are A, the system is identical to the cyclic one 5 ): 

 (1 . 2 . . 5 . . .) ; (1 . 3 . . . 9 . . .) : (1 . . . 6 . . . 11 . . .). 



II. Complement xab, .red, a? 12, x 34, x 56, #78, 



y (( c, yb d, y 13, y 24, y 58, y 67, 



'z a d, z b c, 'z 14, z 23, z 57, z 68, 



x y z. 



J ) Gomp. the author's dissertation; "Biid ragen tot de theorie der configuraties", 

 Amsterdam 1907, § 36. 



-) L. Heffter, Math. Annalen, Vol. 49. 



