( 50i ) 



111 the following the A"^^^ is iindci- closer investigation especially 

 with a view to the Cff. obtainable ont of it l)y omission of certain 

 elements. To this end it is necessarv to construct an oilier diagram 

 than that of eight simplexes, which can only clearly show the 

 Cff. (56,,); (48,,); (40,,); (32,,); (24, J and ( 1 6, j/ formed by 

 omission of the elements of 1, 2, 3, 4, 5, 6 simplexes, all (except 

 the first) in different types, likewise of Cff. (,24J and (32,,), con- 

 structed exclusively of fillings (Sj). 



§ 2. If we isolate in the Cf. of Kummer a point and a plane not 

 incident to it, the remaining fifteen elements of each kind are divided 

 into a sextuple incident to one of the isolated elements and a nonuple. 

 Each one of the two sextuples forms with the 15 elements of the 

 other kind a free Cf. (6,, 15,) which means nothing else but that 

 each of the fifteen right lines connecting the C/.-points in one 

 plane bears another C/'-plane and reciprocally. 



The two nonuples of elements, however, of both kinds together 

 form one Cf. (OJ, the structure of which is identical to that of 

 Cf. (9^) III of the classification of Martinetti "). 



This arrangement can be done in 16 >( 10 := 160 ways. 



We can likewise isolate out of /v'^'ii in 64x36=2304 ways 

 a point and an Sp^, not incident to it, by which the sixty-three other 

 elements of each kind are divided into a group of twenty-eight 

 incident to the isolated element of the other kind and a remaining 

 group of thirty-five. The two groups of twenty-eight form together 

 a scheme (28, J; each group of twenty -eight with that of thirty-five 

 of another kind a scheme (28,5, 35,,); addition of (28,,) and (28,5, 35,,) 

 furnishes a scheme (28,., 63,,); the two groups of thirty-five form 

 together a scheme (35, J. 



This arrangement made for the two elements .41 is shown in the 

 plate, where the same notations are assigned to the elements as in 

 the diagram of simplexes of which it is a transformation. 



We ha\e but to explain how the regular composition indicated 

 by the thicker lines is obtained. 



§ 3. Let us first take into consideration the scheme (28,., 63,,) 

 of points (columns) and Sp^ (rows). Every number of twelve points 

 on a row lying in two different Cf.-Sp^ lies in an -S/?, ; so we 

 can take the (f. to consist of twenty-eight points and sixty- three 

 Spi in Spa ; each of the sixty-four Cf.-Sp^ of the lO'^^ contains such 

 a Cf. (and reciprocally). 



2) Atli della R. Accademia Pel ori tana XV. 



