70 



MEMOIRS NATIONAL ACADEMY OF SCIENCES. 



[Vol. XTV. 



Tlie new groupless systems are formed by interchanging duads of one column wnth those 

 of another column. For example, in the pair of columns ab, the duads de, fg, of column a, 

 may be exchanged with df, eg, of column h; such an interchange involving four elements, con- 

 tained only in two pairs in each cohimn, shall be designated as a quadrangular transformation. 

 The columns d, e, f, g must now be rewiitten in agreement with the new triads introduced 

 into the system, and the imdisturbed nine columns of VII318 with the reconstructed six col- 

 Tuuns form a new system 13. The four duads 12, 34, 56, 78 of column a might be interchanged 

 with tilt! fom- duads on the same elements in column b, forming an octagonal transformation, 

 but tliis is equivalent to the above quadrangular transformation followed by the interchange 

 of the elements a and b. 



In the pair of columns bl the duads df, 68, 57 of column b may be interchanged with the 

 duads /8, 67, d5 of column 1; such an interchange, involving six elements appearing exchi- 

 sivel}' in three pairs in each of two cohimns, shall be designated as a Tiexagonal transformation. 

 New colunnis d,f, 6, 8, 5, 7 must next be constnicted; these eight reconstructed columns, with 

 the seven undistm-bed columns of Vllgi?, form a system 4. The appHcation of the second 

 hexagonal transformation in 61 is equivalent to an application of the first hexagonal trans- 

 formation followed by an interchange of the elements b and 1. 



A transformation on 12 elements simply interchanges the two elements wliich head the 

 colunms. Therefore only the quadrangidar and the hexagonal transformations wliich exist 

 in a system require consideration. 



By means of the operatore of the gi'oup of the system, the maximum number of non- 

 congruent transformations of each of the above types is determined — for example, in Wl^& 

 the eight hexagonal transformations reduce to one, and the 30 quadrangular transformations 

 to fom- noncongruent transformations. 



Each of the noncongriient transformations is now applied to the system VII318, and the 

 sequences and indices are determined for the five transformed systems. 



The 35 triads of the system 4, arranged in classes according to their indices, are shown 

 in the following table: 



The enumeration of the elements in the 17 classes shows that the sets of transitive ele- 

 ments are a; h; c; d; e; f; g; 1; 2; 3; 4; 5; 6; 7; 8; hence the group for the system is identity. 

 Therefore under this hexagonal transformation the system YII3P, with a group of order 4, 

 is changed into a system 4 with the group identity. The four quadrangidar transformations 

 apphed to VIlj/S jaeld four noncongruent systems. One of these is a new groupless system 

 13; the remaining three are the known systems Vl2^e, VI2,5, VI2^7, with groups of orders 2, 2, 

 and 6, respectively. 



The headless, groupless system 4 which has been derived by a hexagonal transformation 

 from a headless system VI 1 3/3 with a group of order 4, may also be derived by a quadrangidar 

 transformation from the headed system IC with a group order 3. Hence a quadrangular 

 transformation may alter the number of systems A 7 in a A15 and change a headed system 

 into one without a head. Hexagonal transformations, on the contrary, leave unchanged the 

 number of systems A, in a A 15 and, therefore, always transform systems with or mthout 

 heads into systems with or without heads, respectively. 



