72 MEMOIRS NATIONAL ACADEMY OF SCIENCES. ivol. xiv. 



The 77 noncongrueiit systems thus far derived are interconnected by quadrangular and by 

 hexagonal transformations, and in general a system is not united uniquely to another system 

 but is derivable from several systems by different transformations; for example, the system 7 

 possesses the folio wmg interconnections: 7 = I C, {b3)e8g6; 7=F4al, (dg) a3 ef 7 6;7 =11, 

 1, (a 1) & c 4 e 6 2. 



After this point in the investigation many other systems which were transformed by this 

 process fiu-nished only repetitions of the 77 noncongruent systems thus far determined, showing 

 that the number of groupless systems was probably not much in excess of 33. An exhaustive 

 determination of all groupless systems by this empirical method requires that every new system, 

 as it appears, shall be subjected to each one of its possible quadrangular and hexagonal transfor- 

 mations. The enumeration of these transformations for the 14 new systems derived above 

 from the system 27 shows that a complete investigation even of these systems would necessitate 

 the application of more than 435 transformations. Hence while this empirical method for 

 generating new systems is productive, the amount of work involved in an exhaustive investi- 

 gation is prohibitive. Therefore it is evident that a new starting point and a new method are 

 requisite to insure a complete detei'mination of the groupless systems. This desideratum is 

 fully met in the following Part 4. 



