38 MEMOIRS NATIONAL ACADEMY OF SCIENCES. [vol. xiv. 



VI, 2^7, a new system; VI, 2^1, a new system. 



VI, 3, =VIA by s=/A B C a^ a^ a, a, 05 Og 6, h^ fr, 64 65 iA 



Vc98125a66473(ic/ 

 VI, 830, a new system; VI, 837, a new system. 

 VI, 838 , a new system; VI, 847, a new system. 

 VI, 34a=VII by s=/A B C a, a^ a, a, 05 o, fc, 63 63 5, 65 66\ 



U55rrfa47/2361ec8/ 

 VII, 1, =IIIAbys=/l 23456T AaBbCcDd\ 



\abcdegfl234b687J 

 VII, 1, =IIIB by s=/l 234567^aB6 Cci)d\ 



Vo6crfe5r/1284568 7>' 



VII, It =IIIC by 8=/l 234567 AaBbCcDd\ 



\abcdegfl2345eS7J 

 VII, 1. =IIIDbys=/'l 284567^aB6CcZ»<^ 



Va6crfe(7/1234568 7/ 

 VII, 23 =IIEby s=/l 234567J[aB6CcDrf\ 



^ca642138576d/jf«/ 

 VII, 2. =IVA by s=/l 2 34567^aB6CcZ)(A 



Va(fe/s?c534785612/' 



VII, 2, =IVBbys=/l 234567 AaBbCcDd\ 



\adegfbc21G5783 4/ 

 VII, 4, =IIA by s=/l 234567^aB6Cc/)rf\ 



Uca«yd/3124857 6/ 

 VII, 43 =IIFby s=/l 284567vlaB6CcDrf\ 



\6ca/d{?e2481856 7y 

 VII, 4^ =IIB by s=/l 284567^aB6CcX>cf\ 



\6ca/d(;e2431856 7/ 

 VII, 45 =IIDby s=n 234567 AaBbCcDd\ 



ycbadfgeS56742l3J 



VII, 4e =IICby s=/l 284567AaB6CcDd\ 



•V6ca/d(!^8124678 5/ 



VII, 5^ =VA by 8=/! 2 3 4567 AaBbCcDd\ 



\abcdgf€8521374 6/ 

 VII, 5. =VCby ss/l 234567 AaBbCcDd\ 



Kb c a g f d e 1582634 7>' 

 VII, 5a =VB by s=n 2 34567 AaBbCcDd\ 



\>ibcgdefl25S647 3J 



9 d e f 

 4 5 6' 

 a / g d 



Trains foe Triad Systems on 15 Elements whose Group is of Order Unity. 



VII, 5, =VDby s=n 2345 6' 7 AaBbCcDd\ 

 \cbaf(ide7436285 1/ 



The trains for each of the 36 noncongruent groupless systems on 15 elements have been 

 determined. Tliese 36 systems furnish 449 distinct types, different from the trains of the sys- 

 tems with a group. Among these appear trains terminating in polygoms of 4,6, 11, 12, 13, 

 and 14 sides, respectively. 



Hence the 80 noncongruent systems applied as transformers to the 455 triads on 15 ele- 

 ments, produce 665 distinct covariants or trains. 



