16 



MEMOIRS NATIONAL ACADEMY OF SCIENCES. 



[Vol. XIV 



there are three possibilities, and these serve to fix uniquely also the subscripts of the remaining 

 sets of triads. There are then three schedules: 



(o) &,Ciai (/3) iiCiflj (7) 61^203 



tiCjOj i^Cia^ hiCiO^ 



AhiCs Ab^c^ -46,c, 



BCiUs Bcitts Bc^a^ 



Cafis Cafii CaJ)j^ 



Species in case 4. — With the letters A, B, C, there occur in case 4 twice the pairs ah and cd. 

 With either one of these the subscripts can be taken as 11, 11 ; then with the other the two can 

 be, in their order, either 12, 12, or 12, 13. In the third set, ac and hd, we are still free to fix 

 arbitrarily one pair of subscripts, as 6,(Zi, leaving two alternatives for the other pair ac. Four 

 partial schedules result, as follows, for the pairs with A, B, and C. 



Species in case 5. — The same four schedules, for the same reasons, are valid in case 5 as in 

 case 4. 



The foregoing 19 schedules can be completed to actual systems, as is readily seen by inspec- 

 tion, in the following 26 ways only: 



System V, la: ABC, a^a^a^, l^ljb^, c.c^c^, dA(h: Aafi^, Ac,d^, Ba^c,, 5&,J„ Ca^d^, Cb,c,; 



afi^Cs, afiAt «iC2<^3> '^iCa'^a- 



A second system, of course, equivalent to this, differs only in the cyclic order 132 



instead of 123. 

 System V, 1^ is impossible. 

 System V, I7: ABC, a^a^a^, hfi^h^, c^c^c^, dAd^i- Aafi^, Ac A, Ba^c^, BhA, CaA, Ch^c„. 



afi^c.,, afiA, fliC3<^i. &iCi<^3- 



An equivalent system is converted into this by the substitution {ac) (bd) (23); or 



two others similar. 



System V, 15: ABC, a.a^a^, bfijb^, c.c.fi^, dA^z- ^«i^, ■^<^A, -SaiC,, BbA, OaA, Cb.c^. 



afix^, afiAi <''i<^Ai b^cA- 



' System V, 2a: ABC, dA^z: AaA, -^^iCi, BbA, Ba,c„ CcA, CaJ),; a,aA, ^^0-3, ^1^,03/ 



b^c^ds, b^cA- 



System V, 2/3: ABC, dA^^ AaA, ^^i^i, BbA,.Ba,c„ CcA, CaJ),; a,aA, ^A^z, c.c^a^; 

 b^c-fli, b^cA- 



System V, 2t: ABC, dAds,- AaA, Ab,c„ BbA, Ba,c„ CcA, CaA; «iM3. b^a^, c^cm^; 



bicA, biCA- 



System V, 28: ABC, dA^^; AaA, Ab,c„ BhA, Ba^c^, CcA, Cafi^; Oia^d^, b^a^, c^c.a^; 



b^c^d^, bicA- 



System V, 3o: ABC, dA^^; AaA, Ab,c.„ BbA, Ba,c., CcA, ^a^- a^^^A, ^Ms, CiCj^s/ 



System V, 3;3: ABC, dAda,' AaA, Ab.c^, BbA, Ba^c^, CcA, Ca^; "iMs, ^fiA, c,cA; 



System V, 37: ABC, dAdz; AaA, Ab,c„ BbA, -BffliC,, CcA, <^aA; OiMs, &1M3, Ci^A; 



0,6263, ffli&jCj. 



System V, 4a (1 and 2) : ABC, dA^^; 

 Aafii, AcA, (1) aiaA, &1V2) CiMa,' <iJ>3^2, b^cA; 

 Bafi^, BcA, or 



Ca,Ci, Cb,d^, (2) a,aA, ^A^s, c^c^a.,; aJ)A, \cA- 



These two are equivalent; (1) is converted into (2) by the substitution Ahh) (d^dA) 

 (23) (AB). 



