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



§4. CLASS n. EIGHT SYSTEMS INVARIANT UNDER A SUBSTITUTION OF THE TYPE (3)\ 



A first separation into two divisions is found when we distinguish tliree kinds of triads. 

 Denote the elements of the five cycles by the letters a, b, c, d, e, attaching to each in their order 

 the subscripts 1, 2, 3: 



/S=(<i, ttj a,} (6, 62 63) (c, C2 C3) {di d^ d^) (e, Cj eg). 



Denote any three different letters from among these five by Tc, I, m, leaving the subscripts 

 undetermined. Then obviously every triad that can occur is of one of the three types repre- 

 sented by 



Iclvk (Denote their number by u) , 

 IcTcl (Denote their number by 3u), 

 1dm (Denote their number by 3w). 



In a given triad system, every possible pair of elements occurs once. There are in the entire 105, 



15 pairs of type Tck, 



90 pairs of tjrpe Tel, makmg in combination 35 triads. 

 Compare these with the numbers of each found in triads of each of the three types above. We 

 find the necessary relations : 



3u + 3-y = 15, 



6i; + 9w = 90, 



« + 3i; + 3iy = 35 



We can have, therefore, the two kinds of S3'stems, divisions 1 and 2 : 



Division 1. — As u is 2, assume triads a^a^a^ and &1&2&3. Now subdivide further, and let 

 Ic represent either an a or a h, and Z, m represent any two of the letters c, d, c, subscripts being 

 disregarded. Since 311 = 9, let us distinguish — 



Of tyi)e ahm, 9=3.3 triads. 



Of type IcU, 3x triads, 



Of type Hm, Sy triads. 



Of type Imm, Sz triads, 



Of type cde, Bt triads. 



These numbers have to satisfy the following conditions: 

 Triads, 3a; + 3(/ + 32 + 3< + 2 + 9=35; 

 Pairs H, 18 + 6x + 6y = 54, 

 Pairs ZZ, 32; + 32 = 9, 



Pairs Im, 3?/ + Qz + 9t = 27. 



These conditions admit three solutions, which wUl be taken up in order. 



Family a 

 Family 6 

 Family c. 



