512 



Mr. W. S. B. Woolhouse on certain Triadic 





1. 



2. 



3. 



4. 



5. 



6. 



7. 



~ 



a x b x c x 



(iihh 



a x a i a 3 



a x b 3 b b 



a x c 2 c b 



a x c 4 c 3 



a x a b a 4 



o 



ft 



a<2 b 2 c 2 



b 3 a 2 c 4 



c 2 b 3 a 5 



c 4 c%b x 



a b c 4 b 2 



b x a b a 2 



b 2 b x b 3 



a 3 b 3 c 3 



b b c^a 4 



c 5 c 4 c x 



c 3 a b b 4 



a 4 b x a 3 



c x hh 



b 4 a 2 c b 



a 



a 4 b 4 c 4 



c x a 3 a b 



b 4 b b b x 



H c bh 



h b c 2, a <2. 



c b a 4 b 3 



c 3 c x c 2 



m 



H h b c s 



by C 5 C 3 



b 2 c 3 a 4 



a q a 4 c x 



b 3 c x b 4 



c <2 b 4 a 3 



c 4 a 3 b b 



oi 



b x a x c x 



b x a Q a b 



b x c 2 c 4 



b x a 3 a 4 



b x b 2 b 3 



b x b 4 b b 



b x c b c 3 



o 



Q 1 C 2 *2 



Cc L a 3 b 4 



a 3 b 2 c b 



b^b 4 a x 



b 4 c b « 2 



c b a x c 2 



a x a a a 3 



fl 



a 3 b 3 c 3 



hh c \ 



b 4 c 3 a b 



c 5 c x c 4 



a x a b a 4 



a 2 c 4 b 3 



Cc l a 4 b b 



<B 



b 4 c 4 a 4 



cJa 4 b 3 



a x b 3 b b 



#A C 3 



c 2 c 3 c x 



a 3 c x a b 



b Q a b c 4 



CO 



c S a i b S 



a \ c 4 C 3 



a <2. a 4 c \ 



c*h a s 



a 3 b b c 4 



h c 3 a 4 



b 4 c x b 3 



CO 



OjijCj 



a x a 2 b 3 



a x b 2 b b 



a x a 3 c b 



a } c 2 b 4 



a x a 4 c 3 



a x a b c 4 



o 



a z b 2 c Q 



b 2 a 3 a 4 



a 3 c 2 a b 



Ccia 4 b x 



a 4 a b a 2 



a b b x bc l 



b a 2 a 3 



fl 



«3*3 C 3 



c 2 b b c 4 



a 4 c b c x 



a b b 4 b 3 



b x c 3 b b 



a 2 c 4 c b 



h % c x b 4 



09 



a 4 b 4 c 4 



a b c 3 c x 



b x c 4 b 3 



a 2 c x b b 



b 2 b 3 c b 



a 3 b 5 b 4 



C 2 C 5 C 3 



£ 



a bh c b 



b x c b b 4 



a 2 b 4 c 3 



b^c 3 c 4 



a 3 c 4 c x 



c 2 c x b 3 



a 4 b 3 b b 



Here each of the seven combinations is derived from that 

 which immediately precedes it by a fixed law of succession, which 

 on continuation will circulate through exactly the same positions. 

 Also the symbols occupying corresponding places, taken in order 

 horizontally in each system, proceed according to two fixed 

 cyclical series, each consisting of seven symbols, viz. — 



No. 1. | 



No. 2. 



No. 3. 



{ 



a, 



h 



«2 



h 



H 



C 4 



a s> 



Cl 



^ 



«a 



^ 



H 



C 3 



H> 



a x 



«s 



H 



«a 



h 



h 



c s> 



c i 



a 5 



C 4 



"4 



h 



h 



C 3> 



h 



Og 



h 



«a 



C 2 



«4 



flflJ 



C] 



^3 



k 



c b 



^4 



C 3 



c 4 . 



The three systems are essentially independent of each other, 

 and cannot be mutually transformed by symbolic substitution. 

 The particular distribution of the primary combination of No. 2 

 has been modified, so as to exhibit the remarkable fact that the 

 systems No. 1 and No. 2 admit of being made up of the same 

 set of thirty-five triads, and so as to comprise the same triads in 

 four of the seven combinations. Thus the 1st, 4th, 6th, and 

 7th combinations of No. 1 correspond to the 1st, 3rd, 2nd, and 

 5th combinations of No. 2. The thirty-five triads of No. 1 or 

 No. 2 being collected, may be symmetrically arranged thus : — 



