514 On certain Triadic Arrangements of Fifteen Symbols. 



nation is successively put under the following twenty-four forms, 

 which for greater simplicity are here represented by numerals : — 



A. 



B. 



C. 



D. 



a. 



0. 



7- 



8. 



1 2 3 



1 2 3 



1 2 3 



1 2 3 



1 



3 2 



1 



3 2 



1 



3 2 



1 3 2 



4 5 6 



7 8 9 



11 12 10 



15 13 14 



4 



6 5 



7 



9 8 



11 



10 12 



15 14 13 



7 8 9 



4 5 6 



5 6 4 



6 4 5 



7 



9 8 



4 



6 5 



5 



4 6 



6 5 4 



10 11 12 



13 14 15 



8 9 7 



12 10 11 



13 



15 14 



10 



12 11 



14 



13 15 



9 8 7 



13 14 15 



10 11 1214 15 13 



9 7 8 



10 12 11 



13 



15 14 8 



7 9 



12 11 10 



A'. 



B'. 



C. D'. 



a'. 



. £'• 



y' 



8'. 



1 2 3 



1 2 3 



12 3 12 3 



1 



3 2 



1 



3 2 



i 



3 2 13 2 



6 4 5 



9 7 8 



10 11 12 14 15 13 



5 



4 6 



8 



7 9 



12 



11 1013 15 14 



15 13 14 



12 10 11 



13 14 15 8 9 7 



11 



10 12 



14 



13 15 



9 



8 710 12 11 



9 7 8 



6 4 5 



4 5 6 5 6 4 



8 



7 9 



5 



4 6 



6 



5 4 4 6 5 



12 10 11 



15 13 14 



7 8 911 12 10 



14 



13 15 



11 



10 12 



15 



14 13 7 9 8 



A". 



B", 



C". 



D". 



-". 



jS", 



y". 



8". 



1 2 3 



1 2 3 



1 2 3 



1 2 3 



1 



3 2 



] 



3 2 



i 



3 2 13 2 



5 6 4 



8 9 7 



12 10 11 



13 14 15 



6 



5 4 



9 



8 7 



10 12 11 14 13 15 



11 12 10 



14 15 13 



9 7 8 



10 11 1215 



14 13 



12 



11 10 



13 



15 14 8 7 9 



14 15 13 



11 12 10 



15 13 14 



7 8 9112 



11 10 



15 



14 13 



7 



9 811 10 12 



8 9 7 



5 6 4 



6 4 5 



4 5 6 9 



8 7 



U 



5 4 



4 



6 5 5 4 6 



In consequence of this flexibility in the disposition of the con- 

 stituent triads of each combination, a solution obtained by a ten- 

 tative process is most likely to belong to the first system. The 

 seven combinations which result from the primaries A', A" follow 

 in the order of those from A when the latter are taken with 

 strides of two and four respectively ; and so of the others. 



To determine the number of synthetic combinations of the 

 fifteen symbols that can be formed out of a given set of thirty- 

 five triads, suppose pqr to be a triad taken as one of a combina- 

 tion : it can be associated only with the sixteen of the remaining 

 triads that do not contain p, q or r. Let p'q'? J , taken from these, 

 be the second triad ; then p'q'r' can be associated only with the 

 six of the sixteen triads that do not contain p', q' or r 1 . Again, 

 let a third triad p"q"r" be taken from these ; then p"q u r" can be 

 associated only with the two of the six triads that do not con- 

 tain p", q" or r" ; and these last will be the fourth and fifth 

 triads of the combination. The number of combinations that 

 can thus be made, comprising the fifteen symbols, will therefore 

 be 35x16x6x2; and as the results will comprehend every 

 form of permutation of the five triads under each combination, 

 the total number of such combinations that can be formed, with- 



.„, 35x16x6x2 

 out permutation, will be -= — -. — 5 — =- 



O X 4 X O X <i 



= 56. These combina- 



tions are stated at length in the ' Diary.' 



