Arrangements of Fifteen Symbols. 

 a x b i c l + ajbfa + a 3 b 3 c 3 + c 4 5 4 e 4 + a 5 b s c s 



+ a 1 



« 2 « 3 H-6, 



J 2 6 3 + C l 



c 2 c 3 + o 2 



b 3 c 4 + b i 



C 3 ff 4 + C 2 



«4«5 



hh 



C 4 C 5 



Vs 



c 4 ff 5 



C 2 C 5 



c 3 c 4 



c 2 c 4 



« 3 « 4 



«2«4 



b 3 b 4 



V» 



*S«3 



513 



•A 



+ «5 J 4 C 3 + h C 4 a 3 + C 5 ff 4^3 5 



and it is a still more remarkable fact that from this same set of 

 triads it is possible to construct no less than 1080 systematic 

 arrangements according to the first and second systems, each of 

 them fulfilling the conditions of the problem. 



The triads contained in the system No. 3 are essentially un- 

 symmetrical, and admit of only one systematic arrangement. 



In the ' Diary ; I have also briefly indicated a direct and effec- 

 tual method of systematizing any given solution when presented 

 under an irregular form, and of thereby ascertaining to which of 

 the three systems it belongs. As an example of the application 

 of this method, the numerical solution given in the c Diary ' for 

 1851 is here transcribed for comparison with the same after being 

 so adjusted and arranged*. 



Solution by Mr. Samuel Bills, of Hawton, near Newark-upon- 

 Trent; Mr. Thomas Jones, Abbey Buildings, Chester; Mr. 

 Thomas Wainman, Burley, near Leeds; and Mr. W. H. Levy, 

 of Shalbourne, near Hungerford. 



1st day. 



2nd day. 



3rd day. 



4th day. 



5th day. 



6th day. 



7th day. 



1 2 3 



4 8 12 



5 11 14 



6 9 15 



7 10 13 



1 4 5 



2 8 10 



3 12 15 



6 11 13 



7 9 14 



1 6 7 



2 12 14 



3 8 11 



4 9 13 



5 10 15 



1 8 9 



2 13 15 



3 5 6 



4 10 14 

 7 11 12 



1 10 11 



2 4 6 



3 13 14 

 5 9 12 

 7 8 15 



1 12 13 



2 5 7 



3 9 10 



4 11 15 

 6 8 14 



1 14 15 



2 9 11 



3 4 7 



5 8 13 



6 10 12 



The same duly arranged: System No. 1. 



5 14 11 



13 10 7 

 8 4 12 



6 15 9 

 3 1 2 



5 10 15 



4 13 9 



1 7 6 



11 8 3 



14 2 12 



5 13 8 

 7 4 3 

 2 9 11 

 15 1 14 

 10 12 6 



5 4 1 

 9 7 14 



12 3 15 

 8 2 10 



13 6 11 



5 7 2 



3 9 10 



6 14 8 

 1 12 13 



4 11 15 



5 9 12 

 14 3 13 

 11 10 1 



2 6 4 

 7 15 8 



5 3 6 

 10 14 4 

 15 13 2 

 12 11 7 



9 8 1 



I have. further to observe that in constructing the first system 

 the seven resulting combinations will be essentially the same. 



though 



in a different order, when the primary combi- 



* If any correspondents should think it worth while to communicate 

 other solutions, I shall willingly systematize them in like manner. 



