350 Mr. W. Spottiswoode on a Problem in Combinatorial Analysis. 



15 14 13 12 11 10 9 



1.2 1.3 1.4 1.5 1.6 1.7 1.8 



13 14 11 12 9 10 



2.3 2.4 2.5 2.6 2.7 2.8 



15 10 9 12 11 



3.4 3.5 3.6 3.7 3.8 



9 10 11 12 



4.5 4.6 4.7 4.8 > 



15 14 13 

 5.6 5.7 5.8 



13 14 

 6.7 6.8 

 15 

 7.8 



15 14 13 " 



9.10 9.11 9.12 



13 14 



10.11 10.12 



(1) 



(2) 



.} 



(3) 



15 

 13.14 



Here it will be observed that the first triangular arrangement 

 gives all the binary combinations of the numbers 1, 2, 3, 4, 5, 

 6, 7, 8, and the combinations of them with the numbers 9, 10, 

 11, 12, 13, 14, 15, so arranged that each of this latter group of 

 numbers is combined once with all of the former group ; the 

 second triangular arrangement gives all the binary combinations 

 of the numbers 9, 10, 11, 12, and the combinations of them with 

 the numbers 13, 14, 15, so arranged that each of this latter group 

 of numbers is combined once with all of the former group ; and 

 finally, the last arrangement, consisting, however, of only one 

 term, is similar with respect to the numbers 13, 14 and 15. The 

 third number in each ternary combination has been written over 

 the numbers forming the binary combinations, in order to exhibit 

 the laws of formation of the latter, and of the position of the 

 former as distinctly from one another as possible. 



To arrange the groups of 5 for each day, select a ternary com- 

 bination from one of the two lower triangles, and omitting from 

 the upper those combinations which the selected combination 

 excludes, arrange the remainder (16 in number) as a determi- 

 nant, omitting from the development of it those terms in which 

 any of the upper numbers recur ; thus, selecting the combina- 



