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



(or, adopting Mr. Sylvester's notation, 



f\ 2 3 4 5 6 7 B-\ 



U 2 3 4 5 6 7 8J 

 with the conditions 



i.j=j.i, 



i and 7 receiving successively all values from 1 to 8 inclusively), 

 being those fourth minors which do not involve the principal 

 constituents ; and the solution of the problem will be given by 

 selecting those terms in the development of the determinant 

 which are perfect squares, and do not involve any of the prin- 

 cipal constituents or repetitions of the superior numbers (sup- 

 posed to be inserted as in the triangle (1)). 



With respect to the general problem of ternary combinations 

 of a given number of persons, it is clear that the number must 

 be divisible by 3, and also uneven, because each person {e. g. 

 No. 1) is to be combined with all the binary combinations of the 

 rest in which no number recurs ; a condition which would be 

 obviously impossible if the number of terms with which No. 1 is 

 to be combined (i. e. the given number less one) be odd, i. e. if 

 the given number be even. From these considerations it is fur- 

 ther observable, the number of binary combinations with which 



each number is to be combined will be ■= (n— 1), n being the 



given number. The first step, then, in the solution of the pro- 

 blem, according to the present method, will be to form a trian- 

 gular arrangement of all the binary combinations of the first num- 

 bers 1, 2 . . -x {n + 1) (in the above problem n = 15, ^ (n — 1) = 7, 

 -i 2 & 



- (ra + l)=8). The next step will be to form a triangular ar- 



2 



rangement of all the binary combinations of the numbers 



l(n+l) + l, |(» + l)+2,..^|(»-H)+lJ-J 



and so on, until the numbers be exhausted. But here it must 

 be again observed, that since the first triangular arrangement 

 has exhausted all the binary combinations of the numbers 1, 2, . . 



^(ra + 1) inter se, and all the combinations of the remaining 



1 1 



numbers ^ {n + l) + 1, g («+ 1) +2, . . n, with them, there now 



2 li 



remains only the combinations of this last group inter se ; and 

 reasoning similar to that by which it was proved that the number 



n itself must be odd, would prove that - (?i— 1) must be odd; 



and similarly, it would be seen that the number jr-< ^ (n — 1) — 1 >■ 



