574 
PROBLEMS AND SOLUTIONS. 
[705 
Observe that there is in the system a single triad of triads A a a.', B/3/3', Gyy, con 
taining all the numbers; viz. for the system with AG, this is 123, 856, 947; for the 
system with AF, it is 145, 837, 926 ; and for the system with AH, it is 167, 824, 935. 
Cor.—It is possible to find a system of nine points such that the points belonging 
to the same triad lie in lined. Such a figure is this:— 
The solution shows that these are the only systems of nine points satisfying the 
prescribed conditions. 
[Vol. xv., pp. 66, 67.] 
3329. (Proposed by Professor Cayley.)—It is required to show that every per 
mutation of 12345 can be produced by means of the cyclical substitution (12345), and 
the interchange (12). 
Solution by the Proposer. 
It is sufficient to show that the interchanges (13), (14), (15) can be so produced; 
for then, with the interchanges (12), (13), (14), (15), we can, by at most two such inter 
changes, bring any number into any place. 
Writing P = (12345), a = (12), we have 
(12) = «, 
(13) = aPaP 4 a, 
(14) = a Pa P 4 a P 2 a P 3 a Pa P 4 a, 
(15) = P 4 «P, 
as can be at once verified; and the theorem is thus proved. 
I remark that, starting with any two or more substitutions, and combining them 
in every possible manner (each of them being repeatable an indefinite number of 
times), we obtain a “ group ”; viz. this is either (as in the problem proposed) the