THINGS. 
[323 
1 out of the 
Kings, there are 
"re is no triad 
le system there 
them occurs in 
been given of 
solutions which 
s a solution of 
solution which 
at least a 6 
Df solutions, viz. 
in 8 without 3. 
which is known 
ds involving all 
bhout 3, that is, 
rhis will be the 
d each of them 
11.13 = ) 5005, 
d of the system 
= 4.5.11 = 220 
ntaining a triad 
For, first, con- 
triads, each of 
fact if e.g. 123 
lads, must have 
itaining 3, and 
Hence we have 
tern; and since 
nt triads, these 
them the same 
pairs such as 
ich as a/3y, a8e. 
'• 
323] ON A TACTICAL THEOEEM EELATING TO THE TEIADS OF FIFTEEN THINGS. 
And for any such pair, combining with oc/3ySe any one of the remaining 10 things, 
we have 10 hexads a/3<y8e£, each of them derivable from either of the triads a/3y, aSe; 
that is, we have (315 x 10 =) 3150 hexads which present themselves twice over among 
the 7700 hexads. The hexads not belonging to one or other of the foregoing classes 
are derived each of them from a single triad only of the system, and they present 
themselves once among the 7700 hexads. This number is consequently made up as 
follows, viz. 
280 hexads each twice = 560 
3150 
840 
4270 
„ = 6300 
once = 840 
7700 
or there are in all 4270 distinct hexads; and since this is less than 5005, it follows 
that there are hexads not containing any triad of the system: there must in fact be 
(5005—4270=) 735 such hexads. The theorem in question is thus shown to be true. 
2, Stone Buildings, W.G., November 24, 1862. 
C. V. 
PI 
. : & 
muni