366 Proceedings of Royal Society of Edinburgh. [suss. 
That this must in all cases hold is evident from considering 
that the interchanges, which when made on 00' produce the nine 
products of the first triad, are 
rf\ 
( g ) 
^ a)’ 
Q 
(0, 
0, 
and that these when taken in columns are exactly the interchanges 
which need to be performed on 00' to produce the products of the 
second triad. 
(4) The ten groups of such sets of six expressions may thus be 
compactly exhibited as follows : — 
00 ' 
11' 
44' 
77' 
88' 
22' 
55' 
66' 
99' 
33' 
11 ' 
99 / 
33’ 
00' 
-44' 
-77' 
- 66' 
55' 
22' 
-88' 
33' 
99' 
00' 
- 55' 
-88' 
-44' 
66' 
33' 
-99' 
11' 
77' 
00' 
-66' 
-99' 
-55' 
44' 
11' 
-77' 
22' 
88' 
44' 
55' 
66 ' 
00' 
-22' 
-99' 
-11' 
33' 
55' 
-77' 
66' 
go 
00 
00' 
-33' 
-77' 
- 22' 
11' 
66' 
-88' 
44' 
99' 
00' 
- 11' 
-88' 
-33' 
22' 
44' 
-99' 
55' 
77' 
77 
88 ' 
99' 
00' 
-11' 
-44' 
-33' 
22' 
88' 
-55' 
99' 
66' 
00' 
-22' 
-55' 
-11' 
33' 
99' 
-66' 
77' 
44' 
00' 
-33' 
-66' 
-22' 
11' 
77' 
-44' 
88' 
55' 
