368 
Proceedings of Royal Society of Edinburgh. [sess. 
IT — 22' = 55'- 66 / = 99'-77' \ 
22' - 33' = 66' - 44/ = 77' — 88' V 
33' - 1 1' = 44' - 55' = 88' - 99' j 
11' - 55' = 22'- 66' = 33' -44', 
11' - 99' = 22' - 77' = 33' - 88', 
44' -88' = 55' -99' = 66' -77'. 
It will be seen that the second line is derivable from the first,, 
and the third from the second, by the cyclical substitution : and 
that the number of such triads is four. The last three lines are 
not so related : the cyclical substitution if performed on any one 
of these would simply reproduce that one. 
(6) It is interesting to note that to each of the foregoing fifteen 
sets of three equivalents a fourth equivalent of a different form 
may be added. Thus for the seventh line w T e have the additional 
equivalent 
I «A I I a A I I a A I 
l/l&l f/u&l l/a* I 
I C f l 2 I I C 2^3 i I C 3^1 I p 
for this can be shown equal to 
I a A&i I I a A/3 1 I j 44 ' + 11 ' 
I C A#3 I I C A/3 I I 
and as the interchanges 
alter only the sign of the three-line determinant,* the latter must 
also be equal to 
and 
\fi9-fiz 1 
l/iS'aS 1 
! 1 
| ! 
1 <W J 3 1 
1 1 
1/isAI 
i.e. 3'3 + 5'5 
i.e. -77' + 00. 
* The other similar interchange 
gives nothing new. 
