248 Proceedings of Royal Society of Edinburgh. [sess. 
A Peculiar Set of Linear Equations. By Thomas Muir, 
LL.D. 
(Read December 3, 1900.) 
(1) It is easily seen that each of the equations of the set 
X 1 
+ 
g 2 x 2 + g 3 x 3 
+ 
9i 
= 0 ) 
9i x i 
+ 
x 2 + g 3 x s 
+ 
92 
= 0 \ 
9i x i 
+ 
92 X 2 X 3 
+ 
9 3 
= 0 ) 
remains unaltered 
for 
each of the three interchanges 
x i ‘-r 9i » 
(i) 
*2 9i, 
(2) 
x ?, > 
(3) 
and that the set as a whole is not altered by the simultaneous 
performance of the cyclical substitutions 
If therefore we solve for x, in terms of g 1 , g 2 , g 5 , and obtain 
x i ~ > d 2 > 9$) > 
it must follow from (4) that 
x 2 = ’H.Oi > .? 3 . 9l) , 
and * 3 = <£(53 • 
Prom this set of three, by the use of (1), we deduce 
