1900-1901.] Dr Muir on a Peculiar Set of Linear Equations. 249 
from the same, by the use of (2), we deduce 
x i = ,9s), "I 
ffs = <i>( x 2>9s,gi), h 
j X„) • j 
and from the same, by the use of (3), we deduce 
x i ~ $(9\ 5 92 > x z) > 1 
X 2 = $(92’ X 3>9l) , h 
9 S = <K X 3 > 9i , 9 2 ) '> J 
In the next place, by using simultaneously a pair of the three 
interchanges, the following three sets of results are obtained, viz. : — 
9\ — $( x i » x 2 j 9z) > 1 
92 = $( X 2 J 93 5 x \) ) r 
X 3 ~ 4 > (93 i X 1 ) X 2) ) ) 
X 1 = <K9i , x 2> x s)> 1 
92 = <K X 2> X 3i9l), h 
93 = J 
^1 = ^(^1 5 #2 > ^ 3 ) > "j 
X 2 = $(92 > X 3 > X l) > r 
93 = $( X 3 i X 1 > 92) • J 
Finally, by using all the three interchanges at the same time 
we obtain 
9\ = $( x l J x 2 1 X 3) 1 
92 ~ $( x 2 i X 3 i X l) i 
93 = $( X 3 1 X 1 J ^ 2 ) * 
These eight sets of three equations may also be advantageously 
arranged as six sets of four, viz. : — 
x i = $(9n 92> 93) ~ $(9 1 1 x 2 1 93) = $(9i, 92i x 3) = $(9i, x 2 > ^ 3 ): 
X 2 = $( 92 i 93 1 9 1 ) = $( 92 1 X 3 1 9i ) = $( 92 1 93 1 X l) = $( 92 1 X 3 > ^ 1 ) * 
X 3 ~ $(93 1 9\ i 92 ) = $( 9s 1 X \ 1 92) — $(93 5 9l i X ‘a) = $(93 i X 1 j X 2) • 
91 ~ $( X 1 } X 2 > X 3 ) = $( X 1 1 92 i X 3 , ) = $( X 1 i X 2 > 9 3 ) ~ $( X 1 i 92 1 93 ) : 
92 ~ $( X 2 5 X 3 1 X l) ~ $( X 2 1 93 i ^ 1 ) = $( X 2 > X 3 1 9l) — $( x 2 i 93 1 9}) • 
93 ~ $( X 3 l X 1 1 X 2 ) = $( X 3 1 9\ 1 X 2) ~ $( x 3 ) x \ ) 92 ) = $( X 3 l 9\ l ^ 2 ) • 
