1900-1901.] Dr Muir on a Peculiar Set of Linear Equations. 259 
where t/q + ij/ 2 = T) f /(g 3 - 1), then by using the three other equa- 
tions we have in all 
From these by cyclical substitution we shall obtain four similar 
expressions for g 2 , g 3 , and by the interchange of z’a and g’s four 
similar expressions for each of the three x 1} x 2i x 3 . In regard 
to this interchange however it is important to note that the 
expressions obtained for any x are exactly those obtained for the 
corresponding g , the reason for which is apparent on looking at 
the above quadratic equation in g 1 , where, on account of the 
original equations being symmetric with respect to g x and x 1 , it 
is legitimate to substitute x 1 for g x . If g x and x 1 be supposed 
different, the above twenty-four results may therefore be arranged 
as twelve pairs, viz. : — 
From this there follow four expressions for each of the sums g y + x 1 
g 2 x 2 , g 3 -t* x 3 , viz. i- — — 
pq -f aq = D (g 3 , g 2i x 2 ) -r (g 3 — 1) , 
= T>'(x g, g 3i x 3 ) -r (x 2 - 1) , 
= D ( x 3 , g 2 , x 2 ) — ( x 3 — 1 ) . 
(16) Writing the original set of equations in the form 
9 \ — Cliffs > 9-2 i x 2) or ^2(^3 1 9-2 > ^2) 1 
9 i = ^1(^2 > ^ 3 ' x s) or ^2(^2 » 93 1 x 3 ) 1 
9i = 1 1 (*2> fl r s» *3) or ^2(^2 >^3 » x s ) 1 
9 \ ~ $l (, X 3 j 92 > ^2) or 2(^3 > 92 i ^2) • 
01 = ^1(93 > 0 2 > *2) 01 921 x 2)i 
x \ ~ ^2(^3’ 92 1 X 2 ) 0r *W03 > 02’ ^2)’ 
and eliminating aq and aq we have 
03 02^2 d* ^7i 
0i 93 x 2 + 9 2 = 0 > 
£7i 1 92 x 2‘ 3r 9% 
