1900-1901.] Dr Muir on a Peculiar Set of Linear Equations. 251 
are different ; and the property of it which lies nearest the surface 
is that it is a symmetrical function of all its variables. In proof 
of this we have only to note that the transposition of the p ih and 
2 th rows, followed by the transposition of the p th and columns, 
has the effect of interchanging the two variables a p and a q and yet 
makes no alteration in the value of the determinant. This means, 
of course, that the order of the variables in D (a 1 ,a 2 ,...,a 7l ) is of 
no consequence. 
(5) From this and the fact that, as the determinant form shows, 
the function D is linear in each of its variables, we should expect 
that D must he expressible in terms of the fundamental symmetric 
functions 2a 2 , 2a 4 a 2 , Soqa^a o , . . . . As a matter of fact it is 
found that 
D = 1 — Soqc^ + 22a ] a 2 a 3 — 32cqa 2 a 3 a 4 + . . . , 
where it has to be noticed that the only missing member of the 
series is 2a r By way of proof of this second property we may 
proceed as follows, a special order, the 5th, being taken merely for 
the sake of brevity in writing : — 
D(a 1 ,a 2 ,a 3 ,a 4 ,a 5 ) 
1 
a 2 
a 3 
a 4 
a 5 
+ 
1 
a 2 
a 3 
a 4 
a l 
1 
a 3 
a 4 
a 5 
a l 
1 
a 3 
a 4 
a l 
a 2 
1 
a 4 
a 5 
a l 
a 2 
1 
a 4 
a l 
a 2 
a 3 
1 
a 5 
a l 
a 2 
a 3 
1 
a l 
a 2 
a 3 
a 4 
• 
1 
a 2 
a 3 
a 4 
1 
a l 
1 
a 3 
a 4 
1 
a l 
a 2 
1 
a 4 
1 
a l 
a 2 
a 3 
1 
1 
a i 
a 2 
«3 
a 4 
+ D(a 1 ,a 2 ,a 3 ,a 4 ) . 
If the subsidiary determinant which here arises as the co- 
factor of a 5 , and which therefore is the differential-quotient of 
D(a 1} a 2 ,a 3 ,a 4 ,a 5 ) with respect to a 5 , be expressed in terms of 
