262 Proceedings of Boy al Society of Edinburgh. [sess. 
The coefficient of a r in the latter expression is 
( — 1) £n-lQ- — {n — l)n-lC r _iJ 
i-e, -(-1)>-1),A. 
Hence 
n C r A r = -(-l)>-l) w C rJ 
i.e., Ar — -(-l) r (r-l). 
The expansion of the Determinant is therefore 
1 — Sajag + 22a 1 a 2 a 3 — 32a-|a 2 a 3 a 4 + . . . . 
§ 3. Expansion of N. — The coefficient of a 4 is D(a 2 , a 3 , . . . a n ), 
and the remaining terms form N(0, a 2 , a 3 , . . . a n ). Now this latter 
determinant is, when expanded, a symmetrical function of a 2 a 3 . . . a m 
for the interchange of a p and a q may be effected by the interchange 
of the p ih - and <2 th columns, followed by the interchange of the p ih - 
and 2 th rows ( vide Dr Muir’s paper). Let us note what Types of 
symmetric functions can occur, and let us select those that involve 
a 2 . Now a 2 occurs only in the first and second columns. If the 
type contains a% it must he linear in other variables, and if it con- 
tain a 2 and, say, a 4 as from the first and second columns, then it 
must contain a 4 again, since by taking a 4 from the first column we 
are prevented from taking the constituent 1 from the fourth column. 
This term is also linear in any other variables. Finally, there is 
no term independent of the variables. 
The expansion of N(0a 2 a 3 . . . a n ) must therefore be of the form 
A 2 2a| + A 3 2a 2 a 3 + A 4 2a.2Ct 3 a 4 + 
To determine the coefficients, put a 2 = a 3 = ... =a n = a. The 
determinant then becomes 
0 
a 
a 
... a 
1 
a 
a ... 
a 
1 
a 
a 
a 
a 
1 
a 
... a 
a 
1 
a ... 
a 
a 
1 
a 
. . a 
a 
a 
1 
... a 
= 
a 
a 
1 ... 
a 
- 
a 
a 
1 . 
a 
a 
a 
a 
... 1 
a 
a 
a 
1 
a 
a 
a 
.. 1 
n n n—1 . 
in which, by § 2, the coefficient of a is 
r b» C,.-r„_ 1 C r 
-(-!)' 
