1900-1901.] Mr Tweedie on the foregoing Paper by Dr Muir. 261 
Note on Dr Muir’s Paper on a Peculiar Set of Linear 
Equations. By Charles Tweedie, M.A., B.Sc. 
(Read December 17, 1900.) 
§ 1. In Dr Muir’s Paper on a Peculiar Set of Linear Equations 
( communicated December 3, 1900) there occur two Determinants 
of the order, the expansions of which are given by Dr Muir. 
As the paper in question has so much to do with Symmetric 
Functions, the following simple method of obtaining their expan- 
sions may not prove uninteresting, based, as it is, upon the ele- 
mentary theory of Symmetric Functions and the so-called Principle 
of Indeterminate Coefficients. The two Determinants given are : — 
1 a 0 a a 
D - 
and 
N = 
§ 2. Expansion of D. — As Dr Muir points out, D is a symmetric 
function of a 15 a 2 , . . . a M for the interchange of a p and a q may 
be effected by interchanging first the p th and 2 th columns, and 
then the £> th and 2 th rows, and the result of these operations on 
the determinant is to leave it unaltered in value. Moreover, the 
expansion must be linear in each of the a’s. It must therefore be 
of the form, — 
1 + AjFcq + A 2 Fa 1 a 2 + AgFoqcqag + . . . . 
To determine the coefficients, put a x = a 2 = . . . = a n = a . The 
expansion then becomes 
1 + ftC-^ A^a + n C 2 A 2 a^ + . . . + n C r A r a r + . . . , 
while D is clearly (1 - a) n-1 (l + n - la) . 
