1887.] Dr T. Muir on Simple Alternants. 439 
the triads are 
0, 2, /? + y + S + l; 1, 2, f3 + y + S : 
and consequently the number of dovetailings 3, as has been 
asserted. 
For the case of ^a a lTcy, there is only one duad 
0, y + 1 ; 
two triads 
0, 2, (3 + y + 1 ; 1, 2, /? + y , 
and therefore, as is at once seen, only 1 dovetailing. And this 
establishes the theorem. 
These two theorems and one well known before this (v. Theory 
of Determinants, § 123) constitute an interesting group. 
Denoting by cr n the sum of all the symmetric functions whose 
terms involve n letters and are of the s th degree, we may write the 
third theorem referred to in the form 
<x(0, 1, 2, 3 + s) — oq + <t 2 + cr 3 + <x 4 . 
But by the two new theorems 
cc(0, 1, 3, 2 + s) = cr o + 2 <r 3 + 3?r 4 , 
a(0, 2, 3, 1 + s) = or 3 + 3 <t 4 . 
And again by the third theorem 
a( 1 , 2, 3, .?) == d £ . 
We can thus express oq, o- 2 , cr 3 , or 4 , in terms of the four functions 
on the left, the result evidently being 
oq = a(0 1 2 3 + s) - a(0 1 3 2 + s) + a{0 2 3 1 + s) - a( 1 2 3 s ) , 
o- 2 = a(0 1 3 2 + s) - 2 . a(0 2 3 1 +*) + 3 . a(l 2 3 *) , 
q- t = a(0 2 3 1 + s) - 3 . a( 1 2 3s), 
c r 4 = a(l 2 3 «) . 
Of course if these last could be established independently, we 
should have a very simple additional method of obtaining the 
identities from which they are here derived. 
The third theorem is — The expansion of 
a{ 0, 1, q, r) - a( 0, 2, q, r - 1) + a( 1, 2, q, r - 2) 
