Dr T. Muir on the Theory of Determinants. 
525 
the theorem regarding the sign of a permutation which is got from 
another permutation by the interchange of two elements. If the 
under-indices of the one term whose sign is s be 
and of the other whose sign is Z * be 
a'aV" a(^'-l)aWa,h’+l), , , . . 
it is shown that 
(71) 
M) 
cf>(a: 
- a') - a}) 
i+l 
cf>{a^ - a^) «^(g^ - a^+^) 
</)(a* — a^) * — a^) 
and there being here 2k - 2^ - 1 quotients each = 
arrived at is 
</)(a' - ak) 
<h{ak-ak'^) 
1 , the result 
or 
2= -Z, 
as was to be i^roved. (iii. 23.) 
The body of the Appendix contains, along with other matter 
which falls to be considered later, the statement and proof of pro- 
positions identical in essence but not in form with the following : — 
(1) 
where 
1 
2 
71 — 1 
71 
a 
a .. . 
a 
a 
1 
1 
1 
1 
1 
2 
71 — 1 
71 
a 
a .. . 
a 
a 
2 
2 
2 
2 
1 
2 
71- 1 
71 
a 
«... 
a 
a 
n-l 
71-1 
77-1 
72 - 
1 
1 
2 
71 — 1 
71 
T 
T . . . 
. T 
T 
1 
1 1 
1 
T 
= m a m a . 
. . . + 
m 
a 
1 
1 2 2 
71-1 
71 
-1 
2 
2 2 
2 
T 
= m a + m a + . 
. . . + 
m 
a 
1 
1 2 2 
71-1 
71 — 1 
(XLIX.) 
^^More than a page is occupied in writing the expressions for z and Z. 
