324 
Proceedings of the Poyal Society of Edinburgh. [Sess. 
but that for higher orders the result is 0. He next notes the identity 
4> X 
, x </> 
4> 4> 1 _ j # 
*' X 
+ / ,# 
I x </> 
x'x' 1 XX 
adding “ etc. for determinants of any order ” ; * and then from this set of 
identities and the previous set he concludes that if any four adjacent 
columns of a quaternion determinant be transposed in every possible 
manner, the sum of the determinants thus obtained vanishes — a property 
which, he says, is much less simple than the analogous one for the rows, 
this last being the same that holds in the case of determinants with ordinary 
elements. Lastly, he gives the important warning that the eliminant of 
ttI I + (£><[> = 0 I 
Tr'n + = 0 j 
is neither i r<f>' — ir'ty nor 7 T 0 ' — , but 
TT- 1 ^ - 7 r'-y. 
Tissot, A. (1852, May). 
[Sur un determinant d’integrales definies. Journ. {de Liouville) de 
Math., xvii. pp. 177-185.] 
The subject here is the evaluation of the determinant of the (^ + l) th 
order whose {r, s) th element is 
[ a$ _ x x r dx 
Ja 6 MU*)’ 
Ja s-1 
where 
= (x - a 0 ) m °(x - a Y ) mi .... (x - aj) mi (a i+1 - x) mi+1 . . . ( a n -x) mn , 
and where therefore 
123 + 132 + . . . = | xy'z" I . {ij + ij + ij — ij - ij + ij) = 2 k I xy'z" I , 
and so on. The multiplication table of i, j, k, it may be recalled, is 
( ii 
V 
ik ) ( 
; -i 
fc -j ) 
P 
jj 
jk 
# 
-k 
-i i 
ki 
kj 
kk 
j 
-i -i 
* Very probably the next case is the identity 
0 
X 
0 
0 
0 
X 
X 
0 
0 
0 
X 
0 
0 ' 
x' 
0' 
+ 
0' 
0' 
X 
+ 
X 
0' 
0' 
+ . . . + 
0' 
X 
0' 
0 " 
x" 
0" 
0" 
0" 
X 
X 
0" 
0" 
0" 
x" 
0" 
0 
0 
0 
0 
0 
0 
0' 
0' 
0' 
0" 
0" 
0" 
X 
X 
X 
- 
X 
x" 
X 
- 
X 
X 
X 
+ . . . - 
X 
x' 
x' 
0 " 
r 
0" 
0' 
0' 
0' 
0" 
0" 
0" 
0 
0 
0 
where, as in the other cases, the r th determinant on the left is equal to the aggregate of the 
r th terms of all the determinants on the right. 
