20] 
ON CERTAIN RESULTS RELATING TO QUATERNIONS. 
125 
but 
3733' — ot'oT, 4=0 
3X , 3X 
/ / 
■nr, rar 
•(8). 
that is, a quaternion determinant does not vanish when two vertical rows become 
identical. One is immediately led to inquire what the value of such determinants is. 
Suppose 
ox = x + iy +jz + lew, sx' — x' + iy' +jz' + lav, &c., 
it is easy to prove 
3X 
3X 
— — 2 
i , 
j * 
h 
(9), 
/ 
'US 
/ 
US 
X , 
y> 
z 
x', 
y'> 
z' 
ox, 
3X , 
3X 
= — 2 
3 
i 
j 
, £ 
(10), 
3x', 
ox' > 
/ 
US 
X 
y 
z 
, w 
ot", 
// 
, 
// 
US 
x' 
y' > 
* 
, w' 
x", 
z", w" 
w , 
3X , 
w , 
37 
= 0 . 
(11)- 
/ 
US j 
ox' , 
ax', 
/ 
US 
// 
3X , 
// 
3T , 
ox", 
// 
US 
/// 
3X , 
/// 
3X , 
/// 
3X , 
/// 
US 
or a quaternion determinant vanishes when four or more of its vertical rows become 
identical. 
Again, it is immediately seen that 
OX, (f> + 
<£, 3X 
= 
ax, 
3X — 1 
3t / , 
3X', </>' 1 
0', 3x' 
</>> 
<t> 
(12) 
&c. for determinants of any order, whence the theorem, if any four (or more) adjacent 
vertical columns of a quaternion determinant be transposed in every possible manner, 
the sum of all these determinants vanishes, which is a much less simple property 
than the one which exists for the horizontal rows, viz. the same that in ordinary 
determinants exists for the horizontal or vertical rows indifferently. It is important to 
remark that the equations 
i.e. 
3X , 
</> 
11 
o 
or 
ax, 
3x' 
II 
o 
&c. 
ax', 
3X0' — 
3j'(f> 
= 0, 
or 
37(f)' 
= 0, 
&c. 
(13) 
are none of them the result of the elimination of II, d>, from the two equations 
OT n+0d>=O, (14). 
axil + (fi'Q* = 0,