92 
Transactions of the Royal Society of South Africa. 
columns, and whose square array is that of the r th compound of the original 
determinant. 
(5) When we pass on to the case where the initiating determinant is 
axisymmetric and is axisymmetrically bordered, another previously known 
theorem can be brought into service with advantage, namely, the theorem 
regarding the determinant whose matrix is the sum of an axisymmetric and 
a zero-axial skew matrix. The 3rd order being taken this theorem is # 
a /+ i 
e+ fx d — 
e - ix 
d + A I 
af e 
fbd 
e d c 
a f e 
fbd 
e d c 
Returning then to our first result above, and taking g, h, h for the border 
of the basic determinant here, we have 
a f e . a f e k 
fbd fbdh 
e d c e d c g 
h h g . 
and consequently, by the theorem just recalled 
kh a 
a f e 
fbd 
e d c 
a f+G e - H = 
afe 
afe 
f - Gr b d + K 
fbd 
fbd 
e + H d - K c 
e d c 
e d c 
a f eh 
fbdh 
e d c g 
Jc h g . 
an unexpected result in pure determinants. 
(6) When the determinant with the peculiar matrix referred to at the 
beginning of §5 is of the 4th order, there is in the expansion an additional 
term of a quite different type, namely, we have — 
a 
g + w 
f-v l+u 
a 
9 
f 
I 
9 — w 
b 
e + 3 h — y 
9 
b 
e 
h 
f+ v 
e — z 
c h + x 
f 
e 
c 
h 
I - u 
Jc + y 
h — x d 
I 
h 
h 
d 
+ P + I w ■ 
-y 
where P is the bilinear whose square array is that of the 2nd compound of 
the 4-line axisymmetric determinant, and whose laterals are 
X, ?/, Z, U, V, w. 
On account of the existence of the said additional term 
(xw — yv + zuf 1 
it might fairly be expected that the deduced theorem would no longer hold, 
or, at least, not in the same form as before. Considerable importance 
therefore attaches to the value which this additional term assumes when 
* 'Trans. R. Soc. Edinburgh/ xxxix (1897), p. 222. 
