* 
294 
Transactions of the Boyal Society of South Africa. 
In the next place, it being impossible that a determinant with one 2 in its. 
symbol — that is, with one column of 5's — can have any of its terms cancelled 
by terms of a determinant with three columns of 6's, we infer that we are 
face to face with two vanishing aggregates, namely, / 
|0 1112|+|01121|+|01211|+|02111|=0 
and 
|01222| + 102122| + |02212| + |02221|=0. 
It is seen, however, on a second glance that either of these is got from 
the other by merely interchanging a's and 5's, so that in reality there is 
only one to be proved, the full-length form of which is 
1 a.T &o a, a- 
O 4- o 
1 «j 
+ 1 «5 ^5 «3 
1 a_^ h-^ tti Oo 
1 «3 &2 <^'5 
0 = 
1 fl^ % ^5 
1 a ^ a2 ci^^i 
1 a-^^ a^ ho 
1 04 «5 ^3 
1 a. a^ a- \ 
+ 
1 a^ «3 «5 
I «i a 2 h- a^ 
1 a- a^ 61 
1 «4 05 ho a.2 
1 O3 ^4 Z>3 Ctj 
1 ^2 «3 «4 ^5 
1 63 ^3 a^ 
\ 1 a^ ao a^ 
1 b^ dr, d]^ ^2 
I 1 6j (X^ a- (Xj 
To effect the proof it would seem as if we^had to show that the cofactors 
of h^, bo, 63, 64, 65 vanish separately : but, as before, although the said five 
cofactors do vanish separately, it is sufiicient only to show that one of them 
vanishes, say the cofactor of b^. Our problem is thus further reduced to 
proving the equality 
0 =- 
and this is final, for the cofactors of the elements in the first columns, it will 
be found, cancel each other in pairs or in triads. As a matter of fact, if we 
call the four determinants P, Q, E, S, and append the suffixes 1, 2, 3, 4 to 
each to obtain a notation for the said cofactors, we have 
1 
a.2 
«3 
«4 
1 
a^ 
a. 
1 
«4 
«5 
1 
^3 
^4 
«5 
1 
^5 
«1 
a.2 
1 
a^ 
^2 
CI4 
1 
a^ 
«.i 
1 
a. 
a^ 
1 
«r 
t> 
«i 
1 
a^ 
a.2 
1 
tto 
1 
a^ 
a^ 
1 
«4 
(^0 
1 
^3 
a^ 
1 
«3 
ar, 
«1 
1 
«5 
H 
Po 
-p. 
— Q3 +E3 =: 
-Ql +So 
+ Ei +84 = 
Q4 ~~^3 "^1 = 
+ E, 
Q2 
0 
0 
0 
0 
0 
0, 
where the vanishing trinomials are instances of a known theorem * regarding 
vanishing aggregates of secondary minors in a persymmetric determinant, 
the persymmetric determinant here being the circulant C(a^, a^, a^, a^, a^). 
* Cazzaniga, T., 'Kendic. . . . Istituto Lombardo ' (2), xxxi, pp. 610-614 j 
Mmr, T., 'Trans. E. Soc. Edin./ xxxix, p. 226; xl, pp. 511-533. 
