'200 
Transactions of the Boyal Society of South Africa. 
In preparation for it it is necessary to recall the fact that a determinant 
with polynominal elements so constituted that all the elements of each 
•coaxial minor have a term in common has nothing but positive terms in 
the final development : for example, 
a + a + /3i -f /33 + a + ft^ 
which is denoted by 
M(a, /3A/33, 7x7273) or M3, 
has 32 positive and no negative terms. 
Now the theorem just referred to is to the effect that the adjugate of 
^3 is the M-unisignant of the 3rd order in which 
a^ , 
(o^^aj)^, l32 = a.b^, f^^ = a^c^. 
73= I {^2 
h 
3; I 
The values of a, y^, y^, 73, it will be observed, are the minors of ^3 got by 
deleting its frame-lines in order : the fact that they constitute a set is in 
accordance with a previously known result, namely, that a is involved in 
the development of M3 exactly as the 7's are. 
Similarly the adjugate of is the M-unisignant of the 4th order 
1^3 ^4 «5) 
where 
have the values 
the /3's the values 
a,\{c, 
and the 7's the values 
J)- 
M(a ; /31/32/33/34 ; 7x7273747576 ; ^1^203^4) 
a, C12, O3, ^4 
! I ^2 a^ (Xg 
a^ a^ a^^ 
\ ^5 
a,\\b.^ 
a, a^^ 
, b,\[a^ aA 
, b,\[a, a,\ 
, {^2 a^l 
, C5 a^ 
, 1^2 «3) 
^5/ 
cJ 
b,) 
^3) 
