102 Transactions of the Royal Society of South Africa. 
and , 
{n-p + \) _ . [ 
— ^11 • • • \'^p-\, p-l ^p, p + l • ' ' Ctw, n + \\ 
— that is to say, the latter is formable from An+i hy deleting the last row 
and the _2:)th column, and the former by deleting the last row and the last 
column. 
(3) With the help of this lemma he arrives at his first important 
result : 
{X,),, = -ii + > ( - 1)^ J^.a_Vl_ . . (X) 
ApH-\ApH 
where 
that is to say, is formable from Aj- by replacing its last column by u^, 
. . . , Ur. 
(4) Next, denoting 
LSp + ^-\£\p + ^ 
he writes the n cases of (X) in the form — 
and having transformed the given w-line set of equations into 
^llC'^l)" + B-^o(x,,)u +....+ Bi,,(Xn)n = 
where 
that is to say, is formable from Ap by replacing its last column by the mth, 
he eliminates the V's and x's, and so obtains his two other important 
results : 
^^ppBpm ~\~ '^p, + m ~^ ... -\- K^Jm-BmjH — 0 . . (Y) 
K^jjhB^j^j -1- + iii^p, p + \ • ' ' ~h '^miit^pm = 0 . . (Z) 
The three equalities (X), (Y), (Z) are what it is desired to draw 
attention to. 
(5) A very slight examination of them suffices to show that they have 
not any necessary connection with a set of simultaneous linear equations, 
but that they are purely determinantal theorems, and, further, that all 
the determinants in them belong to one and the same m-'hj-{m + 1) array. 
