326 
Transactions of the Royal Society of South Africa. 
This can only happen when one of the factors of each term of the left- 
hand member vanishes : and as the vanishing of one factor implies the 
vanishing of the co-factor, it can only happen when 
(X.) 
a^ + a\i 
&i + h\i 
+ c\i 
A,-A[i 
B, - B\i 
-c,r 
a^ -\- a'd 
+ h\i 
■Vc^i 
A3 - A'J 
^3 + a\i 
Z>3 + h',i 
^3 
+ c\i 
A3 - a;. 
-B3-B;.- 
C3 
C2 say, 
in other words, when the elements of each column of jj. are proportional to 
the elements of the corresponding column of M'. But when this is the 
case, each element of fx can be replaced by a multiple of the corresponding 
element of M', with the result that we shall have 
also, the column-by-column multiplication of fx by ^u' would give 
from which the same deduction could be made. 
Similarly the limit 
{a] + + a; + a',^ -t- a: + a';^){h\ + + hi + + hi + h\^){c] + c,^ -f 
will be reached when the elements of each rou^ of are proportional to the 
elements of the corresponding row of M', and row-by-row multiplication 
will give 
Both limits will be reached, and will therefore coalesce when all the 
elements of ^ are proportional to the corresponding elements of M', and 
row-by-row multiplication will then give the same result as column-by- 
column multiplication, namely 
cu 
In this event we may appropriately speak of the determinant having a 
maximum value or heing a maximum determinant. Such is evidently 
possible when )u is axisymmetric or axially skew, because then the two 
