174 Transactions of the Royal Society of South Africa. 
and multiplying it by the outside elements which are in the same column 
or row with it : the function of degree-order 4,2 is 
a h 
which equals 
c d 
e f 
k 
g I 
h I 
a h 
or 
a b 
k I 
c d 
9 i 
/ 
c d 
e f 
+ I 
a h 
c d 
e f 
and therefore stands for 
acgJc + acil + aehJc + aejl + hdgk + hdil + hfhk -f hfjl : 
and so on, the number of terms in the final expansion being n'"-'^ when the 
degree-order is m,7i. 
(4) The most interesting occurrence of the functions in analysis is as 
the elements of product-determinants : for example, 
«1 
Pi P2 
^1 
^2 
^2 
rtj^ rto rtg rt-j rto rtr^ 
?1 % % n ^2 ^'3 
6, 60 63 63 63 
?1 ?2 ^3 
C| C2 
rt| rto rt^ 
Pi -P2 i's 
i^l i^2 
C| Co Co 
1^1 i^2 ^1 % % ^'l ^2 ^*3 
^3 
^2 ^3 
a-i 
rto 
1 ^^1 ^^2 
-JW.^ ?«o 
/J 
I /i^j A^o 
%1 ^^2 
rt| rto 
rtj rtc: 
I 
rtj 5o 
^1 ^2 
l/i-^ %oj . I 2/2 
''■2 
I, 
ho 
Til 
h 
h 
Wo 
h 
ho 
Wo 
h 
h 
ho 
h 
^2 
^1 
h 
h 
^2 
'W.o 
> 
rtj 
rto 
^2 
rt| 
rto 
2/1 
2/2 
A, 
Wj 
7li 
ft, 
W, 
^1 
h 
no 
W2 
% 
?>2 
2/2 
h 
A3 
^1 
Wj 
^1 
h 
^1 ^2 
W3 
^2 
but several other distinct occurrences have been pointed out.* 
(5) Let us now turn to the multiplying of 
rtj + a.yU) + a-Tofi + ... -[- rt-,ia;"~^ 
by another expression of the like kind 
h^ -f 60W -f 63(02 + . . . + 
* For example, ' Proc. Lond. Math. Soc./ xvi, pp. 276-286. 
/ 
