A Sum of Products which Involves Symmetrically the Nth Boots of 1. 177 
(8) As a consequence of the preceding, some interest attaches to 
determinants whose elements are of the type 
+ ctow + (X.^oj- + . . . + ano)^~^. 
Confining ourselves to the third order, merely from considerations of space, 
let us examine the determinant 
di -i- d.jOj + fZ^w" e^ -f- ecihi + ggw^ /j + -f- /gw^ 
By partitioning it into twenty- seven determinants with monomial elements 
it is readily seen to be expressible in the form 
P + Qo; + Ea;2 
where P, Q, R are each the sum of nine determinants, and in particular 
P = 1147| + 12581 + |369| 
+ |159I -f 12671 + |348| 
4. |168| + [2491 4- |357[, 
if \mnr\ stand for the determinant whose columns are the m*^, n^^, of 
the 3-by-32 array 
\ ^2 h ^1 ^2 ^3 
% fi h fs 
9i 92 9s h h h h h K 
^1 ^2 ^3 
d^ dc^ d^ e^ eo 
P = 
= i2 
It follows therefore that, if we use 2 as above we have 
d^ + cZoW 4- fZgW^ ^1 + ec,<^ + egw^ + /ow + /!^a>^ 
^1 + ^2^ + ^3^^ ^1 + ^h^ + ^^3*^^ ^1 + ^2^ + ^3*^^ 
Instead, however, of partitioning the determinant we may take the six 
terms of its ordinary expansion, and, these being of the type 
we can make use of our above theorem regarding such products, and thus 
arrive at the result 
9} + ^2<^ + 93"^^ \ + ^^2^^ + ^h"^"^ \ + ^2^^ + h'^^ 
^1 ^2 ^3 , ^1 ^2 ^3 
^1 ^3 ^2 
^2 ^1 ^3 
63 ^2 ei 
^1 ^2 ^3 
/l /3 /2 
/2 /l /3 
/s f'Z fl 
fi /s A 
/2 /l 
/s /2 fl 
h \ h 
d^ d.^ (^2 
dc^ d-^ cZg 
d^ df) d^ A/2 
^1 
9:^ 
92 
+ 
^2 *^3 
c/j^ cZg (^2 
d^ d-^ d^ 
d^ dc^ d~i 
^1 ^2 % 
63 ^2 
^2 ^1 ^3 
63 62 ^1 
^1 
hey 
9i 
9s 
92 
