35 
1913-14.] The Theory of Bigradients from 1859 to 1880. 
The number of bigradient arrays associated with the two sets of 
elements 
’ ' • ’ K ^1’ • ' ’ ’ 
is evidently n : thus, in the case where m,n = 7,4 the arrays are 
(%< • • 
• . «r)x 
( a o> • • 
• > ^7)2 
(®o > . . 
. , oq) 3 
«o>* ■ 
. . , & 7)4 
(K ■ ■ 
•.M4 
f 
■ ■ 
• . *4)5 
> 
(K ■ ■ 
• 5 
I ? 1 
K • • 
■ ..sa 
the last, where r = n, being square, and therefore preferably written in 
the form 
I (« 0 » • • • » I 
I (&0. • ' • 5 ^4)7 I ’ 
These and other preliminaries being settled, he is in a position to deal 
with an important theorem on the subject of what we may call the con- 
densation of a bigradient array. The proof given is, unfortunately, not 
at all so simple as it might have been. We shall therefore substitute for 
it one of our own, which Trudi himself would probably have devised had 
he been aware of Cayley’s work of 1845. Taking, first, a case in which 
m = n, say the case 
a 0 
“1 
«2 
*4 
• 
a 0 
“2 
«B 
a 4 
«o 
a i 
a 2 
«3 
“4 
\ 
h 
*4 
h 
h 
^4 
% 
h 
K 
h 
h 
(«o> • • 
. , a 4 ) 3 
or 
00. • • 
• 5 ^ 4)3 
we multiply the determinant 
1 
1 
0 
r-c> 
1 
1 
~h 
-h 
~ ^2 
a 0 
cq 
a 2 
a 0 
a i 
a 0 
by the given array in column-by-column fashion, obtaining {Hist., ii. p. 34) 
^0 ^2 
a B 
«4 
a 0 
a 2 
«3 
« 4 
• 
\ («0. • • 
. . , a 4 )g 
. . a 0 
a i 
a 2 
«8 
a 4 
1 00. • ■ 
■ • . £ 4)3 
r 
1 “(A 1 
1 «0 6 2 1 
KVI 
1 a A 1 
1 <*A 1 
I S 6 3 1 + 1 a A 1 
1 «0 6 4 1 + 1 « A 1 
l | ! 
«0 & 3 1 
1 «o 6 4 1 + I af 3 I 
1 «A 1 + 1 ,% & 3 i 
1 « 2 & 4 1 1 
