36 
Proceedings of the Royal Society of Edinburgh. [Sess. 
Of the seven identities included in this the first two are Trudi’s, and these 
he writes in combination, thus — 
a o 
a 4 
a 2 
‘h 
• 
a o 
a j 
« 2 
a s 
a 4 
% 
a \ 
a 2 
a 3 
a 4 
h 
b B 
b 4 
b» 
h 
h 
h 
K 
\ 
h 
h 
b s 
b\ 
1 a o h i 1 
1 a 0^2 1 
1 a 0^3 i 
1 %b i \ 
1 a J , 2 1 
1 a 0^3 I 4* 1 a i^2 ! 
1 Vk 1 + 1 a A 1 
I «A 1 
1 a (fo 3 j 
| a (fi 4 1 4 | | 
| a x b 4 1 + ! a. 2 b B j 
1 «2 & 4 1 1 
meaning thereby that the determinants got by leaving out the 7 th column 
on the left and the 4 th on the right are equal to one another, and also 
those determinants got by leaving out the 6 th column on the left and the 
3 rd on the right.* 
He then draws attention to the fact that the two-line determinants 
involved in the array on the right are principal minors of the array 
d rv CL-y Cic) Clo d* 
and he formulates a mnemonic rule like Sylvester’s {Hist., ii. p. 340) for 
the formation of the condensed array. His own illustrative examples are 
a 
b 
c 
d 
. 
. a 
b 
c 
d 
au - bt 
av - d 
ax - dt 
a 
b 
c 
d 
ax — dt 
= 
av - d 
bx - du 
. 
t 
u 
V 
X 
+ bv — cu 
. t 
u 
V 
X 
ax- dt 
bx - du 
cx - dv 
t 
u 
V 
X 
a 
b 
c 
d 
a 
b 
c 
d 
au - bt 
av - d 
ax - dt 
t 
u 
V 
X 
av — d 
ax - dt 
bx — du 
t 
u 
V 
X 
+ bv — cu 
a 
b 
c 
d 
_ 
|| au - bt. 
, av — d, 
ax - dt | 
t 
u 
V 
X 
* With Cayley the assertion 
Ii % 
« 2 
a B 
a 4 1 
= \\ x i 
x 2 
x 3 
x 4 
II \ 
b 3 
\ II 
II Vi 
V2 
V 3 
Vi 
included 6 equations, whereas with Trudi it only includes 3, namely, the first 3 of Cayley’s 
6 : and with Cayley the assertion 
a l 
a 2 
C*3 
a 4 
x i 
x 2 
Xo j 
*1 
b 2 
h 
\ 
— 
! Vi 
V2 
y 3 1 
C 1 
C 2 
C 3 
c 4 
was meaningless, whereas with Trudi it includes 2 equations. Since in the former case 
Trudi’s 3 equations are known to necessitate the other 3, there is clearly no good reason for 
refusing to profit by the new usage. What is common to any two arrays which Trudi may 
equate is the excess of the number of columns over the number of rows : and evidently if 
his excess be 5, the number of included equations is 5 + 1. 
