38 Proceedings of the Royal Society of Edinburgh, 
where the c s are determinants defined by the postulated identity 
[Sess. 
a {) x'" + • • • 
b^x n + • • • 
q 0 x r ‘ 
+ ... Hr 
b 0 x" 
(see under Recurrents ) and where a- = r + 1 + f (m — n)(m — n — 1). For 
example, when m = 5, n = 4, r = 2 the identity is 
a o 
a l 
Ctcy 
® 3 
a 4 
a 5 
\ 
b 
h 
b 
b 
• 
a 0 
a i 
a s 
a 4 
°5 
\ 
b 
b 
_ (-1)* 
a o 
a , 
b 
b 
b 
« 3 
b 
b 
«4 
&4 
«B 
7, 
c o 
c o 
C 1 
C 1 
C 2 
C 2 
C 3 
C S 
V 
\ 
c o 
C 2 
C 3 
b 
b 
b 
b 
b 
b 
b 
b 
Trudi’s proof consists in evolving the second member from the first, but 
here again it is simpler to use Cayley’s multiplication-theorem of 1845. 
Thus, taking the second array as multiplier and the determinant 
1 ..... 
1 
1 .... 
. . - qi 1 . . . 
-0i "0o * 1 • • 
- 01-00 • . . 1 . 
- 00 1 
as multiplicand we at once find the product to be 
c o 
c ] 
C 2 
C 3 
• 
c o 
C 1 
6 2 
C 3 
C 0 
C 1 
C 2 
C 3 
b 
b 2 
h 
b 
. 
b 
b 
h 
b 
b 
b 
\ 
• 
b 
b 
b 
\ 
which is equal to 
c 0 
C 2 
CO 
. c 0 
C 1 
^2* 
C 3 
. 
c o 
C 1 
C 2 
C 3 
. \ 
h 
b 
^3 
b 
\ \ 
b 
V 
(b ■ 
■ • ^ 4)2 
( c o • ■ 
• • 5 ^ 3)3 
as was to be proved. 
