201 
1907-8.] Dr Muir on Compound Determinants. 
and the theorem stands 
cq d 2 a 5 a 6 a 7 
b i b 2 b b b Q b 7 
cq a 3 a 5 a 6 d 7 
bi b 2 b b b 6 b 7 
cq a 2 a b a 6 d 7 
b i h b h b e b 7 
CL i CL^ CLq Ct*j 
b l b 3 h b 6 b 7 
a 3 a 4 a b a 6 a 7 
bi b 2 b b b 6 b 7 
a z a 4 a b a 6 a 7 
b i b s b o b 6 b 7 
a i a 2 a b a 6 a 7 
b s b 4 b b bf. b 7 
CL 2 CLq CL ^ CL^ CL\j I 
b S b 4 b 5 b 6 b 7 I 
CL g CL^ CL ^ CLq CLyj 
b 3 b 4 b 5 b 6 b 7 
“5 
a 6 
(1 l ^ 
a x 
a 2 
a 3 
a i 
«5 
a 6 
a 7 
3 
h 
b 7 | ' 
h 
\ 
h 
h 
• 
No proof is given by Sylvester: attention is merely drawn by him to 
the fact that when r is put equal to 1 we obtain the theorem with which 
his paper commences. It is rather remarkable that he should not have 
singled out the case where n — 0. For then the theorem becomes 
<*n 
®21 ' 
, d rl 
a !2 
a 22 . 
. . a r2 
a w- 
. d rfX 
bc L\ ' 
K i 1 
b \2 
b 22 * ' 
. . b r2 
K 
K ■ 
b rfJ- 
a i 
cl 2 . . 
. a m 
C'wi-l, 1 — 1 
b 2 .. 
• b m 
, 
where, as before, the subscript pq denotes the p th integer of the • g th set of 
r integers taken from 1, 2, . . . , m, and /ul stands for C m n ; and this is the 
theorem well known at a later date in the form : The v th compound of a 
determinant of the m th order is a power of the said determinant , the 
index of the power being C w _ 1( r _ v Cauchy, it will be remembered, only got 
the length of a similar theorem in reference to the product of two com- 
plementary compounds. Now, since the complementary of the r th com- 
pound is the (m — r) th compound, the product of the two must be that 
power of the original determinant whose index is 
C^m— 1, r— 1 C'm— 1 ,m—i — 1) 
i.e. I'm— 1, r — 1 "t C TO _x ( r? 
l.e. '-'m, r } 
which agrees of course with Cauchy’s result. 
We thus learn that Sylvester’s general result may be accurately 
described in later phraseology as the " extensional ” of the theorem regard- 
ing the r th compound of a determinant, and that the discovery of both the 
said theorem and of its “ extensional ” is almost certainly due to him. At 
the same time it is hard to believe that this “ extensional ” is the all-embrac- 
ing theorem referred to by him in a previous paper : for by no stretch of 
