1887.] Dr T. Muir on the Theory of Determinants. 515 
where c is a constant. But if we take the very special case where 
= 1 5 = 1 j — 0, a^. v 0, 
and where consequently 
“1? wv„ — ^ ) 
M„=l, D n =l, S B =1, 
c = l . 
we see that 
and that therefore 
Hence the final result is 
M n = DA . 
(xvn. 5). 
This, the now well-known multiplication-theorem of determinants, 
Cauchy puts in words as follows (p. 82) : — 
Lorsqu’un systeme de quant it es est determine symetriquement 
au moyen de deux autres systemes, le determinant du systeme 
resultant est toujours eg at au produit des determinans des deux 
systemes composans. (xvn. 5). 
It is quite clear, from what has been said above, that it was dis- 
covered independently, and about the same time, bv Binet and 
Cauchy, and ought to bear the names of both. Binet has the 
further merit of having reached a theorem of which Cauchy’s is a 
special case, and then made an additional generalisation in a dif- 
ferent direction ; and Cauchy has the advantage over Binet of 
having produced, along with his special case, a satisfactory proof 
of it. 
From the theorem Cauchy goes on to deduce several results 
equally important. Substituting for the system (a Vn ) the system 
(b rn ) adjugate to ( a Vn ) so that 
S W =S( + ’ ‘ * ^n'v) ~ B„ J 
we know that then 
WW = D n and m (L . v = 0; 
that consequently M w consists of but a single term, viz. 
m ri m 2 . 2 . . . m n . n , i.e. D n n : 
and that therefore by the theorem 
d;=bj> m , 
b^d;; 1 . 
whence 
(xxi. 2). 
