486 
Proceedings of Boyal Society of Edinlurgli. 
ou aura done par suite 
0 = s'’(a'$ 3). 
On aurait de meme, en supposant le signe S relatif a I’indice 
/X, et en designant par (t) une nouvelle combinaison differente 
de (tt) 
0 = •” 3)- 
As this theorem is twin with the preceding, it is best to illustrate 
it by the same special case. By so doing, indeed, both theorems 
become more readily grasped and their details better understood. 
Taking then as before = 5, p = "2 and tt = 7, we first form the 
determinants which Cauchy would have denoted by 
7 , (^2.7 > ^10.7 J 
and which we denote by 
1^12^25! 5 kl2'^35l 5 • • • • J 1*^42^55! * 
Next, for cofactors, we form the determinants which are comple- 
mentary, not of these, as in the preceding theorem, but of the 
members of one of the nine other groups corresponding to the 
values 1, 2, 3, 4, 5, 6, 8, 9, 10 of tt, — say the group 
^(2) ^(2) ^(2) 
*^1.6 ’ ^^2.6 ’ ’ ‘^10.6 * 
These complementaries being 
l%1^43%5l ’ 1 ^ 21 ^ 43 ^ 55 ! 5 5 1 ^ 11 ^ 23 ^ 35 ! J 
we have the desired identity 
6 = 1^12^251- l%1^43^55l “ 1^12%5N'^21^43^55l 1^42^551* l^ll^23%5l ’ 
the right-hand side of which is nothing more than an expansion of the 
zero determinant which arises from the determinant l«ii<* 22 % 3 ^ 44 % 5 l 
“ lorsqu’on y remplace un des indices par un autre,” viz., the second 
4 by 5. 
With the help of these two theorems a third theorem of almost 
equal importance is derived, viz., regarding the product of the 
