Dr T. Muir on the Theory of Determinants. 
487 
determinants of two complementary systems, 
minant of the system 
by 
D 
{p) 
Denoting the deter- 
and that of the complementary system 
and multiplying the two determinants together, we see with Cauchy 
that by (xiv. 4) the principal “ termes ” of the resulting deter- 
minant are each equal to 
D., 
and by (xii. 7) all the other “termes” are equal to zero, 
quently 
D</) . = (DJ"" 
Conse- 
(XLII.) 
As an example of this theorem, it may he added that the product 
of the two determinants printed above (p. 482-3) to illustrate the 
notation 
that is to say, the determinants of the systems 
(«2o) > («2o) , 
is equal to 
l^l*1^2-2^3*3^4'4^5‘5l ‘ 
In connection with all the three theorems, the special case, p= 1, 
is given, so that their relation to previously well-known theorems 
(vi., XII., XXI.) may be noted. It is also pointed out, that when in 
the third theorem n is even and ^ the result takes the interesting 
form 
= (XLII. 2). 
This brings us to the last section of the memoir, the fourth, 
bearing the heading 
Des Systemes (T Equations derivees et de lour 
Determinans. 
What it is concerned with is the relations subsisting between a 
“ derived system ” of the product-determinant 
