4 T. MUIR ON THE LAW OF EXTENSIBLE MINORS IN DETERMINANTS. 
| ce, | | C1 es | | ede, | | dyes | | Bye, | | Dyes | | dyes | | byes | | Dyes, | 
€ | | dydees| |b,dye,| | Oydoe, | J] | eres | | cee | Legs | $e} lere, | | coe, | | ope.) 1: > &&) 
| Becaes | | B,0,8, | | Bye,e, | | d,e;| |d3e,| | d4es| \dye5| |doe5| | dyes | 
In corroboration of these, we observe that from (4) and (6) we deduce 
| aye5| | @e@,| | ages | ase, | 
| Byes | | Byes | | byes | | Byes | 
| Ces | | C2€;, | | eyes | | exes | 
| dye5| | dee,| |dyes| | dye, 
==" | aibaeaae, || ee : . . . (8) 
which is the extensional of the manifest identity 
a 
b, Yb, ~b, 6 
: ; : ‘ — | Abt, | . 
6; "Cy ty) 2, 
d, dy dy 
§ 3. Thus in theory of determinants every general theorem in the form of an 
identity has its complementary and its extensional. The exact relation between 
the two latter is seen from the proof which has been given above, and may be 
formulated as follows :—J/ the Complementary of (A) with respect to a certain 
determinant be (B), its Complementary with respect to a determinant of higher 
order is the Extensional of (B). Consequently, if, as sometimes happens, the 
Complementary of (A) with respect to a certain determinant be (A) itself, its 
Complementary with respect to a determinant of higher order is its Extensional. 
By the two laws the theorems of determinants are knit together in a way 
which is interesting theoretically, and which at the same time has the practical 
advantage of making the remembrance of the whole body of theorems a very 
simple matter. 
