EXTENSIBLE MINORS IN DETERMINANTS. 3 
hastouals Vey, | Osc 1.80): eal, 5 ,° “WO, 1, Say 
into , 
| dybocgdy |, | rdy|, DyCgdy|,  |AoFy|, | Pricsdy|, | aad |, | O64, |, 
and noting that without further change the two sides would not be of the same 
degree, we annex the factor d, to the left hand side, thus, as it were, extending 
the process of elevation of order to an imaginary determinant of order 0. The 
result is the identity 
d, | dyboegd,| = | ad, || b,c,4, | — | aod, | | Dycatly | + |agdy| | dycod, | . a, (3) 
This is verified by observing that 
| Gy A, Az A, Ay [Gy “Gy, Gg Gy Gy 
| 
| by Op By Op OY by By BB, 0 
anette | GAC, €., yO = G, Gy) 6,-6.1 @ 
| dy dy dy d, O-| bd, dy dy d, 0 
i hoy ha edd ee 
and then expressing the last determinant in terms of products of complementary 
minors, one factor of each product.being formed from the first and last. line. 
‘Taking the identity numbered (1) above and choosing the extension e¢,, we 
have 
| @yb9¢5 | |@abses | | aybses | 
[ re, | | Moeses | Lscaes | | =| rbyes@ses| | a2c,| | ages | - . . (4) 
| aye, | | a dge,| | agclye; | 
The corresponding extensional of (2) is 
lege; | | edges; | | cydyes | 
|bgd se, | | Bdge, | | Pydoes | = | dycgd es | | dycyalyes | es . . ; HH °) 
| bscyes | | dyeyes | | dyes | 
The identities (1) and (2) however admit each of two forms of Extensional, 
according as we look upon the letters in the right hand members as being mere 
elements, or as being determinants of order 1. Thus from (1) we have 
| ayes | | does | | ages | | aye; | 
| yes | | does | | Bses | | Dyes | 
C5} | @yC2@5| | docses | | agcyes | | = | aoe, |-| ages | - ~ ©) 
| és | | coes | | C20 | | C4es | 
| dye, | | does | | dees | | dyes | 
| @1b,¢5 | | &odses | | agbse, | 
| ayy, | | agase5| | Ag 4ee | 
and from (2) 
