2 THOMAS MUIR ON THE LAW OF 
and substituting for each determinant its complementary minor in the deter- 
minant | @,b,c,d,|, we have 
lcs] led, | | cd, | | b, 0, 0) | Op & 2b, | 
| b5ty| |Prdy| [Ordol P=] a ey ey | Cy Cy Cy | . : ; . @ 
|dgcy | | Prey | | Pree | | d, ds d, , ad, dy d, 
a special case of a theorem of SYLVESTER’s in regard to compound determinants. 
It is thus seen that in virtue of the Law of Complementaries the theorems 
of determinants range themselves in pairs, like pairs of theorems in geometry 
in virtue of such a law as that of Reciprocal Polars. 
§ 2. We come now to 
THE LAw oF EXTENSIBLE MINORS. 
Tf any identical relation be established between a number of the minors of a 
determinant or between the determinant itself and a number of its minors, the 
determinants being denoted by means of their pr incipal diagonals, then a new 
theorem is always obtainable by merely choosing a line of new letters with new 
suffixes and annexing it to the end of the diagonal of every determinant, including 
those of order 0, occurring in the identity. 
The proof is dependent upon the Law of Complementaries, and upon the 
simple fact that every minor of a given determinant is also a minor of any 
determinant of which the given determinant itself is a minor. Let (A) be 
the established identity, and |a,b,c,...4,| the determinant whose minors are 
involved in it. Then taking the complementary of (A) with respect to 
| %bc,...2,| we have an identity, (B) say, likewise involving minors of 
a,b.c,...4,|. But these minors are also’ minors of |@,b,c,... ,7°,.8.... 2, |, and 
therefore it is allowable to take the complementary of (B) with respect to this 
Doing this we pass, not back to (A), but to a new 
theorem (A’) which is seen to be derivable from (A) by ees to the end of 
the diagonal of every determinant i in it the line of letters 7,s,...2,. The law 
is thus established. 
The clause “ including those of omer 0” is necessitated by the last clause 
in the enunciation of the Law of Complementaries. 
Taking as an example the simple identity 
| aybq¢3 | = ay | does |— ay | Dye; |, | Be, , 
and using only one new letter d and one new suffix 4, we change 
