1894 - 95 .] Dr Muir on the Acljugate Determinant. 327 
For if to each of the two terms in question, T^, T 2 say, there be 
prefixed as a factor one and the same term S of the complementary 
minor, we obtain two terms of the full adjugate, the difference of 
which, STj - ST 2 , not only contains the original determinant A as a 
factor, but, as the proof of this in § 2 shows, contains S as well, S 
in fact remaining unaltered throughout the various steps of the 
proof. Consequently - T 2 is a multiple of A. 
8. Of course this latter theorem does not really depend upon the 
former. A more natural mode of procedure perhaps would have 
been to enunciate them from the first as one theorem, beginning 
with the minor of the 2nd order of the adjugate and proceeding 
upwards to the full adjugate. 
It is, however, interesting to note that the case for the full 
adjugate of one order leads to that for certain minors of adjugates 
of higher orders. Thus for the full adjugate of the 3rd order we 
have 
ffilC .2 I’l M V 3 1 - 1 «iC 3 li ai &2 11 hh 1 = 1 r{ - I ^2^3 I “ ^3 I 
and from this, by the Law of Extensionals, the additional letter and 
suffix being taken, we derive 
I I ‘I <^2^3^4 I1 I ~ 1 II ^^2^4 II \ 
= I «i&2C3t^4 I X { - I %^^4 11 1 - 1 M 4 1} , 
which is a particular case of the theorem for a minor of the adjugate 
of the 4th order. 
Similarly, by taking the additional letter and suffix e^, we have 
1 I I a^^dj^e^ \ *1 h-^egd^er^ j - | a^c.^d-^Q^ ] 1 a^j^d^e^ 1*1 h.^c^d^e^ | 
= 1 ®l^2*^3d4^5 1 * { ~ I a^d^e^ | 'j &2^3^4^S I ~ I ^3^4^5 1 1 I } J 
which is a particular case of the theorem for a lower-ordered minor 
of the adjugate of the 5th order ; and of course the series may be 
continued indefinitely. 
