1894-95.] Dr Muir on the Adjugate Determinant. 
323 
On a Theorem regarding the Difference between any 
Two Terms of the Adjugate Determinant. By 
Thomas Muir, LL.D. 
(Read December 3, 1894.) ' 
1. If we take the set of equations 
a^j?M + a^zw + agryw + a^xyz = 0 
\yzw + + h^xyiu + h^iyz = 0 
e^yzw + e^zw + c^xyw + e^yz = 0 
d^yzw + d^xzio + d^xyw + d^yz -• 0 J ^ 
the eliminant of which is 
\a^h^e^d^, 
and from the 1st, 2nd, and 3rd eliminate xyw and xyz, from the 
2nd, 3rd, and 4th eliminate xyz and yzio^ from the 3rd, 4th, and 1st 
eliminate yziu and xzw, and from the 4th, 1st, and 2nd eliminate 
xzw and xyw, we obtain the set 
1^1 ^3 1^2 ^3 ^ 4 !^ ~ ^ 1 
\b^e^d^\z + \b^e^d^\y = 0 
\cgi-^a^io + \c^d-^^a^z = 0 
\d^a^h^x + \d^a2b^w = 0 J , 
the eliminant of which is clearly 
1^2^3^4H^3^4^lH^4^1^2l’l ^1^2^31 I ^1^3^4H ^2^4^1 H%^1^2H ^4^2^31 ’ 
We are thus led to conclude that the first form of the eliminant is 
a factor of the second. 
The attempt to prove this, and the investigation of the form of 
the quotient, have brought to light a new theorem in determinants, 
which promises to he of some considerable importance. 
2. The theorem is to the effect that the difference between any 
two terms of the adtjugate determinant is divisible by the original 
determinant. 
Taking, for shortness’ sake, the determinant of the 5th order 
\af)^c^d^e^\ or A 
IA 1 B 2 C 3 D 4 EK!, 
with its adjugate 
