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ADDITIONAL NOTE ON THE EESOLYABILITY OF THE MINOES 
OE A COMPOUND DETEEMINANT. 
By Sir Thqmas Muie, LL.D. 
(1) One way in which compound determinants arise naturally in the 
course of work is in connection with the simple matter of elimination in the 
case of a set of homogeneous linear equations when the elimination is effected 
by what we may call the method of instalments. Eor example, if our set 
of equations be 
a-^u + a.yV + a-.iv + a^x + a-y + a^z = 0 ^ 
h^u + hoV + h.vj + h^x -f- h-y + h(.^z = 0 
we may as a first step eliminate n and v, obtaining thus 20 equations of the 
type 
b.2 b.^w + b^^x + b-ij + b^z = 0, 
Co c.;(w + CjX + c^y 
c^z 
I. e. 
ctiboCj^ \ X -\- \ ciib^Cx, j y + 
ajboC^ 
z =:0; 
and then talcing four of this derived set eliminate the remaining unknowns, 
and so arrive at the desired resultant in a form of the type 
1 ^i^^H 1 
i ^hh'^A. i 
1 ^1^2^:) 1 
1 <^i^2C^3 1 
1 ci^^d^ 1 
1 ^ih^h i 
aib^clQ i 
1 ^1^2^3 1 
1 1 
1 ^1^2^.5 i 
dib^eQ 1 
i ^1^2/3 1 
1 Ci'lhfA 1 
1 ^^1^2/5 1 
a^b,f^ 1 
(2) Such a determinant is readily seen to be a minor of the compound of 
the determinant of the original set of equations ; for example, the resultant 
just reached is a 4-line minor of the 3rd compound of | n-fioC-^dj^erfQ \ . Now 
the true resultant being known to be the said determinant of the original 
set, it follows that the compound determinant obtained by the method of 
instalments must in general contain the true resultant as a factor. This 
simple observation is of considerable value in connection with the problem 
of the factorisation of the minors of a compound, the reason being equally 
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