1894 - 95 .] Dr Muir on the Acljugate Determinant. 325 
3 . Of course a different series of interchanges of indices may be 
taken to effect the transformation of A5BJC4D3E2 into A2B4C4D5E3 ; 
and, if so, a different form of the cofactor of A will be obtained. 
For example, we may interchange the indices of A and E, the 
indices of B and C, and then the indices of D and E, the inter- 
mediate terms thus being 
A2B4C4D3E3, 
A2B4C4D3E3. 
This will give us 
A5B1C4D3E0 - AoB4CiD3E3= A3B1C4D3E2 - A2B1C4D3E, 
+ A2B4C4D3E5 — A2B4C4D3Eg 
+ A2B4C].D3Eg — A2B4C]DgE3 , 
= - B4C4D3-I A 2 Eg 1 - 1 - A 2 D 3 Eg-l B 4 C 4 1 -f A2B4Ci-| D 3 Eg ( , 
= { - BjC4l)3‘j + A2D3Eg*j ^2^365 1 + A 2 B 4 Cj‘| « 4 & 2^4 I } ‘ A 
The two results are not at variance, for the difference between 
them is 
- I -^ 2^5 I 1 ^1^4 I ’ 
i.e., D3 • { — 1*1 h^egd^ |* A -|- | h-^c^d^ j*l a^d^e^ 1 ' 
f.e., 0 . 
4 . To obtain the result in any given case it is not necessary to go 
through the whole process of proof. The cofactor may be written 
down with ease as soon as the so-called “ intermediate terms ” have 
been ascertained. Thus, if the cofactor of A in AJB2C3D4E3 
- A3B4C2D3 Ej be wanted, we write down the given terms with the 
“ intermediate terms ” placed in order between them — viz., 
A1B2C3D4E3, 
A5B4C3D2E1 , 
“^ 5 ^ 4 ^ 2 ^ 3^1 ’ 
and ask ourselves what factor is common to every consecutive two. 
The answer being B2C3D4 , A5C3Ej , A^B^E^ , the required cofactor 
is seen to be 
^2^3^4*1 ^2^3'^4 1 "b "^5^3®1*1 1 -^5^4^1*1 '^1^4% i > 
