326 
Proceedings of Royal Soeiety of Edinburgh. [sess. 
the minus sign in the last term being due to the fact that the last 
of the three common factors occurs first along with CgDg instead 
of C2D3. 
5. In the case where the original determinant is of the 3rd order, 
the cofactor takes a peculiar form which is worth noting. For 
example, by the theorem we have 
~ ” ^1^1 ~ -^2^2} * ^ * 
But 
— — ^2^2 ~ ~ 1^2^31 “■ ^2 1^1^31 ’ 
= - ^^ 1 « 2^3 + ^ 1 ^ 3^2 - ^ 2 ^ 1^'3 + «^ 2 ^ 3^1 ? 
= af>2^i ~ • 
And so in other cases : Consequently, in the case of determinants of 
the ?>rd order, the difference of any two terms of the adjugate is 
divisible by the original determinant, the quotient being the difference 
of the corresponding terms of the original determinant. 
6. This special theorem is easily seen to be its own Comple- 
mentary. There is thus suggested a different mode of proving the 
general theorem — viz., by means of the Law of Complementaries.* 
Taking the case of the difference A5BJC4D3E2 - A2B4C1D5E3 
above dealt with, we begin with the corresponding difference in 
the original determinant, and proceed as follows : — 
+ af-^cfl^e^ - af^eff^^e^ 
+ ^2^4^1^3% ~ ^2^4^1^5% > 
= - b^cff^f ^265 I + %^3%-| ^1^4 I + ^2^4^l1 I; 
whence by the Law of Complementaries 
A5biC4D3E2 — A2B4CiDgE3 = A ■ { — B 4 C 4 D 3 * 1 \ -f A2l)3E5‘l \ + A2B4C4*| }, 
as was to be shown. 
7. The theorem applies, however, not only to the adjugate 
itself, but to any minor of the adjugate ; that is to say, the difference 
of any two terms of any minor of the adjugate determinant is a 
multiple of the original determinant. 
* Trans. Boy. Soc. Edin., xxx. pp. 1-4. 
