The **Adjoint** or Reciprocal Determinant arises from A = (a11a22a33 ...a nn) by substituting for each element A ik the corresponding minor Aik so as to form D = (A 11 A 22 A 33 ï¿½ï¿½ï¿½ A nn).

Its value is therefore O n and we have the identity D.0 = A n or D It can now be proved that the first minor of the **adjoint** determinant, say B rs is equal to An-2aï¿½.

The **adjoint** determinant is the (n - I) th power of the original determinant.

The **adjoint** determinant will be seen subsequently to present itself in the theory of linear equations and in the theory of linear transformation.

It is easy to see that the **adjoint** determinant is also 'symmetrical, viz.

For the second order we may take Ob - I - A, 1 1 +A2, and the **adjoint** determinant is the same; hence (1 +A2)x1 = (1-A 2)X 1 + 2AX2, (l +A 2)x 2 = - 2AX1 +(1 - A2)X2.

Similarly, for the order 3, we take 1 v Ab= -v 1 A =1 +x2 + 1, 2 + ï¿½ - A 1 and the **adjoint** is 1+A v +Aï¿½ -ï¿½ +Av -v +Aï¿½ 1+11 2 A +/-tv pt+AvA +ï¿½v 1 +1,2 leading to the orthogonal substitution Abx1 = (1 +A 2 - / 22 - v 2) X l +2(v+Aï¿½)X2 +2(/1 +Av)X3 1bx2 = 2(Aï¿½ - v)Xl+(1 +ï¿½2 - A2 - v2)X2 / +2(Fiv+A)X3 Abx3 = 2(Av +ï¿½)X1 +2(/lv-A)X2+(1+v2-A2- (12)X3.

The same year he obtained the position of **adjoint** to Baudon, one of the farmers-general of the revenue, subsequently becoming a full titular member of the body.