Let k be an algebraically closed field and let x1, x2, . . . , xn be indeterminates. Denote by Rn the ring k[[x1, x2, … , xn]] of power series in the xi With coefficients in the field k. Let and be two ideals in this ring. Then and will be said to be analytically equivalent if there is an automorphism T of Rn such that T() = . and will be called analytically equivalent under T.The above situation can be described geometrically as follows. The ideals and can be regarded as defining algebroid varieties V and V' in (x1, x2, … , xn)-space, and these varieties will be said to be analytically equivalent under T.The automorphism T can be expressed by means of equations of the form :where the determinant is not zero and the fi are power series of order not less than two (that is to say, containing terms of degree two or more only).