From circle to circle correspondence in mapping theory of the linear function Z(ah+b)/(ch+d), it is known that Z also describes a circle when its variable h describes a circle or a straight line. From this point of view, the linear functional expression of bio-electrical impedance having a circular locus was discussed in detail. As a result, it became clear that three types h0, h1 and h2 exist of the variable h and the characteristics of bio-impedance expressions for each of h0, h1 and h2 are follows. For the expression by h0: (1) A polarization impedance having a constant phase angle appears. (2) The most original equation of bio-impedance can be proposed. (3) The linear or nonlinear bio-impedance can be deduced from this original equation. (4) The equivalent circuits satisfying Cole-Cole's circular arc are of two types. For the expression by h1 and h2: (1) These expression correspond to the expression using the combination in group theory of the linear function. (2) The circuit elements of equivalent circuits can not be uniquely determined. (3) Taking account of combination, countless equivalent circuits satisfying Cole-Cole's circular arc exist.

The principle of measuring with random vibration which is well known in industrial vibration testing is applied to measuring the mechanical properties of bio-materials. In this method it is possible to measure in short time (about 10 seconds) and measuring values are not disturbed by noise and the region of the body vibrated is not limited.