Basing on the method of Büttiker et al., the analytic expressions of low temperature transmission amplitude for two kinds of normal metal ring with non-symmetric branches are obtained. The interesting resonances in the transmission probability for such structures are discussed. The results derived from the general analytic expression in two special limits are similar to those in reference 11 . In addition, we point out that there are some mistakes in equations (4.24) and (4.39) of reference 11 .