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## BRANCHING FRACTION AND MASS SPECTRUM CALCULATION 

The numbers of candidate events in the τ − → π $^{−}$π $^{+}$π $^{−}$ν$_{τ}$ , τ − → K $^{−}$π $^{+}$π $^{−}$ν$_{τ}$ , τ − → K $^{−}$K $^{+}$π $^{−}$ν$_{τ}$ , and τ − → K $^{−}$K $^{+}$K $^{−}$ν$_{τ}$ modes, after applying all selection criteria, are sum- marized in Table II. Possible backgrounds in these signal candidate samples include (a) cross-feed from the signal modes and (b) other processes such as τ − → π $^{−}$π $^{+}$π $^{−}$π $^{0}$ν$_{τ}$ , τ − → K 0 $_{S}$π $^{−}$ν$_{τ}$ , as well as continuum e $^{+}$e − → q ¯ q . The fractions of backgrounds coming from other processes are 5% to 12%, as summarized in Table II (fourth column) for each signal mode. The fifth column shows the fraction of the main background component as determined from MC. The mode τ − → π $^{−}$π $^{+}$π $^{−}$π $^{0}$ν$_{τ}$ dominates for the τ − → π $^{−}$π $^{+}$π $^{−}$ν$_{τ}$ and τ − → K $^{−}$π $^{+}$π $^{−}$ν$_{τ}$ modes, while the background from continuum e $^{+}$e − → q ¯ q is dominant for the τ − → K $^{−}$K $^{+}$π $^{−}$ν$_{τ}$ and τ − → K $^{−}$K $^{+}$K $^{−}$ν$_{τ}$ modes. 

To take into account the cross-feed between the decay channels, the true yield N true i ( i is the mode number and the values from 1 to 4 correspond to τ − → π $^{−}$π $^{+}$π $^{−}$ν$_{τ}$ , τ − → K $^{−}$π $^{+}$π $^{−}$ν$_{τ}$ , τ − → K $^{−}$K $^{+}$π $^{−}$ν$_{τ}$ , and τ − → K $^{−}$K $^{+}$K $^{−}$ν$_{τ}$ ), is obtained using the following equation: 

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