  The decay $\eta \to 3 \pi$ proceeds exclusively through isospin violating
operators and is therefore an excellent probe to examine the strength of
isospin breaking in the strong interaction. The latter can be expressed through
the quark mass ratio $Q^2 = (m_s^2 - \hat{m}^2)/(m_d^2 - m_u^2)$. The main
object of this thesis is to analyse the decay $\eta \to 3 \pi$ using dispersion
relations in order to extract $Q$ as well as other physical quantities. The
dispersion relations are a set of integral equations that lead to a
representation of the decay amplitude in terms of four unknown subtraction
constants. We apply two different methods to determine these. Following the
procedure of Anisovich and Leutwyler, we match our dispersive representation to
the one-loop result from chiral perturbation theory, thus updating their old
analysis. In addition, we also make use of the recent experimental
determination of the Dalitz distribution by the KLOE collaboration to obtain
information on the subtraction constants. In this way we find $Q =
21.31^{+0.59}_{-0.50}$.
