  The mechanism of superconductivity in Pr$_2$Ba$_4$Cu$_7$O$_{15-\delta}$ is
studied using a quasi-one dimensional double chain model with appopriate
hopping integrals, on-site $U$, and off-site repulsion $V_1$. Applying the
fluctuation exchange method to this model and solving the Eliashberg equation,
we obtain the doping dependence of superconductivity that is consistent with
the experiments. The superconducting gap has an extended s-wave-like form,
which gives a temperature dependence of the spin lattice relaxation rate that
does not contradict with the experimental results.
