  An NQR experiment revealed superconductivity of
Pr$_2$Ba$_4$Cu$_7$O$_{15-\delta}$ (Pr247) to be realized on CuO double chain
layers and suggests possibility of novel one-dimensional(1D) superconductivity.
To clarify the nature of the 1D superconductivity, we calculate the band
dispersions of Pr247 by using the generalized gradient approximation(GGA). It
indicates that Fermi surface of CuO double chains is well described to the
electronic structure of a quasi-1D system.
  Assuming the zigzag Hubbard chain model to be an effective model of the
system, we derive tight binding parameters of the model from a fit to the
result of GGA. Based on the Tomonaga-Luttinger liquid theory, we estimate
transition temperature ($T_c$) of the quasi-1D zigzag Hubbard model from the
calculated value of the Luttinger liquid parameter $K_{\rho}$. The result of
$T_c$ is consistent with that of experiments in Pr247 and it suggests that the
mechanism of the superconductivity is well understood within the concept of the
Tomonaga-Luttinger liquid.
