  The paper is an essentially extended version of the work math.CA/0601371,
supplemented with an application. We present new results in the theory of
classical $\theta$-functions of Jacobi and $\sigma$-functions of Weierstrass:
ordinary differential equations and series expansions. We also give the
extension of canonical $\theta$-functions and consider an application to the
sixth Painlev\'e equation (P6). Picard--Hitchin's general solution of P6 is
represented explicitly in a form of logarithmic derivative of a corresponding
$\tau$-function (Painlev\'e's form).
