def EpoEpp(s, P, Q):
    """ 
    we need E'/E'' need for Newton's method
    this expression can be evaluated analytically:
        E'  = (P'Q - Q'P)/Q**2 = N/Q**2
        E'' = (N'Q**2 - 2QQ'N) / Q**4
        
        E'/ E'' = NQ / (N'Q - 2Q'N)
        
        with N' = P''Q - Q''P
    """
    QQ = Q(s)
    PP = P(s)
    Pp = P.deriv(1)(s)
    Qp = Q.deriv(1)(s)
    Ppp = P.deriv(2)(s)
    Qpp = Q.deriv(2)(s)
    N  = Pp*QQ - Qp*PP
    Np = Ppp*QQ - Qpp*PP
    return (N*QQ) / (Np*QQ - 2*Qp*N)