  We study the theoretical minimum emittance for a non-uniform bending magnet
with the bending radius linearly ramped from the dipole center to its end. We
derive the expression for the minimum emittance as a function of the bending
angle and expand it into a power series with respect to a small angle. The
first term of the expansion gives the TME minimum emittance while the
high-order terms are responsible for its modification. On the contrary of the
vague and entangled closed-form solution, the coefficients of the power series
are simple and clearly indicate conditions and limitations for emittance
reduction below the TME value. With the help of analytical predictions we
design a lattice cell with longitudinally varied bends demonstrating the
emittance less than that for the TME structure of the same bending angle.
