  In [AHL] Hardt, Lin and the author proved that the defect set of minimizers
of the modified Ericksen energy for nematic liquid crystals consists locally of
a finite union of isolated points and H\"older continuous curves with finitely
many crossings. In this article, we show that each H\"older continuous curve in
the defect set is of finite length. Hence, the defect set is locally
rectifiable. For the most part, the proof follows the work of De Lellis,
Marchese, Spadaro and Valtorta [DLMSV] on harmonic $\mathcal{Q}$-valued maps
closely. The blow-up analysis in [AHL] allows us to simplify the covering
arguments in [DLMSV] and locally estimate the length of line defects in a
geometric fashion.
