We present measurements of the far-infrared (FIR) fine structure lines [S III] (33 mircon), [Si II] (35 mircon), [O III] (52, 88 mircon), [O I] (63 mircon), and [C II] (158 mircon) and the adjacent continua at these wavelengths in a strip crossing the E2 and W1 thermal radio filaments in the Galactic Center ``Arc''. The properties of the two filaments are similar: the line and continuum fluxes peak in the vicinity of the radio ridges, and the contrast of the ridge fluxes relative to the off-ridge fluxes is greatest for the FIR continuum and doubly ionized lines, lower for the single-dish radio measurements (Sofue \etal. 1986) and smallest for the low excitation lines. The near spatial coincidence of the line and continuum emission maxima with the radio filaments demonstrates that the same excitation source must be responsible for both the line and the continuum emission. The peak FIR luminosity of ~ 2.5*10^5 L_sun. per 30'' beam and the association of [O III] intensity with the filaments pose difficulties for shock and MHD models. Excitation by a string of stars is also unlikely, but photoionization of molecular cloud edges by a quasi-uniform distribution of stars is the most plausible excitation source proposed. The peak ionized line intensity is shifted slightly eastward relative to the FIR continuum emission. The deduced excitation, as measured by the [O III] to [S III] line flux ratios, also peaks slightly east of the radio ridges. Together, these two measurements suggest a substantial contribution from an ionizing source to the east. The strengths of the [O I] and [C II] lines are consistent with their production in photodissociated molecular material adjacent to the photoionized gas, but the FIR luminosity and the [Si II] emission appear to originate predominantly in or very near to the ionized gas.