  Theoretical emission-line ratios involving Fe XI transitions in the 257-407 A
wavelength range are derived using fully relativistic calculations of radiative
rates and electron impact excitation cross sections. These are subsequently
compared with both long wavelength channel Extreme-Ultraviolet Imaging
Spectrometer (EIS) spectra from the Hinode satellite (covering 245-291 A), and
first-order observations (235-449 A) obtained by the Solar Extreme-ultraviolet
Research Telescope and Spectrograph (SERTS). The 266.39, 266.60 and 276.36 A
lines of Fe XI are detected in two EIS spectra, confirming earlier
identifications of these features, and 276.36 A is found to provide an electron
density diagnostic when ratioed against the 257.55 A transition. Agreement
between theory and observation is found to be generally good for the SERTS data
sets, with discrepancies normally being due to known line blends, while the
257.55 A feature is detected for the first time in SERTS spectra. The most
useful Fe XI electron density diagnostic is found to be the 308.54/352.67
intensity ratio, which varies by a factor of 8.4 between N_e = 10^8 and 10^11
cm^-3, while showing little temperature sensitivity. However, the 349.04/352.67
ratio potentially provides a superior diagnostic, as it involves lines which
are closer in wavelength, and varies by a factor of 14.7 between N_e = 10^8 and
10^11 cm^-3. Unfortunately, the 349.04 A line is relatively weak, and also
blended with the second-order Fe X 174.52 A feature, unless the first-order
instrument response is enhanced.
