Flare loops form an integral part of eruptive events, being detected in the range of temperatures from X-rays down to cool chromospheric-like plasmas. While the hot loops are routinely observed by the Solar Dynamics Observatorys Atmospheric Imaging Assembly (SDO/AIA), cool loops seen off-limb are rare. In this paper we employ unique observations of the SOL2017-09-10T16:06 X8.2-class flare which produced an extended arcade of loops. The Swedish 1-m Solar Telescope (SST) made a series of spectral images of the cool off-limb loops in the Ca II 8542 r{A} and the hydrogen H$beta$ lines. Our focus is on the loop apices. Non-LTE spectral inversion is achieved through the construction of extended grids of models covering a realistic range of plasma parameters. The Multilevel Accelerated Lambda Iterations (MALI) code solves the non-LTE radiative-transfer problem in a 1D externally-illuminated slab, approximating the studied loop segment. Inversion of the Ca II 8542 r{A} and H$beta$ lines yields two similar solutions, both indicating high electron densities around $2 times 10^{12}$ cm$^{-3}$ and relatively large microturbulence around 25 kms$^{-1}$. These are in reasonable agreement with other independent studies of the same or similar events. In particular, the high electron densities in the range $10^{12} - 10^{13}$ cm$^{-3}$ are consistent with those derived from the SDOs Helioseismic and Magnetic Imager white-light observations. The presence of such high densities in solar eruptive flares supports the loop interpretation of the optical continuum emission of stars which manifest superflares.
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