  We present deep spectroscopic observations of the classical T Tauri stars DF
Tau and V4046 Sgr in order to better characterize two important sources of
far-ultraviolet continuum emission in protoplanetary disks. These new Hubble
Space Telescope-Cosmic Origins Spectrograph observations reveal a combination
of line and continuum emission from collisionally excited H2 and emission from
accretion shocks. H2 is the dominant emission in the 1400-1650 A band spectrum
of V4046 Sgr, while an accretion continuum contributes strongly across the
far-ultraviolet spectrum of DF Tau. We compare the spectrum of V4046 Sgr to
models of electron-impact induced H2 emission to constrain the physical
properties of the emitting region, after making corrections for attenuation
within the disk. We find reasonable agreement with the broad spectral
characteristics of the H2 model, implying N(H2) ~ 10^{18} cm^{-2}, T(H2) =
3000^{+1000}_{-500} K, and a characteristic electron energy in the range of ~
50 - 100 eV. We propose that self-absorption and hydrocarbons provide the
dominant attenuation for H2 line photons originating within the disk. For both
DF Tau and V4046 Sgr, we find that a linear fit to the far-UV data can
reproduce near-UV/optical accretion spectra. We discuss outstanding issues
concerning how these processes operate in protostellar/protoplanetary disks,
including the effective temperature and absolute strength of the radiation
field in low-mass protoplanetary environments. We find that the 912-2000A
continuum in low-mass systems has an effective temperature of ~10^{4} K with
fluxes 10^{5-7} times the interstellar level at 1 AU.
