  We use all available fields with deep NICMOS imaging to search for J dropouts
(H<28) at z~10. Our primary data set for this search were the two J+H NICMOS
parallel fields taken with the ACS HUDF. The 5 sigma limiting mags were 28.6 in
J and 28.5 in H. Several shallower fields were also used: J+H NICMOS frames
available over the HDF North, the HDF South NICMOS parallel, and the ACS HUDF.
The primary selection criterion was (J-H)>1.8. 11 such sources were found in
all search fields using this criterion. 8 of these were clearly ruled out as
credible z~10 sources, either as a result of detections (>2 sigma) blueward of
J or their colors redward of the break (H-K~1.5). The nature of the 3 remaining
sources could not be determined from the data. The number appears consistent
with the expected contamination from low-z interlopers. Analysis of the stacked
images for the 3 candidates also suggests contamination. Regardless of their
true redshifts, the actual number of z~10 sources must be <=3. To assess the
significance of these results, two lower redshift samples (a z~3.8 B-dropout
and z~6 i-dropout sample) were projected to z~8-12 using a (1+z)^{-1} size
scaling. They were added to the image frames, and the selection repeated,
giving 15.6 and 4.8 J-dropouts, respectively. This suggests that to the limit
of this probe (0.3 L*) there has been evolution from z~3.8 and possibly from
z~6. This is consistent with the strong evolution already noted at z~6 and
z~7.5 relative to z~3-4. Even assuming that 3 sources from this probe are at
z~10, the rest-frame continuum UV (~1500 A) luminosity density at z~10
(integrated down to 0.3 L*) is just 0.19_{-0.09}^{+0.13}x that at z~3.8 (or
0.19_{-0.10}^{+0.15}x including cosmic variance). However, if none of our
sources is at z~10, this ratio has a 1 sigma upper limit of 0.07. (abridged)
