  An $^{17}$O NMR measurement was performed on nonoriented polycrystalline
Na$_{0.35}$CoO$_2 \cdot y$H$_2$O with superconducting (SC) transition
temperature $T_\text{c} = 4.6$ K. A weak temperature dependence was observed in
the Knight shift at the O site ($^{17}K$). The spin part of $^{17}K$
($^{17}K_{\rm spin}$) is estimated from the plot of $^{17}K$ against bulk
susceptibility $\chi$. The $^{17}K_{\rm spin}$ decreases in the SC state,
indicative of the decrease in the in-plane component of the spin
susceptibility. The nuclear spin-lattice relaxation rate $1/T_1$ at the O site
$^{17}(1/T_1)$ shows a good scaling with $1/T_1$ at the Co site $^{59}(1/T_1)$.
This indicates that the spin fluctuations at the O site originate from the Co
spin dynamics. The relationships between $^{17}(1/T_1T)$ and $^{17}K_{\rm
spin}$ and between $^{17}(1/T_1T)$ and $^{59}(1/T_1T)$ show the development of
incommensurate fluctuations at {\bf q $\sim$ 0} other than {\bf q $=$ 0} below
30 K. A clear indication of ferromagnetic correlations at {\bf q $=$ 0} was not
observed from the present $^{17}$O-NMR studies.
