William accompanies an older, senile man through the garden, and though he never speaks in the story, we still get a pretty good idea of his character. For instance, he's really patient—Woolf lets us know more than once. He wears "an expression of perhaps unnatural calm" (11) and, later, "the look of stoical patience" (14). Sounds like someone needs to liven up.
The poor guy is clearly exercising all the patience he can because he's dealing with a senile old man. He attempts to direct and guide him, but this is clearly a taxing effort for William. He is pretty cut-off from the other characters too, and we can only guess at what's going on behind that calm, patient expression.