Wanting a break from her husband's nonsensical ravings, Lucrezia Smith steps away from him in the park and tries to admire the fountain. But she can't, because she compares it to a fountain in her home city of Milan, and it pales in comparison. She says, "'For you should see the Milan gardens,'" but there's nobody around to hear her. The narrative then uses a simile that accentuates her feeling of loneliness:
There was nobody. Her words faded. So a rocket fades. Its sparks, having grazed their way into the night, surrender to it, dark descends, pours over the outlines of houses and towers; [...].
This simile compares her unheard words to a rocket that has faded in the sky, most likely referring to fireworks and the way they "fade" into darkness after briefly igniting. In turn, her words are presented as incandescent and bold, suggesting that simply speaking aloud in this moment is a very emotional thing for her to do—she needs, it seems, to express herself, but she has nobody in her life who will truly listen.
This simile also subtly becomes more of a metaphor, as the narrator expands on the comparison and talks about the "sparks" of a rocket "surrender[ing]" to darkness, which "pours over the outlines of houses and towers." The night itself (and darkness) thus takes on some metaphorical significance, since the mention of "houses and towers" hints at a certain widespread feeling of unacknowledged sorrow. The city is full, in other words, of private woes. The idea here is that there are many private sentiments that go unheard or unacknowledged, though this doesn't mean the feelings don't still exist—like explosions in the sky, private feelings ignite and expand within people, but then they're swallowed by the world at large, which, like the vast and all-encompassing darkness of the night's sky, makes people like Lucrezia feel all the more isolated and alone.
After Richard Dalloway and Hugh Whitbread leave Lady Bruton's luncheon, she lies down to take a nap, and the narrative metaphorically focuses on what happens to their relational connection as she sinks into sleep and the two men recede into the distance:
And they went further and further from her, being attached to her by a thin thread (since they had lunched with her) which would stretch and stretch, get thinner and thinner as they walked across London
This metaphor suggests that Richard and Hugh are "attached" to Lady Bruton by a "thin thread" that becomes progressively "thinner"—or weaker—the farther they get from her house. This is an illustration of the way certain experiences feel like they slowly fade away after ending, lingering in the minds of the participants but progressively becoming less and less immediate or relevant. What's more, the idea of being "attached" to somebody after having lunch with them aligns with the novel's interest in the characters' social associations. According to this metaphor, sharing lunch with people in this social context (British upper-class society in the 1920s) is like forming an attachment to them.
To that end, the passage goes on to use a simile that cements the idea of relational connections as actual physical attachments:
[...] as if one’s friends were attached to one’s body, after lunching with them, by a thin thread, [...]
The passage also uses some auditory imagery, saying that this "thin thread" slowly becomes "hazy with the sound of bells, striking the hour or ringing to service"—a description that emphasizes the fading feeling of connection while also hinting at the "hazy," satisfyingly faint feeling of nodding off to sleep. After all, this entire image of Richard and Hugh being connected to Lady Bruton by a thread ultimately comes from Lady Bruton's half-asleep mind, and the passage begins to reflect the strange, highly figurative ways that people tend to think when they're drifting out of consciousness. In keeping with this, there's yet another simile that reminds readers that all of these thoughts are playing out in Lady Bruton's sleepy mind:
[...] as a single spider's thread is blotted with rain-drops, and burdened, sags down. So she slept.
Basically, the whole progression vividly puts readers into the feeling of falling asleep, a state in which it's possible to continue thinking about something that has just happened before inevitably losing hold of the entire thought and "sag[ging] down" into a full slumber.
When Richard Dalloway leaves Lady Bruton's house after having lunch with her and Hugh Whitbread, Lady Bruton begins to fall asleep and metaphorically thinks about how both Richard and Hugh are connected to her by a "thin thread" as they move through London. As this metaphor develops, the "thin thread" (which is also compared to a spider's thread) eventually breaks when Lady Bruton falls fully asleep. At this point, the narrative focus shifts to Richard's perspective, picking up the thread (pun intended) by using a simile comparing Richard's attention to a loose, disconnected spider thread wavering in the air until it attaches to something new:
And as a single spider’s thread after wavering here and there attaches itself to the point of a leaf, so Richard’s mind, recovering from its lethargy, set now on his wife, Clarissa, [...].
This simile illustrates the wandering nature of human thought by suggesting that Richard's mind needs to latch onto something new. But the simile also creates a sense of continuity in the novel, allowing the narrative to seamlessly transition from one character to another without creating a disjointed feeling. Richard and Lady Bruton are connected by a metaphorical "thread" after having lunch together, but this eventually breaks when Lady Bruton falls asleep, and the idea that Richard's mind is now like "a single spider's thread" that is "wavering" in the air suggests that there was a certain shared sense of connection between him and Lady Bruton—now, though, it has been long enough since he left her house that their lunch is no longer at the forefront of his mind. In the same way that she has nodded off and stopped thinking about Richard and Hugh, then, Richard's thoughts have now moved on, and he has latched onto the thought of his wife, Clarissa. This simile therefore provides insight into the patterns of human thought while also creating a sense of continuity as the novel transitions to focus on a new character.
As Miss Kilman complains about people like Clarissa Dalloway to Elizabeth, the narrative uses a simile to describe Elizabeth's reaction. The simile sheds light on what it's like for Elizabeth to sit there and listen to her tutor say such bitter things about people like her own mother (even if Miss Kilman never actually mentions Clarissa's name):
Like some dumb creature who has been brought up to a gate for an unknown purpose, and stands there longing to gallop away, Elizabeth Dalloway sat silent. Was Miss Kilman going to say anything more?
The novel has already established that Clarissa and Miss Kilman are foils for one another—whereas Clarissa is a well-off atheist who throws parties and has an active social life, Miss Kilman is a devoutly religious tutor who is highly scornful of Clarissa's lifestyle, which she thinks is silly, unserious, and overly privileged. And yet, the source of Miss Kilman's bitterness seems rooted in jealousy, as she goes on at length about how she doesn't care whether or not she's invited to parties like the one Clarissa is throwing: a clear sign that she's jealous of Clarissa and wants a life like hers, even if she also supposedly disapproves of her.
On the whole, Elizabeth seems to have a special connection with Miss Kilman, but this passage implies that she's well aware of the tension between her tutor and her mother. The simile comparing Elizabeth to "some dumb creature" suggests that, although Miss Kilman hasn't explicitly singled out her mother, Elizabeth knows exactly what her tutor is talking about when she speaks disparagingly about throwing and attending parties. And yet, Elizabeth remains silent like a "dumb creature"—that is, an animal incapable of speech that has been led somewhere for an "unknown purpose" and longs to "gallop away." Although Elizabeth has a good relationship with Miss Kilman, then, it's obvious that she doesn't like being forced to listen as her tutor blatantly insults her family.
