Lines 15–25 of T. S. Eliot's “The Love Song of J. Alfred Prufrock” contain a delightful metaphor that compares the yellow fog and smoke of a large city to a cat. The poet begins by asserting that the “yellow fog” “rubs its back” up against the windowpanes. We can picture a yellow cat arching its back as it rubs against the side of a window, marking the building with its scent. The “yellow smoke,” the poet continues, “rubs its muzzle” on those same windows. We cannot help but think of a purring cat turning around again and again, scraping its jaw against the windows. This metaphor gives us such a vivid image of how the fog and smoke behave that even if we have never seen such things ourselves, we can easily picture them.

The poet then says that the fog/smoke licks its tongue “into the corners of the evening” and lingers near standing water in the drains. Once again, we imagine a feline trying to lick up every bit of milk or sniffing around to see what is interesting. The fog allows soot to fall upon its back (although we can envision a cat shaking it off) and then slips alongside a terrace, “slides along the street,” and leaps up into the night. This, too, is typical catlike behavior. Cats often glide along silently almost invisible and then suddenly launch themselves into the air. The fog, propelled by air currents, does much the same, according to the poet. Finally, like a tired cat, the fog curls up around a house and falls asleep.

We can see, then, at least nine details that the poet uses to compare the yellow fog or smoke to a roaming feline, and this metaphor helps us easily picture the varied motion of the fog or smoke as we compare it to the antics of a typical cat.

Lines 15–25 focus on a single, extended metaphor:

The yellow fog that rubs its back upon the window-panes,

The yellow smoke that rubs its muzzle on the window-panes,

Licked its tongue into the corners of the evening,

Lingered upon the pools that stand in drains,

Let fall upon its back the soot that falls from chimneys,

Slipped by the terrace, made a sudden leap,

And seeing that it was a soft October night,

Curled once about the house, and fell asleep.

And indeed there will be time

For the yellow smoke that slides along the street,

Rubbing its back upon the window-panes

As was mentioned in the previous post, T.S. Eliot uses an extended metaphor that compares the fog to an alley cat throughout the poem "The Love Song of J. Alfred Prufrock." Eliot associates the movements of the fog with those of an alley cat, and indirectly compares the two because he does not use the word "cat" throughout the extended metaphor. Similar to fog, cats are silent, mysterious creatures that are able to inhabit places that other animals cannot. Eliot provides details which reveal that an alley cat is being indirectly compared to the fog by depicting how the fog

"rubs its back...rubs its muzzle...Licked its tongue...Lingered upon the pool...made a sudden leap...Curled once about the house" (15-25).

The movements of the fog mirror those of an alley cat. Cats often rub their backs, lick their tongues, and linger around drains. References to a muzzle, and the creature's leaping around the house, further reinforce the idea that Eliot is comparing fog to an alley cat. It is unusual to compare an animal to a weather element, and this displays T.S. Eliot's expertise in creating an atmosphere. 

In these lines, Eliot indirectly compares the fog that covers the streets at night to a cat; it is an indirect comparison because he never directly states the word cat.  However, if you look close at the details and put them all together like a puzzle, the clues are pretty clear.  For example, he uses the following words and phrases to describe the fog:  "rubs its back upon the window-panes," "rubs its muzzle," "licked its tongue," "slipped by," "made a sudden leap," and "curled once about the house, and fell asleep."  All of these details, put together, paint a pretty clear picture of cats.  They rub their backs and muzzles on things, slip by as they walk, and before going to sleep, often do a circle or two before settling in.

So, the fog is compared to a cat, using detailed word-choices and clues.  It is a unique and apt description, one that gives the fog a clear personality and life to it, instead of having it be an inanimate object.  I hope that helped; good luck!