This is the 7911th most frequent Japanese word.
垂らし
"垂らし" (tarashi) refers to "dripping" or "letting something hang or drip downward."
In this sentence, '垂らし' refers to the action of letting something drip or fall, metaphorically describing tears flowing down.
彼は涙を垂らしながら話していた。
He was speaking while letting tears drip.
Here, '垂らし' describes something hanging or dangling, emphasizing the suspension of the rope from the branch.
木の枝から垂らしたロープを使って登る。
Using the rope dangling from the tree branch to climb.
In this context, '垂らし' is used to describe the decorative pouring of a liquid, specifically sauce in this case.