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"to let water in or out" [Johnson], late 14c., from Middle Dutch leken "to drip, to leak," or from Old Norse leka, both of them related to Old English leccan "to moisten" (which did not survive into Middle English), all from Proto-Germanic *lek- "deficiency" (cf. Old High German lecchen "to become dry," German lechzen "to be parched with thirst"), from PIE root *leg- "to dribble, trickle." The figurative meaning "come to be known in spite of efforts at concealment" dates from at least 1832; transitive sense first recorded 1859. Related: Leaked; leaking.
late 15c., from leak (v.) or Old Norse cognate leki. Sense of "revelation of secret information" is from 1950. Meaning "act of urination" is attested from 1934 ("Tropic of Cancer"); but the verb meaning "to piss" is from 1590s: "Why, you will allow vs ne're a Iourden, and then we leake in your Chimney." ["I Hen. IV," II.i.22]
With a qualifier, one of a class of resource-management bugs that occur when resources are not freed properly after operations on them are finished, so they effectively disappear (leak out). This leads to eventual exhaustion as new allocation requests come in.
One might refer to, say, a "window handle leak" in a window system.
See memory leak, fd leak.