Trippingly on the tongue
Hamlet:
Speak the speech, I pray you, as I pronounc'd it to you,
trippingly on the tongue; but if you mouth it, as many of our
players do, I had as lief the town-crier spoke my lines.
Having penned some lines for a play he hopes will pique the king's interest [see THE PLAY'S THE THING], Hamlet now assumes the role of director. He instructs the "players" (performers) on proper delivery, apparently fearing that they may smother his lines with the sort of bombast common on the stages of Shakespeare's London. Rather than "mouth" his speech—declaim it with the whole mouth—he would have them deliver it "trippingly on the tongue." "Trippingly" seems to mean "liltingly" or "nimbly," using only the delicate tongue rather than the full throat. This, Hamlet believes, will make for a more effective delivery, because it will be more like real speech.
Shakespeare coined the word "trippingly" in an earlier play, A Midsummer Night's Dream. There, Oberon the king of fairies commands all his elves and spirits to "Hop as light as bird from brier,/ And this ditty, after me,/ Sing, and dance it trippingly" (Act 5, scene 1). "Trippingly," as Oberon uses it, refers back to the light hopping of birds and is applied to the physical act of dancing. Hamlet attributes the adverb to the physical action of speech, but lends the word a more abstract meaning, imagining that the words themselves can "trip."