subs. phr. (common).—An imaginary product in quest of which fools are sent: cf. STRAP-OIL, SQUAD UMBRELLA, &c.—GROSE (1785). Hence TO MILK THE PIGEON = to attempt impossibilities. [The idea is old: cf. Aristophanes in Aves (line 1672).]

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  1883.  J. H. FRERE, The Birds of Aristophanes, iii. p. 75.

        Here you shall domineer and rule the roast,
With splendour and opulence and PIGEON’S MILK.

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