Green’s Dictionary of Slang

cony n.

also coney, conie, conny
[SE cony, a rabbit]

1. [mid-16C–mid-17C] a term of affection for a woman.

2. in sexual senses (cf. cunny n.) [the stereotyped sexuality of rabbits + the pubic hair is supposedly reminiscent of the rabbit’s tail + play on cunt n.].

(a) [mid-16C–1920s] (also cony-field, cony-skin) the vagina; thus coney-skin merchant, a pimp.

(b) [17C] a prostitute.

(c) [1920s] (US) attrib. use, pertaining to sexuality or promiscuity.

3. [late 16C–19C] (also cunny) a dupe, the victim of a confidence trick, of card-sharping etc.

In compounds

cony...

see also under cunny and its combs.

cony-burrow (n.) (also coney-burrow, connie-burrow, cony-berry, cony-borough, cunniborough, cunny-borough)

1. [late 16C–17C] a brothel.

2. [mid-17C–early 18C] the vagina; thus a woman considered as a sexual object; thus cony-burrow ferret, the penis.

conycatch/-catcher/-catching

see separate entries.

cony-dog (n.) [lit. a dog that catches rabbits]

[late 17C] one who assists a confidence trickster in cony-catching n.

cony-skin (n.)

see sense 3a above.