So here's what Carroll said (rather ungrammatically, I thought) about syzygies in his first "Lady" article.

"The puzzle consists in linking together two given words by a change of words, called links, such that every two consecutive words may contain a syzygy, and the longer the syzygies are, the more marks do they obtain. Thus, supposing the two given words were door and window, the following would be a chain linking them together.

DOOR
(oor)
POOREST
(res)
RESOUND
(und)
UNDO
(ndo)
WINDOW

He then follows with a list of Rules, and the challenge to make a CONSERVATIVE into a LIBERAL and get a VERDICT from a JURY.

The prize for the best links was a guinea.

By the way, I found out that the name of this game is derived from the Latin for "conjunction" and it is often used in astrology in connection with discussing coordination or alignment.

The illustration is not of a syzygy, but of another bit out of "The Lady"s small-ads. Those stockings clearly had plenty of room for growth.

Gloves and Fans?