First, it should be noted that the "Transit Circle" refers to just one Transit in particular, and that there is no other built instrument in the world with that name. The term should only refer to the original Airy Transit Instrument, built by George Biddell Airy in 1850 for the Royal Greenwich Observatory which became the Greenwich Meridian in 1884.

Airy´s Transit Circle.
http://www.ast.cam.ac.uk/~ipswich/History/Ransomes/Airy_Transit_2.jpg

During the International Meridian Conference in 1884, several countries voted that the cross hairs of the Airy Transit Instrument should define the Initial Meridian of the world (0°).

The instrument observes the "times of passage" across the Meridian of Greenwich of "clock stars" which were standard stars whose positions in the sky are well known to astronomers, thus the term "transit of stars". These times are used to find the small deviations from perfect time-keeping to which the standard clock may be subject. There is also an elongated opening on top of the observatory roof, the Meridian Opening, which opens in sections for the telescope measure viewing.

This standard clock, located in a contiguous chronograph room, keeps "sideral time" and it is by comparison of this with the mean solar clock that the mean solar clock is corrected. The Transit can turn only in the meridian time; it also has a finely graduated silver band, on which a giant declination circle can be read by six equidistant microscopes, therefore the name "Transit Circle".

The circle is divided into 4320 divisions, each about 1/20 of an inch apart, with twelve divisions to every degree (12 x 360 = 4320), and each division equals five minutes of arc. Using all six microscopes, and taking their mean, the observer can read up to 0.01” In practice, instrumental and personal errors reduced this figure. The giant declination circle is positioned to one side of the main telescope body which enables the instrument to be correctly set for viewing a clock star. Such was the accuracy of this instrument, and the volume of excellent measurements that it produced, that Greenwich was chosen as the basis for the Meridian Line system.

According to "www.ast.cam.ac.uk":

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The telescope was mounted on two piers, one of granite and the other of Portland stone, via two 6" diameter pivots. The whole assembly ran on an axis of 6 feet overall and weighed in at 1,890 lbs. The roof shutter mechanisms are a classic example of Airy's eye for minute detail, just before the shutters reach the fully open or closed positions a hammer strikes against a bell as a warning to the person who turns the winch that he must move it gently.
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According to "labbey.com":

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The circle, along with its accompanying chronograph, was in service for over a century. When the Royal Observatory relocated to Herstmonceaux in the 1960's the circle's work was assigned to more modern equipment there. But Airy's transit circle remained in place, because the old observatory had become an astronomical museum.
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Illustration on how the Trasit Circle was used.
http://www.ast.cam.ac.uk/~ipswich/History/Ransomes/Airy_Transit_1.jpg
Source(s):
http://www.ast.cam.ac.uk/.../Ransomes/Ransomes.html

http://labbey.com/Telescopes/Airy.html

http://atschool.eduweb.co.uk/bookman/library/ROG/ROG07.HTM

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