Tall and Thin
Originally the shape of the chimney lamp shade was designed such that they could fit onto devices known as chimney holders. A chimney holder is a small, circular metal piece that's a "cup" for the modern day light bulb or the old-fashioned gas flame. In fact, the chimney lamp shade wasn't thought of as a shade at all but rather a tall glass enclosure that housed the flame to prevent it from being blown out.
Nowadays chimney lamp shades may be used for anything from electrical to kerosene-powered light sources. They are almost always accompanied by the holder. The most important role played by the holder is supporting the shade "enclosure", as well as permitting a good flame that feeds on a lamp fuel source sitting right below the chimney holder. Seeking a replacement chimney shade should be easy given the easy profile and relatively non-ornate design. Just gather up your pieces (as long as they aren't too small) and bring them to a lamp shop or glass blowing shop. Skilled artisans will be able to produce a replica of the shade quickly.
Feeding the Flame
The chimney shade is constructed with a tall, thin shape which directs the flame of a gas-fired lamp toward the top opening, funneling post-combustion air and drawing in fresh air to continue fueling the process. A poorly fitting holder and shade will cause some sort of inadvertent air leakage that leads to inconsistent or unsteady flames. The height of the glass tube may affect how the degree of efficiency of the burn.
Because it houses a flame, a chimney shade should be made of high quality glass that doesn't deteriorate under heat. Pyrex is a good example of the kind of high quality robust glass material for use in these products. They can also be placed on chandliers to enclose light bulbs. But as with all electrical lighting, a chimney lamp shade serves little functionality except for decor. The parameters that define the shade are the height, the base diameter, and the widest diameter along the height axis. In general the heights of the chimney shade range from 4 inches to 12 inches.
The picture we supply above seems to be the most common shape, but there seems to be some diagreement in the community on how it should be shaped. Some stores call the wide cylindricals by the "chimney" name, whereas others strictly interpret the concept as a tall, thin shade.