Abstract
SOME thirty or forty years ago geometricians were much interested in the endless band of paper to which one half twist had been given before joining the ends. This gave the figure having only one surface and one edge. At that time those who studied this figure were so obsessed with the consequence of cutting down the middle line of such a band or of a band with two or more half twists that I believe no one noticed the result which I wish now to describe. It is the doubling up in a proper manner of an endless band to which four half twists have been given so as to produce the endless band first described but of double thickness. The first band is shown at A, as an endless belt connecting two crossed shafts for which, as is well known, it is exactly fitted. B shows the band with the four half twists all on one side in the form of two complete loops, and to be uniform with the other it is shown as an open band connecting two parallel shafts. The only object of putting in the shafts and pulleys is to assist the perspective. They are not wanted in making the experiment. Now if B, which appears sufficiently uncompromising, is folded up properly it becomes A but of double thickness.
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