But even if observations of this kind are continued throughout all the hours of darkness, we do not obtain complete circles, but only those portions of circles traced out on the sky between sunset and sunrise. If the night is twelve hours in length, we get half-circles on the plate; if it is eighteen hours long, we get circles that lack only one-quarter of being complete. In other words, we get a series of circular arcs, one corresponding to each close polar star. There are no fewer than sixteen stars near enough to the pole to come within the range of a photographic plate, and bright enough to cause measurable impressions upon the sensitive surface. The fact that the circular arcs are not complete circles does not in the least prevent our using them for ascertaining the position of their common centre; and that centre is the pole. Moreover, as the arcs are distributed at all sorts of distances from the pole and in all directions, corresponding to the accidental positions of the stars on the sky, we have a state of affairs extremely favorable to the accurate determination of the pole's place among the stars by means of microscopic measurements of the plate. It will be perceived that this method is extremely simple, and, therefore, likely to be successful; though its simplicity is slightly impaired by the phenomenon known to astronomers as "atmospheric refraction." The rays of light coming down to our telescopes from a distant star must pass through the earth's atmosphere before they reach us; and in passing thus from the nothingness of outer space into the denser material of the air, they are bent out of their straight course. The phenomenon is analogous to what we see when we push a stick down through the surface of still water; we notice that the stick appears to be bent at the point where it pierces the surface of the water; and in just the same way the rays of light are bent when they pierce into the air. Fortunately, the mathematical theory of this atmospheric bending of light is well understood, so that it is possible to remove the effects of refraction from our results by a process of calculation. In other words, we can transform our photographic measures into what they would have been if no such thing as atmospheric refraction existed. This having been done, all the arcs on the plate should be exactly circular, and their common centre should be the position of the pole among the stars on the night when the photograph was made.