=Lens Testing and Tolerances for Aerial Work.=—Simple and rapid comparative tests of lenses may be made by photographing a _test chart_, consisting of a large flat surface on which are drawn various combinations of geometrical figures—lines, squares, circles, etc.—calculated to show up any failures of defining power. For testing aerial lenses the chart should be as large as possible, so that it may be photographed at a distance great enough for the performance of the lens to be truly representative of its behavior on an object at infinite distance. This means in practice a chart of 4 or 5 meters side, to be photographed at a distance 20 to 30 times the focal length of the lens. [Illustration: FIG. 15.—Photograph of a lens testing chart, showing failure in defining power outside area for which the lens is calculated.] A typical photograph of such a chart is shown in Fig. 15. It reveals at a glance the more conspicuous lens errors. At the sides and corners the concentric circles show the lens's astigmatism, by the clear definition of the lines radial to the center of the field and their blurring in the tangential direction. The falling off in illumination with increasing distance from the center is also exhibited; and the blurring of all detail outside the rectangle for which the lens was calculated shows that spherical, chromatic, and other aberrations have become prohibitively large. But the only complete test of a lens is the quantitative measurement of errors made on an optical bench. A point source of light, which may at will be made of any color of the spectrum, is used as the object and its image formed by the lens in a position where it can be accurately measured for location, size, and shape by a microscope. A chart giving the results of such a test is shown in Fig. 16. In the upper left-hand corner is shown the position of the focus for the different colors of the spectrum. Below this is recorded the lateral chromatism at 21 degrees, in terms of the difference in focus for a red and a blue ray. Below this again comes the distortion, or shift of the image from its proper position, for various angles (plotted at the extreme right) from the lens axis. To the right of this is the image size, at each angle, and finally, to the right of the diagram, are plotted the distances of the two astigmatic foci from the focal plane, together with the mean of the two foci, which practically determines the shape of the field.