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OcTO3ER 26, 1883.] THE PHOTOGRAPHIC NEWS. 683 add half the thickness of the lens from the theoretical focal length measured from the back surface of the lens. [In Lesson IV., it was stated that with a lens of no thickness— 1=(1-1)(-I) where f is the principal focus, and r1 and r the radii of the spherical surfaces (reckoned negative, if on the opposite side of the lens to the incident ray), u the index of refrac tion. It can also be shown, where the thickness of a lens is small, that if f be the practical focal distance of the lens, the thickness of the lens being t — /=/+ () , which shows that the thickness of a lens alters the focal And it also shows that it is very material as to which surface the rays fall on, as only r' is involved in the expression. Example.—Let us take as an example a lens whose radii of curvature are + 6 inches and co , or a plano-convex lens, and let the light in one case fall on the plane surface, and 1=15 Then by the theoretical formula 7=(1-1)(-1) =(1 -1)*5=-1 Or the theoretical focal length is 12 inches on the opposite side of the lens to the incident ray. Now let the thickness in the centre be *25 inch, then f‘=12+ (3) X -.=12+0=- 12 inches. Or the focal distance is not altered, the focus being measured from exterior or concave surface of the lens. Next let the ray of light fall on the convex side, the theo retical focal length remains the same, but /‘=12+()x1X1 12+166=12-166 So the focal distance is lengthened by 166 inches, reckoning the focus from the exterior or flat surface of the lens. If reckoned from the optical centre which lies on the convex side of the lens (see Lesson III.), the length of focus would be 12166 +25= 12'416 inch, '25 being the thickness of the lens. Distortion caused by the curvature and thickness of a lens.— We have shown how astigmatism and distortion run to gether in a single lens. There is also increased distortion produced by the fact that a lens must have thickness, and cannot be treated altogether as a lens having none. The following example will illustrate the distortions due to thickness and to the curvature. Let us take a plano convex lens L,* fig. 36. At right angles to the axis, and at some distance from it, place a square figure, of which the central point is K, lying in the axis. Let us also place a diaphragm, 0, in the axis, and consider the effect of the lens on two points a 1 and b', a 1 being in the same hori zontal plane as the centre of the axis. The rays of light from a' and b', passing through 0,t will strike the lens at a and b respectively, and let f b and d a c be the section of the lens at those points. If we were to look down on the lens as in a plan, we should see, as in fig. 37, the rays b' 0 b, and a' O a, as one straight line, O a b, and the sections of the lens as shown. The ray a' O a, b' O b, fig. 36, would stop at a and b in * The lens is supposed to be seen in perspective, the convex face next the diaphragm. + We may take these central rays as approximately representing the pencil of rays passing through O. If O bo not very small, the result would be complicated by having to consider the astigmatism. fig. 37. At these two points, fig. 37, draw M M and N N tangent to the circles. Then, as far as the two rays are concerned, we have two prisms, through which the rays JFii). 36. O a and 0 b are refracted. Now the prism of which M M and d c forms two sides has a less vertical (i. e., refracting) angle than the prism of which N N and f N form the sides 5 therefore, the ray O a will be less refracted towards the Fig. 37. perpendicular than the ray O b, and if S be the screen on which a focus is obtained, the image of the point b' (from which the ray O b proceeds) will usually be nearer to the axis horizontally than the image of the point a' (from which the ray O a proceeds), and be respectively at B and A. In this case it is clear that the distance of B from the axis is dependent on the distance of P from Q, which is itself partly dependent on the thickness. If we trace the course of the rays in a vertical plane, we shall still have the same difference in the positions of the images of a' and b', and the image of the square will have the form given in fig. 32. If the diaphragm were placed behind the lens, we should get distortion, as in fig. 31. Let us take a numerical example. Let the distance from O to a is 2-5 inches, the radius of the spherical surface be 7'7 inches, the thickness of the lens be -inch, and the index of refraction 1-6. Let two rays, passing through the stop, come from two distant points, each making an angle of 30° with the axis in the horizontal plane, one lying in this plane, and the other making the same angle in the verti cal plane. These two rays will correspond with the points a 1 and b 1 , fig. 36. If the path of these two rays be traced, it will be found that the ray corresponding to a’ O will emerge from the flat side of the lens in the horizontal plane at an angle of 21% 29 , whilst the ray corresponding to b' O will emerge at an angle of 20° 37'. Now the direction in which there would be no distortion would be 30° from the point opposite the optical centre (the point where the axis cuts the spherical surface). Suppose the screen to be placed at the true focus for direct parallel rays: using the formula in the preceding paragraph, we shall find that distance to be 13'1 inches. The image of the two points, if there were no distortion, would be 7'6 inches from the axis on the screen 8 in the horizontal plane. The ray Oa' would cut at a distance 6-72 inches, and the ray Ob' 6'5 inches from the axis on the screen S in the horizontal plane. This shows that the distortion is barrel-shaped. This indicates that the further a point is away from the axis, the less proportionally is it displaced from the axis. If the diaphragm were behind the lens, the point a' would