26 THE MODERN KEFLECTING TELESCOPE. 



from the illuminated pinhole strikes B, is reflected to A, thence back to B, 

 thence to a focus close beside the illuminated pinhole. 



When using the knife-edge test the optician sees the mirror B brilliantly illu- 

 minated, and in ellii)tical outline, the hoi'izontal diameter appearing foreshortened 

 by an amount depending upon the angle at which the mirror is viewed. With the 

 knife-edge test the sui'face of B is seen in relief, as a whole ; any zonal errors appear 

 enormously exaggei'ated, and their chai'acter and position are readily determined, 

 just as when a spherical rairi'or is tested at its center of curvatui-e ; these zonal 

 errors, of coui-se, appear elliptical, on account of their foreshortening ; their effect is 

 doubled in intensity on account of the two reflections from B (assuming that the 

 illumination is as brilliant as the eye requires). 



The test, as already described, is all that is necessai-y for the detection and loca- 

 tion of zonal errors. But something more is necessary in order to detect general 

 curvature, i. e., convexity or concavity, in B. Let us assume that the mirror, when 

 fine-ground and polished, is so nearly flat that no curvature can be detected with a 

 Brown and Sharpe steel straight-edge of the finest quality; and for convenience in 

 description let us also assume that the surface is free from zonal eri'ors. Let the 

 knife-edge be moved across the reflected cone from the left; a focal point is found 

 at which the right and left sides of the mirror darken simultaneously ; this focal 

 point we will call/j. Now let the knife-edge be moved across the cone from 

 above, instead of from the left; a focal point will be found at which the upper and 

 lower parts of the mirror darken simultaneously ; this focal point we will call^g. 

 It is only when the mirror B is a perfect plane that fj and fj coincide with each 

 other and with the point /'(see figure). If B is slightly convex, /j and/", are out- 

 side oif(i. e., farther from the mirror than/") and /"j is outside oi f^. If ^ is 

 slightly concave both/j and^g are inside of/, and/j is inside of y,- I^ practice, 

 the exact position of /"is not found (except incidentally when the plane mirror is 

 finished), for this would involve the very accurate measurement of the large dis- 

 tance cm + ^'/- The determination of the positions of/', and/g with reference to 

 each other is all that is needed. 



That/"j and/'g do not coincide when B is convex or concave is due to the fact 

 that the cuivation of B is apparently increased or exaggerated in the direction of 

 the horizontal diameter of the mirror, on account of its foreshortening in this direc- 

 tion, as seen from/'; while the curvature in the direction of its vei'tical diameter is 

 not thus exaggerated. Tiie effect is precisely as if the spherical mirror A were astig- 

 matic, the parts of the sui'face adjacent to the horizontal diameter having a different 

 radius of curvature from those adjacent to the vertical diameter. This effect is so 

 marked that an exti'emely small deviation of B from a ti'ue plane can be detected. 

 For example, if A and B are each two feet in diameter, the radius of curvature of 

 A being fifty feet as before, and if the angle which the line/m subtends with the 

 surface of B is 45°, a deviation from a true plane of 3 j,/,^tnr '^^^' "^ ^'^^ surface of 

 B is readily detected. If the angle of the mirror B be changed to 30°, as shown 

 in Fig. 3, the accuracy of the test for genei'al curvature is about doubled ; the latter 

 position, however, is not usually so convenient for determining the positions of zonal 



