CONCAVE MIRRORS. 15 



except that the focal distance has to be determined by experiment. 

 So, too, if the surface is not spherical, the principal radii of curvature 

 are not very different, and we may investigate the direction of the 

 principal planes by means of the images, and make one of these 

 parallel to the lines of the scale ; the definition of the telescope 

 is not diminished thereby. 



On the other hand, the scales being smaller are easier to 

 illuminate. It will, lastly, be at once seen that for the same 

 illumination, images in the telescope are as bright as with a plane 

 mirror. 



Yet if somewhat large deflections are to be measured, the scale 

 should be curved, and in any case the graduation must be empirical. 



Another arrangement consists in placing the scale at the centre 

 of curvature of the mirror. A reversed image of the scale is then 

 produced of the same size, which can be observed with a lens or 

 with a microscope. The precision only depends then on the 

 diameter of the mirror, and a deflection can be observed equal 

 to half the limiting angle which corresponds to this diameter. 



673. We may, in the preceding cases, replace the concave 

 mirror by a plane one, and a condensing lens at a small distance 

 from each other. 



Let f be the focal length of the lens and B the distance of the 

 mirror from the optical centre. In order that the scale be placed 

 in the principal focal plane of the optical system, the rays from one 

 point, after being refracted in the lens and then reflected from the 

 mirror, must seem to start from the principal focus of the lens. It 

 is readily seen that the scale should be placed at a distance d' from 

 the lens, defined by the equation 



and that the angles a and /3 be in the ratio 



. 



2/-S 



The system is then virtually equivalent to a concave mirror. The 

 hypothesis 8 = 0, would correspond to the case of a plano-convex 

 lens of focal distance / silvered on its plane face. 



With so complicated a construction, the graduation of the scale 

 must necessarily be empirical by means of a divided circle. 



