CONSTRUCTION OF THE MICKOSCOPE. 



15 



fig. 5. 



pendicular to the surface at r, hrc will be the angle of incidence ; and 

 if a circle is described with a radius rh, hm, will be the sine of the 

 angle : the same happens as was shown occurring with parallel surfaces 

 in fig. 3. From a scale on which hm is I'SOO take in the compasses 

 1, and find some point, 

 6, in the circle, where 

 when one foot of the 

 compasses is placed, the 

 other will fall only on 

 one point, n, of the per- 

 pendicular, re; the line 

 r b, drawn through this 

 point, will be the re- 

 fracted ray. By continuing this ray, b r, backwards, it will be found 

 that it meets the axis at /. In like manner it will be seen that the 

 ray h" r' will be refracted in the direction of r r", as if it also diverged 

 from/! Hence /will be the focus of the parallel rays refracted by a 

 single concave surface, and may be found by the following rule : Divide 

 the index of refraction by its excess above unity, and the quotient will 

 be the principal focal distance fc, the radius of the surface. If, by a 

 similar method, we find the refracted ray r c at the emergence of the 

 ray r b from the second surface r r' of the lens, and continue it back- 

 wards, it will be found to meet the axis at /' ; so that the divergent 

 rays r r'', r r'' are rendered still more divergent by the second surface, 

 and c will be the focus of the lens m' n. 



Bays of light falling upon a convex lens parallel to its axis are 

 refracted in precisely the same manner as those falling on a sphere ; and 

 the refracted ray may be found iu the very same way. But as a sphere 

 has an axis in every direction, every incident ray must be parallel to 

 an axis of it ; whereas in a lens, which has only one axis, many of the 

 incident rays must be oblique to that axis. In every case, whether of 

 spheres or of lenses, all the rays that pass along the axis suffer no 

 refraction at all, because the axis is always perpendicular to the 

 refracting surfaces. 



When parallel rays, rl,¥c, r" V, fig. 6, fall upon a double-convex 

 lens, IV, parallel to its axis, r'f, the ray r' c, which coincides with the 

 axis, will pass through without suffering any refraction ; but the other 

 rays, r I, ¥' I', will be refracted at each of the surfaces of the lens ; and 

 the refracted rays corresponding to them, namely If, If", will be 

 found, by the method already given, to meet at some point, /', in the 



