[ 477 ] 



LVI. Investigations in Optics, with special reference to the 

 Spectroscope. By Lokd Kayleigh, F.R.S. 



[Continued from p. 411.] 



§ 5. On the Accuracy required in Optical Surfaces. 



EOUCAULT, in the memoir already referred to, was, I be- 

 lieve, the first to show that the errors of optical surfaces 

 should not exceed a moderate fraction of the wave-length of 

 light. In the case of perpendicular reflection from mirrors, the 

 results of § 4 lead to the conclusion that no considerable area of 

 the surface should deviate from truth by more than one eighth 

 of the wave-length. For a glass surface refracting at nearly 

 perpendicular incidence the admissible error is about four 

 times as great. It will be understood, of course, that the 

 errors of one surface in an optical train may compensate for 

 those of another, all that is necessary being that the resultant 

 error of retardation rise nowhere to importance. 



In the case of oblique reflection at an angle <£, the error of 

 retardation due to an elevation B D (fig. 7) is 



QQ'-QS = BD sec <£ (1- cos SQQ') 



= BD sec $ (1 + cos 2<j>) = 2 BD cos <f>, 



Fig. 7. 



from which it follows that an error of given magnitude in the 

 figure of a surface is less important in oblique than in perpen- 

 dicular reflection. At first sight this result appears to be con- 

 tradicted by experience ; for it is well known to practical opti- 

 cians that it is more difficult to secure a satisfactory perform- 

 ance when reflection is oblique. The discrepancy is explained 

 in great measure when we take into account the kind of error 

 to which surfaces are most liable. No important deviation 

 from a symmetrical form is to be feared; but a surface intended 

 to be plane may easily assume a slight general convexity or 

 concavity. Now in direct reflection, a small curvature is 

 readily and almost completely compensated by a small motion 

 of the eyepiece giving a change of focus ; but the compensa- 

 tion obtainable in this way is much less perfect when the 

 reflection is oblique. In the first case the family of surfaces 



