478 Lord Rayleigh's Investigations in Optics. 



approximating to a plane, which will answer the purpose, coin- 

 cides with the family of surfaces most likely to be produced ; 

 in the second case the family of ellipsoidal or hyperbolic sur- 

 faces capable (with suitable focus) of giving good definition 

 contains only one symmetrical member — the perfect plane. In 

 order to test experimentally the correctness of the theoretical 

 result, it would be necessary to retain the focus suitable to the 

 true surface, and not to allow a readjustment by which its 

 errors may be in greater or less degree compensated. 



A further difficulty, not touched by the preceding conside- 

 rations, still remains to be mentioned. In the ordinary method 

 of testing plane surfaces by measuring the change of focus 

 required when a distant point is viewed through a telescope, 

 first directly, and then after reflection in the surface, the test 

 is found to be more delicate as the reflection is more oblique. 

 The explanation of the apparent inconsistency will be best un- 

 derstood by a calculation of the focal length of mirrors, founded 

 directly upon the principles of the wave theory. Let A C B 

 (fig. 8) be an arc of a (parabolic) mirror, which reflects pa- 

 rallel rays GA, HD, KB Fig. 8. 

 to a focus F. A D = y, 

 DF=/, CD = *. In calcu- 

 lating the retardations of the 

 various rays, we will take as 

 standard the phase at F of a 

 ray coincident with H D, 

 reflected at D (as by a plane 

 mirror A D B) instead of at 

 C, so that the actual retardation at F of the central ray HCF 

 is 2t. The retardation of the extreme ray GrAFisAF — FD, 

 or */ (f 2j ty 2 )~f' Since F is by supposition an optical focus, 

 the phases of all the rays must be the same, and thus 



V(/ 2 +/)-/=2< (1) 



If the aperture (2y) be small in proportion to the focal length, 



V 2 

 ^(f 2 + f)—f=ij approximately, 



so that 



f= y i < 2 > 



In the limit it is a matter of indifference whether / be mea- 

 sured from D or from C. If r be the radius of curvature at C, 



