The Hclmholtz Theory of the Microscope. By J. W. Gordon. 427 



fchat the conditions which produce aplanatism in a centred system will of 

 necessity also produce an undistorted imaire according bo tfre sine law. 

 But even Helmboltz' proof does not explicitly bring out the sine-tangent 

 law, and as this is much less well understood than the sine law at the 

 present time, it is, perhaps, not an impertinence to offer a further proof 

 which will embrace both propositions in one demonstration. 



Postulate. Let it be granted that any image-forming optical system 

 which is capable of producing a regular image of any surface must be 

 capable of forming a similar image of a wave-front which coincides with 

 that surface and moves through the system. 



This proposition is almost, axiomatic, and is here put forward as 

 sufficiently evident without formal proof. 



We start, then, with a plane wave-front e. . . r, in fig. 99 ; we assume 



Fig. 09. 



it to he propagated unchanged as far as A,, and there converted into a 

 spherical wave-front having its focus at C. It is quite immaterial by 

 what apparatus this change of form is brought about provided that it is 

 effected correctly. If correctly made the change must have produced a 

 spherical image of the plane wave-front in which the distances of its 

 parts measured upon the spherical surface are so rearranged that the 

 original axial distances are preserved. That is to say, the law of 

 formation is that all the rays travel in radial paths with equal velocities, 

 and the criterion of resemblance is that every ray preserves, under all 

 changes in the form of a wave-front, its" angular position in the beam, 

 so that at every point its axial distance e is 



ej = sin #! r, 



(9) 



being the angle which it makes with the axis, and r the radius of 

 curvature of the wave-front at the point under observation. 



The spherical wave-front having been formed, converges in con- 



