The HelmJwltz Theory of the Microscope. By J. W. Gordon. 409 



aperture, and <f> for the phase expressed in wave-lengths — the 

 phase value, as we may term it. Therefore 



F = <pXr (r,\ 



* — O I > /i \' J J 



2 K. cos v ' 



Now we have seen that the radius of the corresponding ring in 

 the antipoint = Fcos 6. Therefore writing p for this radius, we 

 have 



<f) \ T </> A. / ~ \ 



2 It 2 sin u 



since — is the sine of the divergence angle u. 

 r 



Main Results of the Helmlioltz Theory, 



From this expression for the radius of the antipoint several 

 inferences may at once be drawn. 



(1) In the first place, we infer that the successive phase-rings 

 of the antipoint must be distant from the centre in the exact pro- 

 portion of their phase-values. For, X and u remaining constant, 

 the value of p is simply proportional to <j>. That is to say, a 

 focussed beam of given angular aperture in a given transparent 

 medium has all its bright rings equidistant from one another, and so 

 with all its dark rings, and has all its rings formed with radii having 

 lengths proportionate to the phase- values of the several rings.* 



(2) In the next place we may note that the equation (5a) 



p = --? is wholly independent of r. "We conclude, therefore, 



Ji sin u 



that the dimensions of the antipoint formed by a focussed beam 

 depend upon the wave-length and the divergence angle only, and 

 are entirely independent of the focal length. This agrees with the 

 result of observation mentioned above (p. 403), and may be ex- 

 pressed by saying that the diameter of the antipoint is inversely 

 proportional to the numerical aperture of the beam by which it 

 is given off. 



(3) In the next place, it can be shown by the same expression 

 that the antipoint which a given aperture produces is, other things 

 being equal, directly proportional to the radius of curvature of the 

 wave-fronts which pass the aperture. 



For equation (5a) can be stated alternatively thus : 



<f>\ <f> X /■ 

 9 ~ 2smu ~ '2K ' 



* It 16 not of any importance in tlie present connection to take notice of Sir George 

 Airy's conection — of wliich Helmholtz is quite aware — of this result as npplkd to the 

 innermost rings of the antipoint produced by a circular aperture. See (Jumb. Philo- 

 sophical Transactions, vol. v. (1835) p. 283. 



