Highly Magnified Images. By J. W, Gordon. 5 



uplanatic and yield a flat image in two planes not conjugate to one 

 another. Now it is manifestly impossible to compute the light 

 phase in a region where the optical system is non-aplanatic, for in 

 that case the phase is wholly indeterminate, and hence it is im- 

 possible to make the image formation in the plane through P 2 

 dependent upon that in the plane through P x , or vice versa. 



Coming now to what has been accomplished in the way of 

 constructing a theory of the image formed by a Microscope, the 

 fundamental proposition is worked out in Sir Geo. Airy's paper, 

 and may be formulated thus : — 



I. The image formed by any aperture of a luminous point is 

 ■an illuminated area, the shape and dimensions of which depend 

 upon the form and size of the aperture. 



II. The focussed image of such a point is an antipoint, the 

 shape of which is derived from the shape of the aperture by a rule 

 of inverse resemblance, so that the antipoint is narrow across any 

 diameter across which the aperture is broad, and vice versa. This 

 rule of inversion results, in the case of a symmetrical aperture, in 

 an approximate reproduction by the disc of the antipoint of the 

 form of the aperture turned through an angle of 90°. 



III. In the case of a circular aperture, transmitting a beam in 

 which the light arrives at the aperture in the form of plane wave' 

 fronts, the form of the antipoint is a circular disc surrounded by 



1*2 A, 



rings, the disc having a radius of „-. , and the dark rings sur- 



° ° 2 sin u ° 



rounding it being situated at radial distances which tend to become 

 equal to . - for the nth dark ring. In these expressions X = 



_j S1H '10 



the wave-length, u = the divergence angle of the focussed beam, 

 and n is any integer. The inner rings have, as here shown in the 



case of the first ring, a somewhat greater radius than^ — = — .* 



°' ° 2 sin. u 



IV. Sir Geo. Airy calculates and gives in the form of a table 

 the comparative amplitudes of the light undulation at selected 

 zones in the circular antipoint. Plotted down, his amplitudes are 

 proportional to the ordinates of the curve in fig. 2, where the 

 calculated results are shown by the points of intersection of the 

 curve with the scale rulings. The intermediate values are deter- 

 mined graphically by carrying a continuous curve through the 

 calculated points. 



This curve, and the table given by Sir Geo. Airy in the paper 

 cited, are open to the criticism that they express only the semi- 



* This is the accepted description of the antipoint formed by a circular aperture, 

 and is given here upon the authority of Sir Geo. Airy. The present writer submits 

 considerations bearing upon it in a note subjoined to this paper (below, p. 25). 



