July 28, 1923] 



NATURE 



MI 



which OP is normal. Consider a pair of such zones 

 in latitudes + Q and - Q (taking the diametral plane 

 as equatorial). Every point in each zone is at a 

 constant distance from P, and the constant difference 

 between PS and PS' is 2OP sin Q. Assuming that 

 the focal length OS is great compared with X, and the 

 conjugate focal length great compared to OS, then 

 the difference of phase in the waves contributed 

 to the image by each pair of zones is (if OS = v) 

 47r(y/\) sin Q. Putting A for the wave amplitude which 

 would exist in the image if all the partial waves 

 arrived in the same phase, and writing <i> for 47r()'/\) sin d, 

 the actual amplitude at the geometrical conjugate 



focus of a point distant r from O is ^ cos {(pl'z)de, 



where 0^ and 6^ define the operative areas of the wave 

 surface SS'. The value of A will be different for each 

 pair of limits, but the ratio between the amplitude at 



O and that at r is cos {<Pl2.)dO. In computmg this 



integral a table was formed for cos </>/2 between the 

 limits o and 2 for rj\, and o and 7r/2 for 6. Fair 

 curves were drawn through the plotted values of 

 cos (l>J2 for each of the chosen values of (see Fig. 2) 



8 20 



Fig. 2. — Horizontal lines measured from the curves to each of the principal 

 verticals are the values of cos ^jz (where <\> — ^n(rlK) sin 6) from e = o to 

 = 7r/2, and the principal verticals refer to values of rj\ from o to 2. 



and the algebraic area of the curves for various limiting 

 values of 6 was measured with a planimeter ^ (see 

 Fig. 3). The intensities of the illumination are of 

 course as the square of the amplitude. 



When two or more luminous points in the focal 

 plane are in proximity, the interference effects 

 occurring between their ring systems are not independ- 

 ent of the nature of the illumination. If the luminous 

 points radiate light proceeding from a single source, 

 there is a definite phase relation among the emitted 

 waves, and in this case the intensity is proportional 

 to the square of the sum of the amplitudes ; if, how- 

 ever, the points are self-luminous it is the sum of the 

 squares which must be taken. 



The change in the appearance in the field of a 

 microscope when a point source is substituted for 

 diffused light is very conspicuous. 



The curves in Fig. 3 indicate that as the aperture 

 of the lens is increased from o to 90° the diameters of 

 the central disc and of the rings are reduced, as well 

 as the relative brightness of the rings, and that when 

 the whole hemisphere of the wave surface is operative 

 the diameter of the central disc — i.e. the radius of the 

 first dark ring — is a little greater than o-4X. 



When the central rays are stopped out the diameter 

 of the disc is still further reduced, but the brightness 

 of the rings is greatly increased. Thus when only 

 the marginal rays are effective the image of a single 

 line will appear multiple. 



It must be remembered that these curves only 

 apply to points in the focal plane, and that the radii 

 of the rings for points slightly out of focus are greater. 



• For a somewhat similar purpose Airy (see his " Intensity of Light in the 

 neighbourhood of a Caustic," Camb. Phil. Trans., vol. 6, pp. 379 et seq.) 

 computed his table numerically by methods much more accurate, but also 

 much more laborious, than the planimeter. The latter, however, is 

 sufficiently good for the purpose of this note. 



The lateral spectra which accompanj^ the image of 

 lines (which may be regarded as the envelope of the 

 ring systems of a series of points) have a considerable 

 effect on the appearance seen in the field of the 

 microscope. 



It is usually held that an object is in focus when 

 the definition is sharpest. This, however, is not 



Limitsof^. 



1-0 1-2 1-4 1-6 1-8 20 



Fig. 3. — The curves are the algebraic integrals giving the areas included 

 between the curves and verticals in Fig. 2 for each value of rjK, and 

 between the limits for e indicated on each of the diagrams, namely : 

 Diagram e, e^ \ A 90"= o 



a 20° o e 90° 20 



b 40" of 90° 40 



c 60° o I g 90' 60 



The ordinates of the curves give the amplitudes of the resultant vibra- 

 tions (expressed as fractions of the amplitude at the geometrical focus) 

 at the various values of rjK. 



really the case. If bands of fine ruling in close 

 proximity to one another are examined, it will be 

 found that a separate adjustment of focus has to be 

 made for each and that the best result is obtained 

 when the focal adjustment makes the spacing of the 

 lateral spectra the same as that of the lines of each 

 band. 



With ordinary test objects (diatoms, engraved 

 lines, etc.) this effect is somewhat disguised owing to 

 the thickness of the objects themselves, which is quite 

 comparable to the wave-length, but in such test 

 plates as I have described in my former letters, where 

 the thickness of the film on which the lines are ruled 

 is only 1/15 to 1/30 of a wave-length, the question of 

 thickness does not arise. 



The high resolving power which has been attained 

 on diatoms and engraved lines should be attributed to 

 variations of thickness in the objects, as these increase 

 the rate at which the length of the optical path 

 changes for points near the geometrical focus ; i.e. 

 for the variation of r. It is customary to mount 

 such objects in media of high refractive index, which 

 has the effect of exaggerating the optical depth of the 

 grooves, etc., and it is worth notice that if an object 

 has no thickness, or a thickness small compared to the 

 wave-length and the only characteristic of which is a 

 difference in opacity from place to place, the refractive 

 index of the mounting medium is without effect on 



NO. 2804, VOI-- I 12] 



