The Helmholtz Theory of the Microscope. By J. W. Gordon. 439 



" Let us now consider the distribution of brightness in the image of 

 a double line whose components are of equal strength and at such an 

 angular interval that the central line in the image of one coincides with 

 the first zero of brightness in the image of the other. In fig. 1 * the 

 curve of brightness for one component is A B C D and for the other 

 OA'C; and the curve representing half 

 the combined brightnesses is E' B E F. 

 The brightness (corresponding to B) mid- 

 Avay between the two central points A, A' 

 is *8106 of the brightness at the central 

 points themselves. We may consider this 

 to be about the limit of closeness at which 

 there could be any decided appearance of 

 resolution. The obliquity corresponding 

 to u = w is such that the phases of the Fig. 109. 



secondary waves range over a complete 



period, i.e. such that the projection of the horizontal aperture upon this 

 direction is one wave-length. We conclude that a double line cannot 

 be fairly resolved unless its components subtend an angle exceeding that 

 subtended by the wave-length of light at a distance equal to the horizontal 

 aperture.'''' 



Now, it is to be observed, concerning this proof, that the light- 

 intensity curve here shown is not the light-intensity curve of a very 

 narrow rectilinear surface — such as would be commonly called a line — 

 but on the contrary, it is the light-intensity curve, calculated for a 

 mathematic point, such being assumed to be the source of light. Such 

 an imaginary source differs, therefore, from any possible source of light 

 in the important particular that it has no radiating surface and its light- 

 intensity curve does not, in fact, exhibit the properties of a light-intensity 

 curve derived from even the smallest imaginable surface. It represents 

 only the infinitesimal element of such a curve from which the curve 

 itself must be derived by integration. Treating it as such, it is easy to 

 show that as a matter of theory, the semidiameter of the antipoint is not 

 the limit of resolving power in a perfect instrument. 



This may be shown without any recourse to abstruse mathematics by 

 a very slight modification of Lord Rayleigh's diagram. Let it be 

 assumed that four "lines" delineated by means of antipoints such as 

 Lord Rayleigh has figured lie side by side, distant t l of a wave-length 

 from one another. We may treat such a group as constituting a quasi- 

 surface. It will be a narrow surface, for its breadth is but \ A, and it 

 will be visibly an unbroken surface if we assume that two lines lying 

 within jL x of one another will be indistinguishably fused together in 

 the optical image. Let us next assume" that in place of the mathematical 

 lines of fig. 109 we have two such narrow surfaces lying side by side with 

 an interval of | X between them. We shall then obtain a result indi- 

 cated by fig. 110. Here, in addition to the antipoint curves, we have five 

 total illumination curves laid down. The two antipoints marked 1 ... 1 

 will yield the total light curve I in fig. 109. The pair marked 2. . .2 

 •will in like manner yield the total light curve II, and so of the re- 



* Fig. 109 in this paper. 



