8 Transactions of the Society. 



affected if certain of the constants employed were affected with a 

 negative sign and others not. For this purpose, therefore, the 

 emendation of Sir George Airy's table is of importance. 



Of the antipoint formed by a star in the image plane ot a 

 telescope, Sir Geo. Airy's theory affords a sufficient explanation 

 subject only to very small corrections, and that, in truth, is all 

 that its author set himself to explain. But as it stands in the 

 'Philosophical Transactions,' the theory is not directly applic- 

 able to the image formed by a Microscope, and this for two 



reasons : . . . , , 



1. The Microscope receives upon its objective not plane but 



spherical wave-fronts of incident light ; and 



2 The object on the stage of the Microscope, even when very 

 minute, is not infinites imally small, like the disc of a star seen in 

 the heavens, but is always of finite dimensions and usually ot 



sensible magnitude. 



In order to adapt the Airy theory to the case ot the Micro- 

 scope both these new conditions must be investigated— that is to 

 say the law of diffraction from spherical wave-fronts must be 

 ascertained and substituted for the law of diffraction from plane 

 wave-fronts as the basis of the theory, and the diffraction fringe 

 formed about a small finite area must be substituted for the anti- 

 point curve as the boundary region between light and dark areas. 



So the problem stood when, in 1874, Prof. Helmholtz; con- 

 tributed his paper to ' Poggendorff's Annalen.' The paper was a 

 composite, put together confessedly under great pressure ot time 

 and apparently comprising, to judge by internal evidence, three 

 constituent fragments tumbled together without sufficient or 

 effectual editing. It is proper to recall these circumstances when 

 discussing Helmholtz' paper, for they explain its limitations; 

 they explain, for example, how it should have come about that 

 Prof Helmholtz, while he solved the first half of the problem, 

 and showed how to adapt Airy's theory to an instrument re- 

 ceiving spherical wave-fronts, left the second half unattached, and 

 incautiously assumed that the diffraction fringe of the smallest 

 visible luminous area would be indistinguishable from the section 

 of an antipoint. Such shortcomings are the results ot precipi- 

 tation, which betrayed even the great Helmholtz into serious 



l °But it is his solution of the first part of the problem which 

 concerns us, and this is so elegant that, formidable as the problem 

 itself looks, the solution can be stated in a few words if we confine 

 ourselves to results, referring the reader to other sources ot 

 information for the demonstration. 



• There is another point affecting the form of the curve discussed in Note I. 

 (below, p. 30 (6) ), in respect of which it may turn out that Sir Geo. Airy « resulta 

 require correction. 



