382 Transactions of the Society. 



thus evolving what may fairly be called the Helmholtz theory of 

 the instrument. 



Stated very roughly, the Helmholtz method of investigation 

 may be described as the use of wave-fronts as instruments of 

 research. It rests on this assumption, almost a self-evident pro- 

 position, that any instrument which is capable of giving a regular 

 image of any surface whatever must be capable of giving a similar 

 image of any wave-front which can be made to coincide with that 

 surface, and pass through the instrument. The ordinary geometri- 

 cal method of investigation deals with separate rays and radiant 

 points only. It results, as we all know to our cost, in the most 

 intricate mathematics before we have proceeded far along any line 

 of research. Helmholtz substitutes for these elements the ele- 

 mentary notion of radiant surfaces which move backwards and 

 forwards through the instrument, and by noting what happens at 

 various points to these travelling surfaces, he investigates the 

 relations of various parts of the instrument to one another. The 

 conception of the travelling radiant surface may perhaps seem a 

 little unfamiliar, but it is fully warranted by the undulatory theory 

 of light, and, in fact, produces the most enormous simplification in 

 the discussion of many problems of optics. 



The radiant surfaces with which we are here concerned, are- 

 not only wave-fronts. A wave-front is defined to be a surface 

 disturbed by wave-motion in such wise that all parts cf the surface 

 move in unison, that is to say, that all parts of the surface exhibit 

 the same phase of the undulation at any one and the same instant 

 of time. It may be described briefly as an uniphasal surface. This 

 is, in fact, strikingly unlike the front of an advancing wave, as we 

 see ic in the ocean, for example. Such a front-of-a-wave is a 

 surface in which a complete half-series of undulation-phases is to- 

 be discovered. At its foot in the trough of the wave the water is 

 m a state of momentary rest in its lowest position ; as we climb 

 the front we find the water getting gradually into a swinging 

 motion, swinging upward, and at a point half-way up the front of 

 the wave this upward motion of the water is at its fastest. From 

 that point to the crest, the rising motion grows less and less rapid, 

 and when the crest is reached the upward movement is exhausted 

 and the water comes momentarily to rest in its topmost position. 

 Now, this is not a " wave-front " in the technical sense of the term,, 

 for it is a polyphasal surface, whereas a wave-front is by definition 

 uniphasal. But with such polyphasal surfaces, the theory of 

 resolution in the Microscope is much concerned, for whereas 

 uniphasal surfaces, or wave-fronts, give rise to the direct, or, as 

 they are termed, the dioptric beams in an optical instrument, the 

 diffracted beams arise from polyphasal surfaces and therefore 

 the law of their refraction is of equal importance with the law of 

 the refraction of uniphasal surfaces. The diagram fig. 76 will 





