The Abbe Diffraction Theory. By J. W. Gordon. 383 



work it and one another more or less. The extent to which they will 

 thus counterwork one another is evidently measured by their differences 

 of phase. A complete oscillation comprises an excursion first in one 

 direction and then back again in the opposite direction, like the 

 forward and backward swing of a pendulum. If two wave-fronts 

 overlap one another, therefore, to the extent of half a period, they 

 will exactly neutralise one another, for the one will tend to elicit a 

 movement the direct opposite of that which the other tends to set up. 

 These equal and opposite forces can produce stress but no movement, 

 and thus we see that by interference the same set of undulations 

 which at p give rise to bright illumination, may at p 2 set up a state of 

 strain of which our eyes can give us no account. Such, in broad 

 outline, is the undulatory theory of shadow formation. 



This explanation must, however, be pursued a step farther, for it 

 is not yet apparent why over one region the successive wave-fronts 

 should always harmonise so as to reinforce one another and produce 

 light, and in another region, sharply divided from the first, should 

 interfere destructively and produce shadow. This, however, will be 

 very clear when we consider how small a magnitude a wave-length 

 of light is. Something like ^ inch comprises a complete set of 

 wave-fronts of light ; that is to say, within that small space, 

 measured radially from any focal point, you may find, at any given 

 instant of time, a set of wave-fronts exhibiting every possible phase of 

 the light oscillation. If all these wave-fronts are brought together 

 at any given point in time and space, they will evidently cancel one 

 another ; for the series will, of course, contain in the second half an 

 impulse which is the opposite of every impulse in the first half. Hence, 

 whenever we find that a given segment of a wave-front occupies 

 a whole wave-period in delivering its consignmeut of energy at a 

 given point, we may infer tbat the discharge is not effected without 

 impediment, that succeeding wave-fronts will enter into competition 

 with it for discharging facilities, and that the visible effect will be 

 diminished accordingly. If the period of discharge be exactly one 

 wave-period, and if the rate of discharge be uniformly maintained 

 throughout that period, we may infer that all possible phases of 

 the luminiferous impulse are concurring at every instant, and the 

 result will therefore be a strained but total darkness. A case may be 

 supposed in which the period of discharge will occupy longer than a 

 complete wave-period, say for example 3 Of- wave-periods. In that 

 case thirty complete sets of undulation phases will at any instant 

 concur and cancel one another. There will be § of a complete set 

 over. But this assortment of undulation phases represented by the 

 fraction f will not be all concordant. On the contrary, the phases of 

 the 5th eighth will be opposite each to each to the phases of the 

 1st eighth, and these two divisions will, in like manner, cancel one 

 another. There remain three eighths of a single set of phases 

 uncancelled, and these, although not mutually destructive, are never- 



