78 Mr. A. Schuster on the Elementary Treatment 



separate parts is very instructive ; and I hope therefore that 

 the following way of treating the subject, which seems to me 

 to be free from objection, and gives numerical results practi- 

 cally identical with those obtained by Fresnel, will be found 

 to be of use. 



For the sake of subsequent reference it is necessary to say 

 a few words on the division of a wave-front into circular 

 zones. I shall only consider plane waves, but the results 

 may easily be extended to other cases. 



In order to calculate the amplitude produced at a point P 

 by a plane wave-front at a distance p from it, circles are drawn 



such that their distance from P is^>+ -^-, n being an integer. 



The plane is thus divided into so-called Huyghens zones, 

 the areas of which are equal as long as rik is small com- 

 pared to p. The effect of a single zone can be obtained 

 by subdividing it into narrower rings of equal areas, when it 

 will be found that the phase of vibration at P due to such 

 elementary rings will vary uniformly over two right angles. 

 The phase of the resultant vibration will therefore be halfway 

 between that due to the extreme portions, and the amplitude 

 is obtained by reducing, in the ratio of it : 2, the amplitude 

 calculated on the assumption that the phase due to each part 

 of the zone is the same (Lord Eayleigh, Phil. Mag. xlvii. 

 1874). This is true also for the first circular area. The 

 effect of two successive zones is therefore strictly opposite in 

 direction, and we can calculate the whole effect by means of 

 a series m ^ — m ^ -f- m 3 — ra 4 + . . . . 



The sum of this series, as maybe shown, is -^-, and the phase 



of the resultant vibration will be one right angle behind that 

 due to the central point. , . 



To calculate m„ let the amplitude of vibration be unity. 

 If a small surface s, taken out of the wave-front, will produce 

 an amplitude ks at P, the central area will produce an ampli- 



tude — . 7H- 2 , where r is the radius ; the factor 2/-7T being 



IT 



applied, as just explained, because the phases due to different 

 parts of the central area range over two right angles. As r l 

 is equal to pX, the effect of the first zone, as regards ampli- 

 tude, will be 2fcp\, and the whole wave-front will cause an 

 amplitude equal to half this value, which must obviously be 

 equal to unity, because a plane wave does not alter in am- 

 plitude during its propagation. It follows that K=l/pX; 

 and hence that if s be a small surface at a distance p from a 



