Study of Huygens s Secondary Waves. 207 



where A is the amplitude in the incident wave, c is a constant, 

 6 the angle of diffraction, and P the path-difference of the 

 effects at f , 77, o£ the element dx dy and o£ an element at the 

 origin in the plane of the aperture. The rigorous expression 

 for P is a little complicated, but if the angle of incidence is 

 not so large that (sin i — sin 6) cannot be put equal to 

 sin (i — 6) cos i, P can be shown to be equal to 



x% cos i-ryrj 



The amplitude of the effect at any point in the diffraction- 

 pattern is therefore 



CCA. 7 7 -r^/' /r • 27r/ Tr , x£ cos i + yn \ 



Jj V sm *" v / ~ G )> 



the double integral being taken over the whole of the aper- 

 ture. Putting xcosi=>> and y = q, p and q being therefore 

 the projections of x and y on the wave-front, the above given 

 expression reduces to 



the integral being taken over the whole of the projection of 

 the aperture on the wave-front. Now, the quantity of energy 

 passing, per unit of time, through the obliquely held aperture, 

 must, at any rate approximately, be equal to the quantity 

 that would pass through an aperture identical with the pro- 

 jection of the first on the wave-front, cut in a screen held 

 parallel to the waves. From this it follows that, prodded 

 we assume F(«, 6) does not vary sensibly throughout the 

 diffraction-pattern, it is equal to cosi *. This, on the assump- 

 tions made, agrees with the expression deduced by Kirchhoff 

 in his ' rigorous ' formulation of Huygens's principle 



A . 2-7T 



— ^-rrr (cos i + cos 0) sin — - (Yt — H)dx dy, 



* From this and the integral given above, it appears that the diffraction- 

 bands due to an aperture held obliquely are, at moderate incidences, 

 identical with those due to its projection held normally : a proposition 

 that might seem otherwise obvious, were it not for the fact that it is not 

 true for very oblique incidences. As an instance, it can be shown 

 experimentally that the diffraction-pattern observed when a circular 

 aperture is held obliquely in front of a telescope directed at a point-source 

 consists of a system of ellipses. 



