52 M. J. Frohlich on a new Proposition in the 



The motion of the diffracted light is of the form 



A sin ( 2?r^ + 8 ) ; 



the general expression which contains its amplitude both for 

 Fresnefs and Fraunhofer's phenomena is 



A 2 =^ 

 in which 



^{(j^-^y+on?^^}*' 



*= vl 2 — Vft + ^-(-+^+w)j' 



and a?, ?/, z are the coordinates of the element d$ of the imagi- 

 nary surface which covers the aperture and possesses its 

 boundary, n the normal to this element ; and 



« = COS a — COS fit]., /3= COS ft — COS ft, 7=008 70—008 7!, 



if a x ft 7i, « ft 7 denote the direction-angles of the incident 

 and the diffracted ray. 



The equation of the surface § can in any case be put 



z = <p(a:,y), from which we get ^- 1 = cos (pin) as a function 



of x and y ; besides, B& is = ^^, the denominator of which 



J u cos (nz) 



can likewise be expressed by x and y ; so that a function of x 

 and y only stands under the integral-symbol of the above ex- 

 pression, and the integration itself can be effected along x 

 and y. 



The amplitude A is entirely independent of the position of 

 the coordinate-system; let us in future give it such a position 

 that the incident ray, and therefore the a 1 fi 1 y 1 direction, will 

 fall very nearly in the plane of ZX ; by this the following 

 considerations will be simplified. 



Upon the carrying-out of the integrations in A the variables 

 x and y vanish ; their place is taken by the constants of the 

 limitation of the aperture ; and A depends on these and also 

 on p ± and p Q} a, ft 7. Yet we have to do only with indefi- 



* The factor -p~ = cos (pin) follows from the principle of the equality 



of the kinetic energies, as will be shown in a subsequent paper. W. Voigt 

 also deduced it from Fresnel's theory ( Wied. Ann. iv. p. 542 &c. ), and 

 likewise from the elasticity theory of light; it has, however, no influence 

 at all upon the deduction or validity of the following proposition. 



