176 Lord Ray lei gh on the Theory of Optical Images, 



where AP, BQ denote the corresponding linear magnitudes of 

 the two images. This is the theorem of Lagrange, extended 

 by Helmholtz so as to apply to finite divergence-angles *. 



We now pass on to the actual calculation of the images to 

 he expected upon Fresnel's principles in the various cases that 

 may arise. The origin of coordinates (£ = 0, 77 = 0) in the 

 focal plane is the geometrical image of the radiant point. If 

 the vibration incident upon the lens be represented by 

 cos(2irYt/\) 9 where V is the velocity of light, the vibration 

 at any point f, 77 in the focal plane isf 



-£$*£{*-/+ ^}**, • • P) 



in which / denotes the focal length, and the integration with 

 respect to x and y is to be extended over the aperture of the 

 lens. If for brevity we write 



MtM=p, tnvf»f=9, (8) 



(7) may be put into the form 



where 



S = jj sin (px + qy) dx dy, .... (10) 



C=$ cos (px + qy)dxdy (11) 



It will suffice for our present purpose to limit ourselves to the 

 case where the aperture is symmetrical with respect to x and 

 y. We have then S = 0, and 



C=jJ cos^cos qy dx dy, .... (12) 



the phase of the vibration being the same at all points of 

 the diffraction pattern. 



When the aperture is rectangular, of width a parallel to 

 x, and of width b parallel to y, the limits of integration are 

 from —^ a to + i a for x, and from — -J b to + ±b for y. Thus 



n_ „ h sin (*"£<*/¥) sin (rnibfrf) 



and by (9) the amplitude of vibration (irrespective of sign) 

 is Cfkf. This expression gives the diffraction pattern due to a 

 single point of the object whose geometrical image is at £ = 



* I learn from Czapski's excellent Theorie der Optischen Instrumente 

 that a similar derivation of Lagrange's theorem from the principle of 

 minimum path had already been given many years ago by Hockin 

 (Micros. Soc. Journ. vol. iv.'p. 337, 1884). 



t See for example Enc. Brit., " Wave Theory," p. 430 (1878). 



