The Theory of Optical Images. By Lord Bayleigh. 455 



near to PESQ and another to P L M Q.] And, since A P, B Q 

 ;are perpendicular to the axis, the optical distance from P to Q is 

 the same (to the first order of small quantities) as from A to B. 

 Consequently the optical distance P R S Q is the same as A It S B. 

 Thus, if fM, ft be the refractive indices in the neighbourhood of A 

 and B respectively, a and /3 the divergence-angles EAL, SBM 

 for a given ray, we have 



fi. A P. sin a = yu/.B Q.sin/3, . . . . (6) 



where A P, B Q denote the corresponding linear magnitudes of the 

 two images. This is the theorem of Lagrange, extended by Helm- 

 holtz so as to apply to finite divergence-angles.* 



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

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

 arise. The origin of co-ordinates (ff = 0, 17 = 0) in the focal plane 

 is the geometrical image of the radiant point. If the vibration 

 incident upon the lens be represented by cos (2 ir V t / X), where 

 V is the velocity of light, the vibration at 11 ny point £, rj in the 

 focal plane is % 



-x/ / -'' sin -r{ v '- /+ ' lL 7 Z -"} f '"'* • (7) 



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 



'^ZjXf = p, 2 7T7] /\f = q, ... (8) 



«(7) may be put into the form 



_ C , sin ^5 (V t - f) - S -,cos 27r (V t - f), . (9) 



where 



S = // sin (px -h qy) dx dy, . . . (10) 



t C = ff cos (px + qy) dx dy. . . . (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 

 liave then S = 0, and 



C = jf cos px cos qy dxdy, . . . (12) 



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

 diffraction pattern. 



* I learn from Czapski's excellent "Theorie der Optisclien Instrumente' tbat a 

 similar derivation of Lagrange's theorem from the principle of minimum path had 

 •already been given many yeard ago by Hockin (Micros. Soc. Journ., vol. iv. p. 337, 

 1884). % See, for example, Enc. Brit., ' Wavo Theory,' p. 430 (1S78). 



