Lord Rayleigh's Investigations in Optics. AX 



If QA, Q'A be also corresponding rays, sin $—fJ> sin (£ /= =0, as 

 appears by supposing 6 = in (3), which then becomes 



(rcos 2 (f>\/ 1 3r6 sin 6\ 

 C0S ^ + __Tj^i_^__j 



/ ., , rcos 2 (f/\/i 3W?sin<£/\ , . 



If we make = in (4), 



, , rcos 2 6 ( ,. , rcos 2 (/A /K . 



cos <ft + ^ y = ^cos f + — -^ J, . (5) 



the usual formula for the primary focus. For our present 

 purpose 6 is small, but is not zero. 



We will now apply the fundamental formula (4) to the case 

 of a lens whose thickness can be neglected in comparison with 

 the radii of curvature and the distances of the foci. The pencil 

 is supposed to fall centrically, so that the angle of incidence 

 at the second surface is equal to the angle of refraction at the 

 first surface. The distance corresponding to PA is the same 

 for both surfaces, and will be denoted by y. Thus, for the first 

 refraction, 



(cos $ cos' 2 </>\ / 1 3y sin (f>\ 

 r u / \ 2u ) 



For the second refraction we have to interchange cf> and ft, 

 writing s for r, u 1 for u, /-6" 1 for jul, and v for u' . Thus 



/cos ft cos 2 ft \ /., By sin ft\ 



=C-^+ £2 ?)( i -^>- • ti 



By addition of (6) and (7), and writing for brevity c for cos</> 

 and d for cos ft, we get 



