FOR A NARROW BEAM OF LIGHT. 389 



either apart from the other. In the case of an optical instrument symmetrical 

 about its axis, 



,= u 2 = v,, fi = g and /,=& (29). 



Let l lt In be the tangents of the angles which the incident and emergent rays, 

 projected on the plane of xz, make with the axis of z, 



dV 



7 ( Z l ~ U l) X t ~Jl X l /q-|\ 



- -" 



If the incident ray is parallel to the axis, Zj = 0, and the equation of the 



emergent ray is 



(z 1 -u i )x 1 -f a x 1 = Q .............................. (32). 



The emergent ray cuts the plane of yz where 



z 3 = tt 3 ...................................... (33). 



This therefore is the position of the second principal focus. 



When z = u t +f,> x^ = x t ................................. (34), 



or the ray is at the same distance from the plane of yz as before incidence. 

 This gives the position of the second principal plane. 



Its distance from the second principal focus is f 3 , which is called the 

 second principal focal length. 



When 2j = i+/ 1 and x t = 0, ^ l., .......................... (35), 



or every ray which passes through this point is equally inclined to the plane 

 of yz before and after passing through the instrument. This point is called the 

 second focal centre. 



The distance of the emergent ray from the axis of z, when z = z 2 , is given 

 by the equation 



/A = (Z - M.) *1 - *1 {( 2 1 - ( Z * - ) -/I/*} .................. ( 36 )' 



When k-tOk-tO-/./. = ............................ ( 3 ?)> 



the term multiplied by ^ vanishes. Hence all the rays which pass through the 

 point (Xj, z,) pass through (x,, z,), whatever their inclination to the axis. The 

 points x lt z 1 and x 2 , z, are therefore conjugate foci, and x, is the image of x v 



