Nul Point of Thin Axial Pencils of Light. 175 



the various points of an optical system. Let F 1 P 1 = ^ lr 

 F 2 P 2 =* 2 , HiF^/j, H 2 F 2 =/ 2 , HjP^^, H 2 P 2 =^ 2 , H 2 H 1= =g, 

 and let m be the magnification. 

 We have 



m _0P 2 _ x 2 h 



' OPx / 2 «*" 

 Hence 

 „ ? _OP 2 _P 2 F 2 _ 0F 2 . 



OHx+Zs-a 



OP! H 2 F 2 OP! + H 2 F 2 

 Therefore 



0^ + *!+/!+/, 



or 



(1 - m)OH, =«-/, + m(/i +/, + tfj) 



= « —fi + "»(/i +/s) — /l 

 = a -(l-m)(/ 1 +/ 2 ), 



and OH 2 =OH 1 - a = I ^-(/ I+ / 2 ). 



If ?n = 0, is at a distance/!+/ 2 to the right of H 2 ; and, 

 if m=co, at a distance /i-f/ 2 to the right of Hi. These 

 positions of are the nodal points of the system. In the case 

 where the index of refraction of the first medium is the 

 same as that of the last, /i+/ 2 =0; and, consequently, 

 coincides with H 2 when m=0, and with H x when m = co. 

 In the former case the incident light is a pencil of parallel 

 rays, in the latter the emergent pencil is parallel. 



The above formulae for OHi and 0H 2 may be used in 

 determining the value of a, or H 2 H l5 by experiment. For 

 suppose we measure the magnifications m and m ! for two 

 positions and 0' of the nul point, we have 



00 



'~ a (l-m 1-m'J 



(l-m)(l— m') 



Thus, by measuring 00', we determine a, and, in the case 

 where /i +/ 2 = 0, the actual positions of Hi and H 2 are also 

 found. 



If fi+fs is not zero, we may proceed to find the positions 

 of H 2 and H 2 as follows. 



