38 Mr. 0. V. Boys on Measurement of 



meet the axis at the principal focus F. It is required to find 

 the length d¥, or, shortly, F — that is, the distance of the 

 principle focus from the surface. Since the line pa has cut 

 the front surface at a point d nearer the axis than p, the in- 

 clination of the normal, bs, at d will be less than 0, and will 

 equal md if 



m^S=l t (2) 



pa 7 v 



t being the thickness of the lens. Now the angle pds == the 



angle dbp + ihe angle bj)d = 0[m + - ); therefore the angle 



f &— 1\ ^ 



bdF=/jL0(m + - ), and the angle 



9 dF=fM6(in + ,J ^)-md=e(l+^iji^l). . . (3) 



Since the lines dF and da leave the same point d with different 

 inclinations, they will meet the axis at distances which are 

 inversely as these angles. Therefore 



da 1 + mu — 1 z 



f= g-i =^+™; 



therefore 



pa - t _ y^-i ; .... (4) 



by (1) and (2). This is the distance of the principal focus 

 from the surface of the lens. 



The distance ef of the apparent centre from the surface may 

 be found in a similar way. The normal at e makes an angle 

 n0 less than 0, such that 



n =-%r ( 5 ) 



The angle pet =(n + 1)0; the angle cef=fj,(n+l)0; and the 

 angle kef= [>(n + 1) - n] . Therefore 

 e f _ 1 , , RR^ 



This is the distance of the apparent centre from the surface. 

 % (6), 



* /(2R-0 ' 



Substitute this value of /* in (4); on simplifying, it will be 



