34 Mr. C. V. Boys on Measurement of 



employed, and by them the curvature of the whole surface, if 

 spherical, determined. 



On the front surface take any point p, and through it draw 

 a radius mR of the back surface. Join p with/, the apparent 

 centre of curvature of the back as seen through the front sur- 

 face. Draw also through p a radius ab of the front surface 

 and a line dc parallel to the axis. Then the angles mpd, dpb, 

 ape are equal to one another. Call these angles 6. The 

 angle apf—f^x angle mpb=/j,20 ; therefore the angle cpf= 

 fi26—d/ But the angle epH=0; 



- = 2/*-l, or p=*-jjr- 



By the property of equiconvex lenses, ^=^ + 1 ; 



R U-f , ^ Ff 



... 2F=1/- and R= F=/ ; 



or, in a thin equiconvex lens, the radius is equal to the 

 product divided by the difference of the principal focal length 

 and the apparent radius of the back as seen through the front 

 surface. 



It might be expected that, as this formula has been deduced 

 from a specially simple case, a more complicated one would 

 be necessary if the two sides of the lens were not equally 

 curved, or if one surface were plane or concave. But such is 

 not the case ; the same formula applies in every possible case, 

 though, as will be shown, experimental difficulty occurs in the 

 case of a diverging meniscus. 



The proof of the formula in the case of a thin lens which is 

 not equiconvex is similar to that already given. Make the 

 same construction as before, and let R 1? R 2 be the centres of 

 the surfaces 1 and 2, and J\ the apparent centre of 1 seen 

 through 2.. Also let R 1 = ??R 2 ' Call each of the angles mpd, 

 cpRi, 6 ; then the angles dp~R 2 and ape will each equal ?i0. 

 As the angle apj^fj, x angle ?npR 2 = ^(l + n)6, .'. angle cpf x = 

 fM(l + n)9—n6. But angle cpB. x =0; 



.'. ^ = p(n+l)-n. 



R 



Substitute ^ for n, and it will be found that 



B,(/, + B,) m 



