36 Mr. C. V. Boys on Measurement of 



equations (5) or (6). As the surface 1 becomes more con- 

 cave, its apparent centre on the other side of the lens will 

 retreat to an infinite distance ; and then the concave side will 

 appear flat when viewed through the convex surface. This is 

 the case when R x = — F, as may be shown by making /' 1 = oo in 

 (4), or as is obvious from a diagram. When E x becomes less 

 than this./'x becomes an imaginary point on the other side of 

 the lens, such that, if rays were sent so as to converge upon 

 it, they would return as though they had come from it. Its 

 position could be determined experimentally by the method 

 given for a convex surface on p. 33 ; but as^the true radius 

 can be determined directly, there is no necessity to find this 

 imaginary apparent radius. 



Let the concavity of surface 1 increase ; the next particular 

 case is that of a watch-glass, where 1^=— R2. Then F be- 

 comes infinite, and the two points f 2 and R x become coinci- 

 dent. When the surface 1 becomes still more concave, F 

 becomes negative and virtual, and R x and/ 2 pass one another. 

 The experimental determination now becomes more difficult ; 

 for neither can F or f x be observed directly; but still the 

 equations (5) and (6) hold. They may each be found by the 

 method for a convex surface, which is less convenient than 

 the direct method. 



If the concavity of the surface 1 continues to increase, 

 another limit will be reached, at which f 2 becomes infinite. 

 This is obviously the case when R 2 = — F; that is, when the 

 focal length has been so shortened by the increasing concavity 

 as to be equal to the radius of the convex surface. When 

 this is the case, the surface 1 seen through 2 appears plane. 

 When the concavity passes this limit, f 2 becomes negative and 

 imaginary, and the experimental difficulty is still further 

 increased, for I?! only can be directly observed ; but still the 

 equations (4), (5), and (6) are true. No further increase in 

 the concavity of 1 will produce any new conditions. Now, 

 the curvature of 1 remaining constant, let 2 become flatter ; 

 when it has become plane, there is no occasion to observe F,f\, 

 or/ 2 to determine the form of the surfaces. When 2 becomes 

 concave also, the curvature of each surface can be directly 

 measured; and all difficulty is removed. Every possible case 

 has now been considered ; and though the equations are always 

 true, experimental difficulty only occurs in the two classes of 

 diverging meniscus. 



If a parallel beam of light falls on the lens, it will be 

 refracted at the front surface, partly reflected from the back, 

 and again refracted at the back surface, and be brought to a 

 focus at a distance from the lens equal to half the apparent 

 radius. 



