Tracing Caustic Curves. 271 



On substituting /O for b in Table II. we get 

 ON or p 2 =/P 2 sin 2 cos 2 ', 



TTT} ^13 cos 2 <f>;/ 

 A P 2 or r =/ P 2 f~- , 



//-COS 2 2 



3p 2 f cos 2 </> 2 ' 



3/>2 f COS" 2 1 ^ 



^ COS 2 00 I A6 2 COS 2 02 J ' 



It should be noted that in tracing the caustic of a lens all 

 the angles are required, for the first surface only /P 2 and 

 sometimes p u but for the second surface YP 2 and p are 

 necessary. 



It will be noticed that the usual form for finding the first 

 focal line (/Pi) has been given for a plane wave-front 

 as this is speedier for logarithmic calculation ; when a is 

 finite and given, the method now suggested is the speedier, 

 for a troublesome preliminary determination of u in terms 

 of a, r, and l5 must be made before the usual method can 

 be applied. 



As an example of class A (2) the caustic is drawn of a plane 

 wave-front incident upon the convex surface of the same 

 planoconvex illustrated in fig. 4. The caustic in fig. 8 in 

 full lines is due to this lens when stopped down to be of the 

 same aperture as the lens in fig. 4, i. e.. when 3 = 41° 2 ! 2S 1 ' : 

 the focal length F"H"= — 9'5602 in each. It is immediately 

 apparent what an advantage is gained by letting the curved 

 surface face an incident plane wave-front. Indeed, if the 

 aperture be increased to admit rays whose incident angle 

 is 60° or even more, the caustic is still much smaller than 

 that shown in fig. 4. 



All the lines which it is necessary to draw are indicated 

 on the diagram ; a set square will indicate the perpendicular 

 (p 2 ) from on an emergent ray, and G'v is drawn parallel 

 to p 2 , its direction and length being given by the sign and 

 figures under p in the explanatory table beneath fig. 8. 



B. The Bispherical Lens. 



The construction of fig. 9 (p. 273) requires little explan- 

 ation : C^Pj is the radius of the first surface, C 2 P 2 that of the 

 second; LSP 1 = 1? (\Y^ = ^, 0^ = 0/, O 2 P 2 T=0 2 , 

 C 2 /'N = 2 , OiMM' and CoN are each at right angles to 

 P 2 PiTM. 



