268 Mr. A. S. Percival : Method of 



by the previous method. To avoid confusion in the diagram 

 /Pi has been made greater than its real length for this angle 

 of incidence. 



Fig. 6. 



Produce /P x to P 2 on the curved surface of the lens, draw 

 P 2 L perpendicular to the axis SCO, and let CM and 01$ 

 (or pi) meet/TP 2 at right angles ; join/C and CP 2 . 



The point / is the radiant point for the second surface, 

 /C is the part of the initial line that is represented by SO or 

 a in fig. 1, and CP 2 / or fa is the angle of incidence, and 

 C/P 2 or 6 2 is the angle which /P 2 makes with the new 

 initial line. As in fig. 5, SO = b, ON=jo b OSP^^, and 

 OTP 3 =<fc'. 



Prom fig. 6 it is readily seen that j 



(i.) CM=rsin^ 2 =TCsin^ = (T(|)-CO)sin;^ 



= (6 tan fa cot fa' — r + t) sin <£/. 



.*. ?*sin fa = bt&n$i cos fa r + (t — r) sin fa' 



or piH- (t — r) sin fa'. 

 (ii.) PiP 2 cos fa'~OL = t—r vers (fa' + fa). 

 CM_ r sin fa 



(iii.) tan # 2 = ^tf 



/P 1 + P 1 P 2 — r cos fa' 



(iv.) /C or a 2 =/Msec0 2 = (/P 2 — rcos </> 2 ) sec # 2 - 



In the diagram all the magnitudes are positive except fa, 

 so to make the formulae of universal application when 

 direction-signs are added it is only necessary to change the 

 sign of 2 as has been done in the table. It has been shown 

 in fig. 4 with what extreme simplicity the caustic can be 

 traced when the light presents a plane wave-front with 

 normal incidence. If the plane wave-front be incident at an 

 angle a on the plane surface and refracted at an angle a! , 

 the initial line for the spherical surface will make an angle 



