by 



„>x 



.o\ 



500 Mr. T. Chaundy: Method of Line- Coordinates for 



(3) Ihe Symmetrical Optical System. 



The optic axis we take as axis of x, the direction of 

 ongoing light being the direction of x 

 increasing. The frame of reference will Fig. 2. 



always be a right-handed set of axes 

 arranged as marginally indicated. 



Deviations from the optic axis will be 

 regarded as small quantities ; quantities 

 measured along the axis as of appreciable 

 magnitude. 



Hence, m, n, y, z, are small quantities >S ^z 



of the first order ; x, I are not small 

 quantities, I differing little from unity. 

 In fact, 1 — Z = (m 2 + n 2 )/(l + Q is a small quantity of the 

 second order. 



Again, b = lz — nx and c = mx — ly are evidently small 

 quantities of the first order, while a = ny — mz is a small 

 quantity of the second order. 



Consider refraction at a spherical surface of radius r 

 whose vertex is at the origin and whose centre is accordingly 

 at the point (r, 0, 0), if we adopt the usual convention 

 that surfaces convex to the oncoming light are of positive 

 curvature. 



The normal at (x, y, z) is the line proceeding from that 

 point to the centre of the sphere. Its direction cosines are 

 accordingly 



{ — {a?-r)/r, -y/r, — z/r\. 



Substitution of these values for L, M, N in equations (5) 

 gives for refraction at a spherical surface 



/* \ fju J r 



i m I nr COS 0\ V 



m = ( cos 6 £ ; 



fju \ /x J r 



, n I al cos 0\ z 



n = I cos 6' ) - . 



fx V fJL J r 



>. 



(6) 



It follows that 

 n'y 



Vz 



, ny — 7nz 

 ■ mz = — : 



n x = 



Iz 



■nx 



+ zl cos 6' 



[ a, cos V\ 



/ n/ cos 0\ 



-n oos * ir)- 



