Investigating Aberrations of Symmetrical Optical System. 499 



Thus LL' + MM ; + NN' = 0, 



since these directions are at right angles. 

 Also ZL' + mW + »N' = sin 0, 



since these directions make an angle 90° — #. 

 Finally VL' + m'W + n'W = sin 0\ 



since these directions make an angle 90° — 0\ 



Fig. 1. 



raW 





'l,m,n 



















L,m.n 



l^v\ 







=> 



L,M,N 



If then we multiply the equations (4) in order by U , M', N' 

 and add, we obtain, in virtue of the results just established, 

 that 



sin0' = Asin#. 



But 



Hence 



sin =fju sin 0' . 

 A=l/ /t . 



Again multiply the equations in order by L, M, N and 

 add. We obtain 



L/'4-Mm' + N^ = A(LZ + M772 + N/0+B(L 2 + M 2 + N 2 ) ; 



i. e. cos 0' = A cos + B ; 



i. e. B= cos 0' — cos 01 \x. 



Thus the fundamental equations for refraction at a single 

 surface are 



m 



, I ( a cos0\ T 

 = — hi cos I L ; 



rc'= - + ( 



COS0'- 



cos # 



)"■ 



