358 Mr. S. D. Chalmers on the Sine Condition 



If we take ^ = or 180°, 



— = 3it sii^Ax + 5i 2 sin^H- By = ; 



if ^=90°, 



— = i*! si^Ai + ig sin 4 A 1 + % = 0. 



OS 



Refraction of Rays not m one plane. 



If the direction cosines of any ray be l m n before and 

 l 2 m 2 n 2 after refraction at the surface (1), the normals being 

 LxMiNi, we have : 



/Wo — 1^2 _ AW^o--/^^ _ /Wjo —/^2^2 /T x 



LT ~ Mi . Ki ' * * • * w 



where fi and /jl 2 are the refractive indices of the two media. 

 Then /^(^Mx— moLj) is invariant on refraction. 



We may choose the coordinate axes and origin so that 



Lift, M^, and ]*!>!+ ft 



represents the point on the surface at which the refraction 

 takes place. To specify the ray independently of these co- 

 ordinates we can take the points where the ray cuts the 

 planes os — and ?/ = 0. 



Let these points be 0, m p , d ; l p , 0, D , then 





Po(= 



d — D ^ 

 wo ) 





is the distance along the ray 

 We have 



between these points. 





M^ — m p 

 m 



Ni*i+G8- 



rc 



-<?o) 



giving 









(II.) 



(LxWq— Z Mi) r 1 =—l o m o p , 



or fi l m p is invariant on refraction. 



The condition that L^, M^, and N^ + ft represents the 

 point on the surface is satisfied, with different values of r x 

 and fti, for all points on a surface of revolution about the axis 

 of z ; and since /Vo w oPo * s independent of the position of the 

 origin it is invariant throughout any system having a common 

 axis. 



This invariant may be written /*(#iwi— -yiQ, where X\ and y x 



