of Lippich s in Geometrical Optics. I'll 



<£. &' being the angles made by the projections with the 

 normal at their point of intersection. These are equivalent 

 to two equations. The second expression is indeterminate 

 since yJ simjr' = /z sin yjr, so the fourth is used instead. Two 

 conditions are necessary in this case, because projection did 

 not take place on a single plane as in the tangential case, so 

 we have to show that the rays and normal are coplanar, 

 besides satisfying the angular relation. 

 Now 



COS = — COS VT -f - = — COS -Ur, 



T cos a. J 



since N, z' , and a are small. Similarly, cosc/>' = — cosi/r'. 

 Hence one condition holds, namely 



fJL cos yjr' — /a cos i|r= — /x f cos <£/ + /jL cos (£. 

 The remaining condition is 



, , i , COS a COS a 

 jJL cos y — fl cos y = , . 



The equations of refraction of the original ray give 





— cz 



Eliminate z\ since the result is to hold for all small values 

 of z' \ and the required condition becomes 



fl'Jj — fJih yU-'N' — flN 



fju' cos yjr' — p cos yjr /x'N' /mN 



COS a COS (X 



This evidently holds, since a, a,' are small angles so that 



, = 1 approximately," 



cos a cos a. L l '' 



and differentiation of the identity /ul' sin \Jr' = fjLs'm yjr gives 



yu/L' — yltL = yu/ COSl// — yU-COSi/r. 



More exactly, differentiating p sin yfr' =y sin yjr we get 

 fjb COS yfr' . cZi/r' =yu, COS i/r . £?^r ; 

 also L = COS (>|r + «), L' = COS (i// + a'), 



