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

 Equations (7) are rewritten 

 a' = a/fi 



ul + v{cos0'-(cos0)/i j l\:\. . . (9) 



v! '=zu/fi+ y {cos 0' — (cos 6)//ul} 



v' = v/fJL+z\cOS 6' — (COS 0)/fl\ 



.! 



Now it is known that if <f> is the angle between the 

 directions (l l9 m l5 rii) and (Z 2 , m 2 , n 2 ), 



cos </> = y 2 + m 1 m 2 + n 1 n 2 , 

 and that 



sin 2 cj) = (m 1 n 2 —m 2 n l ) 2 + (n 1 l 2 — n 2 li) 2 + (^m 2 — £ 2 m i) 2 - 



Hence, since # is the angle between (I, m, n) and 



{(#— r)/r, y/r, «/r}, 



sin 2 # = { (mz—ny)/r}' 2 + {(W — Iz — nr)/r} 2 + { (ly—mx + m7 j )/V} 2 



= {a 2 + (tf 4- wr) 2 + (w + mr) 2 }/r 2 



= { a 2 + (w 2 -|- v 2 ) + 2r (raw + wi') + r 2 (ra 2 -f n 2 ) }/r 2 , 



while sin 0' = (sin #)//* gives 0'. (10) 



(4) The First Approximation. 

 Collecting our four fundamental equations of refraction 

 m' = m/fjL — {cos 0'— (cos 0)'fi\y/r, "" 

 n f = ?i/fjL— {cos 6' — (cos 6)/fjb}z/r, 

 u'=u/fi+ {cosO' —{cos 6)1 fjb}y, 

 v' — vj/ub-h {cos 0' — (cos 6) I fjb}z, J 

 with the defining equations 



u = ly — mx ; v = lz — nx; 



we may now proceed to our first fipproximation. 



In equation (11) ra', n\ w', v\ ; ra, n. w, v, are all small of 

 the first order, as also are 0, 6'. We may therefore replace 

 cos 6, cos 6' by their first approximation, i. e. we may write 

 cos — - 1 = cos ! . 



This leads to 



> 



(ii) 



m! = m/fjb — ( 1 — l/fjb)y/r 



n' — nlfju — (1 — l/fi)z/r ; 

 v' = v/fju+(l-l/fi)z. 



Now, since (w, y, z) is a point on the refracting surface 

 which has its vertex at the origin, x is a small quantity (in 

 point of fact of the second order). Certainly then to the 

 first order # = 0, 1=1. 



