266 Mr. A. S. Percival : Method of 



Before dealing with such a case, or with that of the plane 

 wave-front in fig. 8, it will be necessary to consider refraction 

 at a plane surface. 



Refraction at a Plane Surface. 



Let OP be the plane surface and OT the initial line normal 

 to it, on which is situated the source S (fig. 5) . 



Let iSP incident at P be refracted as PR, where = LPS 

 or OSP, and 0' = LPN = OTP = PON, and let ON or p be 

 perpendicular to PT. 



Then if SO = 5, p = OPcos <j> f = b tan <j> cos <£' and n or 

 NP = OP sin <£' = b tan <j> sin 0'. 



As </>' now replaces yfr, p' is obtained by differentiating 

 p with regard to </)'. 



'=6 (cos <£' sec 2 - ^, —tan <£ sine// j. 



But as sin <f>=fi sin <£', -^ = //, sec $ cos <j>', 



r 

 1 



so p' = b{fx sec 3 <£ cos 2 <£/ — tan<£ sin <£'), 



and p ,f =zb\ 3/x cos 2 <// sec 3 <£ tan <£ ^, — 2//. sec 3 <f> cos <£' sin 



sin (f)' sec 2 </>;t7/ —tan </> cos 



= b (3yu, 2 cos 3 <£' sec 4 <£ tan (/> — 3 cos 0' sec 2 </> tan <£ 



— tan<£ cos<£'), 



p =^ +p" = 6 tan cos <p' (1 -f 3yu- 2 cos 2 (/>' sec 4 c/> 



— 3 sec 2 — 1) 



= Zb tan (/> cos 0' sec 2 <£ (//, 2 sec 2 cos 2 <£' — 1) , 



or 3p sec 2 (/a 2 sec 2 <f> cos 2 0' — 1). 



As before /NP is the tangent to the caustic at the point /, 

 and /T=/N-fNP=^' + ^, and as all the magnitudes in the 

 diagram (b, cf>, <f>') are of the same sign, the formula? tabu- 

 lated below are universally correct when signs of direction 

 are introduced. 



Table II. 



Refraction at a plane surface. 



p=b tan cos 0' 

 it — b tan sin0 



fP = bfi sec 3 <p cos 2 0'. 



p = %> sec 2 0(/x 2 sec 2 cos 2 0' — 1). 



