p 



-,[: 



260 Mr. A. S. Percival : Method of 



Differentiate (a) again with regard to i/r : 



v" ^ in - n dd u o , . , dxj) 1 9 . , . ,,d<b r 



£r- = - sec 2 (9 sin #— — -+- -sec 2 6sm<p~r s ^ c 9 sin 9 Vr 



i' 2 a d-v/r .^ r t T ^ r t ^ 



a a a r r r r r r 



-*-— = — iL~. \u?r z sec 3 9 + fi 3 a? sec 3 6 — a 3 sec 3 6'~\ . 

 p aV L r r r J 



(iii.) p=p+p l '=p(l + ^ 



/jpr 3 sec 3 6 + //, 3 a 3 sec 3 </> — a 3 sec z cf> r 

 (jjur sec # + yu-a sec 9 — a sec <£') 3 . 



Now if C be regarded as the origin of the p, ty equation 

 to the Caustic (which it is unnecessary to find) as/NP is the 

 tangent to the Caustic at the point /, and as fN=p* and 

 iSTP = ?i, the distance /P is given by p' ~{-n where n=r cos <j>' . 

 Hence any point / can be readily found on the Caustic and 

 the radius of curvature at that point is given by (iii.) . 



On now introducing signs o£ direction so that the formula© 

 shall hold universally when the symbols carry signs of 

 direction, it will be noted that in the diagram every magni- 

 tude is positive except <£, <£' and p or r sin <£' ; consequently 

 in (i.) the signs of </> and <fi' must be reversed, and so we 

 obtain 



(1) ^ = 0-<fr + f. 



It will be convenient to regard p as measured always in 

 the direction CN, whatever that may be : for p (Q'f in 

 fig. 1) the radius of curvature is always parallel to CN, and 

 when p carries the same sign as p the radius is drawn in the 

 same direction as CN, but when p carries a sign opposite to 

 that of p it is drawn in the direction NC. 



When the incident light presents a plane wave-front, 

 = and a becomes infinite, but in that case 



1 /m , a , sec 4>' 



-= — ^ + ^sec<£ -> 



p a cos r r 



r 



SO p = 7 T7 



1 fi sec 9 — sec q> 



f /a 3 sec 3 <b — sec 3 <fi' \ 

 and p = P l l- {/ , seC(f> _ sec(f>r j- 



If the Caustic by reflexion at a spherical surface be con- 

 sidered, we have only to substitute —1 for fi, and put 



