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differentiation, d( =: — rf.f — 1. We have also, by the same section, 



\dx^ ' dy^ ~\dx.dy) \ ' "d/i -'^•\'^- dx^ *' dx.dz )' 



d'V _ £V_ 

 dxdy ' dx-dz 



d*V 1 d^V 1 db 



which when we put «-0, y=l, dy=0, -^-^ = 0, ^^;^=0, 



X* -~ i,' dy- - «,' rf^ «^' 



gives by differentiation 



■ *© = -.^<^)=''^- + "'• 



dy' 



and therefore d^^ — {Bd» + Cd^). If then we denote by x^, y„ z„ the coordinates of the 

 caustic surface, considered as functions of a and b, we have 



X, = a + «{, y, == i + /3{, z, = c -^ yj, 

 rfar, = efa -f- ^dot, = t(f«, rf^^, = 0, rfz, = rff aa — (firf* + Crf/3) 

 fi;«y, = ji.c?'/8+ 2</,8.rfj = A.da" — C,dfi\ 

 so that the focus of a near ray has for coordinates 



X, = »•«,. y, = U^u,'-C/3,''), z, = e, — (B«, + CA); 

 eliminating »„ /8„ we find for the approximate equation of the caustic surface, 

 2i'C.y,+ {i.(2; — «J + ^^,| '- ACx,' =0, 



which shews, that the radius of curvature ^ of a normal section of thjs surface, is given by the 

 following equation, 



i'C 



—5- =: i*. cos. «« + 2iB. sin. «. cos. u + (B* — AC), sin. «<«, 



{«) being the angle which the plane of the section makes with the plane o({xz). Making « =0, 

 we get il = C ; and the maximum and minimum of R, are given by the equation 



- A.R' +(B' — AC+i'). R—C.i^ =0; 



from which it follows that C is the radius of curvature of the caustic curve, and that if we 

 denote by (») the angle at which this curve crosses either line of curvature on the caustic surface, 

 we shall have 



