= A. 



266 Mr. Moseley on Caustics. 



Eliminating x and j/ between equations (2) and (3) and the 

 given equation to the reflecting curve, we have the equation in 

 X and Y to the caustic. 



On Caustics by Refraction. 



The oval whose equation is 



(x- + 7f)^ + m{{a — xf + y}s = A 

 will refract rays accurately from the origin to a point in the 

 axis at distance {a) from it *. Let the coordinates x, y be 

 transferred to an axis inclined at an angle (5) to the former; 

 then 



{ (.r cos 6 — j/ sin fl)- + (.r sin fl + j/ cos 6)^j 2' 

 + m {a + y sin fl — jr cos &f + {x sinS + y cos Qf}^ 

 Which equation involves four arbitrary constants or if (?m) be 

 given three ; hence the curve may be made to have from a 

 given origin a contact of the second order with any given re- 

 fracting curve, and the medium being the same, both curves 

 will refract a small pencil of rays to the same point ; the locus 

 of this point will therefore be the caustic required. Let X 

 and Y be the coordinates of this point ; then we have, 

 {j:^ +y}^ + m{{X -xf + (Y -yfY^ = A 

 And proceeding as in the last case, we may obtain equations 

 determining the caustic. 



In the case in which parallel rays are incident on a refract- 

 ing curve : let there be taken an ellipse having its axis pa- 

 rallel to the direction of incidence, and its ellipticity, or the 

 ratio of its semiaxis major to its eccentricity, equal to the ratio 

 of the sine of incidence to the sine of refraction ; and let it be 

 made to have a contact of the second order with the refracting 

 curve in any point P. Then it is evident that a small pencil of 

 rays incident at P, will be made to intersect in the further focus. 



Calling (X, Y) the coordinates of this focus, {x,y) those of 

 the point P, and [a) the semiaxis major of the ellipse ; we have 



^J^ + l^-Y-«"^-'f = «^ (1) ^ 



- riS + y {i^ - Y - «?«-• \ = (2) ' 



\ 



I 



J 



__!_ + ^f + y I ^ _ Y- «,„-■ \ = (3) 



•. by equa. (3) y -Y - a ni' = - ^-±SLz_!^vl 



•.by (2) ^^^ - y(t + (i- m-^)y°-} 



y (1 -7«i) 

 * Coddington's Optics. 



.'. X —'K 



