Mr. Moseley on Caustics. 267 



.-. -^ - X = -J^l 1 + (1 - nn2/"\ («) 



by (1) .-. y-{x + (i-m-°)y'' }^ {j_+ (i-m-py^- _ 



{l + ( l-m-^ )y-4^ 



— y (1 -TO^) 



• ,, V l+(l-m-^)/^ , {i+(i-^-.)y-.} f 



7/ (1 — W-) ^ (1 — 7?l-) 



also,;c-X = ^^l +(1 -?«■")/" ^ («) 



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

 given equation to the refracting curve, we obtain an equation 

 in X and Y to the caustic. 



If ?K = 1, and the sine of incidence becoming equal to the 

 sine of refraction, we pass from a refracting to a reflecting 

 medium. Calling 4 j> the parameter of the parabola into which 

 our ellipse will have resolved itself, we have since 1 — vi'^ = 



a'J — aP-m^ 2p 



a~ a 



neglecting powers of — above the first 



„_Y=-— 5-j - ^--^yi 



^-Y=-^^ 



2y 1 



also, T — X = -^ by equ" 

 The above equations coincide with those already determined 



* This expression for the semiaxis major of the osculating ellipse evi- 

 dently reduces itself, as it ought, to the known expression for the radius of 

 curvature, when «''= 0. 



2 M 2 for 



