Mr. Moseley on Caustics. 269 



Therefore differentiating the equation ( 1 ) with regard to x 

 and 3/. (^_X)-y(.r/-Y) = ^i^ 



■^-(^ +i/y) - (X + Yy) = (2) 



In the case of parallel rays, the equations (1) and (2) become 



(^-xr- + (Y-3/r = -^ 



^.; + i/y-(X + Yy) = o. 



The equations (1) and (2) together with the given equation 

 j/ = J'x of the refracting curve determine by the elimination 

 of X and i/, the involute to the caustic. 



Thus when the lefracting curve is a straight line taking the 

 axis of {x) perpendicular to it, we have 



X = constant = a i/ = x^y 



hence Yy — yy^ — = 



-IT m" — 1 



••• ^ =y--^- 



( } m* TO- 



hence Y^ = {vf- l)\{X-aY- (-^)' J . 



The equation to an ellipse or hyperbola according as {m) 

 is less or greater than 1. According to this condition, there- 

 fore, the caustic is the e volute to an ellipse or an hyperbola. 

 We may establish geometrically a relation between SP and 

 SQ, and the perpendiculars from S on the tangents at P and 

 Q by means of which the 

 involute to the caustic may be 

 readily determined in terms of 

 the radius vector, and the per- 

 pendicular on the tangent. 

 For let PM and QN be tan- 

 gents at P and Q, SM and 

 SN perpendiculars upon 

 them from S. 



Let SP = r, SQ = R 

 SM = 2J, SM = P 

 Draw PV perpendicular to PM 



XT sin PQS rS 



sin PSQ PQ 



Also PQ is perpendicular to QN, and .-. 

 .-. sin PQS = sin QSN 

 and PSM = SPY = <of inc". 



NSM = VPQ =< of reft" 



parallel to SN 



