Mr. Moseley on Caustics. 

 ^-Y =(g)'- l 



265 



(2) 



As before, the equations (1), (2) here proved with regard to 

 the parabola only may be shown to hold also in the reflecting 

 curve. By eliminating x andj/ between them and the given 

 equation in x and ij, the required equation in X and Y is 

 readily determined. 



To investigate the equation to a caustic formed by rays di- 

 verging from a point at a finite distance in terms of its rect- 

 angular coordinates. 



Let S be the radiating 

 point, AP the reflecting 

 curve, VPV an ellipse hav- 

 ing a contact of the second 

 order with the reflecting 

 curve in P, and having its 

 focus in S. The given po- 

 sition of the focus determin- 

 ing two of the arbitrary con- A' "V^ 

 stants, and the contact, the remaining three. 

 is the caustic required. 



X, Y coordinates of PI "1 on any axis OX passing 



Xy y P / through S, 



VV = 2«. 

 .-. [(X - xf + ( Y - yf\ ^ + [^« + y]i = 2a. 



Now by supposition, y and j/" or -^, -^, and also x and 



y are the same in the reflecting curve and in the ellipse .'. dif- 

 ferentiating twice with respect to x and y, the resulting equa- 

 tions will hold in both curves 



The locus of H 



( X-x) + (Y-yy _ yy' ^ x 



Hence, taking the h. logs, and differentiating again, 



(2) 



-i-y» + (Y-.v)y' 



+ 



(X-x) + (Y-y) 



(x-x) + (Y-3,)y 



+ , 



t \-y''--\- yy" 

 yy + X 



^ +yy' 



i^-'l' + i^-yy J ^ ^'+y' 



whence by elimination in equation (2) and by reduction, we 



X + Yy x-\-yy' 



■ -^ I \+Yy' S (x" + yO 



New Series. \'()l.2. No. 10. Oct. 1827. 



0.'- + '/' 



( .»/ - xyy 



2M 



. (3) 

 Elimi- 



