264 Mr. Moseley on Caustics. 



Then we have by a pi-operty of the eUipse 



^ _ /S=r 

 ^ ~ 'la — r 



.: .'. log. — = log. /3- — log. (2« — r) 



7)2 

 rfloo 



Now p and ?• are the same in the reflecting curve and the 

 ellipse; and p, being in fact a differential expression in x 



and V of the first order, -^ is one of the second ; it follows 



therefore, that since the differential coefficients of the first 

 and second orders are the same in the two curves, r, p, and 



— ^ are the same, and therefore the expression ^-—^ is the 



d log. — 



r 



same. Taken therefore in the reflecting curve it will deter- 

 mine the length of the reflected ray. 



The above equation is given (deduced on other principles) 

 in Mr. Coddington's Optics. The reader is referred to that 

 ingenious work for a method of determining from it the egtia- 

 tion to the caustic in terms of r and p. 



In that particular case of the caustic by reflexion, in which 

 the radiating point is situated at an infinite distance, and the 

 incident rays are therefore parallel to one another. The equa- 

 tion to the caustic may be determined immediately in terms of 

 its rectangular coordinates, by considering it as traced out by 

 the focus of an osculating parabola, having its axis parallel to 

 the direction of incidence. 



Calling X, y the coordinates of any point P of the reflect- 

 ing curve, X, Y the coordinates of the corresponding point S 

 of the caustic (or of the focus of the parabola), and 4 « the 

 parameter of the parabola ; we have, 



(.r-X)2 = 4.« [j/- Y + «]' 



••• — ^=2^(4^) 



... ^. _X =-74^ (1) 



\ dxi ) 



y-\^ 



