on we-ha-ve tlio following equations, 



d u _j_ d v o. 

 uf. cos 



also v 



/cos <p ' 

 du 



2 dp __ rfr* dl S- -~ 



p u 



2. Given the radiant point and the reflecting surface, to find the cau- 

 stic. 



Let p and w be the perpendicular and radius vector of the reflecting 

 curve, p' and u' = do. of the caustic, the rest as before, then 



and p' 2p A/ j __ P 9 . 



u* 



For v put its value U ~ ^ S ' ^ or -- and for p the proper func- 

 c-./cos? dlog PL_ 



tion of ^* given by the equation to the original curve, let u be then eli- 

 minated, and we shall have an equation in u' and p' t which will be that 

 of the caustic. 

 Ex. Let the reflecting curve be the log. spiral. 



Here p = mu, v 



_ 

 p' =2m u' Vl m* ; u't = 4 w8 _ 4 m 



u =r 4 wz (1 *) ; hence 

 u 

 p 1 = mu' ; the caustic is therefore another log. spiral. 



An equation in rectangular coordinates may also be obtained, but the 

 method is too long for insertion here. 



There are some simple cases in which it is easy to determine the nature 

 of the caustic by geometrical investigation ; for instance, when the re- 

 flecting curve is a circular arc, and parallel rays are incident in the plane 

 of the circle ; or when the focus of incident rays is in the circumference 

 of the circle, the caustic in either case may be proved geometrically to 

 be an epicycloid. A^ien the reflecting curve is a common cycloid, and 

 the rays are incident parallel to its axis, the caustic is also a common 

 cycloid. 

 47 



