39 



curve, the pencil point will describe another curve, which I 



call the Unipolar Convolute of the given curve. 



In fig. 1 let VV be the given curve, the fixed point, and 



PP' the convolute. Then it follows that if 



OP = r, ?Y = l, and YY' = d>r, dl + dr^da 1 



Hence we get a 



geometrical method 



of describing the 



curve which, with a 



given radiant point 



shall produce a given 



caustic by reflexion- 

 All theorems respecting caustics are thus capable of a 



geometrical solution. 



It has been noticed and is easily proved that if r, p be 



the coordinates of the curve (P), and p, tt those of (V) 



d)' 

 dp 



v/r2-7r2+v^p2_^2^. 



dr 



y.p — n-T- 



:i' - p 



2p 



djo 



and 7r=-^^/;-2-;/^ 



from which equations 

 the convolute may some- 

 times be found. But, iii 

 order to remark the pro- 

 perties of the curve (P) 

 with regard to (Y), 1 

 have used pedal coordi- 

 nates. Let the coordinates of P be (r, B) as before and those 

 of D be (tt, o)). 



It may be observed that the relation above remarked 

 follows from regarding the curve (P) as the envelope of a 

 family of ellipses, whose fixed focus is 0, and whose instanta- 

 neous focus lies on the curve (V). 



