268 Mr. Moseley on Caustics. 



for the equation to the caustic by reflexion in the case of pa- 

 rallel rays. 



On the Involutes of Caustics hy Reflection and Refraction. 



Let the consecutive 

 rays SP and SP' inci- 

 dent fi'om S on the re- 

 fracting curve AB, be 

 made after refraction 

 to intersect in M. 



Now a curve, such 

 that for every point of 

 it SP + ?«PM = con- /a 



stant, may be taken so as to have a contact of the second 

 order with the curve AB in P, and therefore so as to coincide 

 with it in the consecutive points P and P'. 



Also in this curve, and therefore in AB 



SP -f ?«PM = SP + /«P'M 

 .-. ^ + PM = — = FM. 



m iu 



SP 



Produce MP and MP' to Q and Q', so that PQ = - - 

 P'Q' = -^, 



then PQ + PM = P'Q' -f PM, 



or, QM = Q'M; 



therefore the locus of Q is the involute to the locus of M, i.e. 

 it is the involute to the caustic. 



Let (X, Y) be coordinates of Q, {x, y) of P, 



then ^,-^Y+lY-yY = ^^ (1) 



Now I have demonstrated in the Annals of Philosophy for 

 July 1826, that if there be two curves such that the position 

 of the tangent to any point in the one curve is given in terms 

 of the coordinates to a corresponding point in the other curve, 

 then establishing any relation whatever between the coordi- 

 nates of these two points, we ma}- differentiate the equation 

 expressing this relation with regard to the coordinates of the 

 latter point, considering those of the former as constant. 

 This principle is manifestly applicable to the case we are in- 

 vestigating, since the position of QM and the distance PQ 

 are given in terras of the coordinates of P, the quantity m, 

 and the position of S ; also the position of the normal QM 

 and of the point Q determine the position of the tangent 

 at Q. 



Therefore 



