384 Mr airy, on THE INTENSITY OF LIGHT 



and therefore V - V, which in (5) was found = X'P + PP - X P, is 

 = PP — QP. Let a; be measured nearly perpendicular to the caustic 

 at P; put p for the radius of curvature of the caustic at P, and <{> for 

 the small angle made by PX and PX'. Then Sx = PX.(p, therefore 



<P=^- ^"^^'^-Q'^ = ^-li-S = (Ay-r^3; Making this 

 equal to the corresponding term in Taylor's series for T^', we find 



_ c^ifn _ p 



da? ~ {PXf ■ 



9. Now take a point near the caustic, whose co-ordinates are p + S]j 

 and q {Sp being measured from the convexity of the caustic parallel 

 to X, or nearly perpendicular to the caustic at P). Let F be the length 

 of the path of light from the origin to any point of the reflector and 

 thence to the point p + Sp, q. Then we have 



V = Vx- +f + V{x -pY +(y- q)\ 



V^ = Vx'' + f + \/{x-p-^pf + {y-qy; 

 or V,= V + V{x-p-^pr + {y-qy - V{x-py + {y-q)\ 

 which, if we expand to the first power of ^p, becomes 



r = r - -^=£^£== ^;>. 



V{.x-py + {y-qy 



and therefore, in the general case of measuring V through any point 

 of the reflecting surface, 



dx dx ^' 



dx- dx^ 



A, B, and D, being finite functions of a-, y, p, and q. 



