134 Mr Larmor, On the Laws of the [Jan. 26, 



or say 



which 



( [sin j 2 ^ (t - ~) + Act + Bx 2 + cA dx\ 

 = lJ)~ 1 (sin ]-—(* — const.) + \k (w a — mw) dw, 



where w = D ( x + ^ n J , so that £77 D 3 = (7, 



3(7, 



and - \irm = i ^ - ^J = (£ir)* 40-* (l - ^J ; 

 which gives on reduction, writing unity for j/2a, 



- m = 6"* (|j* 2af |l - £ (K - &a» " 1/3) I j , 



so that 



2a£ = m (i\f b h {1 + \rno-* (\\f (\d l - ^a 2 - $0) £}, 

 or in terms of the radius of curvature (R = l/2a) of the wave-front 

 at the origin and the radius of curvature p of the caustic 



3. Neglecting the term of the second order in this expression 

 we have £ = — m ( ~ J , 



showing that the course of the ray caustic is bordered by a series 

 of fringes which remain similar to each other throughout, and 

 therefore are of the same type as the asymptotic fringes of Airy's 

 supernumerarjr rainbow; but they come closest together at places 

 of greatest curvature of the caustic according to the law that 

 their separation at any place is proportional to the cube root of 

 the radius of curvature of the caustic at that place. 



The investigation shows that unless for fringes at a considerable 

 distance from the ray-caustic their form is not sensibly affected by 

 the varying intensity in the wave-front. 



As we proceed along a caustic, the curvature gradually in- 

 creases and the fringes therefore come together when we approach 

 a cusp. At the cusp itself 6 = 0; and very near to it b is very 

 small, so that to determine the state of matters for an unlimited 

 beam another term would have to be included in the equation 

 of its front; but if the beam is limited in any way the fringes 

 produced by this limitation will there rise in importance, and 

 practically obliterate the ones now under discussion 1 . 



1 See Rayleigh, "Investigations in Optics," Phil. Mag., Nov. 1889, pp. 408—10. . 



