1891.] Diffraction at Caustic Surfaces. 135 



4. The results of this analysis indicate how we may proceed 

 in the general case where the wave is not cylindrical, but is curved 

 in two dimensions. 



Referred to the normal as axis of z, and the tangents to the 

 arcs of principal curvature as axes of x and y, the equation of its 

 front is 



z = ax* + bf + px* + Sqx 2 y + 3rxy* + sif 4- ... 



It is required to find the disturbance propagated to the point 



1 



6 0,. 



m 



Here 



-(r+e-sf+'+v-i 



= y" - 7 jf* + (i - £) y" - v* - H«?y - 3» V - «/} , 



so that I It sin ( — -^jdxdy 



will be complicated. 



But the considerations already mentioned show that the value 

 of the integral is practically settled by the elements in the neigh- 

 bourhood of the origin, for which x and y are small. We may 

 therefore consider only a small rectangular portion of the wave- 

 front bounded by arcs parallel to the axes of x and y. To deter- 

 mine the diffraction in the plane z = l/2a, we may consider only 

 the plane problem presented by a wave of the form 



z = ax 1 + px 3 ; 



for the uniform curvature in the perpendicular plane represented 

 by the coefficient b will not affect the result at all, as is also 

 obvious on continuing the general calculation. The variation of 

 that curvature represented by the coefficient r will slightly dis- 

 place the fringes, as it will alter the mean value of y%. 



The dissymmetry indicated by the coefficients q and s will on 

 the average produce no effect on the disturbance at a point in the 

 plane xz; these coefficients introduce odd powers of y which 

 integrated over equal positive and negative range leave no 

 appreciable result. 



Thus the illumination in the plane z= l/2a is determined by 

 the values of a and p only. 



