1801.] Diffraction at Caustic Surfaces. 133 



and the radius of curvature of the evolute or caustic is 



_dR = Sb 

 P ~dcf>~ (2af 



We have to determine the disturbance, at a point (£, l/2a) in 

 the focal plane of the origin, due to the propagation of this wave. 

 If the amplitude of the motion in the wave-front is t sin 2tt£/t 

 per unit length, the value required for the point in question will 

 be 



where for the part in the neighbourhood of the origin 



«-{«-7+(iH? 



= 7~* [1 " 7"f« + *7 4 ?V - i (j'ba" + 7 4 f + yT) «'+••■) 

 correct as far as terms in x 3 ; where y~ 2 — (4a) -2 + £ 2 . 



This integral is to be taken throughout the extent of the 

 wave-front. The phenomena of optics show however that it is 

 only the parts of the wave-front in the neighbourhood of the 

 normal that are efficient in producing illumination along the 

 normal, for the more remote parts may be blocked out without 

 affecting it. The integral may therefore be confined to the im- 

 mediate neighbourhood of the origin, and we may proceed by 

 approximation. Taking £ to be small of the same order as b, we 

 have as far as cubes 



r = 7 _1 — y%x — ^yba~ l x 3 , 

 ds=dx(l + 2aV + §abx s ), 

 and i is of the form i = i (1 + ax + fix 2 + yx 3 ), 

 t being the amplitude at the origin. 



J V t \y A, a\ J l 



Writing x' = x + \ax 2 + £ (2a 2 + /3) a fi , 



so that x = x'-\ ax' 2 - i (2a 2 + /3) x' 3 + } aV 3 , 



) ( 27rf _ 2tt + 2^y| ^ _ tttoI ^ 



( T X7 X \ 



+ i [- %iry%{2a 2 -f/3) + 7T 7 £cr + tt 7 6] a*"} aV, 



A, J 



12—2 





this becomes 



