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



a more exact view of the phenomena is that of the theory in- 

 vestigated by Sir George Airy for the case of the rainbow, on 

 the basis of Fresnel's theory of diffraction. It was shown by 

 him that, outside the real caustic of maximum concentration, 

 the energy of the undulations gradually fades away, so that with 

 very minute wave-lengths the boundary of the caustic is quite 

 sharp ; but that inside the caustic there is presented a series of 

 successive maxima and minima, in bands running parallel to the 

 absolute maximum or caustic surface. 



As the calculations of Sir George Airy had reference chiefly 

 to the phenomena of supernumerary rainbows, he only cared to 

 obtain the relative distances and illuminations of the succession 

 of bands along the asymptote of the caustic. 



But the peculiarity of this case of diffraction is that there is 

 no question of an aperture limiting the beam of light, so that the 

 degree of closeness and other relations of the bands must depend 

 only on the character of the caustic surface itself, along which 

 they run. The law, connecting these elements, which is thus sug- 

 gested for investigation, comes out to be very simple. It appears 

 that for homogeneous light the system of bands is similar to 

 itself all along the caustic, as regards relative positions and 

 relative brightness, and that they are therefore similar to the 

 supernumerary rainbows calculated by Airy and verified experi- 

 mentally by W. H. Miller; while the absolute breadths at different 

 parts vary inversely as the cube root of the curvature of the 

 caustic surface along the direction of the rays. For different 

 kinds of light the breadths vary as the wave-length raised to the 

 power two-thirds. These laws are exact for the first few bands, 

 usually all that are visible, owing to the extreme closeness of the 

 subsequent ones; they form in fact the physical specification of 

 the nature of caustic surfaces. 



2. These statements will be verified in the course of the 

 following analysis of the diffraction near the surface of centres of 

 a wave front, which forms the natural extension of Sir George 

 Airy's investigation for that portion of the caustic which sensibly 

 coincides with its asymptote. 



In the first place, taking a cylindrical wave-front, and referring 

 it to the tangent and normal as axes, we have for its equation in 

 the neighbourhood of the origin 



z = ax 2 + bx 3 + 



The inclination of the tangent at the point x is <£ = 2ax, the 

 radius of curvature is 



R = O'llV = 1 ( x _ ^ x 

 \dx 2 / 'la V 2a 



