380 Mr airy, on THE INTENSITY OF LIGHT 



are then to be compounded,) should not apply to those cases. For 

 though, strictly speaking, we ought to consider the wave to be thus 

 broken up where it leaves the first surface, in order to find the inten- 

 sity of vibration at every point of the second ; yet it seems clear, that 

 those reasonings which establish the definite reflection or refraction of 

 a wave, (and which are founded upon the consideration above alluded 

 to,) point out that there will be as to sense a mutual destruction of 

 all vibrations at the second surface, (supposed to be not distant from 

 the first,) excepting those which woidd be fully taken into account on 

 the ordinary laws of Geometrical Optics. Where the light meets the 

 second surface in the state of convergence, this conclusion perhaps is 

 not so clear : but even there I believe that it may easily be shewn to 

 be correct. I have mentioned these points because one of the most 

 interesting cases of natural caustics (the rainbow) is affected by them; 

 the exterior bow involving the first-mentioned condition, and the in- 

 terior bow involving both the first and the second. 



1. The notion of a caustic, and its mathematical definition, are 

 essentially founded upon the laws of Geometrical Optics ; and to these, 

 therefore, we must refer in order to discover a representation of the 

 conditions adapted to the investigations of Physical Optics. For sim- 

 plicity we shall confine our diagrams to the plane of reflection, and 

 shall consider the reflecting surface as symmetrical (to a sensible 

 extent) with respect to that plane, so that the portion of the caustic 

 formed by that part of the surface will be in the same plane. 



2. In fig. 1., let the origin of light S be the origin of co-ordi- 

 nates ; X, y, the co-ordinates of a point X of the reflecting surface ; 

 jj, q the co-ordinates of a point P in the reflected ray ; V the length 

 of the path of light from S to any point of the reflecting surface and 

 thence to the point P. The ordinary law of reflection informs us that 

 the angles of incidence and reflection are equal; and therefore, that, 

 if we take a point X' on the reflecting surface very near to X, and 

 join it with the origin and the point P, the lengthening ZX' of one 



