IN THE NEIGHBOURHOOD OF A CAUSTIC. 391 



The extent of this table for the positive values of m is not so great 

 as I could wish ; but it goes far enough to enable us to point out the 

 most remarkable circumstances of the distribution of illumination. 



18. From m = — 4*0 to m = — 1-6, the illumination is almost in- 

 sensible. (In fact it appears to diminish, as the negative value of m 

 increases, in a nearly geometrical or perhaps hyper-geometrical progres- 

 sion). It then increases rapidly, and acquires its maximum value when 

 m = + 108 nearly; its value is then nearly 1001. It then diminishes 

 rapidly till m = + 248 nearly, when the illumination is zero. It then 

 increases till m = + 347, when the illumination is nearly 0-615, or about 

 three-fifths of its former maximum. It then diminishes rapidly to the 

 end of the table : and appears likely to become zero for a value of m 

 differing little from + 4'4. 



19. One of the most important points to be remarked is, that the 

 maximum illumination does not take place at the Geometrical Caustic, 

 or where m = 0, but where w = + 1-08, that is, on the external side 

 of the convexity of the caustic, or on the luminous side of the geome- 

 trical position of the rainbow, that is, (for the primary bow,) within it. 

 The following rule derived from the numbers above, will suffice, in 

 practice, to determine the geometrical position. When the first spurious 

 bow is visible, measure the distance of its maximum intensity from 

 that of the brilliant bow ; then the geometrical bow is exterior to tlie 



brilliant bow by — - of this distance. 



•^ 24 



20. It is a matter of curiosity to ascertain the relation of the 

 intensities, or at least of the places of maximum and minimum inten- 

 sity, as determined thus by a complete investigation on the theory of 

 undulations, with those which would be found on the imperfect theory 

 that light proceeds in straight rays according to the laws of Geome- 

 trical Optics, and that rays of light are capable of interfering ac- 

 cording to the simple rules of interference. We have first to discover 

 the position of the two rays which interfere at any point. Now the 



