144 



Mr potter, on THE PROBLEM OF THE RAINBOW, 



I regret my inability to deduce results perfectly rigorous by the 

 method which I have followed on account of the complicated and tran- 

 scendental nature of the relations between the quantities to be expressed, 

 but I have pushed the mathematical part of the investigation to as 

 close an approximation as the general discussion of the problem may 

 require. 



I have adopted the method, of first finding the caustic; because 

 this and a very numerous class of interferences is produced, not by a se- 

 paration of the original luminiferous surface into two separate surfaces, 

 as in the cases ordinarily considered, but by a reduplication of the 

 surface upon itself after reflection, or refraction, or both. In these 

 cases, as I have shewn in a paper read before the Physical Section, 

 at the meeting of the British Association at Cambridge, and published 

 at Brussels in M. Quetelet's ' Correspondance Mathenmtique et Phy- 

 sique,' there is an arete de rebroussement of the luminiferous surface 

 at the caustic surface, or, in the usual sections a cusp in the section 

 of the luminiferous surface at the caustic, and the former curve is 

 always an involute of the latter. Having once found the caustic, this 

 consideration enables us to proceed to the calculation of these compli- 

 cated effects with a close approximation to the accurate result. 



Proceeding first to find the expressions for the caustic when pa- 

 rallel rays have been twice refracted and once reflected in a transparent 

 sphere, as in the primary rainbow, and using the ordinary mode of 

 determining the caustic by considering it the locus of the intersections 

 of consecutive rays. 



Let (p= I oi incidence, 



(^'= / of refraction, 



