IN THE NEIGHBOURHOOD OF A CAUSTIG. 387 



If we take only the principal part of each coefficient (which for any 



practical case will be abundantly sufficient, observing that ^ will 



b - q 

 probably never, in observation, amount to tan 2"), this becomes 



i b-q 2 f^_^ ^Sa' 



In the applications of this, it will be important to notice that the 

 coefficient of x' is independent of p and q, (depending only on the 

 dimensions of the rain-drop or other refracting or reflecting body,) and 

 that the coefficient of x depends only on the angle made by PX with 

 the asymptote. 



13. It appears, therefore, that in both the cases considered, the form of 

 the expression for the length of the path of the wave to the point under 

 consideration near the caustic, passing through the general point of the 

 reflecting surface or of the front of the wave, is that of a general formula 

 of the third order; in which the coefficient of the first power of the 

 ordinate of the point on the front of the wave is proportional to the 

 distance of the illuminated point from the caustic or the asymptote, and 

 in which the coefficient of the third power is independent of that dis- 

 tance. If in the first instance we make » + /^ = %' the first 



P 

 expression becomes (putting E for a term independent of %, and in 

 the coefficient of z omitting the term involving Sp" in comparison 

 with Sp,) 



And if in the second instance we make x + ^. , " = v and ob- 



b — q 



serve that for the rainbow a is a very small fraction of an inch while 



p may be many feet, and that a may therefore be omitted in comparison 



with p, 



PX=F+1^^ {x'^-p^p,a:'\. 

 3a b — q'^ ' 



Vol. VI. Pakt III. 3D 



