IN THE NEIGHBOURHOOD OF A CAUSTIC. 385 



In the particular case of measuring V through the point which 

 reflects rays (according to the ordinary rules) to p, q, 



or, as we shall always suppose ^p small, 



dKH-r 

 da? ~ ^• 



10. Consequently, if V] be put for the length of the path through 

 X + lx, y + Sfj, to p + Sp, q, (Sp being entirely independent of Sx), 



r = r, + ASp — + BSp.^^ + c. ^-^ , 



or, putting a; for Sx, 



r=r-ASpr^.B^p.^^.c.^^, 



omitting the following terms. 



11. The value of C has been found: that of A (the only other 

 quantity which interests us) will be obtained by actual differentiation 

 of the expression for F", : Thus we find 



CttJ 



{x-p)(y-q)--£.-{y-qy 

 ^^P-^P- Kx-prHy-q)V ■ 



1 1 



Or, as x—p is supposed to be very small, A = — 



y-q PX 



12. There is one case so peculiar, and which seems so likely to 

 cause a failure of these expressions, that it merits a particular investi- 

 gation : the more so as it occurs in the rainbow. It is the ca.se in 



