133 



a"'^ the density at the mirror, we shall have for the density at the point x, y', on the plane of 

 aberration, the following approximate expression 



an expression which shews that at the caustic surface the density is infinitely greater than at 

 the mirror ; and that near the caustic surface the density is not uniform, but varies nearly in- 

 versely as the square root of the perpendicular distance from that surface ; so that we may con- 

 sider this density as constant in any one of the little parabolic bands comprised between two 

 infinitely near parallels to the enveloping curve (R") [60.] 



[68.] To treat this question, respecting the variation of density upon the plane of aberration, 

 in a more accurate manner, let us take into account the remaining terms of the developments of 

 x' and y, as given by the general theory, which we have explained at the beginning of the pre- 

 ceding section. For although we were at liberty to neglect these terms, when we were only in 

 quest of approximate formulae to represent the manner in which certain of the near rays are dif 

 fused over the plane of aberration ; yet, when we are returning from this latter plane to the per- 

 pendicular surface at the mirror, it is not safe to neglect any term on account of its smallness, 

 until we have examined whether, in thus returning, its influence may not be magnified in such 

 a manner as to become sensible. 



Let us then resume the general series (K") [57.] 



dY dY ^ d'^Y rl?-Y d^V 



in which we have at present X=0, Y = 0, ~- = ?, — = 0, — =0, — - 0. 



da ' d/i ' dx. rf/3 ~ 



d^X , dW d"-X „ d'V d"-X . dW 



d'Y _ J _^ .^_ 2 rf^F d'Y _ 3 d'V 

 d^ «' '^"dx\dy' d».dl3 "«'•«' • dx.df ' ~dF~^^ '~df ' 



V being the characteristic function ; so that 



d^X d^Y , d^X d^Y 



and 



dit.di3 6?«» ' c?/3» d».dli ' 

 We have in like manner, 



^* I da , , , rd""* , . «P« - d^a ..-i 



d^ dfi z , <d'fi „ ,, d'^H . d^» ,„7 



