120 



^. + r^. = '' (M") 



and this area is equal to the product of the focal distances (5— .^j), (j^j^), multiplied by 

 w. ()% «• being the semicircumference of a circle whose radius is equal to unity. 



[59.] As a second application, let us take the case where the plane of aberration passes 

 through one of the two foci of the given ray, for example, through the second, so that { = g^* 

 In this case the formulae (L") become 



so that if we continue to neglect terms of the second order, the points in which the near rays 

 cross the plane of aberration, are all contained on the axis of {x), that is on the tangent to the 

 caustic surface. But if we take into account the aberrations of the second order, that is, if we 

 do not neglect the squares and'products of «„ $„ which enter into the general expression (K") 

 for y, then the rays will cross the plane of aberration at a small but finite distance from the 

 axis of (x'); that is, y will have a small but finite value, which we now propose to investigate. 

 For this purpose, that is, to calculate the coefficients in the expression 



f d^Y (PY fPY ^ 



I observe that the general formula, [57.] 



d^ y=4^. d^u + ~. rf^/3 + -5^. d»'+2^. dcc.dii + ~. d/i\ 

 dec ^ d/i ' rf«^ ^ d*d/i dfi^ ' 



(in which a, /3, Y, are considered as functions of the independent variables (a, b), and which is 

 equivalent to three distinct equations) gives, in the present case, 



d^Y _ , £Y_ d^ _ d'^Y £Y_ „ £Y_ 



rf«2 ~ ^^ ' da" ' d».dfi ~ *'**■ da.db ' rf/3' " *'"' db^ ' 



because 



d"^ ^ ^y ^ ^ da _, db 

 -r-=0, -r-=0, d* = , d/3 = . 



a» d/3 gi gj 



Again, the equation Y = b +ii g, gives d'Y ss {a. d^/3, when we put 



we have therefore 



d'Y _ d^ £Y_ _ (f/3 d~Y _ d^fi 



da^ «'■ da"' da.db ~^*' da.db' db^ ~ ^^' W 



and the question is reduced to calculating these partial differential coefficients of (/3). Now, the 

 equation 



