164 



+ 2^2^ 



' ^"o 2 ilt!!^ (^-«or iiizAT', (Kffl) 



in which m > 1, »n'> 1, 



7.' ~2' 



m'" 







(Lffl) 



dao"'.dbo'"' ' da"'+'"".db'''+'""' [«"]"'" [m'"]"' 



o, b, being supposed to vanish after the differentiations in these second members : and it is easy 

 to see that by means of these equations we can develope a — «o> ^ — ^o' and therefore also a, I, 

 according to the positive integer powers and products of x", y S'. With respect to the coefficients 

 of these developments, they may be calculated by differentiating the equations that we have just 

 established ; they may also be deduced from the coefficients of the series (T'"), by relations which 

 will be elsewhere indicated. 



In the mean time let us remark, that instead of measuring the distance from the caustic sec- 

 tion in a direction parallel to the axis of y", we may measure it parallel to any other line upon 

 the plane of aberration. If for instance, to simplify our remaining calculations, we resume the 

 rectangular coordinates x', y' , of which the former is a tangent, and the latter a normal to the 

 section ; if from the point (x', y) we draw a line parallel to this normal, and denote by j/o, y^ 

 the coordinates of the point where this line meets the section, and by 3 the intercepted portion, 

 so that 



«' — ^0 = 0, y — yo = 5 ; (M(7)) 



jf also we call a^, b^, the coordinates of the point in which the ray that passes through (x'o, y'o) 

 is crossed by ihe perpendicular plane at the mirror ; we shall have the equations 



dx' cT+^V 



m 



da ' "^ "' da"'.db"'' [w]"' ' ["i']"' 

 rf'"+'"'y a'" b"'' 



(S'") [68.] 



dx' , d'"+'"'x' Co"- bo"-' 



(NW) 



r,vo = 



da da'".db'"' ' [m]"' ' Im'y 



da"'.db"" [wi]" • [m'}" 



