Mr. Lubbock on the Double Achromatic Object Glass, 165 



A 



or, for parallel rays, that is, when A is infinite and — = 0, 



± - * m ^ • f - a 



A 4 A A 2 J * 1 — tff 



f A may be assumed at pleasure, and /% being obtained from 

 the last equation, f^ and f A may be considered as given. It 

 remains to determiney^j^, in doing which another arbitrary 

 condition is admissible. 



If m { jF-tf*fi T ' p lel rays ' 



= a { {f 4 -/ 3 y+ {A- ^} {/ 4 - 2 / 3 +/j }. 



The last equation coincides with the equation (z) of Sir 

 John Herschel in the Phil. Trans, for 1821, p. 258. 



Since 



1 ?w-l 



A 2 r 2 



m 1_ _ m f J_ ljl 



A, r 2 W2— 1 LAj wA 2 J 



A 3 " m'r 3 + w'A 2 r 3 " m' — l\A 3 w'Aj 



if the radii of the second and third surfaces are equal 



/_L _L~V- m ' SI l 1 



7»— 1 LAj 



Bi-lV 1 m J " w'-l l/ 3 wi' J " 



This condition was originally suggested by Clairaut ; but 

 according to Sir John Herschel, " when the average values of 

 the indices of refraction, such as are likely to occur most fre- 

 quently, are employed, the construction becomes imaginary 

 for the more dispersive kinds of glass ; and within the limits 

 for which it is real, the radii change so rapidly as to render it 

 difficult to interpolate between their calculated values." 



Hence, instead of this condition, Sir John Herschel recom- 

 mends another, which results from making equal to zero the 



coefficient of — in the quantity 



