174 PROCEEDINGS OF THE AMERICAN ACADEMY 



These numbers we will now compare with the two solutions of Her- 

 schel's equations, using the notation I, r and ?-', to denote the reciprocals 

 of the compound focal length and of the radii of the front surfaces of the 

 two lenses. The substitution of the values of the indices of refrac- 

 tion and of the dispersive powers which have been used by Gauss for 

 computing the system I. gives the relations * : — 



= 2.3200 r- — 21.31 Ir -f 59.57 P + 3.5792 I r> — 1.4233 r'^ 

 = 6.6400 r — 24.95 ^ — 4.1119 r' 



From which we have 



= — 1.3917 r" + 12.37 I r — 14.56 P 

 with the roots 



y:= 7.4922, and [=1.3964, 



which afford the subjoined two sets of values. 



IV. HerscheVs Curves. 



7 = 7.4922 



1st surface of the crown lens, convex, radius, : 

 2d " " " concave, " : 



1st " " flint lens, convex, " 



2d " " " concave, " 



Compound focus, " 



V. ~ = 1.3964 



1st surface of the crown lens, convex, radius, 

 2d " " " convex " 



1st " " flint lens, concave, " 



2d " " " convex, " 



Compound focus, 



With these radii the figures in the accompanying Plate have been 

 constructed, representing sections of the different object-glasses, each 

 having a focal length of two feet, and an aperture of nearly four 



* In the equation [z] Phil. Trans. 1821, p. 258, the coefficient of zj^ has been 

 corrected from -^-i-i to -^4:^^ F/Je Article on Light, Encyc. Met., p. 424. 



