480 M. Biot on certaiji Points of Mathematical Optics. 



that it is confined within extremely narrow limits, between 

 absolute equality of the two radii (which would put the poste- 

 rior surface of the crown-glass in contact with the anterior of the 

 flint), and a very small difference of length (which would sepa- 

 rate by a very minute quantity the margins of the two sur- 

 faces). The combinations comprised between these two limits 

 are therefore the only ones which it is suitable to choose, and 

 it appears that they must be nearly equivalent in effectiveness, 

 when we thus confine ourselves to the destruction of the first 

 term of the two aberrations. All give the flint concave in the 

 interior and convex on the exterior. This is precisely the con- 

 figuration which Fraunhofer adopted, and which he always 

 combined with the nullity of the interval between the two 

 lenses. But the agreement of analytic theory with the prac- 

 tical combinations of this great artist, is seen to be still much 

 more close v/hen it is followed out in numbers. For, starting 

 from the same physical data which he employed for the con- 

 struction of an object-glass of this kind, all the peculiarities of 

 which he himself indicated numerically, it is not only found to 

 be comprised within the limits of the relations assigned above 

 for the stability of achromatism, but, by adopting the propor- 

 tion of inequality which Fraunhofer established between the 

 radii of the opposite surfaces, the radii of the four curvatures 

 calculated by my formulae have been numerically almost iden- 

 tical with his. We may therefore hope that, by following the 

 course which I point out, we shall obtain directly and surely, 

 in all similar cases, the combinations of spherical curvatures 

 which will apply with the greatest advantage to the physical 

 data assigned for the execution. 



The object-glass being thus completely calculated, it is 

 necessary to be able to verify, by an exact calculation, whether, 

 in fact, the spherical and chromatic aberrations are sufficiently 

 destroyed in it, with the adopted combinations of thicknesses 

 and curvatures, for the effective aperture which we propose to 

 give it. For this purpose I propose a method of trigonome- 

 trical calculation, by which we obtain strictly the values of 

 these aberrations in the different directions in which they have 

 effect; and as the equation of condition which destroys the 

 most sensible parts of it yet admits a slight inequality in the 

 radii of the surfaces which face one another, we may, by va- 

 rying these elements by a slow gradation, ascertain the direc- 

 tion as well as the extent of the modifications which must be 

 made to render the final values of the aberrations insensible, 

 or at least as small as possible. By these definitive correc- 

 tions we ought to obtain from the spherical curvatures all the 

 best effects which they can give. 



