172 PROCEEDINGS OF THE AMERICAN ACADEMY 



disposition of the surfaces of two or more lenses, composed of materials 

 of different dispersive powers, which shall most effectually destroy the 

 aberrations of color and of figure. The problem, in the form in which 

 it has been practically presented, is indeterminate, so that, for instance, 

 in the case of lenses of crown and flint glass, " For every lens of 

 crown-glass of positive focus, whatever the radii of its surfaces may 

 be, a lens of flint-glass can be computed which will form, when united 

 with it, an achromatic object-glass," — achromatic, that is to say, in the 

 limited sense in which the term is commonly accepted. 



This allows, of course, of a great range in the choice of curves, and 

 a variety of conditions have been proposed for determinhig the selec- 

 tion. In one respect only has there been a general consent of authori- 

 ties. The front lens has always been convex on both surfaces. But 

 it would seem that in this particular the direction given to the investi- 

 gation has not been fortunate. It is at least an oversight, that the 

 relative importance of the two principal sources of indistinctness has 

 not been kept prominently in view. For while it is admitted that the 

 chromatic dispersion is the chief source of indistinctness, the arbitrary 

 condition has not been determined with special reference to this cir- 

 cumstance. 



This omission has been supplied by Gauss, who has given attention 

 mainly to the more complete elimination of the aberration of color, 

 while, at the same time, his expectations that this could be done with- 

 out sensibly increasing the spherical aberration, have been fully re- 

 alized in the performance of the new object-glasses. Indeed, it de- 

 serves notice that the resulting curves bear a considerable resemblance 

 to one of the systems which has been designed with express reference 

 to the correction of the spherical aberration. Allusion is here made 

 to the forms deduced by Herschel * for the elimination of the spherical 

 aberration of diverging, as well as parallel rays. From the compari- 

 sons subjoined, it will be seen that one of the solutions satisfying 

 his equations approximates nearly to Gauss's system, while the other 

 approaches to a form employed by Frauenhofer. So far, therefore, 

 as this holds good, each fulfils the conditions proposed in Herschel's 

 theory. 



As Gauss has published neither the mathematical investigation of the 

 subject, nor even the final equations from which his curves were com- 

 puted, we have not the means of deciding with entire certainty, whether 



* Phil. Trans., 1821, p. 258. 



