M. Biot on certain Points of Mathematical Optics. 4-79 



that for a like total even number of surfaces the number of 

 terms is reduced to one-half. Under this new form all the 

 effects of any dioptric system are found still to depend solely 

 on four })rincipai coefficients, connected by one equation of 

 condition. I deduce these four coefficients, both in the gene- 

 ral and in the simple case, from one amongst them, by a simple 

 process of differentiation ; and I afterwards lay down a gene- 

 ral analytical rule, by which we directly obtain this coefficient 

 from which the three others are derived. The path of the 

 rays in any optical system is thus completely determined by 

 the explicii expressions of the four principal coefficients, in 

 which we have only to substitute numbers for each system of 

 given construction. I have collected these results, relative to 

 systems purely dioptric, in a table which expresses explicitly 

 all their effects, and in which may be introduced immediately 

 all the particular conditions to which we may desire to sub- 

 ject them. 



I avail myself first of these expressions to develope the con- 

 ditions which would establish perfect achromatism in an astro- 

 nomical object-glass of two lenses. As these conditions could 

 not be completely fulfilled without fear of an excessive com- 

 plexity, I discuss carefully their explicit form, and by means 

 of it I analyse their physical effects, in order to distinguish 

 the most influential of them, and to ascertain the degree of 

 approximation to which they ought to be satisfied. We see, 

 then, in the first place, that there is great danger in leaving 

 a sensible interval between the two lenses of crown- and flint- 

 glass, which was readily evident from physical considerations; 

 so that it is desirable to make this interval nothing, or nearly 

 so, as Fraunhofer always did. When this restriction is effected, 

 we discover the possibility of establishing between the radii of 

 curvature certain relations, which, leaving still a very great 

 liberty of choice in the fixing of their values, have the effect 

 of rendering the achromatism stable, when it shall have been 

 established approximately; that is to say, it will be preserved 

 sensibly exact to the eye, even when, in the practical execu- 

 tion, there may be some slight deviation from the precise va- 

 lues which these relations suppose in the radii of curvature. 

 Having made this remark, I combine the conditions of approxi- 

 mate achromatism with those which destroy the first term of 

 the spherical aberration, to form the final ecjuation which con>- 

 pensates them simultaneously; and as it leaves still disposable 

 the relation of the radii of curvature of the two opposite sur- 

 faces, I extract from them ihe real values of this relation, which 

 approach the nearest possible to the relations previously found 

 for the stability of the chromatic compensation. I thus find 



