478 M. Biot on certain Points of Mathematical Optics. 



synthesis of muriatic acid, we have deduced the fact of latent 

 tithonicity, and the definite action of rays of that principle; 

 we have also seen the totally distinct functions which chlorine 

 and hydrogen respectively discharge; we have gathered the 

 reason why water will not decompose under the most brilliant 

 radiation, nor oxygen and hydrogen unite. And, lastly, we have 

 alluded to the probable reason of the coincidence of the maxi- 

 mum point of decomposition of carbonic acid by the leaves of 

 plants, and the maximum point of illumination for the human 

 eye, in the yellow space of the spectrum; — that it originates in 

 the physical peculiarities of the carbon atom. 



LXVIII. On certain Points oj Mathematical Optics, 

 By M. Biot*. 

 A S the second volume of the third edition of my Traite 

 -^^ d' Astronomic will appear in a few days, I request per- 

 mission to lay before the Academy various applications of a 

 new mathematical theory of optical instruments which are in- 

 cluded in it. 



The problem which this theory solves consists in determi- 

 ning, by general and explicit formulae, the motion of luminous 

 rays through any number of spherical surfaces, either refract-- 

 ing or reflecting, centred on the same axis, and separated by 

 singly-refracting media, of whatever nature, when the inclina- 

 tions of the rays to the central axis are very small. This enun- 

 ciation comprises all the conditions to which optical instru- 

 ments are subjected in the central part of their field of view, 

 where perfection is most important: and the explicit formulae 

 at which I have arrived serve to establish them directly by 

 simple substitutions of numbers for each given construction of 

 the instrument under consideration. 



In xwy first volume I presented these formulae in all their 

 generality, and I deduced from them the conditions common 

 to all classes of instruments. But I remarked that, in the 

 special application to purely dioptrical systems, they became 

 susceptible of an abridgement, which, while leaving them the 

 same form, rendered their numerical employment much more 

 simple. It remained for me to prove the truth of this asser- 

 tion, and to develope its consequences. 



For this purpose, resuming the general formulae which I 

 had established, I limit the systems to which I apply them to 

 any number ol spherical lenses, centred on the same axis and 

 surrounded by the same singly-refracting medium. The peri- 

 odical return of each ray to the same velocity, at every second 

 surface, then allows the general formulee to be simplified, so 



* From the Comptes Renclus de I* Acad, des Sciences, Sept. 9, 1844. 



