II. On the Reflexion and Eefraction of Light at the Surface of an Uncrystallixed 

 Body. By the Rev. M. O'Brien, late Fellow of Caius College. 



tRead Nov. 28, 1842.] 



The object of the present paper is to determine completely the Laws of Reflexion and Re- 

 fraction of Light, without introducing any empirical considerations, or omitting to take into account 

 the normal vibrations which are generated in cases of oblique incidence. Though several eminent 

 niathematicans have written upon this subject, I believe that most of what is here contained is new. 

 I must state, however, that I have not been able to procure a Memoir by M. Cauchy, which 

 he constantly refers to in his Exercices d^Analyse et de Physique Mathimatique (for 1840), 

 and in which he has given a general method of arriving at the equations of condition relative 

 to the limits of bodies. I can therefore only guess at the physical principles upon which he 

 obtains his equations of condition, which equations, in the form he has given them in the 

 Exercices for 1840, are particular cases of those obtained in the present paper. As M. Cauchy 

 states that he has made use of some new principles in obtaining his equations of condition 

 (see the Nouveaux Ewercices, Prague 1835, p. 203), I am justified in assuming that the 

 method employed in the present paper is different from his ; for I have deduced my equations 

 of connection, not from any new physical principal, but from an old and obvious one, which 

 has been either directly used, or tacitly assumed by all the writers upon the refle.xion and refraction 

 of Sound and Light, that I am acquainted with. This principle is very clearly stated by 

 Poisson, in the Memoires de Vltistitut, Tom. x. p. 320. 



The following is a brief outline of the course pursued in the present paper. 



In Section I. I have proved some very simple theorems by means of which I have after- 

 wards deduced the laws of reflexion and refraction, without assuming the integrals of the 

 equations of motion, or supposing the waves to be plane. 



In Section II. I have deduced the equations of connection of the vibratory motion of two 

 media, separated by a plane, from the principle above alluded to. These equations of con- 

 nection are apparently the same as those given by Mr Green in the Cambridge Transactions, 

 Vol. VII. p. 11.; but they differ from them very materially with respect to the constants in- 

 volved in them, and on that account they, and the results deduced from them, are perfectly 

 free from difficulties* which seem to me to be fatal to the correctness of Mr Green's equations, 

 and which he appears to have felt himself. I shall not however enter into this subject now, 

 as I shall be obliged to do so on a future occasion. 



I have shewn that these equations of connection are considerably simplified when we 

 suppose the ether to have the same constitution as ordinary gases, and neglect the variation of 

 temperature. 



In Section III. I have applied these equations of connection to determine completely the 

 laws of reflexion and refraction of polarized light, both as regards direction, colour, and in- 

 tensity, taking fully into account the production of normal as well as transversal waves in the 



" One difficulty I have mentioned a little farther on. Another difficulty is this, that there are just the same constants (A) and 

 (/?) in Mr fjreen's Equations of Connection, as those in his Equations of Motion : which arises, first, from an error in the form 

 of the function V»2 {Cuinbriilye Trdusactions, Vol. vii. p. 7), and secondly, because <p2 is not symmetrical round the axes of y and 

 « at the plane of separation, as Mr Green assumes it to be. 



