THEORY OF THE DISPERSION OF LIGHT. 



257 



On expanding the value Qf y. .j. a r neglecting the squares, substituting in the 



above equations (12.), observing that they will involve the expressions (9.) for equili- 

 brium, which vanish, and writing for abridgement, 



while we have 



these equations of motion (12.) will ultimately become 



^ = 2|^(r)A| + ^(r)Aa^(A^A| + A3/A;? + As;AOJ 



f^ = 2|^(r)A;7 + ^}/(r)A3/(A^A|-t-A^/A;^ + AsJAOj [. . (13.) 



^=2|?)(r)A^ + ^|.(r)A;s(Aa^Ai + Ai/A^ + A;?^A0J. 



These equations form the common basis of all the investigations of the subject as 

 originally pursued by MM. Navier and Cauchy. They apply generally to all elastic 

 media, or systems of molecules, affected only by their mutual attractions and repul- 

 sions, and slightly disturbed. 



(9.) In applying this hypothesis to the case of light, agreeably to what has been 

 already observed (§. 6.), we have 



A I = and ^2 = 0, 



so that the equations (13.) are reduced to 



^ = 2 {p (r) A, + 4- (r) Aj, [Ay A, + A^ AC]} 



fJ=2{?.(r)AC + ,|,(r)A2[A3,A, + A!.Aa} 



The general direct integration of these forms, and even of the preceding (13.), has 

 been effected by the writers already referred to ; but only upon the suppositions 



2{A3/A;z} = 0, and 2 {sin /c A 07} = (15.) 



the import of which will be considered in the sequel. Mr. Tovey by a skilful analysis 

 gives a particular solution corresponding to the case of elliptically polarized light, 

 without introducing those conditions. The investigation, however, appears susceptible 

 of being simplified ; and we shall here pursue a method at once attaining this object, 

 and enabling us more clearly to trace out some interesting results, as follows : 

 Taking the finite differences of the expressions (5.), we have 



A?7 = a [2 sin^ (— ^— j sin (w # — /r J?) — sinA Aa?cos (w^ — A:a?)J . . (16.) 



(14.) 



MDCCCXXXVIII. 



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