«272 PROFESSOR POWELL ON THE ELLIPTIC POLARIZATION OF LIGHT 



imum) from the rhomb observations without a knowledge of the change of plane: 

 employing these latter data in combination with the former (3.), I have further 

 estimated the ellipticity at different incidences for four principal metals. 



7. For the application of the undulatory theory to these phenomena, we ought to 

 be able to assign the law of metallic retardation, but this has not yet been done. 

 The theory as here given indicates the conditions of the maximum, and shows in general 

 a change, but not its amount. 



Professor MacCullagh has however proposed, in accordance with a remarkable 

 mathematical analogy, certain modifications of Fresnel's formula, which he has 

 reduced to calculation in the case of steel. 



My theoretical formula gives rise to an expression for the change of plane, but in- 

 volving undetermined functions of the retardation. 07i deducing the corresponding 

 terms from Professor MacCullagh's data, and introducing them into my formula, I 

 find it gives a very close representation of the observed results for steel. Thus Professor 

 MacCullagh's empirical expression receives an additional confirmation in accordance 

 with a direct deduction from the undulatory theory. 



The rest of this paper is devoted to the details of the observations, and of the ana- 

 lytical investigation. 



Theoretical Investigation. 

 (I.) The original vibration in a plane P being 



a^ivi — {vt—x), 



in general on reflexion in a plane R inclined to P by an angle i (R' being the plane 

 perpendicular to R) it is resolved into 



a cos I sin — (vt—x) ... in R, 



a sin I sin — (vt — x) . . . in R'. 



(2.) But in the case of metallic reflexion, one of these components is accelerated in 

 phase by a quantity §, and at the same time for the greater generality, supposing the 

 coeflScients unequal, or changing a in the 2nd formula to b, and writing for brevity 



a=acos^, |3 = ^sin|; 



after metallic reflexion the component vibrations will be 



us'm (— (vt—x)) =R, 

 /3sin(^(t;^~aO+f) = R'. 



(3.) Here we may remark that these formulas give directly the equation to the elliptic 

 vibrations, the ratio of whose axes is that of a to j3, which vary at different incidences 



