SURFACE OF ELASTICITY. 217 



right angles to tlie plane of polarization. This settles the first of the questions 



proposed above. The second is less simple. 



If a polarized ray, Avhose molecular movements arc 

 in the direction 01*, in the annexed figure, fall upon 

 a lamina of doubly reftacting crystal, wliose principal 

 section is MM', it undergoes double refraction in every 

 case except that in which OP coincides with MM' 



<;^ ^ or NN', the conjugate plane. The effect is the same, 

 ^ in seeming, as if the lamina alio 



^ in seeming, as if the 'amina allowed no free passage 

 for movement, except in these directions. The im- 

 aginary structure presented in the figure is in ac- 

 cordance with this idea. The undulations of which 

 OP is an element, encountering such a structure, 

 would be necessarily resolved into two movement*, 

 taking the directions of the open passages ; and 



according to the laws of the resolution of forces, we should have OP^^Ot^^-f-OlF. 



Or putting I for the total intensity of the light, and a for the angle between OP 



and MM',— 



I = Icos^a-j-Isin^a, 



which is the law of Malus. 



This illustration is given merely to focilitate the conception of the constant 

 determination of the ethereal vilirations in crystalline bodies to fixed directions. 

 The cause must be one more general than such a mechanical structure could pos- 

 sibly be. The theory of Fresnel, embracing all the cases of double refraction, 

 is founded on the assumption that the elasticity of the ether may be different 

 in the directions of three rectangular axes ; and among the conclusions mathe- 

 matically deducible from this assumption, is the proposition that, in a medium 

 so constituted, the molecular movements of an incident ray will be unstable 

 except in two determinate azimuths at right angles to each other. If they are 

 not in those azimuths, or one of them, on entering the medium, they will be 

 instantly turned into them; and thus the ray will be polarized in planes havin"- 

 different directions in the crystal. 



If we take a crystal of two axes, and form from it prisms, of one of which 

 the edges shall be perpendicular to the plane containing the axes, while the 

 others have their edges respectively parallel to the lines which bisect the angles 

 between the axes, we shall find that, in the planes of refraction of these prisms, 

 one of the rays follows the law of Sncllius ; but that the indexes of refraction 

 for these arc different. These rays thus ob(>ying the ordinary law are moreover 

 polarized in their several planes of refraction. Their molecular movements are 

 therefore perpendicular to those planes, or parallel to the edges of the prisms, 

 that is to say, parallel (by construction) to three determinate fixed lines iu the 

 crystal, each at right angles to the other two. These velocities, then, determine 

 the ela-'ticities in the direction of three rectangular axes. From these as constants, 

 Presnel derived an equation expressing the elastic force in all intermediate 

 directions. The three velocities are distinguished by the letters a, b and c, iu 

 the order of their magnitude — that denoted by a being greatest. And elasticities 

 being as the squares of the velocities which they generate, the three elasticities 

 are c^, li^, and c^. Now if any line be taken which makes with the directions 

 of the elasticities a^, l?, t^, angles represented by A, 13 and 0, and if It denote 

 the velocity which the elasticity in the direction of that line is capable of gen- 

 ei-ating, then we shall have the equation — 



R2z=«2cos2A-l-//cos2B-|-62cos2C. [44.J 



Giving A, B, and C all possible values. It will have all possible directions ; 

 aaid. considered as a radius vector, its extremity will describe a surface the 



