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tional to those of the former, so that the rectangle under any coin- 

 ciding pair of semiaxes, is equal to A;* : then if a plane, touching this 

 second ellipsoid in Q, cut OR perdendicu- 

 larly in P, the line OP will be a third pro- 

 portional to OR and A*, (lem. 2.) and if Or 

 intersect the second ellipsoid in q, the semi- 

 axes of the section QOq will be OQ and Oq 

 (lem. 4.) Draw OT perpendicular to QOq and 

 equal to OQ, and conceive the surface which 



is the locus of the point T to be described, and a tangent plane, to 

 which the line OS is drawn perpendicular, to be applied at the point T. 

 Then OS will be perpendicular to the plane ROr and equal to OP 

 (lem. 5.): and hence the point T always lies in the surface of the 

 wave. Similar things may be proved with respect to the other semi- 

 axis Or of the ellipse ROr. Hence we deduce the following con- 

 struction for the surface of the wave : — 



" Describe an ellipsoid whose axes are in the directions of the 

 axes of elasticity, the squares of their lengths being directly as the 

 elasticities in their respective directions ; cut the ellipsoid by a plane 

 through its centre, as QOq, and in a perpendicular to that plane take 

 two portions OT and OV equal to the semiaxes OQ and Oq of the 

 section. The double surface which is the locus of the points T and V 

 is the surface of the wave." 



As to the planes of polarisation of the rays belonging to the two 

 parts of the wave, Fresnel has shewn how to find them by means of 

 a surface which he calls the surface of elasticity. But it is desirable 

 to be able to find them by means of the same ellipsoid which serves 

 to find the surface of the wave. Now TS, parallel to OR, is the 

 direction of the vibrations of the ray OT, and the tangent plane at Q 

 is perpendicular to OR, and therefore parallel to the plane of polari- 



