On a Mode of deducing the Equation o/Tresnel's Wave. 381 



according to descending powers of cos 6 (where n is a posi- 

 tive integer), it seems extremely doubtful whether anything 

 would be gained by its introduction in an elementary work ; 

 the method given in the ordinary treatises on trigonometry 

 depending on the logarithmic series, is on every account pre- 

 ferable. 



In the same article in your Number for August there is given 



a proof that A" x m = when m < n. 



If the author of it will only turn to art. 885 in Lacroix, I 

 am convinced that he will regret having offered his new de- 

 monstration. Believe me, yours sincerely, 



Trinity College, Cambridge, J. E. 



Aug. 24, 1841. 



LVI. On a Mode of deducing the Equation of FresnePs Wave. 

 By Sir William Rowan Hamilton, LL.D., P.R.I.A., 



Member of several Scientific Societies at Home and Abroad, 

 Professor of Astronomy in the University of Dublin, and 

 Royal Astronomer of Ireland*. 



r r , HE following does not pretend to be the best, but merely 

 •* to be one way, of deducing the known equation of Fres- 

 nePs wave, from a known geometrical construction. It re- 

 quires only the first principles of the application of algebra to 

 the geometry of three dimensions, and does not introduce any 

 of the geometrical properties of the auxiliary ellipsoid em- 

 ployed, except those which are immediately suggested by the 

 equation of that ellipsoid. It has, therefore, in the algebraical 

 point of view, a certain degree of directness, although it might 

 be rendered easier and snorter by borrowing more largely 

 from geometry. 



1. The known construction referred to is thus enunciated 

 by Sir John Herschel, in his Treatise on Light, Encyclopaedia 

 Metropolitana, article 1017. " M. Fresnel gives the fol- 

 lowing simple construction for the curve surface bounding 

 the wave in the case of unequal axes, which establishes an 

 immediate relation between the length and direction of its 

 radii. Conceive an ellipsoid having the same semiaxes a, b, 

 c; and having cut it by any diametral plane, draw perpen- 

 dicular to this plane from the centre two lines, one equal to 

 the greatest, and the other to the least radius vector of the 

 section. The loci of the extremities of these perpendiculars 

 will be the surfaces of the ordinary and extraordinary waves." 



2. The coordinates of the wave being x y z, and those of 

 the ellipsoid X Y Z, we have the six equations, 



* Communicated by the Author, 



