FBESNELi ON DOUBLE REFRACTION. 305 



are composed of a circle and an ellipse; in fact, if we, for 

 example, suppose ^ = in it, we find 



(O^ X^ + ^2 y^) [x^ + 2/2) _ «2 (^2 + c^) X-i - U" (fl^ + c^) tf + O^ 6^ C« = 0, 



r 



[a^x'' -\-b-7/- tt" U") {x^ + / - c2) = 0, 



an equation compounded of the equation to a circle whose radius 

 is (c), and of that to an elHpse whose semi-axes are (a) and (6). 



The equation of the Wave Surface cannot be decomposed into 

 two rational factors of the second degree, except ivhen two of 

 the axes of elasticity are equal. 



But the general equation to the surface of the wave is not, 

 like those of its intersections, always decomposable into two 

 rational factors of the second degree, as I have assured myself 

 by the method of indeterminate coefficients ; this decomposition 

 can only be effected when two of the axes are equal. Suppose, 

 for example, that b = c, the equation (D.) then becomes 



[a^ a?2 + £2 (^2 ^ .S'lJ (^2 + 2/2 + ^2) _ 2^2 ^2^2 



- 62 (a2 + ^2-) (^2 + .2-) ^ ^2 ^4 ^ 0; 

 or 



(a;2 + 2,2 + ^2) [-^2^2 ^ ^2 (^/S ^ ^2) _ ^2 42J 



- A2 [a2^2 ^ ^,2 (y2 ^ ^2) ^ ^2 ^2J ^ ; 



or, lastly, 



(^2 + 2/2 _j. _j2 _ ^2) [-^2^2 + ^2 (^2 ^ ,2) _ ^2 ^2"] ^ Q, 



an equation which is the product of that of a sphere by that of 

 an ellipsoid of revolution. * 



The construction of Hwygens, which determines the path of 

 swiftest arrival, or the direction of the refracted ray, is appli- 

 cable to bi-axal crystals as to calcareous spar, and in general 

 to all waves of any form whatever. 



It is to these two surfaces that a tangent plane is successively 

 drawn, in the construction given by Huygens for Iceland spar. 

 In the general case of bi-axal crystals, that is to say, when the 

 three axes of elasticity are unequal, we must dra\v a tangent 

 plane to each of the two sheets of the surface represented by the 

 equation (D.) ; and by joining the points of contact with the 

 centre of the surface, we shall have the directions of the two 

 paths of swiftest arrival, and consequently of the ordinary and 

 of the extraordinary ray, I employ here the received expression 



