Prof. M'Cullagh on Crystalline Reflexion and Refraction. 231 



and if the constants in the equation of each plane denote the cosines 

 of the angles which it makes with the coordinate planes, we shall 

 have A for the length of the wave, and s for the velocity of propaga- 

 tion ; while the rapidity with which the motion is extinguished, in 

 receding from the second plane, will depend upon the constant r. 

 The constants p and q may be any two conjugate semidiameters of 

 the ellipse in which the vibration is performed ; the former making, 

 with the axes of coordinates, the angles a, (3, y, the latter the 

 angles a', |3', y'. 



As vibrations of this kind cannot exist in any medium, unless 

 they are maintained by total reflexion at its surface, we shall sup- 

 pose, in order to contemplate their laws in their utmost generality, 

 that a crystal is in contact with a fluid of greater refractive power 

 than itself, and that a ray is incident at their common surface, at 

 such an angle as to produce total reflexion. The question then is, 

 the angle of incidence being given, to determine the laws of the dis- 

 turbance within the crystal. 



The author finds that the refraction is still double, and that two 

 distinct and separable systems of vibration are transmitted into the 

 crystal. He shows that the surface of the crystal itself (the origin 

 of coordinates being upon it at the point of incidence) must coincide 

 with the plane expressed by equation (8.), a circumstance which 

 determines the three constants /, g, h. The plane expressed by 

 (7.) is parallel to the plane of the refracted wave; and a normal, 

 drawn to it through the origin, lies in the plane of incidence, making 

 with a perpendicular to the face of the crystal an angle w which may 

 be called the angle of refraction, so that if i be the angle of inci- 

 dence, we have 



sin w = s sin i, 



the velocity of propagation in the fluid being regarded as unity. 



To each refracted wave, or system of vibration, corresponds a par- 

 ticular system of values for r, s,w. These the author shows how to 

 determine by means of the index-surface (the reciprocal of Fresnel's 

 wave-surface) which he has employed on other occasions (Transac- 

 tions of the Academy, vol. xvii. and xviii.), and the rule which he 

 gives for this purpose affords a remarkable example of the use of the 

 imaginary roots of equations, without the theory of which, indeed, it 

 would have been difficult to prove, in the present instance, that there 

 are two, and only two, refracted waves. Taking a new system of 

 coordinates x', y', z' , of which z' is perpendicular to the surface of 

 the crystal, and y' to the plane of incidence, while x' lies in the in- 

 tersection of these two planes, put y' = in the equation of the 

 index- surface referred to those coordinates, the origin being at its 

 centre ; we shall then have an equation of the fourth degree between 

 x' and z', which will be the equation of the section made in the index- 

 surface by the plane of incidence. In this equation put x' = sin i, 

 and then solve it for z'. When i exceeds a certain angle i', the four 

 values of z' will be imaginary, and if they be denoted by 



u±v V — 1, m' + v' */ — \, 



