IN CRYSTALLIZED MEDIA. 333 



2. These values are either two positive and one negative, or two 

 negative and one positive, for if we write the equation under the form 



A' -{a + b + c)A'+ \ab + ac + be- {X' + Y + Z')} A 



- abc + aX' + b Y" + cZ" + 2XYZ= 0, 



it will be evident that the coefficient of A- is equal zero (1). 



The roots then must assume the form A^Ao— (A^ + A2), in which 

 At A-, may be either both positive or both negative: suppose the former. 



3. Now, corresponding to any value of A, a value of P, Q, R, 

 respectively can be determined ; but A is the velocity of transmission 

 of a vibration whose direction makes with the co-ordinate axes the 

 angles cos"' P, cos~' Q, cos"' R respectively, and which is transmitted 

 in a direction making with the same axes other angles 6, tp and \|/. 



^^'e conclude then, that there are in general two directions and 

 no more, in which a vibration taking place, the transmission along a 

 given line is possible. A disturbance in a given direction being re- 

 solved into these two, will give rise to two different rays, transmitted 

 with different velocities. 



4. The third value of A which is negative, will not correspond 

 to a vibration ; the manner in which it may affect the motion, and 

 the probable results to which it gives rise, I have fully discussed in 

 a paper read before this Society a short time since, and shall leave 

 it untouched in the present INIemoir. 



DhciissioH of the Equation for A. 



.5. As a preliminary step towards a complete discussion of this 

 e(;[uation, we will first consider the medium perfectly symmetrical. 



Transform the co-ordinates in such a manner that the axis of a;' 

 shall coincide with the direction of transmission, and that of y lie 

 in the plane of xy. 



Denote the angle between the axes of x and x by the symbol 

 ix' x~), and so on for the others : 



u u 2 



