Crystalline Reflexion and Refraction. 155 



so that the edges connecting their corresponding angles will no 

 longer be parallel to the axis of z', but will be inclined to it at 



an angle K whose tangent is f . 



Now the function Y can only depend upon the directions 

 of the axes of #', y', z' with respect to fixed lines in the crystal, 

 and upon the angle K, which measures the change of form pro- 

 duced in the parallelepiped by vibration. This is the most 

 general supposition which can be made concerning it. Since, 

 however, by our second assumption, any one of these directions, 

 suppose that of #', determines the other two, we may regard V 

 as depending on the angle K and on the direction of the axis of 

 x alone. But from the equations (F) it is manifest that the 

 angle K and the angles which the axis of x' makes with the 

 fixed axes of x, y, z are all known when the quantities JT, Y, Z 

 are known. Consequently Y is a function of X, Y, Z. 



Supposing the angle K to be very small, the quantities 

 X, Y, Z will also be very small; and if Y be expanded 

 according to the powers of these quantities, we shall have 



Y = K+ AX + BY + CZ+DX 2 + EY* + FZ Z 



+ GYZ + EXZ+ IXY+ &c., 



the quantities K, A, B, C, D, &c., being constant. But in the 

 state of equilibrium the value of 8 Y ought to be nothing, in 

 whatever way the position of the system be varied ; that is to 

 say, when the displacements , TJ, , and consequently the 

 quantities X, Y, Z, are supposed to vanish, the quantity 



SY = ASX + ESY + C%Z + 2DXSX + &c., 



ought also to vanish independently of the variations S, Srj, , 

 or, which comes to the same thing, independently of $X, SF, 

 Z. Hence* we must have A = 0, j0 = 0, (7=0; and therefore, 

 if we neglect terms of the third and higher dimensions, we may 

 conclude that the variable part of Y is a homogeneous function 



* See the reasoning of Lagrange in an analogous case, Mecaniquc Analytique, 

 torn. i. p. 68. 



