26 Mr. Mac Cuttaau on the dynamical Theory of 
axes of 2, y, z, are all known when the quantities x, y, z are known. Conse- 
quently V is a function of x, Y, Z- 
Supposing the angle « to be very small, the quantities x, y, z will also be 
very small; and if V be expanded according to the powers of these quantities, 
we shall have 
Vox+ax-+sy+cz+ px’ + EY’ + FZ + GyZ + HXZ + IXY + ete., 
the quantities K, A, B, C, D, &e., being constant. But in the state of equilibrium 
the value of &V ought to be nothing, in whatever way the position of the system 
be varied; that is to say, when the displacements & », ¢, and consequently 
the quantities x, y, z, are supposed to vanish, the quantity 
6V = aox + Boy + coz + 2Dxex + etc., 
ought also to vanish independently of the variations ¢&, oy, &f, or, which comes 
to the same thing, independently of &x, éy, &z. Hence* we must have a = 0, 
8 = 0, c=O; and therefore, if we neglect terms of the third and higher dimen- 
sions, we may conclude that the variable part of V is a homogeneous function 
of the second degree, containing, in its general form, the squares and products 
of x, Y, Z, with six constant coefficients. 
Of these coefficients, the three which multiply the products of the variables 
may always be made to vanish by changing the directions of the axes of .r, y, <. 
For this is a known property of functions of the second degree, when the coor- 
dinates are the variables; and we have shown, in Lemma IL, that the quantities 
x, Y, Z, are transformed by the very same relations as the coordinates themselves. 
Therefore, in every crystal there exist three rectangular axes, with respect to 
which the function V contains only the squares of x, y, z; and as it will pre- 
sently appear that the coefficients of the squares must all be negative, in order 
that the velocity of propagation may never become imaginary, we may conse- 
quently write, with reference to these axes, 
V=—i@x? Py cz), (2) 
omitting the constant K as having no effect upon the motion. 
The axes of coordinates, in this position, are the principal axes of the crystal, 
* See the reasoning of Lagrange in an analogous case, Mécanique Analytique, tom. i. p. 68. 
