nw 
Or 
crystalline Reflexion and Refraction. 
Corollary. When the condition 
a cosa cos a’ + b° cos B cos B’ + c cos y cosy’ = 0 (1) 
is satisfied, each of the angles , w’ isa right angle. Let us suppose, at the 
same time, that the direction of s is perpendicular to that of s’. Then will the 
directions of s and r’ coincide with the axes of the ellipse in which their plane 
intersects the ellipsoid; for s is perpendicular to 7’ and parallel to the tangent 
plane at its extremity. The directions of s’ andr, in the same manner, will 
coincide with the axes of another elliptic section. 
SECT. I1l.—DETERMINATION OF THE FUNCTION ON WHICH THE MOTION DEPENDS. 
PRINCIPAL AXES OF A CRYSTAL. 
- 
We come now to investigate the particular form which must be assigned to 
the function V, in order that the formula (1) may represent the motions of the 
ethereal medium. For this purpose conceive the plane of zy’ to be parallel to 
a system of plane waves whose vibrations are entirely transversal and parallel to 
the axis of y’, so that # =0, ¢’=0. Imagine an elementary parallelepiped 
dx‘dydz’, having its edges parallel to the axes of 2’, y’, 2’, to be described in the 
ether when at rest, and then all its points to move according to the same law as 
the ethereal particles which compose it. The faces of the parallelepiped which 
are perpendicular to the edge dz’ will be shifted, each in its own plane, in a 
direction parallel to the axis of y’, but their displacements will be unequal, and 
will differ by dy, so that the edges connecting their corresponding angles will no 
longer be parallel to the axis of 2’, but will be inclined to it at an angle « whose 
. ay 
tangent 1s —> 
dz” 
Now the function V can only depend upon the directions of the axes of 
x’, y’, 2 with respect to fixed lines in the crystal, and upon the angle «, which 
measures the change of form produced in the parallelepiped by vibration. This 
is the most general supposition which can be made concerning it. Since, how- 
ever, by our second assumption, any one of these directions, suppose that of 2’, 
determines the other two, we may regard V as depending on the angle « and 
on the direction of the axis of 2 alone. But from the equations (F) it is mani- 
fest that the angle « and the angles which the axis of 2’ makes with the fixed 
VOL. XXI. E 
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