154 On a Dynamical Theory of 



will be converted into 



rs (cos a cos a' + cos /3 cos j3' + cos 70 cos /) = rs cos a>, 

 and by the other set into 



r s (cos eti cos a + cos j3i cos /3 + cos y x cos 7) = r' s' cos to' ; 

 so that we shall get 



rs cos w = / s' cos a/ = a? cos a cos a' + 6 2 cos /3 cos /3' 



+ c 2 cos y cos 7'. (H ) 

 Corollary. When the condition 



a 2 cos a cos a' -f Z> 2 cos /3 cos /3' + c 2 cos 7 cos 7' = (i) 



is satisfied, each of the angles w, a/ is a right angle. Let us 

 suppose, at the same time, that the direction of s is perpendi- 

 cular to that of s'. Then will the directions of s and r coin- 

 cide with the axes of the ellipse in which their plane intersects 

 the ellipsoid ; for s is perpendicular to / and parallel to the 

 tangent plane at its extremity. The directions of s' and r, in 

 the same manner, will coincide with the axes of another elliptic 

 section. 



SECT. III. 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 x' y' 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'dy'dz', having its edges parallel 

 to the axes of of, y', z', 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 parallele- 

 piped which are perpendicular to the edge dz' will be shifted, 

 each in its own plane, in a direction parallel to the axis of y f ; 

 but their displacements will be unequal, and will differ by drj ', 



